Convex PBW-Type Lyndon Bases and Restricted Two-parameter Quantum Groups of Type B

Abstract

We construct convex PBW-type Lyndon bases for two-parameter quantum groups U_{r,s}({so}_{2n+1}) with detailed commutation relations. It turns out that under a certain condition, the restricted two-parameter quantum group u_{r,s}({so}_{2n+1}) (r, s are roots of unity) is of Drinfeld double. All of Hopf isomorphisms of u_{r,s}({so}_{2n+1}), as well as u_{r,s}({sl}_n) are determined. Finally, necessary and sufficient conditions for u_{r,s}({so}_{2n+1}) to be a ribbon Hopf algebra are singled out by describing the left and right integrals.

reduced version: 32+(3) pages

References in corpus (7)

• Drinfel'd Doubles and Lusztig's Symmetries of Two-Parameter Quantum Groups
• Two-parameter Quantum Affine Algebra $U_{r,s}(\widehat{{\mathfrak{sl}}_n})$, Drinfel'd Realization and Quantum Affine Lyndon Basis
• Notes on two-parameter quantum groups, (II)
• Notes on two-parameter quantum groups, (I)
• Representations of Two-parameter Quantum Orthogonal and Symplectic Groups
• Two-parameter Quantum Group of Exceptional Type G_2 and Lusztig's Symmetries
• Two generalisations of the Binomial theorem

Cited by in corpus (10)

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• Classification of Hopf superalgebras associated with quantum special linear superalgebra at roots of unity using Weyl groupoid
• On the Center of Two-parameter Quantum Groups $U_{r,s}({\mathfrak{so}}_{2n+1})$
• Yetter-Drinfeld modules under cocycle twists
• Generating Functions with $\tau$-Invariance and Vertex Representations of Quantum Affine Algebras $U_{r,s}(\hat{\mathfrak{g}})$ (I): Simply-laced Cases
• Cyclic Homology of Strong Smash Product Algebras
• Novel isoclasses of one-parameter exotic small quantum groups originating from a two-parameter framework
• Derivations of the Positive Part of the Two-parameter Quantum Group of type $G_2$
• Two-parameter Quantum Affine Algebra of Type ${\mathrm{C}}_n^{(1)},$ Drinfeld Realization and Vertex Representation
• Automorphism group of Suzuki's Hopf algebra