The R-matrix presentation for the rational form of a quantized enveloping algebra
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arXiv
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| Format: | Preprint |
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2023
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| author | Rupert, Matthew Wendlandt, Curtis |
| author_facet | Rupert, Matthew Wendlandt, Curtis |
| contents | Let $U_q(\mathfrak{g})$ denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra $\mathfrak{g}$. Let $λ$ be a nonzero dominant integral weight of $\mathfrak{g}$, and let $V$ be the corresponding type $1$ finite-dimensional irreducible representation of $U_q(\mathfrak{g})$. Starting from this data, the $R$-matrix formalism for quantum groups outputs a Hopf algebra $\mathbf{U}_\mathrm{R}^λ(\mathfrak{g})$ defined in terms of a pair of generating matrices satisfying well-known quadratic matrix relations. In this paper, we prove that this Hopf algebra admits a Chevalley-Serre type presentation which can be recovered from that of $U_q(\mathfrak{g})$ by adding a single invertible quantum Cartan element. We simultaneously establish that $\mathbf{U}_\mathrm{R}^λ(\mathfrak{g})$ can be realized as a Hopf subalgebra of the tensor product of the space of Laurent polynomials in a single variable with the quantized enveloping algebra associated to the lattice generated by the weights of $V$. The proofs of these results are based on a detailed analysis of the homogeneous components of the matrix equations and generating matrices defining $\mathbf{U}_\mathrm{R}^λ(\mathfrak{g})$, with respect to a natural grading by the root lattice of $\mathfrak{g}$ compatible with the weight space decomposition of $\mathrm{End}(V)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_09971 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The R-matrix presentation for the rational form of a quantized enveloping algebra Rupert, Matthew Wendlandt, Curtis Quantum Algebra Mathematical Physics Representation Theory 17B37 (Primary), 17B38 (Secondary) Let $U_q(\mathfrak{g})$ denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra $\mathfrak{g}$. Let $λ$ be a nonzero dominant integral weight of $\mathfrak{g}$, and let $V$ be the corresponding type $1$ finite-dimensional irreducible representation of $U_q(\mathfrak{g})$. Starting from this data, the $R$-matrix formalism for quantum groups outputs a Hopf algebra $\mathbf{U}_\mathrm{R}^λ(\mathfrak{g})$ defined in terms of a pair of generating matrices satisfying well-known quadratic matrix relations. In this paper, we prove that this Hopf algebra admits a Chevalley-Serre type presentation which can be recovered from that of $U_q(\mathfrak{g})$ by adding a single invertible quantum Cartan element. We simultaneously establish that $\mathbf{U}_\mathrm{R}^λ(\mathfrak{g})$ can be realized as a Hopf subalgebra of the tensor product of the space of Laurent polynomials in a single variable with the quantized enveloping algebra associated to the lattice generated by the weights of $V$. The proofs of these results are based on a detailed analysis of the homogeneous components of the matrix equations and generating matrices defining $\mathbf{U}_\mathrm{R}^λ(\mathfrak{g})$, with respect to a natural grading by the root lattice of $\mathfrak{g}$ compatible with the weight space decomposition of $\mathrm{End}(V)$. |
| title | The R-matrix presentation for the rational form of a quantized enveloping algebra |
| topic | Quantum Algebra Mathematical Physics Representation Theory 17B37 (Primary), 17B38 (Secondary) |
| url | https://arxiv.org/abs/2306.09971 |