Geometry of the reduced quantum plane

TL;DR

This paper explores the geometric structure of the reduced quantum plane modeled by NxN matrices, focusing on quantum group actions, coactions, and a differential calculus, especially for odd N values like N=3.

Contribution

It introduces a differential calculus on the reduced quantum plane and analyzes its geometry under quantum group symmetries, with specific focus on odd N cases.

Findings

Defined a differential calculus as a quotient of the Wess Zumino complex.

Analyzed the geometry of the quantum plane under quantum group actions.

Special case N=3 provides explicit insights into the structure.

Abstract

We consider the space M of NxN matrices as a reduced quantum plane and discuss its geometry under the action and coaction of finite dimensional quantum groups (a quotient of U_q(SL(2)), q being an N-th root of unity, and its dual). We also introduce a differential calculus for M: a quotient of the Wess Zumino complex. We shall restrict ourselves to the case N odd and often choose the particular value N=3. The present paper (to appear in the proceedings of the conference "Quantum Groups and Fundamental Physical Applications", Palerme, December 1997) is essentially a short version of math-ph/9807012.