Yangian Double and Rational R-matrix

TL;DR

This paper explores the algebraic structure of the Yangian double, providing a universal R-matrix formula and its application to evaluation representations, with implications for understanding bilinear forms in representation theory.

Contribution

It introduces the triangular decomposition of the Yangian double and derives an explicit universal R-matrix formula, linking it to evaluation representations and gamma-functions.

Findings

Explicit universal R-matrix formula for ${ m D}Y(g)$

Triangular decomposition of the Yangian double

Interpretation of R-matrix factors as bilinear forms in representations

Abstract

Studying the algebraic structure of the double ${\cal D}Y(g)$ of the yangian $Y(g)$ we present the triangular decomposition of ${\cal D}Y(g)$ and a factorization for the canonical pairing of the yangian with its dual inside ${\cal D}Y(g)$. As a consequence we obtain an explicit formula for the universal R-matrix $R$ of ${\cal D}Y(g)$ and demonstrate how it works in evaluation representations of $Y(sl_2)$. We interprete one-dimensional factor arising in concrete representations of $R$ as bilinear form on highest weight polynomials of irreducible representations of $Y(g)$ and express this form in terms of {\it gamma-functions}.