Hom-quantum groups I: quasi-triangular Hom-bialgebras

Abstract

We introduce a Hom-type generalization of quantum groups, called quasi-triangular Hom-bialgebras. They are non-associative and non-coassociative analogues of Drinfel'd's quasi-triangular bialgebras, in which the non-(co)associativity is controlled by a twisting map. A family of quasi-triangular Hom-bialgebras can be constructed from any quasi-triangular bialgebra, such as Drinfel'd's quantum enveloping algebras. Each quasi-triangular Hom-bialgebra comes with a solution of the quantum Hom-Yang-Baxter equation, which is a non-associative version of the quantum Yang-Baxter equation. Solutions of the Hom-Yang-Baxter equation can be obtained from modules of suitable quasi-triangular Hom-bialgebras.

21 pages

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Cited by in corpus (22)

• On n-ary Hom-Nambu and Hom-Nambu-Lie algebras
• Yetter-Drinfeld modules for Hom-bialgebras
• On unitality conditions for Hom-associative algebras
• Paradigm of Nonassociative Hom-algebras and Hom-superalgebras
• Cyclic homology for Hom-associative algebras
• Purely Hom-Lie bialgebras
• Representations and module-extensions of hom 3-Lie algebras
• Hom-Big Brackets: Theory and Applications
• Generalized Derivations of Hom-Lie Superalgebras
• Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation
• Construtions and bimodules of BiHom-alternative and BiHom-Jordan algebras
• On Antipodes Of Hom-Hopf algebras
• Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields
• Hom-Nijienhuis operator and $T$*-extension of Hom-Lie Superalgebras
• Hom-Yang-Baxter equations and Hom-Yang-Baxter systems
• Algebras of quotients of Hom-Lie algebras
• BiHom-Lie brackets and the Toda equation
• Deforming algebras with anti-involution via twisted associativity
• Derivation Hom-Lie 2-algebras and non-abelian extensions of Hom-Lie algebras
• The braided monoidal structure on the category of Hom-type Doi-Hopf modules
• Crossed product Hom-Hopf algebras and lazy 2-cocycle
• The Hom-Long dimodule category and nonlinear equations