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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2306.04770 |
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| _version_ | 1866911941383946240 |
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| author | Terwilliger, Paul |
| author_facet | Terwilliger, Paul |
| contents | In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra $\mathcal A$. The algebra $\mathcal A$ is associative, noncommutative, and infinite-dimensional. It is defined by two generators $A,B$ and two relations called the down-up relations. In the present paper, we introduce the $\mathbb Z_3$-symmetric down-up algebra $\mathbb A$. We define $\mathbb A$ by generators and relations. There are three generators $A,B,C$ and any two of these satisfy the down-up relations. We describe how $\mathbb A$ is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, the $\mathfrak{sl}_3$ loop algebra, the Kac-Moody Lie algebra $A^{(1)}_2$, the $q$-Weyl algebra, the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$, and the quantized enveloping algebra $U_q (A^{(1)}_2)$. We give some open problems and conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_04770 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The $\mathbb Z_3$-Symmetric Down-Up algebra Terwilliger, Paul Quantum Algebra Rings and Algebras 17B37 In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra $\mathcal A$. The algebra $\mathcal A$ is associative, noncommutative, and infinite-dimensional. It is defined by two generators $A,B$ and two relations called the down-up relations. In the present paper, we introduce the $\mathbb Z_3$-symmetric down-up algebra $\mathbb A$. We define $\mathbb A$ by generators and relations. There are three generators $A,B,C$ and any two of these satisfy the down-up relations. We describe how $\mathbb A$ is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, the $\mathfrak{sl}_3$ loop algebra, the Kac-Moody Lie algebra $A^{(1)}_2$, the $q$-Weyl algebra, the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$, and the quantized enveloping algebra $U_q (A^{(1)}_2)$. We give some open problems and conjectures. |
| title | The $\mathbb Z_3$-Symmetric Down-Up algebra |
| topic | Quantum Algebra Rings and Algebras 17B37 |
| url | https://arxiv.org/abs/2306.04770 |