Covariant Dirac Operators on Quantum Groups

TL;DR

This paper constructs a covariant Dirac operator on quantum groups associated with classical Lie algebras, aiming to develop equivariant Fredholm modules and K-homology cycles in noncommutative geometry.

Contribution

It introduces a new Dirac operator on quantum groups using a q-deformed Clifford algebra, generalizing previous work by Bibikov and Kulish.

Findings

Constructed a Dirac operator invariant under quantum group action.

Established a framework for equivariant Fredholm modules on quantum groups.

Extended classical Dirac operator concepts to the quantum group setting.

Abstract

We give a construction of a Dirac operator on a quantum group based on any simple Lie algebra of classical type. The Dirac operator is an element in the vector space $U_q(\g) \otimes \mathrm{cl}_q(\g)$ where the second tensor factor is a $q$-deformation of the classical Clifford algebra. The tensor space $ U_q(\g) \otimes \mathrm{cl}_q(\g)$ is given a structure of the adjoint module of the quantum group and the Dirac operator is invariant under this action. The purpose of this approach is to construct equivariant Fredholm modules and $K$-homology cycles. This work generalizes the operator introduced by Bibikov and Kulish in \cite{BK}.