$$R$$ R -Matrix Realization of Two-Parameter Quantum Group $$U_{r,s}(\mathfrak {gl}_n)$$ U r , s ( gl n )

Abstract

Abstract. We provide a Faddeev-Reshetikhin-Takhtajan's RTT approach to the quantum group F un(GL r,s (n)) and the quantum enveloping algebra U r,s (gl n ) corresponding to the two-parameter R-matrix. We prove that the quantum determinant det r,s T is a quasi-central element in F un(GL r,s (n)) generalizing earlier results of Dipper-Donkin and Du-Parshall-Wang. The explicit formulation provides an interpretation of the deforming parameters, and the quantized algebra U r,s (R) is identified to U r,s (gl n ) as the dual algebra. We then construct n − 1 quasi-central elements in U r,s (R) which are analogues of higher Casimir elements in U q (gl n ).

Two-Parameter Quantum General Linear Supergroups

“…Let V, W be in category O. Consider the vector representation (6). From the proof of Lemma 2 we see that v i is of weight ǫ i and V is in category O.…”

“…Following [4,6], define Drinfeld-Jimbo generators for 1 ≤ i < M + N : e i := s −1 ii s i,i+1 , f i := t i+1,i t −1 ii , k i := s −1 ii s i+1,i+1 , l i := t i+1,i+1 t −1 ii .…”

“…In the non-graded case N = 0, Benkart-Witherspoon [2,3] proved the existence of universal R-matrix, and derived from it a braided structure in the category of finite-dimensional representations; the exact formula of universal R-matrix was unknown. Recently it was shown [6] that U r,s (gl(M )) can be recovered from a special R-matrix in the spirit of Faddeev-Reshetikhin-Takhtajan [5], the RTT realization.…”

Capelli identity on multiparameter quantum linear groups

“…where R ij ∈ End(C n ⊗ C n ⊗ C n ) acts on the ith and jth copies of C n as R does on C n ⊗ C n . Note that it specializes to the two-parameter case with p ij = s, q ij = r −1 and u = s r up to an overall constant [11]. In particular, the one-parameter case corresponds to p ij = q ij = u = q [10].…”

“…It was known that several of their important properties are similar to the one-parameter analog, for example, Artin-Schelter-Tate [1] showed that the multiparameter general linear quantum group has the same Hilbert function for the polynomial functions in n 2 variables under the socalled (p, λ)-condition (see (2.1) or (p, u)-condition in our notation). Further results have been established for two-parameter and multi-parameter quantum groups [21,2,8,14,11] such as the multiparameter quantum determinant, which converts quantum semigroups into quantum groups. Recently it is known that the quantum Pfaffians can be extended to two-parameter quantum groups as well [13].…”