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Register a new userThe Hom-Long dimodule category and nonlinear equations
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In this paper, we construct a kind of new braided monoidal category over two Hom-Hopf algerbas $(H,alpha)$ and $(B,beta)$ and associate it with two nonlinear equations. We first introduce the notion of an $(H,B)$-Hom-Long dimodule and show that the Hom-Long dimodule category $^{B}_{H} Bbb L$ is an autonomous category. Second, we prove that the category $^{B}_{H} Bbb L$ is a braided monoidal category if $(H,alpha)$ is quasitriangular and $(B,beta)$ is coquasitriangular and get a solution of the quantum Yang-Baxter equation. Also, we show that the category $^{B}_{H} Bbb L$ can be viewed as a subcategory of the Hom-Yetter-Drinfeld category $^{Ho B}_{Ho B} Bbb {HYD}$. Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules.
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In this paper, we mainly present some new solutions of the Hom-Yang-Baxter equation from Hom-algebras, Hom-coalgebras and Hom-Lie algebras, respectively. Also, we prove that these solutions are all self-inverse and give some examples. Finally, we introduce the notion of Hom-Yang-Baxter systems and obtain two kinds of Hom-Yang-Baxter systems.
We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings. If the universal covering exists, we prove that this group is isomorphic to the Galois group of the universal covering. The grading deduced from a Galois covering enables us to describe the canonical monomorphism from its automorphism group to the first Hochschild-Mitchell cohomology vector space.
Starting with a k-linear or DG category admitting a (homotopy) Serre functor, we construct a k-linear or DG 2-category categorifying the Heisenberg algebra of the numerical K-group of the original category. We also define a 2-categorical analogue of the Fock space representation of the Heisenberg algebra. Our construction generalises and unifies various categorical Heisenberg algebra actions appearing in the literature. In particular, we give a full categorical enhancement of the action on derived categories of symmetric quotient stacks introduced by Krug, which itself categorifies a Heisenberg algebra action proposed by Grojnowski.
In this paper, we introduce the notion of a derivation of a Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of Hom-Lie algebras. We show that iso- morphism classes of diagonal non-abelian extensions of a Hom-Lie algebra g by a Hom-Lie algebra h are in one-to-one correspondence with homotopy classes of morphisms from g to the derivation Hom-Lie 2-algebra DER(h).
In this paper, we introduce the notions of biderivations and linear commuting maps of Hom-Lie algebras and superalgebras. Then we compute biderivations of the q-deformed W(2,2) algebra, q-deformed Witt algebra and superalgebras by elementary and direct calculations. As an application, linear commuting maps on these algebras are characterized. Also, we introduce the notions of {alpha}-derivations and {alpha}-biderivations for Hom-Lie algebras and superal- gebras, and we establish a close relation between {alpha}-derivations and {alpha}-biderivations. As an illustration, we prove that the q-deformed W(2;2)-algebra, the q-deformed Witt algebra and superalgebra have no nontrivial {alpha}-biderivations. Finally, we present an example of Hom-Lie superalgebras with nontrivial {alpha}-super-derivations and biderivations.
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