The standard generators of the tetrahedron algebra and their look-alikes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911871123062784 |
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| author | Lee, Jae-Ho |
| author_facet | Lee, Jae-Ho |
| contents | The tetrahedron algebra $\boxtimes$ is an infinite-dimensional Lie algebra defined by generators $\{x_{ij} \mid i, j \in \{0, 1, 2, 3\}, i \neq j\}$ and some relations, including the Dolan-Grady relations. These twelve generators are called standard. We introduce a type of element in $\boxtimes$ that "looks like" a standard generator. For mutually distinct $h, i, j, k \in \{0, 1, 2, 3\}$, consider the standard generator $x_{ij}$ of $\boxtimes$. An element $ξ\in \boxtimes$ is called $x_{ij}$-like whenever both (i) $ξ$ commutes with $x_{ij}$; (ii) $ξ$ and $x_{hk}$ satisfy a Dolan-Grady relation. Pick mutually distinct $i,j,k \in \{0,1,2,3\}$. In our main result, we find an attractive basis for $\boxtimes$ with the property that every basis element is either $x_{ij}$-like or $x_{jk}$-like or $x_{ki}$-like. We discuss this basis from multiple points of view. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05504 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The standard generators of the tetrahedron algebra and their look-alikes Lee, Jae-Ho Rings and Algebras 17B65, 17B05 The tetrahedron algebra $\boxtimes$ is an infinite-dimensional Lie algebra defined by generators $\{x_{ij} \mid i, j \in \{0, 1, 2, 3\}, i \neq j\}$ and some relations, including the Dolan-Grady relations. These twelve generators are called standard. We introduce a type of element in $\boxtimes$ that "looks like" a standard generator. For mutually distinct $h, i, j, k \in \{0, 1, 2, 3\}$, consider the standard generator $x_{ij}$ of $\boxtimes$. An element $ξ\in \boxtimes$ is called $x_{ij}$-like whenever both (i) $ξ$ commutes with $x_{ij}$; (ii) $ξ$ and $x_{hk}$ satisfy a Dolan-Grady relation. Pick mutually distinct $i,j,k \in \{0,1,2,3\}$. In our main result, we find an attractive basis for $\boxtimes$ with the property that every basis element is either $x_{ij}$-like or $x_{jk}$-like or $x_{ki}$-like. We discuss this basis from multiple points of view. |
| title | The standard generators of the tetrahedron algebra and their look-alikes |
| topic | Rings and Algebras 17B65, 17B05 |
| url | https://arxiv.org/abs/2405.05504 |