Multiparameter quantum enveloping algebras

Chin, William; Musson, Ian M.

doi:10.1016/0022-4049(95)00062-3

Cited by 24 publications

(28 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first class we discuss are the multiparameter quantum groups U λ,p (gl n ) introduced by [19] (adapting the presentation in [15]). They are asymmetric braided Drinfeld doubles, which is a generalization of the result of [14] showing that two-parameter quantum groups are Drinfeld doubles.…”

Section: 2]mentioning

confidence: 99%

“…(15) (In the Hopf case, the antipode S is required to preserve the natural grading and the subspaces T…”

Section: Definition 3 a Bialgebra (Or Hopf Algebra) A With Triangularmentioning

confidence: 99%

“…For the purpose of this section, let λ ∈ k be generic, and p ij ∈ k for 1 ≤ i < j ≤ n. Assume that p ii = 1 and p ji = p −1 ij . Following [11,15] and to fix notation, we set…”

Section: Multiparameter Quantum Groupsmentioning

confidence: 99%

“…We will provide a variation of the presentation of [11,15] The result that the multiparameter quantum group U λ,p (gl n ) is the asymmetric braided Drinfeld double U kG (F, E) can be seen as a generalization of the result in [14] where the two-parameter quantum groups were shown to be Drinfeld doubles.…”

Section: Multiparameter Quantum Groupsmentioning

confidence: 99%

“…Generation of the maximal triangular ideal by quantum Serre relations for U λ,p (gl n ) follows from Lemma 4.5 in [15]. For this, it is crucial that λ is not a root of unity.…”

Section: Proposition 7 There Is An Isomorphism Of Graded Hopf Algebrasmentioning

confidence: 99%

See 4 more Smart Citations

Pointed Hopf Algebras with Triangular Decomposition

Laugwitz

2016

Algebr Represent Theor

View full text Add to dashboard Cite

In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (Algebr. Represent. Theory 7(3) 2014) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to sl 2 .

show abstract

Section: 2]mentioning

confidence: 99%

“…(15) (In the Hopf case, the antipode S is required to preserve the natural grading and the subspaces T…”

Section: Definition 3 a Bialgebra (Or Hopf Algebra) A With Triangularmentioning

confidence: 99%

“…For the purpose of this section, let λ ∈ k be generic, and p ij ∈ k for 1 ≤ i < j ≤ n. Assume that p ii = 1 and p ji = p −1 ij . Following [11,15] and to fix notation, we set…”

Section: Multiparameter Quantum Groupsmentioning

confidence: 99%

Section: Multiparameter Quantum Groupsmentioning

confidence: 99%

“…Generation of the maximal triangular ideal by quantum Serre relations for U λ,p (gl n ) follows from Lemma 4.5 in [15]. For this, it is crucial that λ is not a root of unity.…”

Section: Proposition 7 There Is An Isomorphism Of Graded Hopf Algebrasmentioning

confidence: 99%

See 3 more Smart Citations