Covariant differential complexes on quantum linear groups

TL;DR

This paper classifies covariant differential structures on quantum groups GL_q(N) and SL_q(N), establishing algebraic frameworks that generalize classical calculus with quantum deformations.

Contribution

It introduces a classification scheme for covariant exterior algebras on quantum groups based on natural postulates and explores their properties.

Findings

Classified covariant exterior algebra structures on GL_q(N) and SL_q(N).

Defined exterior derivatives with nilpotence and deformed Leibniz rules.

Discussed the relation of known calculi to the classification scheme.

Abstract

We consider the possible covariant external algebra structures for Cartan's 1-forms on GL_q(N) and SL_q(N). We base upon the following natural postulates: 1. the invariant 1-forms realize an adjoint representation of quantum group; 2. all monomials of these forms possess the unique ordering. For the obtained external algebras we define the exterior derivative possessing the usual nilpotence condition, and the generally deformed version of Leibniz rules. The status of the known examples of GL_q(N)-differential calculi in the proposed classification scheme, and the problems of SL_q(N)-reduction are discussed.