How Euler would compute the Euler-Poincar\'e characteristic of a Lie superalgebra

TL;DR

This paper extends the classical result that the Euler-Poincaré characteristic of finite-dimensional Lie algebras vanishes to Lie superalgebras by employing a summation method inspired by Euler, combining elementary homological algebra, calculus, and combinatorics.

Contribution

It introduces a novel summation approach, based on Euler's ideas, to define the Euler-Poincaré characteristic for Lie superalgebras, overcoming issues with infinite sums.

Findings

The classical vanishing result extends to certain Lie superalgebras.

A summation method inspired by Euler enables handling infinite sums.

The approach combines elementary homological algebra, calculus, and combinatorics.

Abstract

The Euler-Poincar\'e characteristic of a finite-dimensional Lie algebra vanishes. If we want to extend this result to Lie superalgebras, we should deal with infinite sums. We observe that a suitable method of summation, which goes back to Euler, allows to do that, to a certain degree. The mathematics behind it is simple, we just glue the pieces of elementary homological algebra, first-year calculus and pedestrian combinatorics together, and present them in a (hopefully) coherent manner.