Diffeological Clifford algebras and pseudo-bundles of Clifford modules

Abstract

We consider the diffeological version of the Clifford algebra of a diffeological finite dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). After considering the natural diffeology of the Clifford algebra, and considering which of its standard properties re-appear in the diffeological context (most of them), we turn to our main interest, which is constructing the pseudo-bundles of Clifford algebras associated to a given (finite dimensional) diffeological vector pseudo-bundle, and those of the usual Clifford modules (the exterior algebras). The substantial difference that emerges with respect to the standard context, and paves the way to various questions that do not have standard analogues, stems from the fact that the notion of a diffeological pseudo-bundle is very different from the usual bundle, and this under two main respects: it may have fibres of different dimensions, and even if it does not, its total and base spaces frequently are not smooth, or even topological, manifolds.     

Clifford algebras and vector-valued rational forms II

We demonstrate how the use of Clifford algebras in the theory of vector-valued rational forms leads to practical recurrence relations which do not involve representations of the algebras     

Generalized Grassmann algebras and applications to stochastic processes

In this paper, we present the groundwork for an Itô/Malliavin stochastic calculus and Hida's white noise analysis in the context of a supersymmetry with ${?}_3$-graded algebras. To this end, we establish a ternary Fock space and the corresponding strong algebra of stochastic distributions and present its application in the study of stochastic processes in this context.     

Irreducible representations of general group algebras and their applications

In this paper we obtain some properties of irreducible representations of general group algebras, then as applications, we extend Gluck-Wolf Theorem for ordinary characters to the case of Brauer characters; we investigate the relations between the degree of an indecomposable module and that of its source.     

Decomposability for division algebras of exponent two and associated forms

This article investigates the structure of quadratic forms and of division algebras of exponent two over fields of characteristic different from two with the property that the third power of the fundamental ideal in the associated Witt ring is torsion free.      

On the transposition anti-involution in real Clifford algebras I: the transposition map

A particular orthogonal map on a finite-dimensional real quadratic vector space (V,?Q) with a non-degenerate quadratic form Q of any signature (p,?q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra C?(V*,?Q) of linear functionals (multiforms) acting on the universal Clifford algebra C?(V,?Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of C?(V,?Q). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of C?(V,?Q). We also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [R. Ab?amowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear].     

Flags in zero dimensional complete intersection algebras and indices of real vector fields

We introduce bilinear forms in a flag in a complete intersection local -algebra of dimension 0, related to the Eisenbud–Levine, Khimshiashvili bilinear form. We give a variational interpretation of these forms in terms of Jantzen’s filtration and bilinear forms. We use the signatures of these forms to compute in the real case the constant relating the GSV-index with the signature function of vector fields tangent to an even dimensional hypersurface singularity, one being topologically defined and the other computable with finite dimensional commutative algebra methods. Partially supported by Plan Nacional I+D grant no. MTM2004-07203-C02-02, Spain and CONACYT 40329, México.     

Tensor products of algebras and their applications to the construction of Anosov diffeomorphisms

In this paper, we develop algebraic approaches to the construction of Anosov diffeomorphisms on compact manifolds. Two mutually dual constructions are described, which provide numerous new examples of Anosov diffeomorphisms on nilmanifolds. The basis of the constructions is the operation of tensor multiplication of Lie algebras by appropriate finite-dimensional associative-commutative algebras. Several examples illustrating the general method are given.     

Cohen-Host type idempotent theorems for representations on Banach spaces and applications to Figà-Talamanca-Herz algebras

Let G be a locally compact group, and let R(G) denote the ring of subsets of G generated by the left cosets of open subsets of G. The Cohen-Host idempotent theorem asserts that a set lies in R(G) if and only if its indicator function is a coefficient function of a unitary representation of G on some Hilbert space. We prove related results for representations of G on certain Banach spaces. We apply our Cohen-Host type theorems to the study of the Figà-Talamanca-Herz algebras Ap(G) with p∈(1,∞). For arbitrary G, we characterize those closed ideals of Ap(G) that have an approximate identity bounded by 1 in terms of their hulls. Furthermore, we characterize those G such that Ap(G) is 1-amenable for some—and, equivalently, for all—p∈(1,∞): these are precisely the abelian groups.     

Explicit Classifications of Orbits in Jordan Algebra and Freudenthal Vector Space Over the Exceptional Lie Groups

ABSTRACT

We give the explicit classifications of orbits in the Jordan algebra 𝔍 over the group E _6(?26) and the Freudenthal, R -vector space 𝔓 over the group E _7(?25).

Communicated by E. Zelmanov     

Generic Splitting Fields of Central Simple Algebras: Galois Cohomology and Nonexcellence

A field extension L / F is called excellent if, for any quadratic form over F, the anisotropic part (L)an of over L is defined over F; L / F is called universally excellent if L E / E is excellent for any field extension E / F. We study the excellence property for a generic splitting field of a central simple F-algebra. In particular, we show that it is universally excellent if and only if the Schur index of the algebra is not divisible by 4. We begin by studying the torsion in the second Chow group of products of Severi–Brauer varieties and its relationship with the relative Galois cohomology group H3(L / F) for a generic (common) splitting field L of the corresponding central simple F-algebras.     

An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems

By using Gerstewitz functions, we establish a new equilibrium version of Ekeland variational principle, which improves the related results by weakening both the lower boundedness and the lower semi-continuity of the ob jective bimaps. Applying the new version of Ekeland principle, we obtain some existence theorems on solutions for set-valued vector equilibrium problems, where the most used assumption on compactness of domains is weakened. In the setting of complete metric spaces(Z,d), we present an existence result of solutions for set-valued vector equilibrium problems, which only requires that the domain XZ is countably compact in any Hausdorff topology weaker than that induced by d. When(Z, d) is a Féchet space(i.e., a complete metrizable locally convex space), our existence result only requires that the domain XZ is weakly compact. Furthermore, in the setting of non-compact domains, we deduce several existence theorems on solutions for set-valued vector equilibrium problems,which extend and improve the related known results.