Dictionary on Lie Superalgebras

TL;DR

This dictionary-style survey compiles foundational definitions, structures, and representations of Lie superalgebras, organizing them alphabetically from automorphisms to Z-graded algebras. It distinguishes classical (basic) from Cartan-type and strange superalgebras, detailing Cartan data, Dynkin diagrams, root systems, and the Harish-Chandra framework. The text consolidates representation theory (highest-weight modules, Kac modules, typical/atypical representations, unitary/star representations), and structural tools (PBW theorem, centers, Casimir invariants, and Weyl groups) while illustrating concrete realizations via oscillator constructions and matrix realizations. It emphasizes the interplay between algebraic structure and physical applications, including supersymmetry, superconformal symmetry, and supergroups, to provide a practical reference for researchers and students in mathematics and theoretical physics.

Abstract

The main definitions and properties of Lie superalgebras are proposed a la facon de a short dictionary, the different items following the alphabetical order. The main topics deal with the structure of simple Lie superalgebras and their finite dimensional representations; rather naturally, a few pages are devoted to supersymmetry. This modest booklet has two ambitious goals: to be elementary and easy to use. The beginner is supposed to find out here the main concepts on superalgebras, while a more experimented theorist should recognize the necessary tools and informations for a specific use.