Identities on Clifford algebras

We obtain some estimates of the codimensions and PI-exponent of identities on the Clifford algebras associated to a quadratic form of a given rank, find a hook where the cocharacters of Clifford algebras lie, exhibit an identity that holds on the Clifford algebras of a given rank, and describe the multilinear identities of minimal degree for the Clifford algebras of rank 1.     

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.     

Discriminant and Clifford algebras

The centralizer of a square-central skew-symmetric unit in a central simple algebra with orthogonal involution carries a unitary involution. The discriminant algebra of this unitary involution is shown to be an orthogonal summand in one of the components of the Clifford algebra of the orthogonal involution. As an application, structure theorems for orthogonal involutions on central simple algebras of degree 8 are obtained. Received: 30 January 2001; in final form: 28 May 2001 / Published online: 1 February 2002     

Vector coherent states on Clifford algebras

The well-known canonical coherent states are expressed as infinite series in powers of a complex number z and a positive integer (m) = m!. In analogy with the canonical coherent states, we present a class of vector coherent states by replacing the complex variable z with a real Clifford matrix. We also present another class of vector coherent states by simultaneously replacing z with a real Clifford matrix and (m) with a real matrix. As examples, we present vector coherent states labeled by quaternions and octonions with their real matrix representations. We also present a physical example.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 9–21, April, 2005.     

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 of hyperbolic involutions

For a central simple algebra of even degree with hyperbolic orthogonal involution, we describe the canonically induced involution on the even Clifford algebra of . When , and the interesting part of is isomorphic to the canonical involution on an exterior power algebra of B. As a corollary, we get some properties of the involution on the exterior power algebra. Received January 27, 1998; in final form June 7, 2000 / Published online October 30, 2000     

Idempotent structure of Clifford algebras

Spinor spaces can be represented as minimal left ideals of Clifford algebras and they are generated by primitive idempotents. Primitive idempotents of the Clifford algebras R _{p, q} are shown to be products of mutually nonannihilating commuting idempotent % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaIXaaabaGaaGOmaaaaaaa!3DBD!\[{\textstyle{1 \over 2}}\]2}}\](1+e _T ), where the k=q–r _q–p basis elements e _T satisfy e _T ²=1. The lattice generated by a set of mutually annihilating primitive idempotents is examined. The final result characterizes all Clifford algebras R _{p, q} with an anti-involution such that each symmetric elements is either a nilpotent or then some right multiple of it is a nonzero symmetric idempotent. This happens when p+q<-3 and (p, q)(2, 1).     

Spinor spaces of Clifford algebras

In this paper, we study the structures of Clifford algebras. We represent the pinor and spinors spaces as subspaces of Clifford algebras. With suitable bases of the Clifford algebras, we construct isomorphisms between Clifford algebras and matrix algebras. In doing these we develop some spinor calculus.     

Clifford algebras and geometric algebra

The geometric algebra as defined by D. Hestenes is compared with a constructive definition of Clifford algebras. Both approaches are discussed and the equivalence between a finite geometric algebra and the universal Clifford algebra R _{p, q} is shown. Also an intermediate way to construct Clifford algebras is sketched. This attempt to conciliate two separated approaches may be useful taking into account the recognized importance of Clifford algebras in theoretical and applied physics.     

Orthogonal symmetries and Clifford algebras

Involutions of the Clifford algebra of a quadratic space induced by orthogonal symmetries are investigated.     

Chain spaces via Clifford algebras

In this note we show that the chain space belonging to a quadric can be embedded into the chain geometry over a Clifford algebra via a generalized stereographic projection.     

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.     

Clifford structures on Riemannian manifolds

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r=2 and r=3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r=2,3 and 4 respectively, several classes of 8-dimensional manifolds (for 5?r?8), families of real, complex and quaternionic Grassmannians (for r=8,6 and 5 respectively), and Rosenfeld?s elliptic projective planes OP², (C⊗O)P², (H⊗O)P² and (O⊗O)P², which are symmetric spaces associated to the exceptional simple Lie groups F₄, E₆, E₇ and E₈ (for r=9,10,12 and 16 respectively). As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.     

Generalized Clifford algebras and certain infinite dimensional Lie algebras

The embedding of certain infinite dimensional Lie algebras in generalized Clifford algobras C(N, p) is given. The correspondence between C(N,2) andgl(N, C) asN⇌∞ is pointed out.     

Conjugate harmonic functions and Clifford algebras

We generalize a Hardy-Littlewood inequality and a Privalov inequality for conjugate harmonic functions in the plane to components of Clifford-valued monogenic functions.