The restricted quantum double of the Yangian
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909596106358784 |
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| author | Wendlandt, Curtis |
| author_facet | Wendlandt, Curtis |
| contents | Let $\mathfrak{g}$ be a complex semisimple Lie algebra with associated Yangian $Y_\hbar\mathfrak{g}$. In the mid-1990s, Khoroshkin and Tolstoy formulated a conjecture which asserts that the algebra $\mathrm{D}Y_\hbar\mathfrak{g}$ obtained by doubling the generators of $Y_\hbar\mathfrak{g}$, called the Yangian double, provides a realization of the quantum double of the Yangian. We provide a uniform proof of this conjecture over $\mathbb{C}[\![\hbar]\!]$ which is compatible with the theory of quantized enveloping algebras. As a byproduct, we identify the universal $R$-matrix of the Yangian with the canonical element defined by the pairing between the Yangian and its restricted dual. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_00983 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The restricted quantum double of the Yangian Wendlandt, Curtis Quantum Algebra Mathematical Physics Representation Theory 17B37 (Primary), 81R10 (Secondary) Let $\mathfrak{g}$ be a complex semisimple Lie algebra with associated Yangian $Y_\hbar\mathfrak{g}$. In the mid-1990s, Khoroshkin and Tolstoy formulated a conjecture which asserts that the algebra $\mathrm{D}Y_\hbar\mathfrak{g}$ obtained by doubling the generators of $Y_\hbar\mathfrak{g}$, called the Yangian double, provides a realization of the quantum double of the Yangian. We provide a uniform proof of this conjecture over $\mathbb{C}[\![\hbar]\!]$ which is compatible with the theory of quantized enveloping algebras. As a byproduct, we identify the universal $R$-matrix of the Yangian with the canonical element defined by the pairing between the Yangian and its restricted dual. |
| title | The restricted quantum double of the Yangian |
| topic | Quantum Algebra Mathematical Physics Representation Theory 17B37 (Primary), 81R10 (Secondary) |
| url | https://arxiv.org/abs/2204.00983 |