On q-Deformations of Clifford Algebras

Abstract

Several Clifford algebras that are covariant under the action of a Lie algebra g can be deformed in a way consistent with the deformation of U g into a quantum group (or into a triangular Hopf algebra) U q g , i.e. so as to remain covariant under the action of U q g . In this report, after recalling these facts, we review our results regarding the formal realization of the elements of such "q-deformed" Clifford algebras as "functions" (polynomials) in the generators of the undeformed ones; in particular, the intruiging interplay between the original and the q-deformed symmetry. Finally, we briefly illustrate their dramatic consequences on the representation theories of the original and of the q-deformed Clifford algebra, and mention how these results could turn out to be useful in quantum physics.-Preprint 99-51 Dip. Matematica e Applicazioni, Università di Napoli * Invited talk given at the "5th International Conference on Clifford algebras and their Applications in Mathematical Physics, Ixtapa-Zihuatanejo (Mexico) June-July 1999.

A Hodge–de Rham Dirac operator on the quantum SU(2)

“…Examples of spin c -spectral triples with a Dirac operator given by a non commutative deformation of the Dolbeault -Dirac operator (which classically coincides with the Hodge -de Rham operator on Kähler manifolds) has been introduced and studied on quantum flag manifolds [19], quantised symmetric spaces [20,25]: via a suitable twist a spin c -structure on quantum complex projective spaces CP N q is seen to give a spin-structure for odd N in [3,4]. Within the quantum group formalism, meaningful Clifford algebras and spinors are introduced in [6,8] in terms of the properties of the R-braiding for the FRT approach. This approach is evolved in [1,2,14] for quantum groups equipped with a Woronowicz bicovariant exterior calculus, thus allowing for the definition of a Dirac operator.…”

On Frt– Clifford Algebras

“…In the last years there were made several attempts in this direction. Quantum Clifford algebras are introduced and studied for instance in [11], [4], [10], [1], [9].…”