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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.

First, we compute the finite and affine $R$-matrices. By embedding the nilpotent half into a shuffle superalgebra, we construct dual PBW bases and factorize the reduced $R$-matrix into an ordered product of local $q$-exponents. Applying Yang-Baxterization then yields the explicit affine $R$-matrix $R(z)$.

Second, these $R$-matrices are used to define the RLL realization. To construct a direct Hopf superalgebra isomorphism and establish injectivity—traditionally done using representation theory, which is more subtle in the super setup—we leverage generalized Drinfeld double structures and the non-degenerate skew-pairing between Borel subalgebras.