Papers by Rafal Ablamowicz
Advances in Applied Clifford Algebras, Jan 28, 2014
We present different methods for symbolic computer algebra computations in higher dimensional (≥ ... more We present different methods for symbolic computer algebra computations in higher dimensional (≥ 9) Clifford algebras using the CLIFFORD and Bigebra packages for Maple R . This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.
Birkhäuser Boston eBooks, 2000
It is a well known fact from the group theory that irreducible tensor representations of classica... more It is a well known fact from the group theory that irreducible tensor representations of classical groups are suitably characterized by irreducible representations of the symmetric groups. However, due to their different nature, vector and spinor representations are only connected and not united in such description. Clifford algebras are an ideal tool with which to describe symmetries of multiparticle systems since they contain spinor and vector representations within the same formalism, and, moreover, allow for a complete study of all classical Lie groups. In this work, together with an accompanying work also presented at this conference, an analysis of q -symmetry -for generic q 's -based on the ordinary symmetric groups is given for the first time. We construct q -Young operators as Clifford idempotents and the Hecke algebra representations in ideals generated by these operators. Various relations as orthogonality of representations and completeness are given explicitly, and the symmetry types of representations is discussed. Appropriate q -Young diagrams and tableaux are given. The ordinary case of the symmetric group is obtained in the limit q → 1. All in all, a toolkit for Clifford algebraic treatment of multi-particle systems is provided. The distinguishing feature of this paper is that the Young operators of conjugated Young diagrams are related by Clifford reversion, connecting Clifford algebra and Hecke algebra features. This contrasts the purely Hecke algebraic approach of King and Wybourne, who do not embed Hecke algebras into Clifford algebras.
Birkhäuser Boston eBooks, 2000
Clifford algebras are naturally associated with quadratic forms. These algebras are Z2 -graded by... more Clifford algebras are naturally associated with quadratic forms. These algebras are Z2 -graded by construction. However, only a Zn -gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cℓ(V ) ↔ V and an ordering, guarantees a multivector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the Zn -grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a Zn -grading which we now call Clifford algebras of multivectors or quantum Clifford algebras. It turns out, that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonalizability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cℓp,q which can be decomposed in the symmetric case into a tensor product Cℓp-1,q-1 ⊗ Cℓ1,1. The general case used in quantum field theory lacks this feature. Theories
arXiv (Cornell University), Jun 16, 2012
We present different methods for symbolic computer algebra computations in higher dimensional (≥ ... more We present different methods for symbolic computer algebra computations in higher dimensional (≥ 9) Clifford algebras using the CLIFFORD and Bigebra packages for Maple R . This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.
Algebra and physics
Birkhäuser eBooks, 2000
Dedication to Gian-Carlo Rota, 1933-1999, Bernd Schmeikal. Part 1 Physics - applications and mode... more Dedication to Gian-Carlo Rota, 1933-1999, Bernd Schmeikal. Part 1 Physics - applications and models: multiparavector subspaces of Cln - theorems and applications, William E. Baylis quaternionic spin, Tevian Dray, Corinne A. Manoque Pauli terms must be absent in Dirac equation, Kurt Just, James Thevenot electron scattering in the spacetime algebra, Antony Lewis et al. Part 2 Physics - structures: twistor approach to relativistic dynamics and to the Dirac equation - a review, Andreas Bette fibre with intrinsic action on a 1+1 dimensional spacetime, Robert W. Johnson dimensionally democratic calculus and principles of polydimensional physics, William M. Pezzaglia Jr. a Pythagorean metric in relativity, Franco Israel Piazzese Clifford-valued Clifforms - a geometric language for Dirac equations, Jose G. Vargas, Douglas G. Torr. Part 3 Geometry and logic: the principle of duality in Clifford algebra and projective geometry, Oliver Conradt doing geometric research with Clifford algebra, Hongbo Li Clifford algebra of quantum logic, Bernd Schmeikal. Part 4 Mathematics -deformations: Hecke algebra representations in ideals generated by q-Young Clifford idempotents, Rafal Ablamowicz, Bertfried Fauser on q-deformations of Clifford algebras, Gaetano Fiore Dirac operators, Hopf algebra of renormalisation and structure of space-time, Marcos Rosenbaum, J. David Vergara non-commutative spaces for graded quantum groups and graded Clifford algebras, Michaela Vancliff. Part 5 Mathematics -structures: Clifford algebras and the construction of the basic Spinor and semi-Spinor modules, Johan Gijsbertus Frederik Belinfante on the decomposition of Clifford algebras of arbitrary bilinear form, m Bertfried Fauser, Rafal Ablamowicz covariant derivatives on Minkowski manifolds, Virginia V. Fernandez et al an introduction to pseudotwistors -Spinor solutions versus harmonic forms and cohomology groups, Julian Lawrynowicz, Osamu Suzuki ordinary differential equation - symmetries and last multiplier, Zbigniew Oziewicz, Jose Ricardo R. Zeni universal similarity factorisation equalities over complex Clifford algebras, Yongge Tian.
arXiv (Cornell University), Dec 13, 2011
We introduce on the abstract level in real Clifford algebras Cℓ p,q of a non-degenerate quadratic... more We introduce on the abstract level in real Clifford algebras Cℓ p,q of a non-degenerate quadratic space (V, Q), where Q has signature ε = (p, q), a transposition anti-involution T ε ˜. In a spinor representation, the anti-involution T ε ˜gives transposition, complex Hermitian conjugation or quaternionic Hermitian conjugation when the spinor space Š is viewed as a Cℓ p,q -left and Ǩ-right module with Ǩ isomorphic to R or 2 R, C, or, H or 2 H. This map and its application to SVD was first presented at ICCA 7 in Toulouse in 2005 [3]. The anti-involution T ε ˜is a lifting to Cℓ p,q of an orthogonal involution t ε : V → V which depends on the signature of Q. The involution is a symmetric correlation and it allows one to define a reciprocal basis for the dual space (V * , Q). When the Clifford algebra Cℓ p,q splits into the graded tensor product Cℓ p,0 ⊗ Cℓ 0,q , the anti-involution T ε ˜acts as reversion on Cℓ p,0 and as conjugation on Cℓ 0,q . Using the concept of a transpose of a linear mapping one can show that if [L u ] is a matrix in the left regular representation of the operator L u : Cℓ p,q → Cℓ p,q relative to a Grassmann basis B in Cℓ p,q , then matrix [L Tε˜(u) ] is the matrix transpose of [L u ], see . Of particular importance is the action of T ε ˜on the spinor space. The algebraic spinor space Š is realized as a left minimal ideal generated by a primitive idempotent f , or a sum f + f in simple or semisimple algebras as in . The map T ε ˜allows us to define a new spinor scalar product Š × Š → Ǩ, where K = f Cℓ p,q f and Ǩ = K or K ⊕ K depending whether the algebra is simple or semisimple. Our scalar product is in general different from the two scalar products discussed in literature, e.g., . However, it reduces to one or the other in Euclidean and anti-Euclidean signatures. The anti-involution T ε ˜acts as the identity map, complex conjugation, or quaternionic conjugation on Ǩ. Thus, the action of T ε ˜on spinors results in matrix transposition, complex Hermitian conjugation, or quaternionic Hermitian conjugation. We classify automorphism group of the new product as O(N), U(N), Sp(N), 2 O(N), or 2 Sp(N).
Clifford Algebras and their Applications in Mathematical Physics: Volume 1: Algebra and Physics
... ISBN 0-8176-4183-1 (Volume 2) SPIN 10768155 (Volume 2) ISBN 3-7643-4183-1 (Volume 2) ISBN 0-8... more ... ISBN 0-8176-4183-1 (Volume 2) SPIN 10768155 (Volume 2) ISBN 3-7643-4183-1 (Volume 2) ISBN 0-8176-4182-3 (Volume 1) SPIN 10768121 (Volume 1) ISBN 3-7643-4182-3 (Volume I) Typeset by the edi, ors in Prin, ed and bound by Ham, lton Prin,, ng Company ...
arXiv (Cornell University), Feb 16, 2011
A signature ε = (p, q) dependent transposition anti-involution T ε ˜of real Clifford algebras Cℓ ... more A signature ε = (p, q) dependent transposition anti-involution T ε ˜of real Clifford algebras Cℓ p,q for non-degenerate quadratic forms was introduced in [1]. In [2] we showed that, depending on the value of (p -q) mod 8, the map T ε ˜gives rise to transposition, complex Hermitian, or quaternionic Hermitian conjugation of representation matrices in spinor representation. The resulting scalar product is in general different from the two known standard scalar products [12]. We provide a full signature (p, q) dependent classification of the invariance groups G ε p,q of this product for p + q ≤ 9. The map T ε ĩs identified as the "star" map known [14] from the theory of (twisted) group algebras, where the Clifford algebra Cℓ p,q is seen as a twisted group ring R t [(Z 2 ) n ], n = p + q. We discuss and list important subgroups of stabilizer groups G p,q (f ) and their transversals in relation to generators of spinor spaces.
arXiv (Cornell University), Jun 16, 2012
We present, as a proof of concept, a way to parallelize the Clifford product in Cℓ p,q for a diag... more We present, as a proof of concept, a way to parallelize the Clifford product in Cℓ p,q for a diagonalized quadratic form as a new procedure cmulWpar in the CLIFFORD package for Maple R . The procedure uses a new Threads module available under Maple 15 (and later) and a new CLIFFORD procedure cmulW which computes the Clifford product of any two Grassmann monomials in Cℓ p,q with a help of Walsh functions. We benchmark cmulWpar and compare it to two other procedures cmulNUM and cmulRS from CLIFFORD. We comment on how to improve cmulWpar by taking advantage of multi-core processors and multithreading available in modern processors.
arXiv (Cornell University), Dec 13, 2011
We introduce on the abstract level in real Clifford algebras Cℓ p,q of a non-degenerate quadratic... more We introduce on the abstract level in real Clifford algebras Cℓ p,q of a non-degenerate quadratic space (V, Q), where Q has signature ε = (p, q), a transposition anti-involution T ε ˜. In a spinor representation, the anti-involution T ε ˜gives transposition, complex Hermitian conjugation or quaternionic Hermitian conjugation when the spinor space Š is viewed as a Cℓ p,q -left and Ǩ-right module with Ǩ isomorphic to R or 2 R, C, or, H or 2 H. This map and its application to SVD was first presented at ICCA 7 in Toulouse in 2005 [3]. The anti-involution T ε ˜is a lifting to Cℓ p,q of an orthogonal involution t ε : V → V which depends on the signature of Q. The involution is a symmetric correlation and it allows one to define a reciprocal basis for the dual space (V * , Q). When the Clifford algebra Cℓ p,q splits into the graded tensor product Cℓ p,0 ⊗ Cℓ 0,q , the anti-involution T ε ˜acts as reversion on Cℓ p,0 and as conjugation on Cℓ 0,q . Using the concept of a transpose of a linear mapping one can show that if [L u ] is a matrix in the left regular representation of the operator L u : Cℓ p,q → Cℓ p,q relative to a Grassmann basis B in Cℓ p,q , then matrix [L Tε˜(u) ] is the matrix transpose of [L u ], see . Of particular importance is the action of T ε ˜on the spinor space. The algebraic spinor space Š is realized as a left minimal ideal generated by a primitive idempotent f , or a sum f + f in simple or semisimple algebras as in . The map T ε ˜allows us to define a new spinor scalar product Š × Š → Ǩ, where K = f Cℓ p,q f and Ǩ = K or K ⊕ K depending whether the algebra is simple or semisimple. Our scalar product is in general different from the two scalar products discussed in literature, e.g., . However, it reduces to one or the other in Euclidean and anti-Euclidean signatures. The anti-involution T ε ˜acts as the identity map, complex conjugation, or quaternionic conjugation on Ǩ. Thus, the action of T ε ˜on spinors results in matrix transposition, complex Hermitian conjugation, or quaternionic Hermitian conjugation. We classify automorphism group of the new product as O(N), U(N), Sp(N), 2 O(N), or 2 Sp(N).
arXiv (Cornell University), Jun 16, 2012
We present different methods for symbolic computer algebra computations in higher dimensional (≥ ... more We present different methods for symbolic computer algebra computations in higher dimensional (≥ 9) Clifford algebras using the CLIFFORD and Bigebra packages for Maple R . This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.
CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with qua... more CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C (B) -the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUMbased on Chevalley's recursive formula, and cmulRS -based on a non-recursive Rota-Stein sausage. Graßmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.
Clifford and Gra�mann Hopf algebras via the package for Maple
Computer Physics Communications, 2005
Clifford Algebras and their Applications in Mathematical Physics, 2000
It is a well known fact from the group theory that irreducible tensor representations of classica... more It is a well known fact from the group theory that irreducible tensor representations of classical groups are suitably characterized by irreducible representations of the symmetric groups. However, due to their different nature, vector and spinor representations are only connected and not united in such description. Clifford algebras are an ideal tool with which to describe symmetries of multiparticle systems since they contain spinor and vector representations within the same formalism, and, moreover, allow for a complete study of all classical Lie groups. In this work, together with an accompanying work also presented at this conference, an analysis of q -symmetry -for generic q 's -based on the ordinary symmetric groups is given for the first time. We construct q -Young operators as Clifford idempotents and the Hecke algebra representations in ideals generated by these operators. Various relations as orthogonality of representations and completeness are given explicitly, and the symmetry types of representations is discussed. Appropriate q -Young diagrams and tableaux are given. The ordinary case of the symmetric group is obtained in the limit q → 1. All in all, a toolkit for Clifford algebraic treatment of multi-particle systems is provided. The distinguishing feature of this paper is that the Young operators of conjugated Young diagrams are related by Clifford reversion, connecting Clifford algebra and Hecke algebra features. This contrasts the purely Hecke algebraic approach of King and Wybourne, who do not embed Hecke algebras into Clifford algebras.
Clifford Algebras and their Applications in Mathematical Physics, 2000
Clifford algebras are naturally associated with quadratic forms. These algebras are Z2 -graded by... more Clifford algebras are naturally associated with quadratic forms. These algebras are Z2 -graded by construction. However, only a Zn -gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cℓ(V ) ↔ V and an ordering, guarantees a multivector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the Zn -grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a Zn -grading which we now call Clifford algebras of multivectors or quantum Clifford algebras. It turns out, that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonalizability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cℓp,q which can be decomposed in the symmetric case into a tensor product Cℓp-1,q-1 ⊗ Cℓ1,1. The general case used in quantum field theory lacks this feature. Theories
Czechoslovak Journal of Physics, 2003
A classification of idempotents in Clifford algebras C p,q is presented. It is shown that using i... more A classification of idempotents in Clifford algebras C p,q is presented. It is shown that using isomorphisms between Clifford algebras C p,q and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one sided ideals in Clifford algebras. Some low dimensional examples are discussed.
Advances in Applied Clifford Algebras, 2014
We present different methods for symbolic computer algebra computations in higher dimensional (≥ ... more We present different methods for symbolic computer algebra computations in higher dimensional (≥ 9) Clifford algebras using the CLIFFORD and Bigebra packages for Maple R . This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.
CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with qua... more CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C (B) -the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUMbased on Chevalley's recursive formula, and cmulRS -based on a non-recursive Rota-Stein sausage. Graßmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.
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Papers by Rafal Ablamowicz