Quantum-sl(2) action on a divided-power quantum plane at even roots of unity

TL;DR

This paper introduces a nonstandard quantum plane at even roots of unity, extending a nearly commutative algebra with nilpotents, and constructs a Wess--Zumino-type de Rham complex with quantum group symmetries.

Contribution

It develops a new version of the quantum plane at even roots of unity, including a de Rham complex and quantum group actions, expanding the understanding of quantum symmetries.

Findings

Constructed a nonstandard quantum plane with nilpotent extensions.

Decomposed the de Rham complex into representations of a quantum group.

Defined quantum group actions on the algebra of quantum differential operators.

Abstract

We describe a nonstandard version of the quantum plane, the one in the basis of divided powers at an even root of unity $q=e^{i\pi/p}$. It can be regarded as an extension of the "nearly commutative" algebra $C[X,Y]$ with $X Y =(-1)^p Y X$ by nilpotents. For this quantum plane, we construct a Wess--Zumino-type de Rham complex and find its decomposition into representations of the $2p^3$-dimensional quantum group $U_q sl(2)$ and its Lusztig extension; the quantum group action is also defined on the algebra of quantum differential operators on the quantum plane.