Leibniz Cohomology with Adjoint Coefficients

TL;DR

This paper computes the Leibniz cohomology of the Lie algebra of affine indefinite orthogonal groups with adjoint coefficients, extending understanding of their algebraic invariants in higher dimensions.

Contribution

It provides explicit calculations of Leibniz cohomology for affine indefinite orthogonal Lie algebras, introducing new invariants and formulas in this context.

Findings

Computed Leibniz cohomology for $rak{h}_{p,q}$ with adjoint coefficients.

Identified and calculated indefinite orthogonal invariants relevant to the cohomology.

Expressed cohomology in terms of these invariants.

Abstract

With the Poincar\'e group $\mathbf{R}^{3,\,1}\rtimes O(3,\,1)$ as the model of departure, we focus, for $n=p+q$, $p+q\ge 4$, on the affine indefinite orthogonal group $\mathbf{R}^{p,q}\rtimes SO(p,\,q)$. Denote by $\mathfrak{h}_{p,\,q}$ the Lie algebra of the affine indefinite orthogonal group. We compute the Leibniz cohomology of $\mathfrak{h}_{p,\,q}$ with adjoint coefficients, written $HL^*(\mathfrak{h}_{p,\,q};\,\mathfrak{h}_{p,\,q})$. We calculate several indefinite orthogonal invariants, and $\mathfrak{h}_{p,\,q} $-invariants and provide the Leibniz cohomology in terms of these invariants.