The restricted quantum double of the Yangian
Canadian journal of mathematics, Tome 77 (2025) no. 3, pp. 770-841
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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 $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ which is compatible with the theory of quantized enveloping algebras. As a by-product, we identify the universal R-matrix of the Yangian with the canonical element defined by the pairing between the Yangian and its restricted dual.
Mots-clés :
Yangians, affine quantum groups, R-matrices, Yang–Baxter equation, quantization
Wendlandt, Curtis. The restricted quantum double of the Yangian. Canadian journal of mathematics, Tome 77 (2025) no. 3, pp. 770-841. doi: 10.4153/S0008414X24000142
@article{10_4153_S0008414X24000142,
author = {Wendlandt, Curtis},
title = {The restricted quantum double of the {Yangian}},
journal = {Canadian journal of mathematics},
pages = {770--841},
year = {2025},
volume = {77},
number = {3},
doi = {10.4153/S0008414X24000142},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X24000142/}
}
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