Inhomogeneous quantum groups and their quantized universal enveloping algebras

1993 ◽  
Vol 27 (3) ◽  
pp. 217-222 ◽  
Author(s):  
Michael Schlieker ◽  
Wolfgang Weich ◽  
Ralf Weixler
Keyword(s):  
Quantum Groups ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Quantized Universal Enveloping Algebras ◽  
Inhomogeneous Quantum

scholarly journals Twisted deformations vs. cocycle deformations for quantum groups

2020 ◽  
pp. 2050084
Author(s):  
Gastón Andrés García ◽  
Fabio Gavarini
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Algebra Structure ◽  
Universal Enveloping Algebras ◽  
Theoretical Terms ◽  
Enveloping Algebras ◽  
Coalgebra Structure ◽  
Quantized Universal Enveloping Algebras

In this paper, we study two deformation procedures for quantum groups: deformations by twists, that we call “comultiplication twisting”, as they modify the coalgebra structure, while keeping the algebra one — and deformations by [Formula: see text]-cocycle, that we call “multiplication twisting”, as they deform the algebra structure, but save the coalgebra one. We deal with quantized universal enveloping algebras (in short QUEAs), for which we accordingly consider those arising from twisted deformations (in short TwQUEAs) and those arising from [Formula: see text]-cocycle deformations, usually called multiparameter QUEAs (in short MpQUEAs). Up to technicalities, we show that the two deformation methods are equivalent, in that they eventually provide isomorphic outputs, which are deformations (of either kinds) of the “canonical”, well-known one-parameter QUEA by Jimbo and Lusztig. It follows that the two notions of TwQUEAs and of MpQUEAs — which, in Hopf algebra theoretical terms are naturally dual to each other — actually coincide; thus, that there exists in fact only one type of “pluriparametric deformation” for QUEAs. In particular, the link between the realization of any such QUEA as a MpQUEA and that as a TwQUEA is just a (very simple, and rather explicit) change of presentation.

scholarly journals Recursion formulas for HOMFLY and Kauffman invariants

2014 ◽  
Vol 23 (05) ◽  
pp. 1450024
Author(s):  
Qingtao Chen ◽  
Nicolai Reshetikhin
Keyword(s):  
Universal Enveloping Algebras ◽  
Recursion Relations ◽  
Enveloping Algebras ◽  
Recursion Formulas ◽  
Quantized Universal Enveloping Algebras ◽  
Two Parameter

In this paper, we describe the recursion relations between two parameter HOMFLY and Kauffman polynomials of framed links. These relations correspond to embeddings of quantized universal enveloping algebras. The relation corresponding to embeddings gn ⊃ gk × sln-k where gn is either so2n+1, so2n or sp2n is new.

scholarly journals Characters of infinite-dimensional quantum classical groups: BCD cases

2021 ◽  
Author(s):  
Ryosuke Sato
Keyword(s):  
Representation Theory ◽  
Character Theory ◽  
Compact Groups ◽  
Markov Semigroups ◽  
Inductive Limits ◽  
Universal Enveloping Algebras ◽  
Infinite Dimensional ◽  
Enveloping Algebras ◽  
Quantized Universal Enveloping Algebras ◽  
The Relationship

We study the character theory of inductive limits of [Formula: see text]-deformed classical compact groups. In particular, we clarify the relationship between the representation theory of Drinfeld–Jimbo quantized universal enveloping algebras and our previous work on the quantized characters. We also apply the character theory to construct Markov semigroups on unitary duals of [Formula: see text], [Formula: see text], and their inductive limits.

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