The restricted quantum double of the Yangian

TL;DR

This paper proves a conjecture that the Yangian double provides a realization of the quantum double of the Yangian, connecting it with the theory of quantized enveloping algebras and identifying its universal R-matrix.

Contribution

It offers a uniform proof of the conjecture over complex power series and relates the Yangian double to the quantum double construction, enhancing understanding of its algebraic structure.

Findings

Confirmed the Yangian double as the quantum double of the Yangian.

Identified the universal R-matrix with a canonical element from the pairing.

Provided a proof compatible with quantized enveloping algebra theory.

Abstract

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.