Hom-quantum groups III: Representations and module Hom-algebras

Abstract

We study Hom-quantum groups, their representations, and module Hom-algebras. Two Twisting Principles for Hom-type algebras are formulated, and construction results are proved following these Twisting Principles. Examples include Hom-quantum n-spaces, Hom-quantum enveloping algebras of Kac-Moody algebras, Hom-Verma modules, and Hom-type analogs of Uq(sl 2 )module-algebra structures on the quantum planes.

Comodule Hom-coalgebras

“…We also show that comodule Hom-coalgebras can be deformed from comodule coalgebras via endomorphisms. All of our results are dual to D. Yau's work in [13,15]. The difference between our's and his is that, since we are dealing with comodules and coalgebras, we will find that the Sweedler notions [10,2] are more convenient for us to do the work.…”

“…The universal Hom-associative algebra of a Hom-Lie algebra was studied in [11]. Module Hom-algebras and Comodule Hom-algebras have been studied by D. Yau in [13,14,15]. Some other Hom-type algebras such as n-ary Hom-Nambu algebras and n-ary Hom-Nambu-Lie algebras have been studied in [1,16].…”

Drinfeld twists for monoidal Hom-bialgebras

“…The generalized notions, Hom-bialgebras, Hom-Hopf algebras were developed in [3], [5], [6]. Further research on various Hom-Lie structures and Hom-type algebras by many schlors could be found in [7], [9], [16], [17]. Quasitriangular Hom-bialgebras were considered by Yau ([8], [9]), which provided a solution of the quantum Hom-Yang-Baxter euqation, a twisted version of the quantum Yang-Baxter equation ( [10], [11]).…”

Twisting Operators, Twisted Tensor Products and Smash Products for Hom-Associative Algebras

“…Yetter-Drinfeld modules over Hom-bialgebras and their category have been studied in [36]. For further results about generalizations of quantum groups and related structures see [8,43,44,45]. In [18], Hom-quasi-bialgebras have been introduced and concepts like gauge transformation and Drinfeld twist generalized.…”