Lie structure of the Heisenberg-Weyl algebra

College

College of Science

Department/Unit

Mathematics and Statistics Department

Document Type

Article

Source Title

International Electronic Journal of Algebra

Volume

35

First Page

32

Last Page

60

Publication Date

2024

Abstract

As an associative algebra, the Heisenberg–Weyl algebra H is gen- erated 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 H. We identify a non-nilpotent but solvable Lie subalgebra g of H, for which, using some facts from the theory of bases for free Lie algebras, we give a presentation by gener- ators and relations. Under this presentation, we show that, for some algebra isomorphism φ : H −→ H, the Lie algebra H is generated by the generators of g, together with their images under φ, and that H is the sum of g, φ(g) and [g, φ(g)].

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Digitial Object Identifier (DOI)

10.24330/ieja.1326849

Disciplines

Mathematics

Keywords

Lie algebras; Polynomials; Associative algebras; Combinatorial analysis

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