The centralizer construction and Yangian-type algebras
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914787485548544 |
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| author | Olshanski, Grigori |
| author_facet | Olshanski, Grigori |
| contents | Let $d$ be a positive integer. The Yangian $Y_d=Y(\mathfrak{gl}(d,\mathbb C))$ of the general linear Lie algebra $\mathfrak{gl}(d,\mathbb C)$ has countably many generators and quadratic-linear defining relations, which can be packed into a single matrix relation using the Yang matrix -- the famous RTT presentation. Alternatively, $Y_d$ can be built from certain centralizer subalgebras of the universal enveloping algebras $U(\mathfrak{gl}(N,\mathbb C))$, with the use of a limit transition as $N\to\infty$. This approach is called the \emph{centralizer construction}.
The paper shows that a generalization of the centralizer construction leads to a new family $\{Y_{d,L}: L=1,2,3,\dots\}$ of Yangian-type algebras (the Yangian $Y_d$ being the first term of this family). For the new algebras, the RTT presentation seems to be missing, but a number of properties of the Yangian $Y_d$ persist. In particular, $Y_{d,L}$ possesses a system of quadratic-linear defining relations.
The algebras $Y_{d,L}$ ($d=1,2,3,\dots$) provide a kind of quantization for a special double Poisson bracket (in the sense of Van den Bergh) on the free associative algebra with $L$ generators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_04809 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The centralizer construction and Yangian-type algebras Olshanski, Grigori Representation Theory Mathematical Physics Quantum Algebra Rings and Algebras Let $d$ be a positive integer. The Yangian $Y_d=Y(\mathfrak{gl}(d,\mathbb C))$ of the general linear Lie algebra $\mathfrak{gl}(d,\mathbb C)$ has countably many generators and quadratic-linear defining relations, which can be packed into a single matrix relation using the Yang matrix -- the famous RTT presentation. Alternatively, $Y_d$ can be built from certain centralizer subalgebras of the universal enveloping algebras $U(\mathfrak{gl}(N,\mathbb C))$, with the use of a limit transition as $N\to\infty$. This approach is called the \emph{centralizer construction}. The paper shows that a generalization of the centralizer construction leads to a new family $\{Y_{d,L}: L=1,2,3,\dots\}$ of Yangian-type algebras (the Yangian $Y_d$ being the first term of this family). For the new algebras, the RTT presentation seems to be missing, but a number of properties of the Yangian $Y_d$ persist. In particular, $Y_{d,L}$ possesses a system of quadratic-linear defining relations. The algebras $Y_{d,L}$ ($d=1,2,3,\dots$) provide a kind of quantization for a special double Poisson bracket (in the sense of Van den Bergh) on the free associative algebra with $L$ generators. |
| title | The centralizer construction and Yangian-type algebras |
| topic | Representation Theory Mathematical Physics Quantum Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2208.04809 |