Finite dimensional compact and unitary Lie superalgebras

TL;DR

This paper classifies finite dimensional compact and unitary Lie superalgebras, focusing on their structure and representations, which is essential for understanding unitary representations of Lie supergroups.

Contribution

It provides a classification of real finite dimensional compact simple Lie superalgebras and analyzes the decomposition of reductive Lie superalgebras into ideals.

Findings

Classification of real finite dimensional compact simple Lie superalgebras

Elementary analysis of reductive Lie superalgebra decompositions

Identification of conditions for faithful finite dimensional unitary representations

Abstract

Motivated by the theory of unitary representations of finite dimensional Lie supergroups, we describe those Lie superalgebras which have a faithful finite dimensional unitary representation. We call these Lie superalgebras unitary. This is achieved by describing the classification of real finite dimensional compact simple Lie superalgebras, and analyzing, in a rather elementary and direct way, the decomposition of reductive Lie superalgebras ($\g$ is a semisimple $\g_{\bar 0}$-module) over fields of characteristic zero into ideals.