Lie structure of the Heisenberg-Weyl algebra

Year 2024, , 32 - 60, 09.01.2024

Rafael Reno S. Cantuba

https://doi.org/10.24330/ieja.1326849

Abstract

As an associative algebra, the Heisenberg--Weyl algebra $\HWeyl$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and $B$ are not able to generate the whole space $\HWeyl$. We identify a non-nilpotent but solvable Lie subalgebra $\coreLie$ of $\HWeyl$, for which, using some facts from the theory of bases for free Lie algebras, we give a presentation by generators and relations. Under this presentation, we show that, for some algebra isomorphism $\isoH:\HWeyl\into\HWeyl$, the Lie algebra $\HWeyl$ is generated by the generators of $\coreLie$, together with their images under $\isoH$, and that $\HWeyl$ is the sum of $\coreLie$, $\isoH(\coreLie)$ and $\lbrak \coreLie,\isoH(\coreLie)\rbrak$.

Keywords

Heisenberg--Weyl algebra, commutation relation, free Lie algebra, Lie polynomial, combinatorics on words, Lyndon--Shirshov word, generator and relation

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