Differential calculus and connections on a quantum plane at a cubic root of unity

TL;DR

This paper develops a differential calculus on a quantum plane at a cubic root of unity, exploring quantum group actions, representation theory, and non-commutative connections with applications to space-time geometry.

Contribution

It introduces a quotient differential calculus on a quantum plane at a root of unity and studies quantum group symmetries and non-commutative connections in this setting.

Findings

Constructed a quotient differential calculus on the quantum plane.

Analyzed the decomposition of the differential algebra under quantum group action.

Explored properties of non-commutative connections compatible with quantum symmetries.

Abstract

We consider the algebra of N x N matrices as a reduced quantum plane on which a finite-dimensional quantum group H acts. This quantum group is a quotient of U_q(sl(2,C)), q being an N-th root of unity. Most of the time we shall take N=3; in that case \dim(H) = 27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess-Zumino complex. The quantum group H also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of H. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.