The R-matrix formalism for quantized enveloping algebras

TL;DR

This paper introduces a modified R-matrix approach to construct a quantized enveloping algebra for semisimple Lie algebras, revealing its structure and relations to quantum doubles and invariants.

Contribution

It develops a new R-matrix formalism for quantum groups, defining a quantized algebra with explicit generators and relations, and characterizes its structure in relation to known quantum algebras.

Findings

The algebra $ ext{U}_R(rak{g})$ is isomorphic to a tensor product involving the quantum double and invariant polynomials.

$ ext{U}_R(rak{g})$ is quasitriangular if and only if the representation's summands are distinct.

The paper provides a new perspective on the structure and automorphisms of quantum groups.

Abstract

Let $U_\hbar\mathfrak{g}$ denote the Drinfeld-Jimbo quantum group associated to a complex semisimple Lie algebra $\mathfrak{g}$. We apply a modification of the $R$-matrix construction for quantum groups to the evaluation of the universal $R$-matrix of $U_\hbar\mathfrak{g}$ on the tensor square of any of its finite-dimensional representations. This produces a quantized enveloping algebra $\mathrm{U_R}(\mathfrak{g})$ whose definition is given in terms of two generating matrices satisfying variants of the well-known $RLL$ relations. We prove that $\mathrm{U_R}(\mathfrak{g})$ is isomorphic to the tensor product of the quantum double of the Borel subalgebra $U_\hbar\mathfrak{b}\subset U_\hbar\mathfrak{g}$ and a quantized polynomial algebra encoded by the space of $\mathfrak{g}$-invariants associated to the semiclassical limit $V$ of the underlying finite-dimensional representation of $U_\hbar\mathfrak{g}$. Using this description, we characterize $U_\hbar\mathfrak{g}$ and the quantum double of $U_\hbar\mathfrak{b}$ as Hopf quotients of $\mathrm{U_R}(\mathfrak{g})$ and as fixed-point subalgebras with respect to certain natural automorphisms. As an additional corollary, we deduce that $\mathrm{U_R}(\mathfrak{g})$ is quasitriangular precisely when the irreducible summands of $V$ are distinct.