On the cohomology of the Weyl algebra, the quantum plane, and the q-Weyl algebra

TL;DR

This paper uses deformation theory and Euler-Poincare characteristic invariance to compute the cohomology of the Weyl algebra, quantum plane, and q-Weyl algebra, revealing potential number theoretic insights when q is a root of unity.

Contribution

It applies deformation theory and invariance principles to explicitly compute the cohomology of key quantum algebras, extending understanding of their structure.

Findings

Cohomology of Weyl algebra computed using deformation theory.

Cohomology of quantum plane and q-Weyl algebra determined.

Behavior at roots of unity suggests number theoretic connections.

Abstract

Deformation theory can be used to compute the cohomology of a deformed algebra with coefficients in itself from that of the original. Using the invariance of the Euler-Poincare characteristic under deformation, it is applied here to compute the cohomology of the Weyl algebra, the algebra of the quantum plane, and the q-Weyl algebra. The behavior of the cohomology when q is a root of unity may encode some number theoretic information.