The restricted quantum double of the Yangian

Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with associated Yangian $Y_{\hslash }\mathfrak{g}$. In the mid-1990s, Khoroshkin and Tolstoy formulated a conjecture which asserts that the algebra $\mathrm{D}{Y}_{\hslash }\mathfrak{g}$ obtained by doubling the generators of $Y_{\hslash }\mathfrak{g}$, called the Yangian double, provides a realization of the quantum double of the Yangian. We provide a uniform proof of this conjecture over $\mathbb{C}[[\hslash ]]$ 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.

68 pages

References in corpus (2)

• The formal shift operator on the Yangian double
• Poles of finite-dimensional representations of Yangians

Cited by in corpus (1)

• The R-matrix formalism for quantized enveloping algebras