The R-matrix formalism for quantized enveloping algebras

Abstract

Let $U_{\hslash }\mathfrak{g}$ denote the Drinfeld-Jimbo quantum group associated to a simple 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 $U_{\hslash }\mathfrak{g}$ on any of its finite-dimensional representations. This produces a quantized enveloping algebra ${\mathrm{U}}_{\mathrm{R}}(\mathfrak{g})$ whose definition is given in terms of two generating matrices satisfying variants of the well-known $RLL$ relations. We prove that ${\mathrm{U}}_{\mathrm{R}}(\mathfrak{g})$ is isomorphic to the tensor product of the quantum double of the Borel subalgebra $U_{\hslash }\mathfrak{b}\subset U_{\hslash }\mathfrak{g}$ 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. Using this description, we characterize $U_{\hslash }\mathfrak{g}$ and the quantum double of $U_{\hslash }\mathfrak{b}$ as Hopf quotients of ${\mathrm{U}}_{\mathrm{R}}(\mathfrak{g})$ and as fixed-point subalgebras with respect to certain natural automorphisms. As an additional corollary, we deduce that ${\mathrm{U}}_{\mathrm{R}}(\mathfrak{g})$ is quasitriangular precisely when $V$ is multiplicity free.

arXiv comment45 pages, published version

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