Formalising Lie algebras

TL;DR

This paper details the formalisation of Lie algebras within Lean's Mathlib, including classical and exceptional types, enabling formal proofs of key classification theorems in Lie theory.

Contribution

It introduces a comprehensive formalisation of Lie algebras in Lean, including constructions of classical and exceptional types, facilitating rigorous proofs of fundamental classification results.

Findings

Formalisation of Lie algebras in Lean's Mathlib

Construction of classical and exceptional Lie algebras

Proof of the classification theorem for semisimple Lie algebras

Abstract

Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none is assumed; the intention is that the overall themes will be accessible even to readers unfamiliar with Lie theory. Particular attention is paid to the construction of the classical and exceptional Lie algebras. Thanks to these constructions, it is possible to state the classification theorem for finite-dimensional semisimple Lie algebras over an algebraically closed field of characteristic zero. In addition to the focus on Lie theory, we also aim to highlight the unity of Mathlib. To this end, we include examples of achievements made possible only by leaning on several branches of the library simultaneously.