The category of Yetter-Drinfel'd Hom-modules and the quantum Hom-Yang-Baxter equation

Abstract

Representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the Yang-Baxter equationIn this paper, we introduce the category of Yetter-Drinfel'd Hom-modules which is a braided monoidal category and show that the category of Yetter-Drinfel'd Hom-modules is a full monoidal subcategory of the left center of left Hommodule category. Also we study the equivalent relationship between the category of Yetter-Drinfel'd Hom-modules and the category of Hom-modules over the Drinfel'd double. Finally, the Faddeev-Reshetikhin-Takhtajan (FRT) type theorem for the quantum Hom-Yang-Baxter equation is investigated. C 2014 AIP Publishing LLC.finitesimal Hom-bialgebras, Hom-power associative algebras, and quasi-triangular Hom-bialgebras were introduced in Refs. 1, 4-6, 15-17, 23, and 26. Hom-associative algebras, as an analogue and generalization of associative algebras for Hom-Lie algebras, have been introduced in Ref. 4. Further, some modules and comodules on these Homalgebras structures such as Hom-modules, Hom-comodules, Hom-Hopf modules, and Hom-module algebras were considered, and the fundamental structure of Hom-Hopf modules was investigated in Ref. 23.The first aim of this paper is to study the category of Yetter-Drinfel'd Hom-modules and the center construction for Hom-Hopf algebras. And the second aim of this paper is to study the FRT-type theorem for the QHYBE, which generalize the FRT-type theorem for the quantum Yang-Baxter equation in Ref. 11. This paper is organized as follows. In Sec. I, we recall the basic definitions about Hom-structures containing monoidal Hom-algebras, monoidal Hom-coalgebras, monoidal Hom-bialgebras, monoidal Hom-Hopf algebras, quasi-triangular Hom-Hopf algebras, Hom-modules, and Hom-comodules. In Sec. II, we introduce the concept of Yetter-Drinfel'd Hommodules as a generalization of Yetter-Drinfel'd modules and show that every Yetter-Drinfel'd Hommodule can give a solution to the QHYBE. As the main result of this section, we prove that the category of Yetter-Drinfel'd Hom-modules is a full monoidal subcategory of the left center Z l ( H( H M)) of left Hom-module category. In Sec. III, we study the equivalent relationship between the category of Yetter-Drinfel'd Hom-modules and the category of Hom-modules over Drinfel'd double. And show that the Drinfel'd double of finite dimensional monoidal Hom-Hopf algebra is a monoidal Hom-Hopf algebra. In Sec. IV, we give the FRT-type theorem for the quantum Hom-Yang-Baxter equation (QHYBE).Throughout this paper, we always work over a fixed field k and all vector spaces and tensor spaces are over k. For a coalgebra, we write its comultiplication :x → x 1 ⊗x 2 and denote the comodule structure ρ : m → m (0) ⊗m (1) with the summation symbols omitted. τ denotes the flip map. II. PRELIMINARIESIn this section, we recall from Ref. 23 some information about so-called monoidal Homstructures. Briefly speaking, these structures are objects in the monoidal category H(M k ) (called the Hom-category associated to M k i...

Hom-L-R-smash products, Hom-diagonal crossed products and the Drinfeld double of a Hom-Hopf algebra

“…15),(4.4) = d ⊗ h, finishing the proof.Assume now that we are in the hypotheses of Proposition 4.3 and consider the Hom-L-Rsmash product D ♮ H = D Q ⊗ R H. Since the map Q is bijective, we can apply Proposition 2.4 and we obtain that the map P : H ⊗ D → D ⊗ H, P = Q −1 • R is a Hom-twisting map and we have an isomorphism of Hom-associative algebras Q : D ⊗ P H ≃ D ♮ H.…”

“…Yetter-Drinfeld modules over Hom-bialgebras were studied in [38] and we will introduce the Drinfeld double in this paper. Since Hom-bialgebras and monoidal Hom-bialgebras are different concepts, it turns out that our definitions, formulae and results are also different from the ones in [15,31].One of the main tools to construct examples of Hom-type algebras is the so-called "twisting principle" which was introduced by D. Yau for Hom-associative algebras and since then extended to various Hom-type algebras. It allows to construct a Hom-type algebra starting from a classicaltype algebra and an algebra homomorphism.The twisted tensor product A⊗ R B of two associative algebras A and B is a certain associative algebra structure on the vector space A ⊗ B, defined in terms of a so-called twisting map R : B ⊗ A → A ⊗ B, having the property that it coincides with the usual tensor product algebra A ⊗ B if R is the usual flip map.…”

Drinfeld twists for monoidal Hom-bialgebras

“…A (co)monoid in the Hom-category is a Hom-(co)algebra, and a bimonoid in the Hom-category is a monoidal Hom-bialgebra. Further research on monoidal Hom-bialgebras can be found in [22], [23], and [14].…”

Entwined Hom-modules and frobenius properties

“…Liu and Shen [15] also studied Yetter-Drinfeld modules over monoidal Hom-bialgebras, they called them Hom-Yetter-Drinfeld modules, and showed that the category of Hom-Yetter-Drinfeld modules is a braided monoidal categories. Chen and Zhang [8] introduced the category of Hom-Yetter-Drinfeld modules, which is differs from that of [15], and indicated that it is a full monoidal subcategory of the left center of left Hom-module category. In [10], we defined the category of Doi Hom-Hopf modules and proved that the category of Hom-Yetter-Drinfeld modules is a subcategory of our category of Doi Hom-Hopf modules.…”