The family of quaternionic quasi-unitary Lie algebras and their central extensions

TL;DR

This paper unifies the study of quaternionic quasi-unitary Lie algebras, including simple and non-semisimple types, and fully determines their central extensions, showing the triviality of their second cohomology groups across all dimensions.

Contribution

It provides a unified framework for quaternionic quasi-unitary Lie algebras and completely characterizes their central extensions in arbitrary dimensions.

Findings

Second cohomology group is trivial for all these Lie algebras.

Unified description of simple and non-semisimple quaternionic quasi-unitary algebras.

Central extensions are fully determined regardless of dimension.

Abstract

The family of quaternionic quasi-unitary (or quaternionic unitary Cayley--Klein algebras) is described in a unified setting. This family includes the simple algebras sp(N+1) and sp(p,q) in the Cartan series C_{N+1}, as well as many non-semisimple real Lie algebras which can be obtained from these simple algebras by particular contractions. The algebras in this family are realized here in relation with the groups of isometries of quaternionic hermitian spaces of constant holomorphic curvature. This common framework allows to perform the study of many properties for all these Lie algebras simultaneously. In this paper the central extensions for all quasi-simple Lie algebras of the quaternionic unitary Cayley--Klein family are completely determined in arbitrary dimension. It is shown that the second cohomology group is trivial for any Lie algebra of this family no matter of its dimension.