Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Restricted Cohomology of Heisenberg Lie Algebras

Yong Yang

TL;DR

The paper addresses the problem of computing the adjoint restricted cohomology $H_*^q(\mathfrak h_m^{\lambda},\mathfrak h_m^{\lambda})$ of Heisenberg Lie algebras in positive characteristic. It leverages the restricted cochain framework, the ordinary Chevalley–Eilenberg complex, and the Hochschild six-term exact sequence to relate restricted and ordinary cohomology, deriving explicit dimension formulas and bases. The main contributions are the complete descriptions and dimensions of $H_*^1$ and $H_*^2$ for all $m>1$, $p>0$, and all $\lambda$, including the cases $p>2$ and $p=2$, along with concrete basis elements and the role of Frobenius maps. These results extend the understanding of adjoint restricted cohomology beyond trivial/cohomology and central extension data, providing explicit structure for restricted Heisenberg algebras relevant in modular Lie theory and quantum-mechanical Lie algebraic contexts.

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.

Restricted Cohomology of Heisenberg Lie Algebras

TL;DR

The paper addresses the problem of computing the adjoint restricted cohomology of Heisenberg Lie algebras in positive characteristic. It leverages the restricted cochain framework, the ordinary Chevalley–Eilenberg complex, and the Hochschild six-term exact sequence to relate restricted and ordinary cohomology, deriving explicit dimension formulas and bases. The main contributions are the complete descriptions and dimensions of and for all , , and all , including the cases and , along with concrete basis elements and the role of Frobenius maps. These results extend the understanding of adjoint restricted cohomology beyond trivial/cohomology and central extension data, providing explicit structure for restricted Heisenberg algebras relevant in modular Lie theory and quantum-mechanical Lie algebraic contexts.

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.
Paper Structure (10 sections, 8 theorems, 39 equations)

This paper contains 10 sections, 8 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Results
  3. The Restricted Lie Algebra $\mathfrak{h}^{\lambda}_m(p)$
  4. Cochain Complexes with Adjoint Coefficients
  5. Ordinary Cochain Complex
  6. Restricted Cochain Complex
  7. Main results
  8. Restricted cohomology $H_*^{q}(\mathfrak h_m^{\lambda}(p),\mathfrak h^{\lambda}_m(p))$ for $q = 1, 2$
  9. The Restricted Cohomology $H_*^{1}(\mathfrak h^{\lambda}_m(p),\mathfrak h^{\lambda}_m(p))$
  10. The Restricted Cohomology $H_*^{2}(\mathfrak h^{\lambda}_m(p),\mathfrak h^{\lambda}_m(p))$

Key Result

Theorem 2.1

Over fields of characteristic $p>0$, we have and

Theorems & Definitions (16)

  • Definition 1.1
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • ...and 6 more