Examples of Homotopy Lie Algebras

TL;DR

This paper explores two detailed examples of homotopy Lie algebras, demonstrating their properties through algebraic and differential operator approaches, highlighting the consistency of these methods.

Contribution

It provides a comparative analysis of algebraic and differential operator methods for verifying homotopy Lie algebra structures.

Findings

Both methods verify the homotopy Lie algebra structures successfully.

The algebraic approach uses degree arguments and combinatorics.

The differential operator approach relies on nilpotency of Grassmann-odd operators.

Abstract

We look at two examples of homotopy Lie algebras (also known as L_{\infty} algebras) in detail from two points of view. We will exhibit the algebraic point of view in which the generalized Jacobi expressions are verified by using degree arguments and combinatorics. A second approach using the nilpotency of Grassmann-odd differential operators \Delta to verify the homotopy Lie data is shown to produce the same results.