Drinfel'd Doubles and Lusztig's Symmetries of Two-Parameter Quantum Groups

Abstract

We find the defining structures of two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ corresponding to the orthogonal and the symplectic Lie algebras, which are realized as Drinfel'd doubles. We further investigate the environment conditions upon which the Lusztig's symmetries exist between $(U_{r,s}(\mathfrak{g}),<,>)$ and its associated object $(U_{s^{-1},r^{-1}}(\mathfrak{g}),<|>)$.

25 pages AMSTeX

Cited by in corpus (24)

• Two-parameter Quantum Affine Algebra $U_{r,s}(\widehat{{\mathfrak{sl}}_n})$, Drinfel'd Realization and Quantum Affine Lyndon Basis
• Convex PBW-Type Lyndon Bases and Restricted Two-parameter Quantum Groups of Type B
• Notes on two-parameter quantum groups, (II)
• Notes on two-parameter quantum groups, (I)
• Fermionic realization of two-parameter quantum affine algebra $U_{r,s}(s{l}_n)$
• Two-parameter twisted quantum affine algebras
• Multiparameter quantum groups at roots of unity
• R-matrix realization of two-parameter quantum group U_{r,s}(gl_n)
• On Hopf algebraic structures of quantum toroidal algebras
• On the Harish-Chandra Homomorphism for Quantum Superalgebras
• On the Center of Two-parameter Quantum Groups $U_{r,s}({\mathfrak{so}}_{2n+1})$
• Lusztig isomorphisms for Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type
• Right coideal subalgebras in $U_q({\mathfrak{sl}}_{n+1}).$
• Fusion procedure for Yokonuma-Hecke algebras
• Twisted deformations vs. cocycle deformations for quantum groups
• $RLL$-realization of two-parameter quantum affine algebra in type $D_n^{(1)}$
• Drinfeld double of deformed quantum algebras
• Quantum Supergroup $U_{r,s}(osp(1,2))$, Scasimir Operators, and Dickson polynomials
• Harish-Chandra Theorem for Two-parameter Quantum Groups
• Novel isoclasses of one-parameter exotic small quantum groups originating from a two-parameter framework
• On finite-dimensional representations of two-parameter quantum affine algebras
• Two-parameter quantum vertex representations via finite groups and the McKay correspondence
• Formal multiparameter quantum groups, deformations and specializations
• Two-parameter Quantum Affine Algebra of Type ${\mathrm{C}}_n^{(1)},$ Drinfeld Realization and Vertex Representation