Kyungtak Brian Hong
Purdue Seminar
Title
Orthosymplectic quantum supergroups revisited
Abstract
A fundamental structural property of Drinfeld-Jimbo quantum (super)groups is their realization as a Drinfeld double, implying quasitriangularity via the universal R-matrix. Evaluating this R-matrix on a specific representation gives rise to the RLL (or RTT) realization, a quantum analogue of the matrix realization of classical Lie (super)algebras. This talk discusses the relationship between the Drinfeld-Jimbo and RLL realizations for orthosymplectic quantum supergroups.
This talk consists of two parts. First, we compute the finite and affine R-matrices. By embedding the nilpotent half into a shuffle superalgebra, we utilize combinatorial tools to construct dual PBW bases. This also enables the factorization of the reduced R-matrix into an ordered product of local q-exponents. Furthermore, applying Yang-Baxterization to these finite R-matrices yields the explicit affine R-matrix R(z).
Second, these R-matrices are used to define the RLL realization, constructing a direct Hopf superalgebra isomorphism from the Drinfeld-Jimbo presentation. The primary difficulty is establishing injectivity; classical proofs in the non-super setup (e.g., Ding-Frenkel for finite A-type) relied heavily on faithful representation arguments, which become more subtle in the super setup. To circumvent these difficulties, we leverage the fact that both realizations admit (generalized) Drinfeld double structures. By exploiting the non-degeneracy of the skew-pairing between the Borel subalgebras, we prove injectivity and establish the isomorphism.
