Twin Bent Functions and Clifford Algebras

Journal of Mathematical Physics, 2009

In this paper we address the problem of constructing a class of representations of Clifford algebras that can be named "alphabetic (re)presentations". The Clifford algebras generators are expressed as m-letter words written with a 3-character or a 4-character alphabet. We formulate the problem of the alphabetic presentations, deriving the main properties and some general results. At the end we briefly discuss the motivations of this work and outline some possible applications.

Linear and Multilinear Algebra, 2011

A signature ε = (p, q) dependent transposition anti-involution T ε˜o f 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˜g ives 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 εĩ s 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.

Advances in Theoretical and Mathematical Physics, 2004

The double triangle algebra(DTA) associated to an ADE graph is considered. A description of its bialgebra structure based on a reconstruction approach is given. This approach takes as initial data the representation theory of the DTA as given by Ocneanu's cell calculus. It is also proved that the resulting DTA has the structure of a weak *-Hopf algebra. As an illustrative example, the case of the graph A 3 is described in detail.

2008

The double triangle algebra(DTA) associated to an ADE graph is considered. A description of its bialgebra structure based on a reconstruction approach is given. This approach takes as initial data the representation theory of the DTA as given by Ocneanu’s cell calculus. It is also proved that the resulting DTA has the structure of a weak *-Hopf algebra. As an illustrative example, the case of the graph A3 is described in detail.

Symmetry, Integrability and Geometry: Methods and Applications

Let D ≥ 1 and q ≥ 3 be two integers. Let H(D) = H(D, q) denote the D-dimensional Hamming graph over a q-element set. Let T (D) denote the Terwilliger algebra of H(D). Let V (D) denote the standard T (D)-module. Let ω denote a complex scalar. We consider a unital associative algebra K ω defined by generators and relations. The generators are A and B. The relations are A 2 B − 2ABA + BA 2 = B + ωA, B 2 A − 2BAB + AB 2 = A + ωB. The algebra K ω is the case of the Askey-Wilson algebras corresponding to the Krawtchouk polynomials. The algebra K ω is isomorphic to U(sl 2) when ω 2 = 1. We view V (D) as a K 1− 2 q-module. We apply the Clebsch-Gordan rule for U(sl 2) to decompose V (D) into a direct sum of irreducible T (D)-modules.

2004

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.

Journal of Pure and Applied Algebra, 2008

The algebras Q n describe the relationship between the roots and coefficients of a non-commutative polynomial. I. Gelfand, S. Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this paper we find the Hilbert series of the class of algebras corresponding to the graph K 3 and show that this algebra is Koszul.

Geometry and computing, 2020

The main goal of this chapter is to explain how rotations in R n are induced by the action of a certain group Spin(n) on R n , in a way that generalizes the action of the unit complex numbers U(1) on R 2 , and the action of the unit quaternions SU(2) on R 3 (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and R n). The group Spin(n), called a spinor group, is defined as a certain subgroup of units of an algebra Cl n , the Clifford algebra associated with R n. Furthermore, for n ≥ 3, we are lucky, because the group Spin(n) is topologically simpler than the group SO(n). Indeed, for n ≥ 3, the group Spin(n) is simply connected (a fact that it is not so easy to prove without some machinery), whereas SO(n) is not simply connected. Intuitively speaking, SO(n) is more twisted than Spin(n). In fact, we will see that Spin(n) is a double cover of SO(n). Since the spinor groups are certain well-chosen subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups. This chapter provides a tutorial on Clifford algebra and the groups Spin and Pin, including a study of the structure of the Clifford algebra Cl p,q associated with a nondegenerate symmetric bilinear form of signature (p, q) and culminating in the beautiful "8-periodicity theorem" of Élie Cartan and Raoul Bott (with proofs). We also explain when Spin(p, q) is a double-cover of SO(p, q). The reader should be warned that a certain amount of algebraic (and topological) background is expected. This being said, perseverant readers will be rewarded by being exposed to some beautiful and nontrivial concepts and results, including Élie Cartan and Raoul Bott "8-periodicity theorem." Going back to rotations as transformations induced by group actions, recall that if V is a vector space, a linear action (on the left) of a group G on V is a map α : G× V → V satisfying the following conditions, where, for simplicity of notation, we denote α(g, v) by g • v:

International Electronic Journal of Algebra, 2021

The chromatic polynomial is characterized as the unique polynomial invariant of graphs, compatible with two interacting bialgebras structures: the first coproduct is given by partitions of vertices into two parts, the second one by a contraction-extraction process. This gives Hopf-algebraic proofs of Rota's result on the signs of coefficients of chromatic polynomials and of Stanley's interpretation of the values at negative integers of chromatic polynomials. We also consider chromatic symmetric functions and their noncommutative versions.

International Journal of Theoretical Physics, 2001

The history and immediate future of the International Conferences on Clifford Algebras and Their Applications.