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1. 2 amp c phi2 p climul f1 Y1 climul f2 Y2 climul f1 1 climul f2 2 Now we compute 84 Y while we store the purespinor u under the name purespinor1 Notice that G is invariant under the unitary group U 2 gt beta_plus psi phi f purespinor1 purespinor1 22 22 21 21 Y11 11 12 12 Id 21 622 12 d11 Y11 G12 22 21 e23 Id 26 Rafat Abtamowicz Observe that 64 6 Wid v 2 where denotes complex conjugation whereas a pure spinor in this case is the identity element 13 Finally we compute G_ w while we store the purespinor u under the name purespinor2 Notice that _ is invariant under the complex symplectic group Sp 2 C gt beta_minus psi phi f purespinor2 purespinor2 w12 622 22 612 Y11 21 21 11 Id 22 611 Y12621 Y11 422 21 612 e23 e2 Observe that 8 Y 6 v1d 2 W2 1 whereas a pure spinor in this case is e2 We can easily check now that pure spinors purespinor1 and purespinor2 have the desired commuting properties with the idempotent f gt u purespinori f amp c u u amp c reversion f 0 gt u purespinor2 f amp c u u amp c conjugation f 0 For more information see 3 and 30 9 Continuous Families of Idempotents Low Dimensional Examples In this section we will show how one can discover with CLIFFORD existence of continuous families of idempotents For a complete treatment of this
2. Id e1 e2 e3 elwe2 elwe3 e2we3 elwe2wes gt map reversion cbas B Id e1 e2 e3 elwe2 2 Fi 2 Id elwe3 2 Fi 3 Id e2we3 2 F gt 3 Id 2 F 3 e1 2 F 3 e2 2Fi 2 e3 elwe2we3 If instead of B we use a symmetric matrix g g or the symmetric part of B then gt map reversion cbas g Id e1 e2 e3 elwe2 elwe3 e2we3 elwe2we3 Convert now e1 A e2 to the dotted basis to get e1 A e2 e1 We2 gt convert elwe2 wedge_to_dwedge F el We2 Applying reversion to e1 We2 with respect to F one gets the reversed element in the dotted basis gt reversed_e1We2 reversion e1We2 F reversed_e1 We2 elwe2 F 2 Id 20 Rafat Abtamowicz Observe that the above element is equal to the negative of e1 We2 just like re versing elwe with respect to the symmetric part g of B gt reversed_e1lWe2te1iWe2 0 Finally convert reversed e1 We2 to the undotted standard Grassmann basis to get elwe2 gt convert reversed_e1lWe2 dwedge_to_wedge F elwe2 The above of course can be obtained by applying reversion to e1we2 with respect to the symmetric part g of B gt reversion elwe2 g reversion w r t the symmetric part g elwe2 This shows that the dotted wedge basis is the particular basis which is stable under the Clifford reversion computed with respect to F the antisymmetric part of the bilinear form B This requirement allows
3. The corresponding images y X p 7 and y T and Ch will be assigned to the Maple variables pSigma pSTS and pSST respectively gt pSigma pSTS pSST phi Sigma M phi STS M FBgens phi SST M STS SST pSigma pSTS pSST 5Id 2e1 9I1d 4V5e1 91d 4V5el Computations with Clifford and Grassmann Algebras Al We should be able to verify in Cl2 9 the following two factorizations of AA and ATA II ATA VETEVT 22 AAT U ETUT 23 like this gt evalb pTp simplify pV amp c pSTS amp c pVt gt evalb ppT simplify pU amp c pSST amp c pUt true true Finally we check the SVD of A which is A USV7 in the Clifford algebra language gt evalb p simplify pU amp c pSigma amp c pVt true 11 2 Additional comments In the previous section we have shown that it is possible to translate the ma trix algebra picture of the Singular Value Decomposition of a matrix A into the Clifford algebra language Although we have not abandoned entirely the linear algebra formalism in our example e g we have computed the eigenvalues and the eigenvectors of AT A and AA and only then we have found images of their eigenvectors in the spinor space Clz of these computations including solving the eigenvalue problem can be done entirely in the Clifford algebra language In the following we show how this can be accomplished The first comment is that it is also possible to compute the eigenvalues A1 A2 and t
4. can be discovered as follows gt dim_V 2 B diag 1 1 bas cbasis dim_V clidata real 2 simple a gt Id e2 Id Id e2 As shown above a standard primitive idempotent in C 2 9 is f 5 3e1 We will look however for the most general element f in C 2 9 that satisfies f f gt f add x i bas i i 1 27dim_V f x Id x2 e1 z3 e2 244 12 There are four real solutions gt sol map allvalues clisolve cmul f f f f sol_real remove has sol I sol_real 0 Id a 5 V1 4442 e2 44 e12 a V1 4242 e2 24 el2 Id 1 DTD y1 4g3 4242 el 3 e2 24 e12 id i V1 4r3 424 el 3 e2 24 e12 We verify that each solution is an idempotent of course 0 and Id are the trivial ones gt map x gt is simplify cmul x x x sol_real true true true true true true Observe that all nontrivial idempotents found above are ungraded i e they are neither odd nor even If we set 74 73 0 in the above two idempotents that contain VI 4 3 4x4 we recover the default mutually annihilating primitive pair However 1 1 gta Vit 4a 403 e1 Tez 401 ez 16 gives a two parameter family of idempotents in Cl2 9 as long as 1 4 x4 4 x3 gt 0 The classical discrete idempotents occupy the center 0 0 of that parameterized region in the real x3a4 plane It can be easily checked that the above two idempo tents in general do not ad
5. e2 ef ae2 1 a Id Irrespective of the bilinear form chosen the Grassmann multiplication table will always remain as gt wedgetable matrix 4 4 i j gt wedge cbas i cbas j Id el e el2 el 0 e12 0 e2 el2 0 0 e12 0 0 0 wedgetable Let B g F where g and F are respectively the symmetric and the antisym metric part of B gt g F matrix 2 2 g11 g12 g12 g22 matrix 2 2 0 F12 F12 0 gt B evalm gtF P gli g12 0 F12 9 g12 g22 F12 0 gll g12 F12 g12 F12 g22 6 Rafat Abltamowicz Then the Clifford multiplication table of the basis monomials in C B will be as follows gt MultTable matrix 4 4 i j gt cmul cbas i cbas j MultTable Id e1 e2 e12 e1 g11 Id e12 g12 F12 Id g11 e2 g12 F12 el e2 g12 F12 Id e12 g22 Id g12 F12 e2 g22 e1 e12 g12 F12 el g11 e2 g22 e1 g12 F12 e2 g12 F12 g22 g11 Id 2 e12 F12 Observe that the standard anticommutation relations ejej eje Bij Bji l 2g j1 1 are satisfied by the generators e i 1 2 n irrespective of the presence of the antisymmetric part F in B For example gt cmul g e1 e2 cmul g e2 e1 gt cmul B e1 e2 cmul B e2 e1 2 Id g12 912 F12 Id g12 F12 Id 2912 Id It is well known 29 35 that real Clifford algebras C0 V Q Clp are classified in terms of the sign
6. http math tntech edu rafal 2009 R Ablamowicz and B Fauser Mathematics of CLIFFORD A Maple Package for Clifford and Grassmann Algebras Advances in Applied Clifford Algebras Vol 15 No 2 2005 157 181 R Ablamowicz and B Fauser Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple Computer Physics Communications 170 2005 115 130 R Ablamowicz B Fauser K Podlaski and J Rembielinski Idempotents of Clifford algebras Czech J Phys 53 11 2003 949 955 R Abtamowicz and P Lounesto On Clifford algebras of a bilinear form with an antisymmetric part In Clifford Algebras with Numeric and Symbolic Computations R Ablamowicz P Lounesto and J Parra eds Birkha user Boston 1996 167 188 R Ablamowicz and G Sobczyk Software for Clifford geometric algebras Appendix in Lectures on Clifford Geometric Algebras and Applications R Ablamowicz and G Sobczyk eds Birkhauser Boston 2004 189 209 J L Anderson Green s functions in quantum electrodynamics Phys Rev 171 94 1954 703 11 P Angles Construction de rev tements du groupe conforme d un espace vectoriel muni d uni m trique de type p q Ann Inst Henri Poincare XXXIII 1 10980 33 51 Axiom Computer Algebra System http en wikipedia org wiki AXIOM 2009 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 Computations with Clifford and Gra
7. where x V u v AV and is the Grassmann grade involution Hence we can use the Clifford map yx Chevalley deformation of the Grassmann algebra to define a Clifford product of a one vector x and a multivector u as xu xdgut xAu 2 Analogous formula can also be given for a right Clifford map using the right contraction Lg implemented as the procedure RC The Clifford product cmul or its ampersand form amp c of two Grassmann basis monomials can now be defined as follows A single element from the first factor of the product is split off recursively and then the Chevalley s Clifford map is applied Namely eg A ANenA ec amp c ep A ea A A ep amp c ec Ip ef A Neg Oc Nex A Ag eaA Aen be ec amp c ef A Ae g 3 Specifically for e1 A e2 amp c e3 A e4 we have e1 Ae2 amp c e3 Aes e1 amp ce2 amp c e3 A e4 Ber e2 1 amp c e3 A e4 e amp c B e2 e3 e4 B e2 e4 e3 e2 A e3 e4 B e1 2 1 amp c e3 A e4 6In CLIFFORD the left contraction Ig is given by the procedure LC x u B or simply by LC x u where B is assumed as default The right contraction uLg x of u by x is encoded as a procedure RC u x B or simply RC u x 10 Rafat Abtamowicz and the second recursion of the process gives now B e2 e3 B e1 e4 B e2 e4 B e1 e3 B e2 e3 e1 A e4 B eg e4 e1 A e3 B e1 e2 e3 A e4 B e1 3 2 A e4 B e1 e4 e2 A 3 e1 A e2 A e3 A
8. 1989 I Porteous Topological Geometry Van Nostrand Reinhold London 1969 G C Rota and J A Stein Plethystic Hopf algebras Proc Natl Acad Sci USA 91 1994 13057 13061 J M Selig Geometrical Methods in Robotics Springer Verlag New York 1996 G Strang Introduction to Linear Algebra Wellesley Cambridge Press Wellesley 1998 S Tunney On the Continuous Families of Idempotents in the Clifford Algebra of the Euclidean Plane M S Thesis Tennessee Technological University 2003 Waterloo Maple Incorporated Maple a general purpose computer algebra system Waterloo http www maplesoft com 2009 F Wright Computing with Maple Chapman amp Hall CRC Boca Raton 2002 Rafa Ab amowicz Department of Mathematics Box 5054 Tennessee Technological University Cookeville TN 38505 USA e mail rablamowicz tntech edu
9. 2 2 Computations with Clifford and Grassmann Algebras 33 gt T linalg matrix 2 2 1 e2 e4 2 0 1 e4 e 1 kia 2 2 0 1 gt Tv amp cm T amp cm Tv evalm W W Tv amp cm T amp cm Tv 1 e24 e4 e2 1 e24 e4 e2 2 2 2 2 2 2 2 2 e4 e2 1 ef e4 e 1 e24 of o gt oP Cae 2z gt pseudodet W computing pseudo determinant of W Id Thus the above computation again confirms that W Tv amp cmT amp cmTv and that the pseudo determinant of W is 1 11 Singular Value Decomposition and Clifford Algebra In this section we will show how the Singular Value Decomposition SVD of a matrix can be translated into the Clifford algebra language For the background information on SVD we refer to 42 There are many uses of SVD such as in image processing description of the so called principal gains in a multivariable system 37 or in an automated data indexing known as Latent Semantic Indexing or LSI LSI presents a very interesting and useful technique in information retrieval models and it is based on the SVD 19 While in these practical cases computations are done numerically it may be of interest to ask whether SVD of a matrix can be performed in the framework of Clifford algebras Likewise whether SVD of a Clifford number can be found without using matrices That is if any new insights theoretical or otherwise into such decomposition could be gained when stated in the Clifford algebra language We will explore a
10. E Baylis ed Birkhauser Boston 1996 463 502 R Ablamowicz Matrix exponential via Clifford algebras J of Nonlinear Math Phys 5 3 1998 294 313 R Ablamowicz Spinor Representations of Clifford Algebras A Symbolic Approach CPC Thematic Issue Computer Algebra in Physics Research Phys Comm 115 1998 510 535 R Abltamowicz Helmstetter formula and rigid motions with CLIFFORD In Advances in Geometric Algebra with Applications in Science and Engineering Automatic Theorem proving Computer Vision Quantum and Neural Computing and Robotics E Bayro Corrochano and G Sobczyk eds Birkhauser Boston 2001 512 534 R Ablamowicz J Anderson and M Baswell Clifford algebra space singularities of inline planar platforms In Applications of Geometric Algebra in Computer Science and Engineering L Dorst ed Chapter 36 Birkhauser Boston 2002 463 502 R Ablamowicz and B Fauser On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form In Clifford Algebras and their Applications in Mathematical Physics R Ablamowicz and B Fauser eds Birkhauser Boston 2000 341 366 R Ablamowicz and B Fauser Hecke algebra representations in ideals generated by q Young Clifford idempotents In Clifford Algebras and their Applications in Math ematical Physics R Ablamowicz and B Fauser eds Vol 1 Algebra and Physics Birkhauser Boston 2000 245 268 R Ablamowicz and B Fauser CLIFFORD and BIGEBRA for Maple
11. Qini are the diagonal entries of the quadratic form Q to be precise Qi e i 1 n extended by linearity to the whole algebra Then tp gives the transposition of the corresponding spinor representation For a thorough treatment of the singular value decomposition for Grassmann and Clifford algebras in the Hopf algebraic language see 23 12 Conclusions The aim of this note was to present a few examples of computations with Clifford algebras where symbolic capabilities of Maple and the CLIFFORD BIGEBRA pack ages were utilized In some a better understanding of the Clifford algebra structure and its properties especially of the Clifford algebras C B of an arbitrary bilin ear form B was achieved see also 6 Gaining a computational proficiency with Clifford algebras of a quadratic form notwithstanding such as their spinor rep resentations or computation of bilinear forms on spinor spaces properties of the contraction the reversion the wedge the dotted wedge and the Clifford products all curiously dependent on the antisymmetric form of B have been successfully for mulated In others a discovery of specific elements was accomplished like finding 27 This has been verified by brute force with CLIFFORD in dimensions up to 9 for simple C p q and such that p q 0 1 2 mod 8 That is for the Clifford algebras whose spinor representations are real A Maple code for a procedure tp is shown in Appendix C Com
12. V54 2 o2 V5 2 Computations with Clifford and Grassmann Algebras 39 When we compute the eigenvectors u1 uz of AAT we will not necessarily have Av ciui i 1 2 This is because the choice of u1 uz is not consistent with the choice of v1 V2 gt P assignL sort eigenvects AAT byeigenvals gt for i to N do lambda i P 2 i gt u i map simplify normalize P 3 i end do Notice that Av 01u while Avg ooQu2 gt radsimplify evalm A amp vi sigmal u1 this checks out 0 0 gt radsimplify evalm A amp v2 sigma2 u2 this does not gt radsimplify evalm A amp v2 sigma2 u2 this checks out 14 6 V5 8 4v5 10 2V5 Vi0 2v5 0 0 Let s re define uz as uz and call it u22 and let s also rename uy as u11 gt uii evalm ul u22 evalm u2 Then we will have Avii Cuji i 1 2 In the Clifford algebra Clz 9 we need to perform similar computations with v1 v2 The elements su y uii SU2 y u22 contained in the spinor ideal S Cg of need to be found first gt for i to N do su i convert u i i spinor spinorbasis od a1 CON f2 a2 LEV 5 V10 2v5 V10 2V5 V10 2V5 V10 2V5 The verification of the condition 20 in C 2 9 looks as follows gt for i to N do simplify p amp c sv i sigma ixsu i od 2 0 0 Now we may define the orthogonal matrix U u11 u22 and its image U in Clz 9 which we assign to pU gt U radsimplify augment u
13. for i from 0 to min N2 max_grade j iterate over all i vectors of psety not exceeding max_grade while others are zero begin F2 lt N2 N2 i i number of terms N2 over i for n from 1 to F1 for all j vectors begin for m from 1 to F2 for all i vectors begin res lt res pSgnx pos1 n pSgny pos2 m cup fun1 psetx PN1 pos1 n fun2 psety pos2 m Iname makeclibasmon gt funl gt psetx pos1 n fun2 gt psety PN2 pos2 m end end pos2 lt pos2F2 end posi lt pos1F1 end reorder gt res reorder basis elements in res into standard order end cmulRS Appendix C Appendix Code of the Transposition Procedure tp Here is a code of the procedure tp that accomplishes the transposition anti automorphism in Cp q tp proc xx local x L p co u x displayid xx if type x clibasmon then if x Id then Id else p op cliterms x L extract p integers list L of indices L seq L nops L i 1 i 1 nops L reversed list L u cmul seq B L i L i cat e L i i 1 nops L reorder u end if elif type x climon then p op cliterms x co p coeff x p p co procname p elif type x clipolynom then u clilinear x procname end if end tp 46 Rafat Ablamowicz References 1 R Ablamowicz Clifford algebra computations with Maple In Clifford Geometric 10 11 12 13 14 15 16 Algebras with Applications in Physics Mathematics and Engineering W
14. rq p and r is the Radon Hurwitz number Number k is the number of factors 1 T where T i 1 k is a set of commuting basis Grassmann monomials squaring in Cl Q to 1 whose product gives a primitive idempotent f in C Q Spinor representation for all Clifford lType RHnumber in a Maple session when CLIFFORD is installed for more help Computations with Clifford and Grassmann Algebras 7 algebras C Q 1 lt n p q lt 9 and for any signature p q has been pre computed 3 and can be retrieved from CLIFFORD with a procedure matKrepr For example 1 vectors e and e2 in Ch have the following spinor representation in the basis f e2 amp c f of S Cho f gt matKrepr 2 0 In another example Clifford algebra C 3 of R is isomorphic with Mat 2 C gt B linalg diag 1 1 1 clidata 3 0 complex 2 simple Zu 5 el Id e2 e3 e28 Id e23 Id e2 and its spinor representation is given in terms of Pauli matrices gt matKrepr 3 0 1 0 0 1 0 e23 jel e2 TEIE 0 1 1 0 e23 0 Notice that F span Id e23 e23 e2we3 is a subalgebra of Cl3 isomorphic to C Since Pauli matrices belong to Mat 2 F it is necessary for CLIFFORD to compute with Clifford matrices that is matrices of a type type climatrix with entries in a Clifford algebra gt M1 M2 M3 rhs 4 1 rhs 2 rhs 4 3 1 0 i 0 e23 M1 M2 M3 l l o i 1 0 e23 0 Of course Pauli matrices
15. Let be either of the two anti involutions 6 or G_ of Clp q Following 39 Lounesto argues that when Clp is simple for any spinor ideal V Clp qf generated by a primitive idempotent f there exists an invertible element u in Clp q such that fu u f Let F fC f Then V becomes a right F module Define a map gt A uG A u which is an automorphism of the ring F Then the map Y y uB w right to left F semilinear on V and therefore the map V x V F defined by V o uy Be 14 is a scalar product on V It turns out that the element u can be chosen to be an element e4 of the Grassmann basis for Cp q The first two arguments beta_plus and beta_minus are spinors w and which are Clifford polynomials of type clipolynom The third argument is a primitive idempotent f for simple Clifford algebras or f f for semisimple Clifford algebras The fourth optional argument s will be a placeholder for the invertible element u described above Example 3 Let s compute the two bilinear forms beta_plus and betaminus on S Clz of a spinor space of the Clifford algebra of the Euclidean space R To shorten output procedure makealiases is used 11When computing matrix products one can apply the Clifford product the wedge product or any product to the matrix entries See help in CLIFFORD 12Similar discussion is extended to semi simple Clifford algebras Cey q In that case one considers W Clp qe where
16. a2 al2 ad al Example 2 In this example we consider the Clifford algebra Cla o C 2 In the following we will see how matrices with entries in Cls o and more precisely in K fClz of C are handled Computations with Clifford and Grassmann Algebras 23 gt dim 3 B linalg diag 1 1 1 define the bilinear form B for C1 3 0 clibasis cbasis dim compute Clifford basis for C1 3 0 gt data clidata B retrieve and display data about C1 3 0 Vv data complex 2 simple a a Id e2 e8 e23 Id e23 Id e2 gt f data 4 assign pre stored idempotent to f or use your own here sbasis minimalideal clibasis f left compute a real basis in C1 3 0 f gt Kbasis Kfield sbasis f compute a basis for the field K vV d Id el e23 e123 Kbasis I gt 5 5 a Id e23 gt SBgens sbasis 2 generators for a real basis in S gt FBgens Kbasis 2 generators for K are two since K C FBgens Id e28 gt K_basis spinorKbasis SBgens f FBgens left 3 Id el e2 e12 K_basis ls ia Id e2 left Here are the matrices representing 1 vector basis monomials of C39 Matrices sigma 1 sigma 2 and sigma 3 are the well known Pauli matrices with entries in the field K gt sigma 1 sigma 2 sigma 3 gt op map spinorkrepr e1 e2 e3 K_basis 1 FBgens left 1 0 0 1 0 e23 O1 02 03 0 1 1 0 e23 0 Let s find matrices representing the two b
17. b e14 2e8 T is Vahlenmatriz T true A Vahlen matrix Dil that gives a dilation transformation gt delta 1 4 a positive parameter gt Dil linalg matrix 2 2 sqrt delta 0 0 1 sqrt delta gt isVahlenmatrix Dil isVahlenmatrix Dil 1 5 0 Dil 0 2 is Vahlenmatrix Dil true Finally a Vahlen matrix Tv that gives a transversion transformation gt c 2 el e3 a vector in R 3 1 gt Tv linalg matrix 2 2 1 0 c 1 gt isVahlenmatrix Tv isVahlenmatrix Tv 1 0 2el e3 1 c 2el e3 Ty is Vahlenmatria Tv true 8 we will obtain an element If we now take a product of these four matrices above conf of the conformal group in R gt conf R amp cm T amp cm Dil amp cm Tv el 10 623 4 e123 2 e2 conf 2 2 e123 4 e2 2e12 Since in the product above each matrix appeared exactly once the diagonal entries of conf must be invertible We find the inverses of each element with cinv gt cinv conf 1 1 inverse of conf 1 1 _2e12 4023 401 401 18 cm denotes a matrix multiplication in CLIFFORD with the Clifford product applied to the matrix entries Computations with Clifford and Grassmann Algebras 31 gt cinv conf 2 2 inverse of conf 2 2 e12 es However there are elements in the conformal group of R whose Vahlen matrices do not have invertible elements at all The following example of such matrix is due to Joh
18. non zero and are likely to remain in the final result If all B j are non zero and different so that no cancellation takes place only exactly those terms will be returned Yet no superfluous terms are created like in the recursive procedure cmulNUM that later are canceled in subsequent recursive steps The combinatorial power of the Hopf algebraic approach is clearly demonstrated with this algorithm and its superior behavior shows up in benchmarks 9 10 SBIGEBRA is an extension of CLIFFORD and was specifically developed to work with Hopf alge bras 8 12 Rafat Abtamowicz The two internal Clifford multiplication procedures along with their advan tages and disadvantages are discussed in 9 It should suffice to say here that cmulNUM is fast on sparse numeric matrices and on numeric matrices in general when dim V gt 5 while the procedure cmu1RS was designed for high efficiency with symbolic calculations 4 Dotted and Undotted Grassmann Bases 4 1 The dotted wedge The dotted wedge product was introduced by Lounesto 35 Clifford algebras with the dotted and the undotted wedge products are isomorphic It turns out that in various situations i e in quantum field theory or in a representation theory one is interested in studying isomorphic but not identical algebras in which mathematical objects can be expressed in different bases such as the dotted and the undotted wedge bases 9 It was shown above that CLIFFORD uses the Gras
19. one to distinguish Clifford algebras C g which have a symmetric bilinear form g from those which do not have such symmetric bilinear form but a more general form B instead We call the former classical Clifford algebras while we use the term quantum Clifford algebras for the general not necessarily symmetric case 6 cmul X CEUX D CLO X ClX reversion X amp reversion X CEUX D CLOX reversion X switch cmul X CEUX D CL X ClX Diagram 7 Relation between the reversion X of type X g F B with the corresponding Clifford multiplication cmul X The map called switch is the ungraded switch of tensor factors that is switch A B B amp A Computations with Clifford and Grassmann Algebras 21 7 Spinor Representation of C Q in Minimal Left Ideals See 3 for a complete treatment of symbolic computation of spinor representations of simple and semisimple Clifford algebras Here we provide some basic facts and a few examples We will use a procedure spinorKrepr from CLIFFORD Procedure spinorkrepr finds a matrix spinor representation of any Clifford polynomial in a minimal left ideal S C Q f or a minimal right ideal S fOQ over the field K fC0 q f Depending on the signature of Q the field K is isomorphic to R when p q mod 8 0 1 2 C when p q mod 8 3 7 or H when p q mod 8 4 5 6 In order to compute the spinor representation one needs i a Clifford polynomial p whose matrix of
20. satisfy the same defining relations as the basis vectors e1 2 and e3 3 For example gt M1 amp cm M2 M2 amp cm M1 evalm M1 amp cm M2 M2 amp cm M1 gt fel amp c e2 e2 amp c el e1 amp c e2 e2 amp c el M1 amp em M2 M2 amp cm M1 el amp c e e2 amp c e1 0 We use the sloppy notation 1 1 in Clifford algebra valued matrices which produces a simpler display 3Here amp cm is a matrix product where Clifford multiplication is applied to the matrix entries See amp cm for more information 8 Rafat Abtamowicz gt M1 amp cm Mi evalm M1 amp cm M1 M2 amp cm M2 evalm M2 amp cm M2 gt M3 amp cm M3 evalm M3 amp cm M3 gt fel amp c elf e1 amp c e1 e2 amp c e2 e2 Ke e2 e3 amp c e3 e3 amp c e3 1 0 1 0 M1 amp cm M1 M2 amp em M2 M3 amp em M3 0 1 0 1 0 1 el amp c el Id e2 amp c e2 Id e3 amp c e3 Id The procedure matKrepr gives the linear isomorphism C Q Mat 2 R and in general C0 Q Mat 2 K where K R C H for simple algebras and Cl Q Mat 2 K 6 Mat 2 K where K R H for semisimple algebras In this latter case it is customary to represent an element in C Q in terms of a single matrix over a double field R R or H H rather than as pair of matrices 4 One can easily list signatures of the quadratic form Q for which C Q is simple or semisimple For more information type 7all_sigs For exa
21. the field K needs to be found ii a list of basis elements of the type type clipolynom which give a K basis for S over the field K Among those elements there is the primitive idempotent f used to generate S iii a list of elements of the type type clibasmon which generate the field K and iv a string left or right depending whether S is a left or right minimal ideal Since the steps needed to compute spinor representations are rather involved the user may just want to use already pre computed and stored matrices over K representing 1 vectors Procedure matKrepr uses stored data and can compute matrices representing any Clifford polynomial A few simple examples are shown below Example 1 Clifford algebra Clz 9 of the Euclidean plane R is known to be iso morphic to R 2 the ring of 2 x 2 real matrices gt dim 2 B linalg diag 1 1 define the bilinear form B for C1 2 0 gt clibasis cbasis dim compute a Clifford basis for C1 2 0 gt data clidata B retrieve and display data about C1 2 0 data real 2 simple a Id e2 Id Jd e2 gt f data 4 assign pre stored idempotent to f or use your own here gt sbasis minimalideal clibasis f left compute a real basis in C1 2 0 f f Id el e e12 sbasis l 5 ore Id e2 left gt Kbasis Kfield sbasis f compute a basis for the field K SBgens sbasis 2 generators for a real basis in S gt FBgens Kbasis 2 gene
22. topic we refer to 11 It is well known 30 that any primitive idempotent f in C q is expressible as a product 1 1 1 f A ten 5 Lten 5 Een 15 where e7 i 1 k are commuting basis monomials with square 1 and k q Tq p Where r is the Radon Hurwitz number 4 Furthermore C p q has a com plete set of 2 primitive mutually annihilating idempotents gt each with k factors as shown in 15 In CLIFFORD procedure clidata displays one chosen primitive idempotent to generate precomputed spinor representations of Clifford algebras in dimensions up to 9 13 One should not confuse this complex conjugation with Maple s symbol for multiplication as in phit phi11i Id phi12 e23 above 14The Radon Hurwitz number is defined by recursion as rj g ri 4 and these initial values ro O r 1 r2 r3 2 r4 r5 re r7 3 In CLIFFORD it is given by the procedure RHnumber 15 There are 2 possible sign choices for the k factors in 15 Any two primitive idempotents f and g obtained by selecting different signs in 15 are mutually annihilating that is fg gf 0 Computations with Clifford and Grassmann Algebras 27 We will show how to find continuous families of idempotents in a Clifford al gebra C0 Q by finding a general solution to the equation f f with a procedure clisolve As low dimensional examples we will use Clz9 Cl1 1 and Cso Example 4 Families of idempotents in Clz 9 see also 43
23. under y in Clz 9 we compute them now and store them under the variables pV and pVt respectively gt pV phi V M finding image of V in C1 2 0 gt pVt phi t V M finding image of t V in C1 2 0 pV ois RSs he 5 25 A2 el 20 40 8 20 40 8 1 1 1 1 1 1 55 2V5 gg Y5 5 Ml 2 55 M2V5 gg VS 3 e12 1 V10 2V5 2 V10 2 V5 The fact that V is orthogonal can be easily verified in the matrix language in C29 it can be done as follows gt simplify cmul pVt pV Id We repeat the above steps and apply them to AA In the process we will find its eigenvectors u1 u2 We must make sure that Av ciu where o i Vi i 1 2 This will require extra checking and possibly redefining of the w s gt AAT evalm A amp transpose A computing AAT 13 8 8 5 AAT The image of AAT under y in Cho we denote as ppT gt ppT phi AAT M finding image of AAT in C1 2 0 ppT 9Id 4e14 8e2 In this case the minimal polynomial of ppT and the characteristic polynomial of AA are of course the same gt pol2 charpoly AAT lambda characteristic polynomial of AAT pol2 X 18A 1 gt ppT climinpoly ppT ppT x 18x41 Since matrices ATA and AAT have the same characteristic polynomials their eigenvalues 1 A2 will be the same We define therefore the singular values c and o2 of A as the square roots of A and Ag gt for i to N do sigma i sqrt lambda i od ol
24. well known fact that when p q 4 1 mod 4 the Clifford algebra Clp q is a simple algebra of dimension 2 n p q isomorphic to a full matrix algebra Mat 2 K of 2 x 2 matrices with entries in the division ring K The ring K is a subalgebra of Clp isomorphic to R C or H depending on the signature p q and the dimension n see 3 Thus any operation performed on a matrix A Mat 2 K can be expressed as an operation on the uniquely corresponding to it element p in Clp q The choice of the signature p q depends on the size of A and the division ring K Of course for computational reasons one should find the smallest Clifford algebra Clp such that the given matrix A can be 19 The value k is determined by the formula k g rgq p where r is the Radon Hurwitz number The Radon Hurwitz number is defined by a recursion as rj4g ri 4 and these initial values ro 0 r1 1 r2 r3 2 r4 r5 r6 HS Pee 3 34 Rafat Abtamowicz embedded into Mat 2 K x C y q In the following we will use the same approach as in 2 where a technique for matrix exponentiation based on the isomorphism y was presented In particular we will use a faithful spinor representation of Clp in a minimal left ideal S Clp qf generated by a primitive idempotent f Symbolic computations of such representations with CLIFFORD were shown in 3 Following 42 let A be an m x n real matrix of rank r Then the SVD of A is defined a factor
25. while _ accom plishes conversion back to the undotted basis To illustrate this fact we first con tract from the left an arbitrary element u in C B by 1 e e e e Ae Nex 1 lt i j k lt 3 here we limit our example to dim V 3 and then we extend it to a left contraction by an arbitrary element v in C B gt u add x i w_bas it1 i 0 7 uF convert uw wedge_to_dwedge F gt vi add y i w_bas it 1 i 0 7 Contraction with respect to 1 gt evalb LC Id u B convert LC Id uF B dwedge_to_wedge F Computations with Clifford and Grassmann Algebras 15 10 r Cl B CLUB Cl B 8 Cl B dp Jg JF Ce B COB Diagram 2 Contraction w r t wedge and dotted wedge true Contraction with respect to e gt evalb LC ei u B convert LC ei uF B dwedge_to_wedge F true Contraction with respect to e A e gt evalb LC eiwej u B convert LC eiwej uF B dwedge_to_wedge F true Contraction with respect to e Ae A ex gt evalb LC eiwejwek u B convert LC eiwejwek uF B dwedge_to_wedge F true Finally contraction with respect to an arbitrary element v gt evalb LC v u B convert LC v uF B dwedge_to_wedge F true Once we have the dotted and the undotted Grassmann bases we can build a Clifford algebra C B over each basis set but with different bilinear forms B g or B g F respectively following notation from 12 Let us compute various Cli
26. 1 1987 14 48 T Y Lam The Algebraic Theory of Quadratic Forms The Benjamin Cummings Publishing Company Reading 1973 P Lounesto Scalar products of spinors and an extension of Brauer Wall groups Found Phys 11 1981 721 740 P Lounesto and E Latvamaa Conformal transformations and Clifford algebras Proceedings of the American Mathematical Society 79 4 1980 533 538 P Lounesto R Mikkola and V Vierros CLICAL User Manual Helsinki University of Technology Institute of Mathematics Research Reports A248 1987 P Lounesto CLICAL and counter examples In Clifford Algebras with Symbolic and Numeric Computations R Ablamowicz P Lounesto and J Parra Birkhauser Boston 1996 3 30 P Lounesto and A Springer M bius transformations and Clifford algebras of Euclidean and anti Euclidean spaces In Deformations of Mathematical Physics J Lawrynowicz Kluwer Academic Publishers 1989 79 90 P Lounesto Clifford Algebras and Spinors 2nd ed Cambridge University Press Cambridge 2001 Macaulay 2 A software system for research in algebraic geometry and commutative algebra http www math uiuc edu Macaulay2 2009 J M Maciejowski Multivariable Feedback Design Addison Wesley Wokingham England 1989 48 38 39 40 41 42 43 44 45 Rafa Ab amowicz J Maks Modulo 1 1 periodicity of Clifford algebras and the generalized anti M bius transformations Thesis Technische Universiteit Delft
27. 11 u22 defining matrix U 1 V5 1 V5 viz V1I0 2V5 V10 2V5 i 1 1 V10 2V75 V10 2V 5 25Expressions 1 and 2 showing up in the Maple output for pU are just place holders for V10 2 5 and v10 2 V5 respectively and are as shown at the end of the display 40 Rafat Abltamowicz gt pU phi U M finding image of U in C1 2 0 gt pUt phi t U M finding image of t U in C1 2 0 1 1 1 1 1 1 1 1 1 1 il 1 1 V10 2V5 2 V10 2 V5 The fact that U is an orthogonal matrix can be easily now checked both in the matrix language and in the Clifford language gt radsimplify evalm t U amp U U is an orthogonal matrix o1 Id gt simplify pUt amp c pU Finally we define matrix using a procedure makediag Recall from 42 that has the same dimensions as the original matrix A and that 57D XXT are the diagonal forms of A A and AAT respectively In this example matrices XTS and EST are the same since is a square diagonal matrix Normally these matrices are different although their nonzero diagonal entries are the same Therefore we have ATA VETOVT AAT UEYTUT X fi dct i 21 ETS DDT i 0 o5 Matrices ETX and D we assign to Maple variables Sigma STS and SST re spectively gt Sigma makediag m n seq sigma i i 1 N gt STS SST evalm t Sigma amp Sigma evalm Sigma amp t Sigma sn 60 0 0 V5 2 V5 2 0 V5 2 0 0 v5 2 0 v5 2
28. 2 2 2 2 e4 el 1 elf e4 el 1 elf yo 2 Be Dy 2z gt pseudodet W computing pseudo determinant of W Id Thus the above computation confirms that W Tv amp cemT kem Tv and that the pseudo determinant of W is 1 There is another variation of Johannes Maks example of a Vahlen matrix W without any invertible entries Matrix W represents an element in the identity component of the conformal group of R t gt W evalm 1 2 linalg matrix 2 2 1 e24 e2 e4 e2t e4 1 e24 1 e24 e ek Notice that the diagonal elements of W are non trivial idempotents in C 3 1 hence they are not invertible in C 3 1 gt type W 1 1 idempotent element 1 1 of W is an idempotent gt type W 2 2 idempotent element 2 2 of W is an idempotent true true Notice also that the off diagonal elements of W are isotropic vectors in R hence they are also non invertible gt emul W 1 2 W 1 2 cmul W 2 1 W 2 1 0 0 Finally we verify that W is a Vahlen matrix gt isVahlenmatrix W isVahlenmatrix W is Vahlenmatrix W true However W is an element of the identity component of the conformal group in R1 since its pseudo determinant is 1 and since it can be written as a product of a transversion a translation and a transversion As before W is not a product of just one rotation one translation one dilation and or one transversion gt Tv linalg matrix 2 2 1 0 e2 e4 2 1 1 0 Tv ef e2 1
29. DEPARTMENT OF MATHEMATICS TECHNICAL REPORT COMPUTATIONS WITH CLIFFORD AND GRASSMANN ALGEBRAS RAFAL ABLAMOWICZ MAY 2009 No 2009 4 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookeville TN 38505 Computations with Clifford and Grassmann Algebras Rafat Ablamowicz pene S Abstract Various computations in Grassmann and Clifford algebras can be performed with a Maple package CLIFFORD It can solve algebraic equations when searching for general elements satisfying certain conditions solve an eigenvalue problem for a Clifford number and find its minimal polynomial It can compute with quaternions octonions and matrices with entries in Cl B the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B It uses standard undotted Grassmann basis in C Q but when the antisymmetric part of B is non zero it can also compute in a dotted Grassmann basis Some examples of computations are discussed Mathematics Subject Classification 2000 15A66 68W30 Keywords Quantum Clifford algebra conformal group contraction dotted wedge product grade involution Grassmann algebra Hopf algebra multivec tor octonions quaternions reversion singular value decomposition spinors Vahlen matrix wedge product Contents Introduction Notation and Basic Computations Clifford Product in Ce B Dotted and Undotted Grassmann Bases 4 1 The dotted wedge 4 2 Dotted and undotted wedge bases 4 3 Contraction and C
30. Mat 2 R the output from the procedure spinorKbasis shown below has two basis elements and their generators modulo f gt Kbasis spinorKbasis SBgens f FBgens left 1 1 1 1 Kbasis ls Id 5 el 5 e2 5 e12 Id e2 left Thus the real spinor basis in consists of the following two polynomials gt for i to nops Kbasis 1 do f i Kbasis 1 i od 1 1 1 1 1 Id el 2 e2 e12 f 2 2 f 2 2 and it uses the Gram Schmidt orthogonalization process if necessary to return a complete list of orthogonal eigenvectors makediag makes a diagonal matrix consisting of singular values 21For more information see 3 and help pages in CLIFFORD 36 Rafat Abtamowicz We are in position now to compute matrices M M2 M3 M4 representing each of the four basis elements 1 e1 e2 12 of Cl2 9 in the basis fi fo gt for i to nops clibas do M i subs Id 1 matKrepr clibas i end do We will use a new procedure phi which realizes the isomorphism y from Mat 2 R to Clz 9 This way we can find the image p y A in C2 of any real 2 x 2 matrix A Knowing image y M of each matrix M i 1 4 in terms of some Clifford polynomial in C2 we can easily find p y A as follows gt p phi A M finding image of A in C1 2 0 p 2Id 2e2 e12 gt pT phi t A M finding image of t A in C1 2 0 pT 2 Id 2 e2 e12 Next we compute a symmetric matrix ATA denoted in Maple a
31. als B name of bilinear form begin if nargs lt gt 3 then error exactly three arguments are needed end if if has 0 map simplify al a2 then return 0 end if if a2 Id then return al end if if al Id then return a2 end if L lt indices from al N lt length of L coB nameB lt coefficient of B B to handle B if N 0 then return coeff al Id a2 elif N 1 then L2 lt list of indices from a2 return reorder simplify makeclibasmon L 1 op L2 add 1 i 1 coB nameB L 1 L2 i makeclibasmon subs L2 i NULL L2 i 1 nops L2 elif N 2 then xl lt substring al 1 2 x2 lt substring al 4 5 p2 lt procname x2 a2 B S lt clibilinear x1 p2 procname B return simplify S coB nameB op L a2 end if 44 Rafat Abtamowicz x lt cat e L 1 pl lt substring al 1 3 N 4 p2 lt procname x a2 B S lt clibilinear p1 p2 procname B add 1 i coB nameB L i L 1 procname makeclibasmon subs L i NULL L 1 2 a2 B i 2 N return reorder simplify S end cmulINUM Appendix B Appendix Code of cmu1RS Here is a pseudocode of the procedure cmul1RS based on the combinatorial process of Rota Stein cmulRS x y B x y two Grassmann monomials B bilinear form begin Istx lt list of indices from x lsty lt list of indices from y NX lt length of lstx NY lt length of lsty funx lt function maps integers 1 NX onto elements of lstx keeping their order funy lt funct
32. annes Maks 38 Matrix W defined below represents an element in the identity component of the conformal group of R gt W evalm 1 2 linalg matrix 2 2 1 e14 el e4 e1t e4 1 e14 1 elf el e4 272 oT el e4 1 elf D D Ce D Notice that the diagonal elements of W are non trivial idempotents in C 3 hence as such they are not invertible gt type W 1 1 idempotent element 1 1 of W is an idempotent true gt type W 2 2 idempotent element 2 2 of W is an idempotent true Notice also that the off diagonal elements of W are isotropic vectors in R t hence they are also non invertible In C3 such vectors have zero squares gt cmul W 1 2 W 1 2 cmul W 2 1 W 2 1 0 0 Let s now verify that matrix W defined above is a Vahlen matrix gt isVahlenmatrix W isVahlenmatrix W true However matrix W represents an element of the identity component of the con formal group in R since its pseudo determinant is 1 and since it can be written as a product of a transversion a translation and a transversion Thus in an other words W is not a product of just one rotation one translation one dilation and or one transversion gt Tv linalg matrix 2 2 1 0 e1 e4 2 1 1 0 Tv ef 4a i 2 2 gt T linalg matrix 2 2 1 e1te4 2 0 1 1 4_et 2 2 32 Rafat Abtamowicz gt Tv amp cm T amp cm Tv evalm W W Tv amp cm T amp cm Tv 1 elf ef _ el 1 elf e4 _ el 2 2 2 2 _
33. asis elements in the spinor ideal S Cls of As expected these matrices over K have the following form gt 1 2 K_basis 1 1 K_basis 1 2 ft fare 248 e2 el2 a 2 gt F1 F2 op map spinorkrepr f1 2 K_basis 1 FBgens left 1 0 F1 F2 A 0 0 1 0 Thus a spinor s is a complex vector written in terms of the basis f1 fo and its one column complex matrix with entries in K 1 e2 Ae3 is gt psi 1 psi 2 a Id b e23 ctId d e23 Y Y2 a Id b e23 c Id d e23 24 Rafat Abtamowicz gt s f1 amp c psi 1 f2 amp c psi 2 remember that S is a right K vector space _ ald be2 3 ael be123 cel2 del8 ce2 d e3 2 2 2 2 2 2 2 2 gt sm matKrepr s matrix of s a be23 0 sm c de23 0 Since CLIFFORD can handle computations with matrices in any Clifford alge bra it can also handle spinor representations in quaternionic spinor spaces and in spinor spaces over dual numbers in the case of semisimple Clifford algebras 3 8 Two Scalar Products in Spinor Ideals Scalar products 6 and _ in spinor ideals S Clp qf are discussed and clas sified in 30 39 In CLIFFORD there are corresponding procedures beta_plus and beta_minus that compute scalar products 34 w and G_ w respectively for any two spinors S Recall that G4 denotes the reversion in Clp while G_ denotes the conjugation that is a composition of the grade involution and the reversion in Clp
34. assmann wedge product and its Grassmann co product as shown by the following tangle Here A is the Grassmann exterior wedge product and A is the Grassmann exterior co product which is obtained from the wedge product by a categorial duality To every algebra over a linear space A with a product we find a co algebra with a co product over the same space by reversing all arrows in all axiomatic commutative diagrams Note that the co product splits each input factor x into a sum of tensor products of ordered pairs 1 i 7 2 The main requirement is that every such pair multiplies back to the input x when the dual operation of multiplication is applied i e 7 1 X 2 x for each i th pair The cup like part of the tangle decorated with B is the bilinear form B on the generating space V extended to the whole Grassmann algebra It isa map B Vx V gt kwithB VxV gt k evaluating to B x y on vectors in V Hence cmu1RS computes the Clifford product on Grassmann basis monomials x and y for the given B which is later extended to Clifford polynomials by bilinearity as follows n m cmulRS x Y B gt X ray A Y 2 j B 2 i vaj 5 i 1 j 1 where n and m give the cardinalities of the required splits and the sign is due to the parity of a permutation needed to arrange the factors After reviewing the code of cmulRS given in Appendix B it becomes clear that in this algorithm only those terms are created which might be
35. ature p q of Q and the dimension dim V n p q Information about all Clifford algebras Clp q 1 lt n lt 9 for any signature p q has been pre computed and stored in CLIFFORD and it can be retrieved with a procedure clidata For example for the Clifford algebra Cl2 also denoted as Clz of the Euclidean plane R we find gt clidata 2 0 Clifford algebra of the Euclidean plane real 2 simple iu el Id e2 Id Id e2 The meaning of the first three entries in the above output list is that Cz is a simple algebra isomorphic to Mat 2 R The 4th entry in the list gives a primitive idempotent f that has been used to generate a minimal left spinor ideal S Cla f and subsequently the left spinor lowest dimensional and faithful representation of C y in S In general it is known that depending on p q and n dim V the spinor ideal S Clp f is a right K module where K is either R C or H for simple Clifford algebras when p q 1 mod 4 or R R and H H for semisimple algebras when p q 1 mod 4 27 30 Elements in the 5th entry here Id e2 generate a real basis in S with respect to f that is S span Id amp c f e2 amp c f span f e2 amp c f Elements in the 6th entry span a subalgebra F of Ce Q that is isomorphic to K In the case of Cla we find that F span Id R The last entry in the output gives 2 generators of S with respect to f viewed as a right module over K where k q
36. d up to 1 and do not annihilate each other unless both parameters are zero Example 5 Let s now change the signature from 2 0 to 1 1 and repeat the above computations in the Clifford algebra Cli of neutral signature gt dim_V 2 B diag 1 1 bas cbasis dim_V clidata real 2 simple a Id e1 Id Id e1 28 Rafat Abtamowicz gt f add x i bas i i 1 27dim_V f x Id 2 el z3 e2 24 12 gt sol map allvalues clisolve cmul f f f f sol_real remove has sol I Id Id 1 sol_real 0 Id aes j y1 4x4 e2 24 e12 IG y1 4x42 e2 24 e12 Id 1 DED y1 4g3 4242 el 3 e2 24 e12 a 5 V1 4 4237 Fra el 23 e2 24 e12 gt map x gt is simplify cmul x x x sol_real true true true true true true Thus like in the Euclidean case we find that 1 1 4 V14 4232 4x42 e1 x3 2 z4 1 A 2 17 Qos 22 gives a two parameter family of idempotents provided 1 423 424 gt 0 Like in the Euclidean case we find that the idempotents in the pair 17 do not add up to 1 and do not mutually annihilate unless 73 x4 0 In that case we find graded idempotents 4 e A In the anti Euclidean signature 0 2 we only find as expected trivial idem potents in Cp H In higher dimensions for example in C39 one also finds families parameterized by more than two parameters 10 Vahlen Matrices F
37. dditional vectors are being annihilated by A and AT re spectively that is they are eigenvectors of A and AT or of AT A and AAT that correspond to the eigenvalue 0 care has to be exercised when finding them For example while the eigenvectors of the symmetric matrix AAT are automatically orthogonal provided they correspond to different eigenvalues eigenvectors of AAT that correspond to the 0 eigenvalue don t need to be orthogonal in this case the Gram Schmidt orthogonalization process is used to complete the two sets 11 1 SVD of a 2 x 2 matrix of rank 2 In this section we present a simple example of SVD of a 2 x 2 real matrix of rank 2 The purpose of this example is just to show step by step how finding the SVD of a matrix can be done in the Clifford algebra language Reader is encouraged to perform these computations with CLIFFORD and an additional package asvd gt A matrix 2 2 2 3 1 2 defining A 20Package asvd contains the following procedures used in the text phi that provides an iso morphism between a matrix algebra and a Clifford algebra radsimplify that simplifies radical expressions in matrices and vectors assignL that writes an output from a Maple procedure eigenvects in a suitable form it sorts eigenvectors according to the corresponding eigenvalues Computations with Clifford and Grassmann Algebras 35 2 3 1 2 Since A Mat 2 R we need to find p q such that Clp q x Mat 2 R Procedure all_sigs buil
38. dotted wedge amp dw can be extended to elements of higher grades Its properties are discussed next Procedure dwedge and its infix form amp dw requires an index which can be a symbol or an antisymmetric matrix That is dwedge computes the dotted wedge Computations with Clifford and Grassmann Algebras 13 product of two Grassmann polynomials and expresses its answer in the undotted basis Special procedures exist which convert polynomials between the undotted and dotted bases When no index is used the default is F gt dwedge K e1 2 e2we3 e4 3 elwe2 amp dw eit2 ejwek eit2 e jwek Ki a 6 K2 3 Ki 2 Id _ 6 K1 2 e2we3 ae 6 K2 3 elwe2 2 K2 4 e3 2 K3 4e2 3 K 2 el elwe4 2 e2weSwet 4 eiwejwek 4 Fi k ej 4 F j ek 8 Fj k ejwek 4Fj k Id Observe that conversion from the undotted wedge basis to the dotted wedge ba sis using antisymmetric form F and dwedge F are related through the following convert function dwedge F el e2 en convert elwe2w wen wedge_to_dwedge F which can be shown as follows gt F array 1 9 1 9 antisymmetric gt dwedge F e1 e2 convert wedge e1 e2 wedge_to_dwedge F elwe F1 2 Id elwe2 Fi 2 Id gt dwedge F e1 e2 e3 convert wedge el e2 e3 wedge_to_dwedge F elwe2we8 Fo 3 el Fi 3 e2 Fi 2 e3 elwe2we3 Fo 3 el Fi 3 e2 Fi 2e8 wedge_to_dwedge Cl B Ce B dwedge_to_wedge Diagram 1 Isomorphis
39. e f f Computations with Clifford and Grassmann Algebras 25 Vv B diag 1 1 1 define B for C1 3 0 1 0 0 B 0 1 0 0 0 1 dim coldim B eval makealiases dim data clidata B retrieve and display data about C1 B Vv data complex 2 simple a A Id e2 e8 e23 Id e23 Id e2 gt f data 4 assign pre stored idempotent to f or use your own here gt for i from 1 to nops data 7 do f i data 7 i amp c f od Id el e el2 1 2 f 2 i 2 f 2 2 gt Kbasis data 6 here K C Kbasis Id e23 Let s define arbitrary complex spinor coefficients psi1 psi2 phi1 and phi2 for two spinors y and in S Clz39f C Notice that these coefficients belong to a subalgebra K of Cls o spanned by 1 e23 that is isomorphic to C since e3 1 Recall also that the left minimal ideal S Cl Q f is a right K module That s why the complex coefficients must be written on the right of the spinor basis elements f1 and f2 in S gt psil psil1 Id psil2 e23 psi2 psi21 Id psi22 e23 Wl Pll Id 1223 Y2 W21 Id W22 e23 gt phii phii1 Id phil2 e23 phi2 phi21 Id phi22 e23 p1 11 Id 12 e23 62 21 Id 22 e23 Thus Y f p fewe and f1 1 fz22 which is shown in Maple with a help of an unevaluated Clifford product climul as follows gt psi f1 amp c psil f2 amp c psi2 phi f1 amp c phil
40. e user normally does not use either one Instead the user uses a wrapper function amp c K arg1 arg2 or cmul K arg1 arg2 that passes the name of a bilinear form K to either cmulRS or cmulNUM whichever one has been selected to act on the multivector basis monomials This approach allows the user to compute in the same worksheet simultaneously with different Clifford algebras of different bilinear forms The wrapper function can also act on any number of arguments of type type clipolynom as the Clifford product is associative and on a much wider class of types including Clifford matrices of type type climatrix It can also accept Clifford polynomials in other bases such as the Clifford basis 1 e amp C ej amp C e j ex where amp C denotes the unevaluated Clifford product Clifford basis differs from the Grassmann exterior basis when B is not a diagonal matrix Procedures converting between Grassmann and Clifford bases belong to a supplementary package Cliplus 8 while Clifford polynomials expressed in the Clifford basis are of type type cliprod Type cliprod for more information Computations with Clifford and Grassmann Algebras 11 The procedure cmulRS is encoded a non recursive Rota Stein cliffordization See 10 20 22 24 40 and BIGEBRA help pages for additional references 8 The clif fordization process is based on the Hopf algebra theory The Clifford product is obtained from the Gr
41. e4 B e1 2 3 A e4 with the bold terms canceling out Note that the last term in the r h s was super fluously generated in the first step of the recursion The Clifford product can then be derived from the above recursion by linear ity and associativity The induction starts with a left factor of grade one or grade zero which is trivial i e L amp cegA A ep a 6p In the case when the left factor is of grade one we use the Clifford product expressed by the Clifford map of Chevalley i e Ea keep A AN Cc Ca dp Cp A Nec Hea Nea A A ee We make a complete induction in the following way If the left factor is of higher grade say n one application of the recursion yields Clifford products where the new left factor is of grade either n 1 or n 2 hence the recursion stops after at most n 1 steps The above shows clearly that both the Clifford product and the contrac tion are explicitly dependent on B Furthermore the Clifford product algorithm based on the Chevalley s approach is recursive It has been encoded in a procedure cmulNUM see Appendix A Since the Clifford product provides the main functionality of the CLIFFORD package care has been exercised to select the most appropriate and sound mathe matics Two internal user selectable functions cmu1NUM and cmulRS an algorithm based on a combinatorial approach due to Rota and Stein have been used to en code the Clifford product but th
42. fford products with respect to the symmetric form g and with respect to the full form B using procedure cmul that takes a bilinear form as its index As an example we will use two most general elements u and v in A V when dimV 3 Most output will be eliminated gt u add x k w_bas k 1 k 0 7 v add y k w_bas k 1 k 0 7 We can then define in A V a Clifford product cmul g with respect to the sym metric part g and another Clifford product cmul B with respect to the entire form B gt cmulg proc return cmul g args end proc gt cmulB proc return cmul B args end proc We will illustrate relation between the two Clifford products by chasing the follow ing commutative diagram however most output will be eliminated to save space First we compute the Clifford product cmul g u v in C g in undotted Grass mann basis 16 Rafat Abtamowicz JFS F Cl g n D Cl g a Clg D Cl g a cmul g cmul B ame Clg a Clg A Diagram 3 Clifford multiplications cmul g and cmul B w r t dotted and undotted basis gt uv cmulg u v Clifford product w r t g in Cl g in wedge basis Now we convert u and v to ur and vp respectively expressed in the dotted wedge basis gt uF convert u wedge_to_dwedge F vF convert v wedge_to_dwedge F We now compute the Clifford product of ur and vp in C B in the dotted wedge basis gt uFvF cmulB uF vF Clifford product in C1 B in dwedge basis conve
43. ford algebras and shows how the theory can be applied This computational approach also provides a fast way to enter into the abstract field We exemplify this approach by using a Maple package CLIFFORD a system for computations with Grassmann polynomials that was developed like CLICAL to support mathematical research in Grassmann and Clifford algebras but using a full computer algebra system 1 8 9 More recently this system was vastly extended with BIGEBRA for computations with tensors in a general setting of Hopf algebras and co algebras 9 This approach of course is common to many other systems and fields that became possible with the onset of fast computers capable of non trivial symbolic computations which had been intractable before by hand For example in the area of commutative algebra one such system is provided by Singular 26 and another by CoCoA 18 in the area of algebraic geometry by Macaulay2 36 and in the area of general mathematics a category theory based AXIOM which is explicitly dedicated to research and development of mathematical algorithms 16 Finally a review of existing software for Clifford algebras often developed with a specific computation in mind can be found in 13 A vector space V endowed with a quadratic form Q is common to a vast host of mathematical physical and engineering problems This leads naturally to an algebra structure of the Clifford algebra C V Q and its generalization a qua
44. ge F uu el 3e2we3 3 Fo 3 Id 2 Id vu 3 Id 4elwe3 4F1 3 Id e7 10Tn CLIFFORD ver 6 and higher there are three procedures useful for testing that return a random Grassmann basis monomial a random monomial and a random polynomial respectively See rd_clibasmon rd_climon rd_clipolynom 18 Rafat Abtamowicz r r Cl B CB Cl B CUB A A CJF CUB COB Diagram 5 Relation between A and A products gt out1 dwedge F uu vv dwedge computed w r t F gt out2 convert out1 dwedge_to_wedge F back to undotted basis out 3e1 9 e2we3 6 Id 8 elwe3 elwe7 3 e2weSwe7 2 e7 gt out3 wedge u v direct computation of wedge product out3 3e1 9 e2we3 6 Id 8 elwe3 elwe7 3 e2weSwe7 2 e7 and it can be seen that out2 out3 establishing the relation 13 The dotted and the undotted wedge bases are treated fully in 9 One can also find there a discussion of a dependence of contraction the Clifford product and the reversion on the antisymmetric part F of B in the Clifford algebra C B In the following section we illustrate properties of the reversion in dotted and undotted bases 6 Reversion in Dotted and Undotted Bases Following 9 we proceed to show that the expansion of the Clifford basis elements into the dotted or undotted exterior products has also implications for other well known operations such as the Clifford reversi
45. gt P assignL sort eigenvects ATA byeigenvals N P 1 ee t o 1 P 2 9 4 V5 9 4 v5 h 5 3 f h 5 54 gt for i to N do lambda i P 2 i v i map simplify normalize P 3 i gt end do We can now verify that vectors v1 v2 are eigenvectors of AT A with the eigenvalues M1 A2 gt for i to N do map simplify evalm ATA amp v i lambda i v i od 0 0 0 0 Similar verification can be done in C2 o since one can view the 1 column eigen vectors v1 v2 as one column spinors in S Clg of We simply convert the two vectors U1 V2 to spinors sv v1 Sv2 y ve which we express in the previ ously computed spinor basis f1 fo gt spinorbasis f1 f2 gt for i to N do sv i convert v i spinor spinorbasis od UWR wmo OIR ViI0o 2v5 vVio 2v5 10 25 vio 2v5 Since v1 v2 are the eigenvectors of AT A spinors sv1 svz must be the eigenspinors of pTp AT A This fact can be verified as follows gt for i to N do simplify pTp lambda i amp c sv i od sul 2 0 0 We define now the orthogonal matrix V v v2 whose entries we will simply with the procedure radsimplify gt V radsimplify augment v 1 N defining matrix V 1 1 2 2 TS vV10 2v5 v10 2v5 1 V5 v5 1 V10 2V75 V10 2 5 24The first entry 2 in the output is just the number of eigenvectors 38 Rafat Abtamowicz Since later we will need images of V and VT
46. he eigenspinors sv v1 sv2 v2 of the Clifford number pTp AT A directly in Cl2 o using the procedure clisolve capable of solving algebraic equations gt eligenspinor x1 f1 x2 f2 gt eigeneq clicollect expand cmul pTp eigenspinor lambda eigenspinor eigenspinor x1 H H 2 e2 el2 2 2 5x1 812 Ax1 el1 5x1 8x2 Ax1 Id Ot e A22 1312 8 11 e12 Axe 1322 821 e2 2 2 26 The SVD of A is not unique For example A U V7 is another such factorization 42 Rafat Abltamowicz gt sol remove has map allvalues clisolve eigeneq lambda x1 x2 gt lambda lambda 9 4V 5 22 1322 ery ae A 9 4 V5 22 22 op C245 22 _ 1322 gt i 8 8 sol A 9 4 V5 22 12 t1 3 gt for i from 1 to nops sol do gt lambda i subs sol 1 lambda gt eigenspinor i clicollect subs x1 1 x2 1 subs sol 1 eigenspinor gt end do M 9 4V5 a 5 eigenspinor1 ee vee 1 v8 ef ewe 1 v6 Id we cies 4 4 2 2 d2 9 4V5 et 5 eigenspinor2 aN l az el CLEE ae Li J The second comment is that it is possible to realize an anti automorphism tp Cl Q Ce Q that acts on homogeneous basis multivectors as follows ei A eig N NC O Qiri Qizia Qinin Cir A Cig AtA ein 24 M n n 1 g where e Aei Aei 1 2 i ei Aei is the reversed multivector while Qiii Qisis
47. hematical knowledge about Clifford algebras of a quadratic form in dimensions 1 through 9 Together with a sup plementary package BIGEBRA 8 it can be extended to graded tensor products of 4Procedures adfmatrix and mdfmatrix add and multiply matrices of type dfmatrix over such double fields For more information see matKrepr 5When B 0 then C V B Clo o n AV and computations in the Grassmann algebra A V can then be done with CLIFFORD Computations with Clifford and Grassmann Algebras 9 Clifford algebras in higher dimensions The BIGEBRA package is described in 10 For more information about any CLIFFORD or BIGEBRA procedure type Clifford or Bigebra to see its top level help page in the Maple browser For a computation of spinor representations with CLIFFORD we refer to 3 3 Clifford Product in Cl B The Clifford product in a Clifford algebra C B of an arbitrary bilinear form B is introduced via the Chevalley deformation and the Clifford map 35 The Clifford map yx is defined on u AV as i x u LC a u B wedge x u xlgu xAu ii Ixy Way B x y 1 iii Yax by Ax by where x y V see for example 35 One knows how to compute with the wedge x u and the left contraction xz u of u by x with respect to the bilinear form B Following Chevalley the left contraction has the following properties i x Jg y Bix y ii x dp u v x4pu Avt A xg v iii u A v 4p w utp v 4p w
48. ine aliases in CLIFFORD for the dotted wedge basis monomials similar to the Grassmann basis monomials For example we could denote the element elwe F 1 2 Id by e1We2 e e2 and similarly for other elements gt alias e1We2 eiwe2 F 1 2 Id e1We3 elwe3 F 1 3 Id gt e2We3 e2we3 F 2 3 Id gt elWe2We3 e1we2we3 F 2 3 e1 F 1 3 e2 F 1 2 e3 I e1We2 e1 We3 e2We3 e1 We2We3 and then Maple will automatically display dotted basis in terms of the aliases gt d_bas Id e1 e2 e3 e1We2 e1We3 e2We3 e1 We2Wes3 That is as linear spaces we find the isomorphism Ce B x 1 1 2 3 1 A 2 1 A 3 2 A 3 1 A e2 A 63 a 1 1 2 3 1 A e2 1 A 3 2 A e3 1 A e2 A e3 where e A e2 e1We2 etc 4 3 Contraction and Clifford product in dotted and undotted bases For details we refer to 9 The contraction Ig w r t any bilinear form B works on both undotted and dotted bases in a consistent and essentially the same manner as can be see from the next diagram which utilizes the conversion functions between the two bases Let F be the antisymmetric part of B To read more about the left contraction LC in C B check the help page for LC or see 12 We have the following identity for any two elements u and v in C B expressed in the undotted Grassmann basis v dp u v Jg UF F 11 As before ur is the element u expressed in the dotted basis
49. ion maps integers 1 NY onto elements of lsty keeping their order this is to calculate with arbitrary indices and to compute necessary signs psetx lt power set of 1 NX actually a list in a certain order the i th and 2 NX 1 i th element are disjoint adding up to the set 1 NX psety lt power set of 1 NY actually a list in a certain order the i th and 2 NY 1 i th element are disjoint adding up to the set 1 NY for faster computation we sort this power sets by grade we compute the sign for any term in the power set psetx lt sort psetx by grade psety lt sort psety by grade pSgnx lt sum_ i in psetx 1 sum_ j in psetx i psetx i j j pSgny lt sum_ i in psety 1 sum_ j in psety i psety i j j we need a subroutine for cup tangle computing the bilinear form cup x y B begin cup if x lt gt y then return 0 end if if x O then return 1 end if if x 1 then return B x 1 y 1 end if return sum_ j in 1 x1 1 G 1 B x 1 y j cup 2 1 y y j B end cup now we compute the double sum to gain efficiency we do this grade wise note that there are r over NX r vectors in psetx analogously for psety max_grade 1stx lt convert_to_set union lsty lt convert_to_set res lt 0 posi lt 0 for j from 0 to NX iterate over all j vectors of psetx begin F1 lt N1 N1 j j number of terms N1 over j Computations with Clifford and Grassmann Algebras 45 pos2 lt 0
50. ization of A into a product of three matrices U X V71 where U and V are orthogonal matrices m x m and n x n respectively and X is am x n matrix containing singular values of A on its diagonal A UxV 1 UTU I VIV I 19 The matrices V vi vo Un and U uy u2 um contain orthonormal bases for all four fundamental spaces of A Namely the first r columns v1 v2 Ur of V provide a basis for the row space R A while the remaining n r columns of V provide a basis for the null space A Likewise the first r columns u1 U2 Ur of U provide a basis for the column space C A while the remaining m r columns of U provide a basis for the left null space M AT Vectors v are the normalized eigenvectors of AT A while vectors u are the normalized eigenvectors of AA For i 1 r these vectors can be chosen to be related via the positive singular values g of A which are just the square roots of the eigenvalues of ATA or of AAT Namely Avi ci t 1 7r 20 It is a little tricky to make sure that the above relation is satisfied this is because the choice of vectors u is independent of the choice of vectors v However it is al ways possible to do so as we will see below see also 42 In order to complete the picture the orthonormal set v1 Ur needs to be completed to a full orthonor mal basis for R while u1 ur needs to be completed to a full orthonormal basis for R Since the a
51. lifford product in dotted and undotted bases Se DON OP More on the Associativity of the Dotted Wedge Reversion in Dotted and Undotted Bases Spinor Representation of C Q in Minimal Left Ideals Two Scalar Products in Spinor Ideals Continuous Families of Idempotents Low Dimensional Examples NOW WY 13 14 16 18 21 24 26 2 Rafat Abtamowicz 10 Vahlen Matrices 28 11 Singular Value Decomposition and Clifford Algebra 33 11 1 SVD of a2 x 2 matrix of rank 2 34 11 2 Additional comments 41 12 Conclusions 42 Appendix A Appendix Code of cmu1NUM 43 Appendix B Appendix Code of cmu1RS 44 Appendix C Appendix Code of the Transposition Procedure tp 45 References 46 1 Introduction Some twenty years ago late Professor Pertti Lounesto together with his colleagues at Helsinki University of Technology developed CLICAL a first semi symbolic Clif ford algebra calculator 32 Along with it Pertti brought to the world of Clifford algebraists a concept of experimental mathematics algorithmic understanding and counter examples 33 One could say that he was a pioneer in bringing together theoretical aspects and the computational part His works see for example 30 abound in concrete examples and counter examples This author also believes that learning how mathematical concepts can be handled by a computer provides a combination of theory and practice that as a result gives a better understanding of the theory of Clif
52. mp w p2 e12 e12 eawebwec eawebwec e123 x0 Id 4 500000000 x0 e1 a x0 e123 x12 e12 Following Chevalley s recursive definition a Clifford product can be intro duced in A V by means of a left Jg or right Lg contraction dependent on an arbitrary bilinear form B V x V R This leads to elements of the Clifford algebra C B expanded into multivectors and makes the Clifford multiplication implicitly dependent on B The associative Clifford product is given by a procedure cmul or its infix form amp c gt cmul el e2 amp c e1 e2 cmul ea eb ec el2 B 2 Id el2 B 2 Id eawebwec Br c ea Ba c eb Ba ec Computations in C K and C B can be performed in the same worksheet since the name of a bilinear form can be passed to cmul as a parameter For example gt cmul K e1 e2 amp c K e1 e2 cmul K ei ej ek e12 K 2 Id el2 K 2 Id Computations with Clifford and Grassmann Algebras 5 eiwejwek Kj ei Ki k ej Ki j ek The form B can be numeric or symbolic For example when gt B matrix 2 2 1 a a 1 1 a B a 1 then the Grassmann basis for C B or A V will be gt cbas cbasis 2 cbas Id e1 e2 e12 while the Clifford multiplication table of the basis Grassmann monomials will look as follows gt MultTable matrix 4 4 i j gt cmul cbas i cbas j Id el e2 e12 el Id e12 ald e2 ael MultTable e el2 ald Id ae2 el e12 ael
53. mple C41 3 has a spinor representation given in terms of 2 by 2 quaternionic matrices whose entries belong to a subalgebra F of C 3 spanned by Id e2 e3 e2we3 gt B linalg diag 1 1 1 1 clidata 1 3 quaternionic 2 simple Zu 5 elwe4 Id e1 e2 e8 e12 e18 e23 e128 Id e2 e3 e23 Id e1 gt matKrepr 1 3 quaternionic matrices 0 1 e2 0 e3 0 0 1 jel e2 e8 e4 1 0 0 e2 0 e3 1 0 CLIFFORD includes several special purpose procedures to deal with quaternions and octonions type quaternion and octonion for help and with quaternionic and octonionic matrices In particular following 32 octonions are treated as para vectors in C 9 7 while their non associative multiplication defined via Fano triples is related to the Fano projective plane F see 7omultable or Fano_triples for more information User can select different Fano triples and redefine the octo nionic multiplication table Since the bilinear form B can be degenerate one can use CLIFFORD to perform computations in Clifford algebras Clp q a of a degenerate quadratic form Q of signature p q and the dimension d of its radical For exam ple the Clifford algebra Clo 3 1 of the quadratic form Q x x x3 x where X T1 1 2 2 733 x4e4 Rt is used in robotics to represent rigid motions in R and screw motions in terms of dual quaternions 4 41 Thus CLIFFORD is a repository of mat
54. ms between C B and Ce B For a more complete treatment see 9 4 2 Dotted and undotted wedge bases Symbolic capabilities of the computer algebra system allow for an investigation of properties of the Clifford product contraction and the reversion in the dotted and the undotted bases In this way the CAS allows for a better understanding of these fundamental to any Clifford algebra C B operations Here we show only a few facts and refer to 9 10 For example we expand the basis of the original wedge into the dotted wedge and back using the two conversion functions mentioned above For this purpose we choose dim V 3 and consider C B with the antisymmetric part F The undotted wedge basis for A V is then gt w_bas cbasis dim_V the wedge basis w_bas Id e1 e2 e3 elwe2 elwe3 e2we3 elwe2wes 14 Rafat Abltamowicz Now we map the convert function onto this basis to get the dotted wedge basis gt d_bas map convert w_bas wedge_to_dwedge F gt test_wbas map convert d_bas dwedge_to_wedge F d_bas Id e1 e2 e8 elwe2 Fi 2 Id elwe8 Fi 3 Id e2we3 F2 3 Id elwe2we8 F2 3 e1 Fi 3 e2 Fi 2 e3 test_wbas Id e1 e2 e8 elwe2 elwe3 e2we3 e1we2wes Notice that only the unity 1 and the one vector basis elements e remain unaltered and that the other basis elements of higher grades pick up additional terms of lower grades which preserves the filtration It is possible to def
55. now well known that such structures are related to Hopf algebraic twists later versions of CLIFFORD make an extensive use of a process called Rota Stein cliffordization described in 20 2225 This in turn has necessitated introduction of a new algorithm based on this process for a more ef ficient computation of the Clifford product in C V B As much as Buchberger s celebrated algorithm is indispensable for computing Grobner bases this new algo rithm described in 9 10 is indispensable for the Clifford product Having devel oped this faster algorithm one was able to find the g Young Clifford idempotents 6 possessing a desired symmetry explicitly describe the structure of the Spin 3 group discover continuous families of idempotents in C Q 11 or gain a better understanding of the properties of the dotted wedge product 9 The present paper brings to the reader a few examples of computations and results derived with CLIFFORD It is assumed that the reader is already familiar with Maple 44 a general purpose CAS if not please consult e g 45 The article is intended as a quick computational introduction to the abstract field of the Clifford algebras and especially to the field of quantum Clifford algebras For computations with tensor and Hopf algebras we refer to 10 that describes the supplementary package BIGEBRA intended for such computations 2 Notation and Basic Computations CLIFFORD uses as default a standard Gra
56. ntum Clifford algebra Ce V B for any bilinear form B 6 This formalism as Computations with Clifford and Grassmann Algebras 3 compared to a standard vector calculus can now be applied to solving completely new problems CLIFFORD was developed as a basic tool for all investigations and applications which can be carried in finite dimensional vector spaces equipped with a quadratic form or equivalently with a symmetric bilinear form commonly referred to as an inner or a scalar product The intrinsic abilities of Maple even allow one to use CLIFFORD in projective and affine geometries while visualizing complicated incidence relations that is helpful e g for image processing visual perception and robotics The authors of CLIFFORD and BIGEBRA have been interested in fundamen tal questions about g deformed symmetries and quantum field theory Just asking questions like such as What is the most general element fulfilling has led to unexpected results and new insights 4 6 7 11 Checking the consistency of the software by testing theorems and known results has led them in the foot steps of Pertti Lounesto to counter examples that have made a rethinking and a more careful restatement of those theorems necessary However the most striking abil ity of CLIFFORD is that it is unique in being able to handle Clifford algebras of an arbitrary bilinear form not restricted by symmetry and not directly related to any quadratic form Since it is
57. on anti automorphism CUB gt Ce B uve v6 which preserves the grades in AV but not in A V unless B is symmetric Only when the bilinear form is symmetric we find that the reversion is grade preserving otherwise it reflects only the filtration That is reversed elements are in general sums of terms of the same and lower degrees gt reversion elwe2 B reversion with respect to B gt reversion elwe2 g reversion with respect to g classical result elwe2 2 F 2 Id elwe2 Computations with Clifford and Grassmann Algebras 19 reversion B CL B COB reversion g CLUB a Diagram 6 Relation between reversion g and reversion B and the basis transformation ar We illustrate how the various reversions are related in the following commutative diagram The reader should note that the map depicted by the diagonal arrow in Diagram 6 involves a change of basis induced by the antisymmetric bilinear form 2 F and not F The factor 2 is crucial and appears due to an asymmetry between the undotted and dotted bases This suggests to introduce a symmetrically related triple of bases w r t 43F F 0 and F In such a setting F resp F connects the two dotted bases induced by 4F Observe in the pre last display above that only when B 2 Bo the re sult e A e2 known from the theory of classical Clifford algebras is obtained Likewise gt cbas cbasis 3 cbas
58. ongs to R 4 a b c d belong to Cl 4 and the prod ucts and the inverse are taken in Clp This transformation may be represented by the Vahlen matrix V defined above Rotations translations dilations and transversions will then be represented as follows e Rotations x gt axa where a belongs to Spin p q the identity compo nent of Spin p q and V o o 0 1 ane e Dilations x sx where s gt 0 and V 0 1 e Translations x x b where b R and V G a Vs 2 X 0 y where c RP x c is the dot product 1 2x c x2c in R and V i c 1 Let s consider a few simple examples in the signature 3 1 Our goal is to see how CLIFFORD manipulates with Clifford matrices At the same time we will verify some results from 38 We begin with a Vahlen matrix R that gives a rotation gt B linalg diag 1 1 1 1 bilinear form for the Minkowski space e Transversions x gt 1 0 0 0 0 1 0 0 B 0 0 1 0 0 0 0 1 elwe2 an element of grade 2 in Spint 3 1 linalg matrix 2 2 a 0 0 a Vahlen matrix that gives a rotation isVahlenmatrix R isVahlenmatrix R VV Vv ow e12 0 0 e12 a e12 R is Vahlenmatria R true 30 Rafat Abltamowicz Next we consider a Vahlen matrix T that gives a translation gt b e1 2 e3 vector in R 3 1 gt T linalg matrix 2 2 1 b 0 1 gt isVahlenmatrix T isVahlenmatrix T 1 e14 2e3
59. or the background material on Vahlen matrices and conformal transformations see 15 31 33 34 38 Procedure isVahlenmatrix determines if a given 2 x 2 Clif ford matrix V Mat 2 C Q is a Vahlen matrix and it returns true or false accordingly Any matrix with entries in a Clifford algebra is of type climatrix A Vahlen matriz isa 2x2 matrix V 4 b with entries in a Clifford algebra Clp q such that the following conditions are met 1 a b c d are products of 1 vectors 2 The pseudo determinant of V computed as ad b equals 1 or 1 3 ab bd d and a are all 1 vectors 7 Condition i above implies that a b c and d are elements of the Lipschitz group Lp q of Clpq Recall 35 that this group is defined as follows and Lp g 8 E Clp q 1x857 E RP x RP 16In CLIFFORD it is computed with a procedure pseudodet 17Here denotes the reversion anti automorphism in Cp q In CLIFFORD it is the reversion operation Computations with Clifford and Grassmann Algebras 29 Procedure isproduct is used to determine whether this condition is met Recall that in dimensions n gt 3 sense preserving conformal mappings are restrictions of the Mobius transformations and are compositions of rotations translations dila tions and transversions called also special conformal transformations A M bius transformation in R 1 can be written in the form ax b cx d x 18 where x is a l vector that bel
60. putations with Clifford and Grassmann Algebras 43 unexpectedly continuous families of idempotents or a computation of eigenvalues and eigenvectors of a Clifford number or its singular values In the process a connection between a transposition and the reversion in C Q was discovered Some other computational and theoretical results facilitated with the pack ages and already reported include e Finding generators for Hecke algebras realized as even elements in a Clifford algebra of a suitable non symmetric bilinear form 7 e Finding g Young idempotents in some Hecke algebras of mixed symmetry that generate a representation space for these algebras 7 e Finding necessary and sufficient condition that a Clifford biconvolution that is a Clifford Hopf gebra for a two dimensional real space posses an antipode 25 e Investigation of Wick normal ordering 21 e Verification of Helmstetter formula 28 that expresses an isomorphism be tween two Clifford algebras C B and C B where the bilinear forms share the symmetric part 4 e Explicit description of all elements in Pin 3 and Spin 3 in 4 e Description of space singularities of a robotic platform 5 17 For a complete discussion of the mathematical capabilities of CLIFFORD and BIGEBRA packages we refer to 9 and 10 Appendix A Appendix Code of cmu1NUM Here is a pseudocode of the recursive procedure cmu1NUM cmulNUM al a2 B al a2 two Grassmann monomi
61. rator for K is only one since K R Vv FBgens Id gt K_basis spinorKbasis SBgens f FBgens left K basis for S f Id el e e el2 K_basis le aro h Id e2 left 22 Rafat Abltamowicz Here are matrices representing basis monomials of Cz o gt MO M1 M2 M3 op map spinorKrepr clibasis K_basis 1 FBgens left 1 0 M0 M1 M2 M3 l gi o i 1 0 f 0 Since the spinor representation of C 2 9 is an algebra isomorphism from C29 to R 2 matrix Myo that represents e1e2 e1 ez is a product of matrices M and M with Clifford multiplication applied to their entries Procedure which handles multiplication of such matrices is called rmulm and it can also be entered in its infix form amp cm gt M12 M1i amp cm M2 0 1 1 0 M12 Notice that M and M have the same algebraic properties as the basis elements they represent 1e2 e2 1 0 gt el amp c e2 e2 amp c el evalm M1 amp cm M2 M2 amp cm M1 0 0 0 0 Let s find a matrix representing an arbitrary Clifford polynomial p in C 2 0 gt p a0tal elta2 e2 al2 e12 p a0 ale1 a2e2 4 al2el2 gt spinorKkrepr p K_basis 1 FBgens left matrix of p in S aO al a2 al2 a2 al2 ad al The simplest way to compute that matrix is to use procedure matKrepr that uses pre computed spinor representations of all Clifford algebras Clp p q lt 9 that are stored in CLIFFORD gt matKrepr p aO al a2 al2
62. rd and Grassmann Algebras 17 gt evalb dwedge F dwedge F e1 e2 e3 dwedge F e1 dwedge F e2 e3 true The associativity of the dotted wedge implies that the diagram 4 commutes It was checked with CLIFFORD up to dimension 5 dwedge F 1 CB Cl B CUB Ce B Cl B 1 dwedge F dwedge F dwedge F Ce B Cl B CUB A Diagram 4 Associativity of dwedge F in Ce B For some arbitrary random Clifford polynomials u v z expressed in Grassmann undotted basis we can show associativity as follows gt u 2 Idt e1 3 e2we3 v 3 Id 4 elwe3t e7 z 4 Id 2 e3t elwe2we3 gt evalb dwedge F Id u u evalb dwedge F u Id u true true gt evalb dwedge F dwedge F u v z dwedge F u dwedge F v z true We have therefore the following identity that expresses an isomorphism between two Clifford algebras dotted and undotted For any two elements u and v in Ct B B g F that are by default expressed in terms of the undotted Grass mann basis we find u v up vp F 13 Here up and vp are the elements u and v expressed in the dotted basis with respect to the form F while _ denotes conversion back from the dotted basis to the undotted basis w r t F FT CB and C B denote the modules w r t the two filtrations in use This can be illustrated in CLIFFORD as follows gt uu convert u wedge_to_dwedge F vv convert v wedge_to_dwed
63. rt back the above result back to the undotted wedge basis gt uv2 convert uFvF dwedge_to_wedge F convert result dwedge gt wedge and verify that the results are the same gt simplify uv uv2 shows equality 0 Thus we have shown that the following identity involving cmul g and cmu1 B is true at least when dim V 3 For a general result see e g 14 28 uv u amp cgv ur amp cgvur r ur vF B F 12 This shows that the Clifford algebra C g of the symmetric part g of B using the undotted exterior basis is isomorphic as an associative algebra to the Clifford algebra C B of the entire bilinear form B g F spanned by the dotted wedge basis if the antisymmetric part F of B is exactly the same as F used to connect the two bases r E Homalg C g Ce B B g F 5 More on the Associativity of the Dotted Wedge It was shown above that CLIFFORD uses Grassmann algebra V as the underlying vector space of the Clifford algebra C V B Thus the Grassmann wedge basis of monomials is the standard basis used in the package A general element u Ce V B can be therefore viewed as a Grassmann polynomial Operation dwedge introduced in Sect 4 1 is associative with the unity 1 Id as its unit Here uv g is the Clifford product with respect to g while up amp cg vp and urup p are the Clifford products with respect to B that is in C g and C B respectively Computations with Cliffo
64. s ATA its characteristic polynomial eigenvalues and its orthonormal eigenvectors v1 v2 Vectors v and v2 will become columns of an orthogonal matrix V needed for the SVD of A gt ATA evalm t A amp A finding matrix ATA 8 8 13 ATA gt pTp phi ATA M finding image of ATA in C1 2 0 pTp 9 Id 4e1 8 e2 which is the same as gt pTp cmul pT p pTp 9Id 4el 8e2 The minimal polynomial of AT A whose image in C 2 o is called pTp is computed with climinpoly gt climinpoly pTp v 182e4 1 and it is the same as the characteristic polynomial of AT A gt pol charpoly ATA x characteristic polynomial of ATA pol xr 18r 1 22 Since a similar computation was done in 2 3 we won t display the matrices 23From now on we use Maple s alias t transpose that is t A denotes matrix transpo sition in Maple The second argument to the procedure phi is a table M containing matrices M M2 M3 M4 as its entries Computations with Clifford and Grassmann Algebras 37 In order to find eigenvalues and eigenvectors of ATA we will use Maple s procedure eigenvects modified by our own sorting via a new procedure assignL The latter displays a list containing two lists one has the eigenvalues while the second has the eigenvectors 4 In the following we assign the eigenvalues of AT A to Ay Ag and the not yet normalized but orthogonal eigenvectors of AT A we assign to vi v2
65. smann algebra V as the underlying vector space of the Clifford algebra C V B Thus the Grassmann wedge basis of monomials is the standard basis used in CLIFFORD A general ele ment u in C V B can be therefore viewed as a Grassmann polynomial When the bilinear form B has an antisymmetric part F F it is conve nient to split it as B g F where g is the symmetric part of B and to introduce the so called dotted Grassmann basis 12 and the dotted wedge product A The original Grassmann basis will be referred to here as the undotted Grassmann basis In CLIFFORD the wedge product is given by the procedure wedge and amp w while the dotted wedge product is given by dwedge and amp dw According to Chevalley s definition of the Clifford product amp c we have X amp cu Xg u x amp wu LC x u B wedge x u 6 for a 1 vector x and an arbitrary element u of C B As before LC x u B denotes the left contraction of u by x with respect to the bilinear form B However when B g F then the left contraction splits too x Ip u LC x u B X 4 u xX Jp u LC x u g LC x u F 7 and x amp cu LC x u B X amp wu 8 LC x u g LC x u F x amp wu 9 LC x u g dwedge F x u LC x u g X amp dw u 10 where x amp dwu x amp wu LC x u F That is the wedge and the dotted wedge differ by the contraction term s with respect to the antisymmetric part F of B This
66. ssmann Algebras 47 M Baswell Clifford Algebra Space Singularities of a Redundant Variable Geometry Truss Manipulator M S Thesis Tennessee Technological University 2000 CoCoA System for computations in commutative algebra http cocoa dima unige it 2009 M Berry and J Dongarra Atlanta Organizers Put Mathematics to Work for the Math Sciences Community SIAM News Vol 32 6 1999 and references therein Ch Brouder B Fauser A Frabetti and R Oeckl Quantum field theory and Hopf algebra cohomology formerly Let s twist again J Phys A Math Gen 37 22 2004 5895 5927 B Fauser Clifford geometric parameterization of Wick normal ordering J Phys A Math Gen 34 2001 105 115 B Fauser A Treatise on Quantum Clifford Algebras Habilitationsschrift Univer sitat Konstanz Konstanz 2002 B Fauser Products coproducts and singular value decomposition Int J Theor Phys Vol 45 No 9 2006 1731 1755 B Fauser and P D Jarvis A Hopf laboratory for symmetric functions J Phys A Math Gen 37 2004 1633 1663 B Fauser and Z Oziewicz Clifford Hopf gebra for two dimensional space Miscel lanea Algebraica 2 1 2001 31 42 G M Greuel and G Pfister A Singular Introduction to Commutative Algebra Springer New York 2000 J Helmstetter Alg bres de Clifford et alg bres de Weyl Cahiers Math 25 1982 J Helmstetter Monoides de Clifford et d formations d alg bres de Clifford J of Alg 111
67. ssmann basis Grassmann multivectors in A V where V span e 1 lt i lt n for 1 lt n lt 9 Then AV spanf ei Ne A Nen 0Si lt 5 lt lt kn 4 Rafat Abtamowicz In CLIFFORD these basis monomials are written as strings Id e1 e9 elwe2 elwe elwe2we3 although they can be aliased to Id el e9 e12 e13 e128 to shorten input Here elwe2 is a string that denotes e ez and Id denotes the identity 1 in AV However CLIFFORD can also use one character long symbolic indices as in eiwej which stands for e A ej Thus in principle it can compute with Clifford algebras in dimensions higher than 9 For example when n 3 Grassmann basis monomials are gt W cbasis 3 W Id e1 e2 e8 elwe2 elwe3 e2we3 elwe2we3 but aliases can also be used to shorten input output gt eval makealiases 3 e12 e21 e13 e31 e23 e32 e123 e182 e213 e231 e812 e321 In the above eijk eiwejwek is the wedge product of three 1 vectors e j ex Thus the most general element in the Grassmann algebra A V is a Grassmann polynomial which is just a linear combination of Grassmann basis monomials with real coefficients Notice that symbolic indices are allowed gt pi Id 4 5 ei alpha eliwe2we3 pl Id 4 5 ei a e123 The wedge product is computed with a procedure wedge or its ampersand coun terpart amp w gt wedge e1 e2 e1 amp w e2 wedge ea eb ec ea kw eb amp w ec pl a
68. t into CLIFFORD displays two possible choices for the signature p q such that p q 2 K R and C is a simple algebra gt all_sigs 2 2 real simple A 1 1 2 0 Thus we can pick either C41 or Clz 9 Our choice is Clz 9 We define B as the 2x2 identity matrix and use CLIFFORD s procedure clidata to display information about Clo 0 gt dim 2 B diag 1 1 eval makealiases dim data clidata data real 2 simple sid el Id e2 Id Id e2 The above output means that Cf2 9 is a simple algebra isomorphic to Mat 2 R that the element 461 displayed by Maple as 4Id 5 el is a primitive idempotent that the list Id e2 shown as the fifth entry displays generators of a minimal left ideal C 2 9f considered as vector space over R that the division ring K fCloof Ud r R and that the last list Id e2 gives generators of Clo of over K and since K R it is the same as the fifth entry In the following we define a Grassmann basis in C 2 9 assign the primitive idempotent to f and generate a spinor basis in C 2 of gt clibas cbasis dim ordered basis in C1 2 0 clibas Id e1 e2 e12 gt f data 4 a primitive idempotent in C1 2 0 gt SBgens data 5 generators for a real basis in S gt FBgens data 6 generators for the division ring K SBgens contains generators for a K basis for S Cloof f e2f Since in the signature 2 0 we have K R S R and Clo

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