Lie structure of the Heisenberg-Weyl algebra

Authors : Rafael Reno S Cantuba

Pages : 32-60

Doi:10.24330/ieja.1326849

View : 319 | Download : 102

Publication Date : 2024-01-09

Article Type : Research Paper

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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