# Hochschild cohomology of some quantum complete intersections

## TL;DR

This paper computes the Hochschild cohomology ring of certain quantum complete intersection algebras, revealing dimension invariance and explicit ring structure of the even part modulo nilpotents.

## Contribution

It provides explicit calculations of Hochschild cohomology for a class of quantum complete intersections, including ring structure and dimension independence from parameter a.

## Key findings

- Dimension of HH^n(A) is independent of a
- Explicit ring structure of the even Hochschild cohomology
- Identification of nilpotent elements in the cohomology ring

## Abstract

We compute the Hochschild cohomology ring of the algebras $A= k\langle X, Y\rangle/ (X^a, XY-qYX, Y^a)$ over a field $k$ where $a\geq 2$ and where $q\in k$ is a primitive $a$-th root of unity. We find the the dimension of $\mathrm{HH}^n(A)$ and show that it is independent of $a$. We compute explicitly the ring structure of the even part of the Hochschild cohomology modulo homogeneous nilpotent elements.