On Clifford representation of Hopf algebras and fierz identities

We present a short review of the action and coaction of Hopf algebras on Clifford algebras as an introduction to physically meaningful examples. Some q-deformed Clifford algebras are studied from this context and conclusions are derived.     

Idempotents of Clifford Algebras

A classification of idempotents of 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.     

On the relation of Manin’s quantum plane and quantum Clifford algebras

In a recent work we have shown that quantum Clifford algebras — i.e. Clifford algebras of an arbitrary bilinear form — are closely related to the deformed structures asq-spin groups, Hecke algebras,q-Young operators and deformed tensor products. The question to relate Manin’s approach to quantum Clifford algebras is addressed here. Explicit computations using the CLIFFORD Maple package are exhibited. The meaning of non-commutative geometry is reexamined and interpreted in Clifford algebraic terms. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Real and holomorphic SuSy algebras

We propose a new constructive scheme for real SuSy algebras within the general formalism of real Clifford algebra theory, and use it to characterize a natural holomorphic extension admitted by a subclass of real SuSy algebras in signaturess–t=3 mod 4. The resulting holomorphic SuSy algebras are the abstract models of a new type of supersymmetry algebras, already known to be relevant in the development of canonical or Hamiltonian methods in superspace.Research supported in part by CNPq-Brazil.     

The Dirac equation in exterior form

Using the correspondence between the Clifford and exterior algebras we write the Dirac equation in terms of differential forms. The covariances of the theory are then examined. We show in detail the correspondence with usual matrix methods.     

SO(r+2, r) pure spinors

Chevalley gave a comprehensive treatment of pure spinors for Clifford algebras whose quadratic form has maximal index. We here show how the notion of pure spinor can be extended to the real Clifford algebras associated with quadratic forms with r+2 positive and r negative eigenvalues.     

Isotopic Liftings of Clifford Algebras and Applications in Elementary Particle Mass Matrices

Isotopic liftings of algebraic structures are investigated in the context of Clifford algebras, where it is defined a new product involving an arbitrary, but fixed, element of the Clifford algebra. This element acts as the unit with respect to the introduced product, and is called isounit. We construct isotopies in both associative and non-associative arbitrary algebras, and examples of these constructions are exhibited using Clifford algebras, which although associative, can generate the octonionic, non-associative, algebra. The whole formalism is developed in a Clifford algebraic arena, giving also the necessary pre-requisites to introduce isotopies of the exterior algebra. The flavor hadronic symmetry of the six u,d,s,c,b,t quarks is shown to be exact, when the generators of the isotopic Lie algebra are constructed, and the unit of the isotopic Clifford algebra is shown to be a function of the six quark masses. The limits constraining the parameters, that are entries of the representation of the isounit in the isotopic group SU(6), are based on the most recent limits imposed on quark masses.     

Generalized Clifford algebras: Orthogonal and symplectic cases

The concept of the Clifford algebra of a bilinear form defined on a vector space, is generalized to include both orthogonal and symplectic forms. This new characterization is intimately tied to a particular anti-automorphism defined on the algebras which in the orthogonal case, provides a unified approach to the reversion and principle anti-involutions of an ordinary Clifford algebra. Most attention is concentrated on the physically important symplectic Clifford algebras, including the link between these and Lie superalgebras.Until February, 1982 at the Institüt für Theoretische Physik, Universität Karlsruhe, 75 Karlsruhe 1, Kaiserstr. 12, W. Germany.     

Modulo 8 periodicity of real Clifford algebras and particle physics

After a review of the properties of real Clifford algebras, we discuss the isomorphism existing between these algebras and matrix algebras over the real, complex or quaternion field. This is done for all dimensions and all possible signatures of the metric. The modulo 8 periodicity theorem is discussed and extended. A comment is made about the appearance of “hidden” symmetries in supergravity theories.     

Octonionic representations of Clifford algebras and triality

The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the nonassociativity and noncommutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octoninic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest ₃ ×SO(8) structure in this framework.     

Symplectic reduction,BRS cohomology,and infinite-dimensional Clifford algebras

This paper gives the mathematical foundations for the BRS quantization procedure. We first discuss the classical finite dimensional BRS procedure and relate it to Marsden-Weinstein reduction. This leads to interesting relations between Lie algebras and Clifford algebras and a novel way of computing Lie algebra cohomology in terms of the spin representation. We then discuss infinite-dimensional Clifford algebras and their spin representations. We find that in the infinite-dimensional case, the analog of the finite-dimensional construction of Lie algebra cohomology breaks down, the obstruction (anomaly) being the Kac-Peterson class which is the cohomology class associated to the representation of the Lie algebra on spinors which is now only a projective representation. Tensoring by a projective representation of opposite class kills the obstruction and gives rise to a cohomology theory and a quantization procedure. We discuss the gradings and Hermitian structures on the absolute and relative complexes.     

CPT groups for spinor field in de Sitter space

A group structure of the discrete transformations (parity, time reversal and charge conjugation) for spinor field in de Sitter space are studied in terms of extraspecial finite groups. Two CPT groups are introduced, the first group from an analysis of the de Sitter–Dirac wave equation for spinor field, and the second group from a purely algebraic approach based on the automorphism set of Clifford algebras. It is shown that both groups are isomorphic to each other.     

A note on quantizations of Galois extensions

In Huru and Lychagin (2013), it is conjectured that the quantizations of splitting fields of products of quadratic polynomials, which are obtained by deforming the multiplication, are Clifford type algebras. In this paper, we prove this conjecture.     

Gauge Field Theories on Clifford Algebras

We present a straightforward model of the U(1) gauge equations of Dirac and Maxwell, as well as the U(n) Yang–Mills equations where all fields and gauge transformations take values in a Clifford algebra. When expressed in terms of the Clifford components of the fields, the equations display various gauge symmetries which we intestigate for all Clifford algebras. In particular, for the Pauli algebra, the Dirace CA equations possess the SU(2) × U(1)-symmetry.     

A Series of Algebras Generalizing the Octonions and Hurwitz-Radon Identity

We study non-associative twisted group algebras over (\mathbbZ₂)ⁿ{(\mathbb{Z}_2)^n} with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion.