Mickelsson algebras via Hasse diagrams
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910738119917568 |
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| author | Mudrov, Andrey Stukopin, Vladimir |
| author_facet | Mudrov, Andrey Stukopin, Vladimir |
| contents | Let $\mathcal{A}$ be an associative algebra containing either classical or quantum universal enveloping algebra of a semi-simple complex Lie algebra $\mathfrak{g}$. We present a construction of the Mickelsson algebra $Z(\mathcal{A},\mathfrak{g})$ relative to the left ideal in $\mathcal{A}$ generated by positive root vectors. Our method employs a calculus on Hasse diagrams associated with classical or quantum $\mathfrak{g}$-modules.
We give an explicit expression for a PBW basis in $Z(\mathcal{A},\mathfrak{g})$ in the case when $\mathcal{A}=U(\mathfrak{a})$ of a finite-dimensional Lie algebra $\mathfrak{a}\supset \mathfrak{g}$.
For $\mathcal{A}=U_q(\mathfrak{a})$ and $\mathfrak{g}$ the commutant of a Levi subalgebra in $\mathfrak{a}$, we construct a PBW basis in terms of quantum Lax operators, upon extension of the ground ring of scalars to $\mathbb{C}[[\hbar]]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_01592 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mickelsson algebras via Hasse diagrams Mudrov, Andrey Stukopin, Vladimir Quantum Algebra 17B10, 17B37 Let $\mathcal{A}$ be an associative algebra containing either classical or quantum universal enveloping algebra of a semi-simple complex Lie algebra $\mathfrak{g}$. We present a construction of the Mickelsson algebra $Z(\mathcal{A},\mathfrak{g})$ relative to the left ideal in $\mathcal{A}$ generated by positive root vectors. Our method employs a calculus on Hasse diagrams associated with classical or quantum $\mathfrak{g}$-modules. We give an explicit expression for a PBW basis in $Z(\mathcal{A},\mathfrak{g})$ in the case when $\mathcal{A}=U(\mathfrak{a})$ of a finite-dimensional Lie algebra $\mathfrak{a}\supset \mathfrak{g}$. For $\mathcal{A}=U_q(\mathfrak{a})$ and $\mathfrak{g}$ the commutant of a Levi subalgebra in $\mathfrak{a}$, we construct a PBW basis in terms of quantum Lax operators, upon extension of the ground ring of scalars to $\mathbb{C}[[\hbar]]$. |
| title | Mickelsson algebras via Hasse diagrams |
| topic | Quantum Algebra 17B10, 17B37 |
| url | https://arxiv.org/abs/2306.01592 |