SEMILATTICE GRADED WEAK HOPF ALGEBRA AND ITS QUANTUM DOUBLE

2005 ◽  
Author(s):  
FANG LI ◽  
HAIJUN CAO
Keyword(s):  
Hopf Algebra ◽  
Weak Hopf Algebra ◽  
Quantum Double

Double-bosonization of braided groups and the construction of Uq(g)

1999 ◽  
Vol 125 (1) ◽  
pp. 151-192 ◽  
Author(s):  
S. MAJID
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Cartan Subalgebra ◽  
Positive Root ◽  
Point Of View ◽  
Root Space ◽  
Quantum Double ◽  
Braided Group ◽  
Negative Root ◽  
Quasitriangular Hopf Algebra

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U(B), the double-bosonization, associated to every braided group B in the category of H-modules over a quasitriangular Hopf algebra H, such that B appears as the ‘positive root space’, H as the ‘Cartan subalgebra’ and the dual braided group B* as the ‘negative root space’ of U(B). The choice B=Uq(n+) recovers Lusztig's construction of Uq(g); other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct Uq(sl3) from Uq(sl2) by this method, extending it by the quantum-braided plane. We provide a fundamental representation of U(B) in B. A projection from the quantum double, a theory of double biproducts and a Tannaka–Krein reconstruction point of view are also provided.

scholarly journals Rota-Baxter Leibniz Algebras and Their Constructions

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Liangyun Zhang ◽  
Linhan Li ◽  
Huihui Zheng
Keyword(s):  
Hopf Algebra ◽  
Weak Hopf Algebra ◽  
Associative Algebras ◽  
Leibniz Algebras

In this paper, we introduce the concept of Rota-Baxter Leibniz algebras and explore two characterizations of Rota-Baxter Leibniz algebras. And we construct a number of Rota-Baxter Leibniz algebras from Leibniz algebras and associative algebras and discover some Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra. In the end, we give all Rota-Baxter operators of weight 0 and -1 on solvable and nilpotent Leibniz algebras of dimension ≤3, respectively.

Some ribbon elements for the quasi-Hopf algebra Dω(H)

2020 ◽  
pp. 2150226
Author(s):  
Daniel Bulacu ◽  
Florin Panaite
Keyword(s):  
Hopf Algebra ◽  
Quantum Double

We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra [Formula: see text] defined in [D. Bulacu and F. Panaite, A generalization of the quasi-Hopf algebra [Formula: see text], Commun. Algebra 26 (1998) 4125–4141] and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for [Formula: see text].

On Braided Lie Structures of Algebras in the Categories of Weak Hopf Bimodules

2010 ◽  
Vol 17 (04) ◽  
pp. 685-698 ◽  
Author(s):  
Shuan-hong Wang ◽  
Hai-xing Zhu
Keyword(s):  
Hopf Algebra ◽  
Monoidal Category ◽  
Weak Hopf Algebra ◽  
Commutator Ideal ◽  
Commutative Subalgebras

Let H be a weak Hopf algebra. In this paper, it is proved that the monoidal category [Formula: see text] of weak Hopf bimodules studied in Wang [19] is equivalent to the monoidal category [Formula: see text] of weak Yetter–Drinfel'd modules introduced in Böhm [2]. When H has a bijective antipode, a braiding in the category [Formula: see text] is constructed by the braiding on [Formula: see text], generalizing the main result in Schauenburg [14]. Finally, the braided Lie structures of an algebra A in the category [Formula: see text] are investigated, by showing that if A is a sum of two braided commutative subalgebras, then the braided commutator ideal of A is nilpotent.

scholarly journals A quantum double construction in Rel

2012 ◽  
Vol 22 (4) ◽  
pp. 618-650 ◽  
Author(s):  
MASAHITO HASEGAWA
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Binary Relations ◽  
Monoidal Categories ◽  
Quantum Double ◽  
Closed Category ◽  
Category Of Modules ◽  
Extra Structure

We study bialgebras and Hopf algebras in the compact closed categoryRelof sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras inRel. In particular, for any groupG, we derive a ribbon category of crossedG-sets as the category of modules of a Hopf algebra inRelthat is obtained by the quantum double construction. This category of crossedG-sets serves as a model of the braided variant of propositional linear logic.

scholarly journals Quantum double and κ-Poincaré symmetries in (2+1)-gravity and Chern–Simons theoryThis paper was presented at the Theory CANADA 4 conference, held at Centre de recherches mathématiques, Montréal, Québec, Canada on 4–7 June 2008.

2009 ◽  
Vol 87 (3) ◽  
pp. 245-250
Author(s):  
C. Meusburger
Keyword(s):  
Hopf Algebra ◽  
Simons Theory ◽  
De Sitter ◽  
Quantum Double ◽  
Isometry Groups ◽  
Chern Simons ◽  
Chern Simons Theory

We clarify the role of Drinfeld doubles and κ-Poincaré symmetries in quantized (2+1)-gravity and Chern–Simons theory. We discuss the conditions under which a given Hopf algebra symmetry is compatible with a Chern–Simons theory and determine this compatibility explicitly for the Drinfeld doubles and κ-Poincaré symmetries associated with the isometry groups of (2+1)-gravity. In particular, we show that κ-Poincaré symmetries with a timelike deformation are not directly associated with (2+1)-gravity. The association between these κ-Poincaré symmetries and Chern–Simons theory is possible only in the de Sitter case and the relevant Chern–Simons theory is physically inequivalent to (2+1)-gravity.

scholarly journals Comparison of two notions of weak crossed product

2020 ◽  
pp. 2150059
Author(s):  
Jorge A. Guccione ◽  
Juan J. Guccione
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Crossed Product ◽  
Weak Hopf Algebra ◽  
Crossed Products ◽  
Hopf Algebroid ◽  
Hopf Algebroids ◽  
Weak Crossed Product ◽  
Cleft Extensions ◽  
Weak Hopf Algebras

We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in [Cleft extensions of Hopf algebroids, Appl. Categor. Struct. 14(5–6) (2006) 431–469] with the notion of crossed product with a weak Hopf algebra introduced in [Crossed products for weak Hopf algebras with coalgebra splitting, J. Algebra 281(2) (2004) 731–752].

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