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An introduction to some novel applications of Lie algebra cohomology and physics

J. A. de Azcarraga, J. M. Izquierdo, J. C. Perez Bueno

TL;DR

The paper surveys how Lie algebra cohomology, from basic operators to BRST formalisms, informs both mathematics and physics. It connects invariant polynomials to primitive cocycles, enabling higher-order and SH Lie algebras, and it develops higher-order generalized Poisson structures with BRST realizations. It also ties cohomology to geometry via relative cohomology and coset spaces, illustrating how WZW actions arise on cosets and how descent and anomalies shape effective field theories. Overall, the work lays out a coherent framework linking cohomology, higher algebraic structures, and invariant physical actions in a broad mathematical-physical landscape.

Abstract

After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra cohomology: a brief introduction 4. Symmetric polynomials and higher order cocycles 5. Higher order simple and SH Lie algebras 6. Higher order generalized Poisson structures 7. Relative cohomology, coset spaces and effective WZW actions

An introduction to some novel applications of Lie algebra cohomology and physics

TL;DR

The paper surveys how Lie algebra cohomology, from basic operators to BRST formalisms, informs both mathematics and physics. It connects invariant polynomials to primitive cocycles, enabling higher-order and SH Lie algebras, and it develops higher-order generalized Poisson structures with BRST realizations. It also ties cohomology to geometry via relative cohomology and coset spaces, illustrating how WZW actions arise on cosets and how descent and anomalies shape effective field theories. Overall, the work lays out a coherent framework linking cohomology, higher algebraic structures, and invariant physical actions in a broad mathematical-physical landscape.

Abstract

After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra cohomology: a brief introduction 4. Symmetric polynomials and higher order cocycles 5. Higher order simple and SH Lie algebras 6. Higher order generalized Poisson structures 7. Relative cohomology, coset spaces and effective WZW actions

Paper Structure

This paper contains 20 sections, 24 theorems, 96 equations, 2 tables.

Table of Contents

  1. Preliminaries: $L_X,\ i_X,\ d$
  2. Elementary differential geometry on Lie groups
  3. Lie algebra cohomology: a brief introduction
  4. Lie algebra cohomology
  5. Chevalley-Eilenberg formulation
  6. Whitehead's lemma for vector valued cohomology
  7. Lie algebra cohomology à la BRST
  8. Symmetric polynomials and higher order cocycles
  9. Symmetric invariant tensors and higher order Casimirs
  10. Cocycles from invariant polynomials
  11. The case of simple compact groups
  12. Higher order simple and SH Lie algebras
  13. Multibrackets and coderivations
  14. The complete BRST operator for a simple Lie algebra
  15. Higher order generalized Poisson structures
  16. ...and 5 more sections

Key Result

Proposition 3.1

The Lie algebra cohomology operator $s$ is nilpotent, $s^2=0$.

Theorems & Definitions (48)

  • Definition 1.1: Lie derivative
  • Definition 1.2: Exterior derivative
  • Definition 1.3: Inner product
  • Definition 3.1: $V$-valued $n$-dimensional cochains on $\cal G$
  • Definition 3.2: Coboundary operator (for the left action $\rho$ of $\mathcal{G}$ on $V$)
  • Proposition 3.1
  • Definition 3.3: $n$-th cohomology group
  • Example 3.1
  • Lemma 3.1: Whitehead's lemma
  • Proposition 3.2
  • ...and 38 more