Restricted Cohomology of Heisenberg Lie Algebras

TL;DR

This paper computes the restricted cohomology spaces of Heisenberg Lie algebras over algebraically closed fields of characteristic p > 0, revealing their structure using ordinary cohomology methods.

Contribution

It introduces a method to determine the restricted cohomology of Heisenberg Lie algebras via ordinary cohomology spaces, providing new insights into their algebraic structure.

Findings

Explicit formulas for restricted 1-cohomology spaces.

Explicit formulas for restricted 2-cohomology spaces.

Application of ordinary cohomology to restricted algebra analysis.

Abstract

The Heisenberg Lie algebras over an algebraically closed field F of characteristic p > 0 always admit a family of restricted structures. We use the ordinary 1- and 2-cohomology spaces with adjoint coefficients to compute the restricted 1- and 2-cohomology spaces of these restricted Heisenberg Lie algebras.