2023/06/16 by Rupert, Matthew, Wendlandt, Curtis
#17B37 (Primary) #17B38 (Secondary) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2306.09971
Let Uq(\mathfrakg) denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra \mathfrakg. Let λ be a nonzero dominant integral weight of \mathfrakg, and let V be the corresponding type 1 finite-dimensional irreducible representation of Uq(\mathfrakg). Starting from this data, the R-matrix formalism for quantum groups outputs a Hopf algebra URλ(\mathfrakg) defined in terms of a pair of generating matrices satisfying well-known quadratic matrix relations. In this paper, we prove that this Hopf algebra admits a Chevalley-Serre type presentation which can be recovered from that of Uq(\mathfrakg) by adding a single invertible quantum Cartan element. We simultaneously establish that URλ(\mathfrakg) can be realized as a Hopf subalgebra of the tensor product of the space of Laurent polynomials in a single variable with the quantized enveloping algebra associated to the lattice generated by the weights of V. The proofs of these results are based on a detailed analysis of the homogeneous components of the matrix equations and generating matrices defining URλ(\mathfrakg), with respect to a natural grading by the root lattice of \mathfrakg compatible with the weight space decomposition of End(V).