A Structure Theorem for Leibniz Homology

TL;DR

This paper establishes a structure theorem for Leibniz Homology of Abelian extensions of simple Lie algebras, providing explicit calculations for classical and affine Lie algebras, and relating results to invariants.

Contribution

It introduces a new structure theorem for Leibniz Homology of Abelian extensions of simple Lie algebras, linking it to invariants and computing specific cases.

Findings

HL_* of Abelian extensions can be described via g-invariants.

Explicit calculations for HL_* of affine extensions of classical Lie algebras.

HL_* of the Poincare and affine Lorentz Lie algebras are determined.

Abstract

Presented is a structure theorem for the Leibniz Homology, HL_*, of an Abelian extension of a simple real Lie algebra g. As applications, results are stated for affine extensions of the classical Lie algebras sl_n(R), so_n(R), and sp_n(R). Furthermore, HL_*(h) is calculated when h is the Lie algebra of the Poincare group as well as the Lie algebra of the affine Lorentz group. The general theorem identifies all of these in terms of g-invariants.