Représentations des algèbres de clifford des formes binaires et ternaires cubiques

Clifford algebras of polynomial forms of degreed>2 defined by N. Roby are infinite dimensional when the module has rank bigger than 1. So the study of their representations differs from that of Clifford algebras of quadratic forms. Some authors have established several results in this domain, especially in the case of binary cubic forms, which can be found in the bibliography. In the first section of this paper, we consider the case of binary cubic forms and the base fieldK is algebraically closed. We define an explicit family of irreducible representations of their Clifford algebras, indexed by only one parameter, such that every 3-dimensional representation is equivalent to an element of this family. When the form is ternary cubic, Revoy and Tesser on one hand, Van den Bergh on the other, proved in two different ways, in [7] and [9] respectively, the existence of 3-dimensional linearizations. In the last section, we show directly that the number of classes of 3-dimensional representations is finite and non zero.     

Factorisations des extensions galoisiennes

ABSTRACT

In general, Clifford algebras of quadratic forms are finite dimensional; therefore, their representations are easy to describe. However, for homogenous polynomial forms of degree dbm > 2, the situation is different because their Clifford algebras are infinite dimensional. In this article, we get a finite set of pairwise orthogonal idempotents of sum 1 in these algebras. This permits us to obtain interesting properties for d-dimensional representations of polynomial forms of degree d; for example, we show that the image C of the Clifford algebra by such representation is an endomorphism algebra of finitely generated projective Z(C)-module of d-rank, direct sum of finitely generated projective Z(C)-module of 1-rank. Before establishing this, we give a new proof of the Poincaré-Birkhoff-Witt theorem for these algebras with the help of a general composition lemma. At the end of this work, we give a linearization of diagonal binary and ternary forms of degree dbm > 3.     

The Graded Structure of Nondegenerate Meson Algebras

Meson algebras are involved in the wave equation of meson particles in the same way as Clifford algebras are involved in the Dirac wave equation of electrons. Here we improve and generalize the information already obtained about their structure and their representations, when the symmetric bilinear form under consideration is nondegenerate. We emphasize their parity grading. We calculate the center of these meson algebras, and the center of their even subalgebra. Finally we show that every nondegenerate meson algebra over a field contains a group isomorphic to the group of automorphisms of the symmetric bilinear form.      

The design of linear algebra and geometry

Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with a simple projective interpretation. This opens up new possibilities for coordinate-free computations in linear algebra. For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar. This approach also reveals deep relations between the structure of the linear geometries, from projective to metrical, and the structure of Clifford algebras. This is apparent in a new relation between additive and multiplicative forms for intervals in the cross-ratio. Also, various factorizations of Clifford algebras into Clifford algebras of lower dimension are shown to have projective interpretations.As an important application with many uses in physics as well as in mathematics, the various representations of the conformal group in Clifford algebra are worked out in great detail. A new primitive generator of the conformal group is identified.     

Isomorphism groups in Clifford algebras

A set of anticommuting multivectors in Clifford algebras can be taken as orthonormal basis set. The Clifford algebra generated by this basis is isomorphic to the original algebra. The non linear transformations between orthonormal basis sets form a group. In the four dimensionnal case six sets of five anticommuting multivectors are found. These sets yield 30 matrices defining basis sets. These matrices are representatives of left cosets, members of these cosets are related by permutation of rows. From the equivalence of all basis sets of multivectors it can be concluded that there is no canonical set of basis vectors in Clifford algebras.     

Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields are isomorphic as algebras to a multivariate skew polynomial ring whose ring morphism from the field to the ring of square matrices is diagonal, and whose derivation is the zero derivation. Furthermore, we prove that two such representations only differ in a permutation on the field automorphisms appearing in the corresponding diagonal. The algebra isomorphisms are given by affine transformations of variables and preserve evaluations and degrees. In addition, ours proofs show that the simplified form of multivariate skew polynomial rings can be found computationally and explicitly.     

Extended Grassmann and Clifford Algebras

This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated extended algebras, and to explore these concepts also from the counterspace viewpoint. The presented formalism explains how the concept of chirality stems from the bracket, as defined by Rota et all [1]. The exterior (regressive) algebra is shown to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, if we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem of Clifford algebras context, wherein the Clifford and extended Clifford algebras can be embedded in which is shown to be exactly the extended Clifford algebra. We present the essential character of the Rota’s bracket, relating it to the formalism exposed by Conradt [25], introducing the regressive product and subsequently the counterspace. Clifford algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior products of 1-forms. R. da Rocha is supported by CAPES     

Development of the Method of Quaternion Typification of Clifford Algebra Elements

In this paper we further develop the method of quaternion typification of Clifford algebra elements suggested by the author in the previous papers. On the basis of new classification of Clifford algebra elements, it is possible to reveal and prove a number of new properties of Clifford algebras. We use k-fold commutators and anticommutators. In this paper we consider Clifford and exterior degrees and elementary functions of Clifford algebra elements.     

Line-bundle-valued ternary quadratic forms over schemes

We study degenerations of rank 3 quadratic forms and of rank 4 Azumaya algebras, and extend what is known for good forms and Azumaya algebras. By considering line-bundle-valued forms, we extend the theorem of Max-Albert Knus that the Witt-invariant—the even Clifford algebra of a form—suffices for classification. An algebra Zariski-locally the even Clifford algebra of a ternary form is so globally up to twisting by square roots of line bundles. The general, usual and special orthogonal groups of a form are determined in terms of automorphism groups of its Witt-invariant. Martin Kneser’s characteristic-free notion of semiregular form is used.     

Skew Clifford algebras

We introduce a generalization, called a skew Clifford algebra, of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincaré-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the ${\mathbb{Z}}_2$-graded PBW deformations of quantum exterior algebras. We also determine the possible dimensions of skew Clifford algebras and provide several examples.     

Affine cellular algebras

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras.We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.     

Clifford-like calculus over lattices

We introduce a calculus over a lattice based on a lattice generalization of the Clifford algebras. We show that Clifford algebras, in contrast to the continuum, are not an adequated algebraic structure for lattice problems. Then we introduce a new algebraic structure, that reduces to a Clifford algebra in the continuum limit, in terms of which we can develop a formalism analogous to the differential geometry of the continuum, also in the sense that we have intrinsic expressions. The differential operator is given by the graded commutator of an operator that generalizes the Dirac operator. We also discuss the applications of this formalism in lattice gauge theories, with particular attention to the fermion doubling problem. On leave of absence from Department of Applied Mathematics, State University at Campinas (UNICAMP), 13081-970 Campinas, SP, Brazil — vaz@ime.unicamp.br     

On some relations between spinor and orthogonal groups

In this article we consider Clifford algebras over the field of real numbers of finite dimension. We define the operation of Hermitian conjugation for the elements of Clifford algebra. This operation allows us to define the structure of Euclidian space on the Clifford algebra. We consider pseudo-orthogonal group and its subgroups — special pseudo-orthogonal, orthochronous, orthochorous and special orthochronous groups. As we know, spinor groups are double covers of these orthogonal groups.We proved theorem that relates the norm of element of spinor group to the minor of matrix of the corresponding orthogonal group.     

Cyclic Structures of Cliffordian Supergroups and Particle Representations of Spin+(1, 3)

Supergroups are defined in the framework of \({\mathbb{Z}_2}\) 2-graded Clifford algebras over the fields of real and complex numbers, respectively. It is shown that cyclic structures of complex and real supergroups are defined by Brauer-Wall groups related with the modulo 2 and modulo 8 periodicities of the complex and real Clifford algebras. Particle (fermionic and bosonic) representations of a universal covering (spinor group Spin ₊(1, 3)) of the proper orthochronous Lorentz group are constructed via the Clifford algebra formalism. Complex and real supergroups are defined on the representation system of Spin ₊(1, 3). It is shown that a cyclic (modulo 2) structure of the complex supergroup is equivalent to a supersymmetric action, that is, it converts fermionic representations into bosonic representations and vice versa. The cyclic action of the real supergroup leads to a much more high-graded symmetry related with the modulo 8 periodicity of the real Clifford algebras. This symmetry acts on the system of real representations of Spin ₊(1, 3).     

Unitary Spaces on Clifford Algebras

For the complex Clifford algebra (p, q) of dimension n = p + q we define a Hermitian scalar product. This scalar product depends on the signature (p, q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These representations take into account the structure of unitary space on Clifford algebra. The work of N.M. is supported in part by the Russian President’s grant NSh-6705.2006.1.     

Cyclotomic Solomon algebras

This paper introduces an analogue of the Solomon descent algebra for the complex reflection groups of type G(r,1,n). As with the Solomon descent algebra, our algebra has a basis given by sums of ‘distinguished’ coset representatives for certain ‘reflection subgroups.’ We explicitly describe the structure constants with respect to this basis and show that they are polynomials in r. This allows us to define a deformation, or q-analogue, of these algebras which depends on a parameter q. We determine the irreducible representations of all of these algebras and give a basis for their radicals. Finally, we show that the direct sum of cyclotomic Solomon algebras is canonically isomorphic to a concatenation Hopf algebra.     

Infinite-dimensional 3-Lie algebras and their connections to Harish-Chandra modules

We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.     

Geometric interpretation of Murphy bases and an application

In this article we study the representations of general linear groups which arise from their action on flag spaces. These representations can be decomposed into irreducibles by proving that the associated Hecke algebra is cellular. We give a geometric interpretation of a cellular basis of such Hecke algebras which was introduced by Murphy in the case of finite fields. We apply these results to decompose representations which arise from the space of submodules of a free module over principal ideal local rings of length two with a finite residue field.