Inverses of Cartan Matrices of Lie Algebras and Lie Superalgebra

TL;DR

This paper provides explicit formulas for the inverses of Cartan matrices across simple Lie algebras, classical Lie superalgebras, and their infinite generalizations, enhancing understanding of their algebraic structures.

Contribution

It introduces explicit formulas for the inverses of Cartan matrices for various Lie algebras and superalgebras, including infinite cases, which were previously unknown.

Findings

Explicit inverse formulas for simple Lie algebra Cartan matrices

Inverse formulas for classical Lie superalgebras

Extensions to infinite Lie algebra generalizations

Abstract

We derive explicit formulas for the inverses of the Cartan matrices of the simple Lie algebras and the basic classical Lie superalgebras, as well as for their infinite generalizations.