Lie structure of the Heisenberg-Weyl algebra
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929203031572480 |
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| author | Cantuba, Rafael Reno S. |
| author_facet | Cantuba, Rafael Reno S. |
| contents | As an associative algebra, the Heisenberg-Weyl algebra $\mathcal{H}$ 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 $\mathcal{H}$. We identify a non-nilpotent but solvable Lie subalgebra $\mathfrak{g}$ of $\mathcal{H}$, 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 $φ:\mathcal{H}\longrightarrow\mathcal{H}$, the Lie algebra $\mathcal{H}$ is generated by the generators of $\mathfrak{g}$, together with their images under $φ$, and that $\mathcal{H}$ is the sum of $\mathfrak{g}$, $φ(\mathfrak{g})$ and $\left[ \mathfrak{g},φ(\mathfrak{g})\right]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_11930 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Lie structure of the Heisenberg-Weyl algebra Cantuba, Rafael Reno S. Rings and Algebras 17B60, 16Z10, 81R50 As an associative algebra, the Heisenberg-Weyl algebra $\mathcal{H}$ 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 $\mathcal{H}$. We identify a non-nilpotent but solvable Lie subalgebra $\mathfrak{g}$ of $\mathcal{H}$, 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 $φ:\mathcal{H}\longrightarrow\mathcal{H}$, the Lie algebra $\mathcal{H}$ is generated by the generators of $\mathfrak{g}$, together with their images under $φ$, and that $\mathcal{H}$ is the sum of $\mathfrak{g}$, $φ(\mathfrak{g})$ and $\left[ \mathfrak{g},φ(\mathfrak{g})\right]$. |
| title | Lie structure of the Heisenberg-Weyl algebra |
| topic | Rings and Algebras 17B60, 16Z10, 81R50 |
| url | https://arxiv.org/abs/2207.11930 |