Action of a finite quantum group on the algebra of complex NxN matrices

TL;DR

This paper explores how a finite quantum group acts on the algebra of NxN complex matrices, revealing indecomposable modules and actions on differential forms, with implications for quantum algebra structures.

Contribution

It demonstrates the action of a finite quantum group on matrix algebras and differential forms, reducing the algebra to indecomposable modules for the quantum group.

Findings

Matrix algebra M_N(C) can be viewed as a module algebra for a finite quantum group.

The algebra M_N(C) decomposes into indecomposable modules under this quantum group action.

The quantum group also acts on the space of generalized differential forms associated with M_N(C).

Abstract

Using the fact that the algebra M := M_N(C) of NxN complex matrices can be considered as a reduced quantum plane, and that it is a module algebra for a finite dimensional Hopf algebra quotient H of U_q(sl(2)) when q is a root of unity, we reduce this algebra M of matrices (assuming N odd) into indecomposable modules for H. We also show how the same finite dimensional quantum group acts on the space of generalized differential forms defined as the reduced Wess Zumino complex associated with the algebra M.