The families of orthogonal, unitary and quaternionic unitary Cayley--Klein algebras and their central extensions

TL;DR

This paper provides a unified description of families of Cayley--Klein algebras over R, C, and H, including their contractions and central extensions, covering simple and non-simple real Lie algebras.

Contribution

It introduces a unified framework for quasi-simple Cayley--Klein algebras and fully characterizes their central extensions across dimensions.

Findings

Unified description of Cayley--Klein algebras over R, C, H

Complete determination of their non-trivial central extensions

Inclusion of simple and non-simple real Lie algebras

Abstract

The families of quasi-simple or Cayley--Klein algebras associated to antihermitian matrices over R, C and H are described in a unified framework. These three families include simple and non-simple real Lie algebras which can be obtained by contracting the pseudo-orthogonal algebras so(p,q) of the Cartan series $B_l$ and $D_l$, the special pseudo-unitary algebras su(p,q) in the series $A_l$, and the quaternionic pseudo-unitary algebras sq(p,q) in the series $C_l$. This approach allows to study many properties for all these Lie algebras simultaneously. In particular their non-trivial central extensions are completely determined in arbitrary dimension.