Author
Zhen Xiong
Bio: Zhen Xiong is an academic researcher from Jilin University. The author has contributed to research in topics: Cellular algebra & Vector space. The author has an hindex of 1, co-authored 2 publications receiving 37 citations.
Papers
More filters
[...]
TL;DR: The notion of omni-Hom-Lie algebras was introduced in this paper, which is an algebra associated to a vector space and an invertible linear map.
Abstract: In this paper, first we show that is a Hom–Lie algebra if and only if is an differential graded-commutative algebra. Then, we revisit representations of Hom–Lie algebras and show that there are a series of coboundary operators. We also introduce the notion of an omni-Hom–Lie algebra associated to a vector space and an invertible linear map. We show that regular Hom–Lie algebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-Hom–Lie algebra. The underlying algebraic structure of the omni-Hom–Lie algebra is a Hom–Leibniz algebra, or a Hom–Lie 2-algebra.
40 citations
Posted Content•
[...]
TL;DR: In this article, it was shown that a regular hom-Lie algebra structure on a vector space can be characterized by Dirac structures in the corresponding omni-hom-Lie algebras.
Abstract: In this paper, first we show that $(\g,[\cdot,\cdot],\alpha)$ is a hom-Lie algebra if and only if $(\Lambda \g^*,\alpha^*,d)$ is an $(\alpha^*,\alpha^*)$-differential graded commutative algebra. Then, we revisit representations of hom-Lie algebras, and show that there are a series of coboundary operators. We also introduce the notion of an omni-hom-Lie algebra associated to a vector space and an invertible linear map. We show that regular hom-Lie algebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-hom-Lie algebra. The underlying algebraic structure of the omni-hom-Lie algebra is a hom-Leibniz algebra, or a hom-Lie 2-algebra.
6 citations
Cited by
More filters
TL;DR: In this article, the notions of a Manin triple for hom-Lie algebras and a purely hom-lie bialgebra were introduced. And they were shown that there is a one-to-one correspondence between Manin triples for Hom-Lie algebra and purely Hom-lie algebra.
Abstract: In this paper, we first show that there is a Hom-Lie algebra structure on the set of $(\sigma,\sigma)$-derivations of an associative algebra. Then we construct the dual representation of a representation of a Hom-Lie algebra. We introduce the notions of a Manin triple for Hom-Lie algebras and a purely Hom-Lie bialgebra. Using the coadjoint representation, we show that there is a one-to-one correspondence between Manin triples for Hom-Lie algebras and purely Hom-Lie bialgebras. Finally, we study coboundary purely Hom-Lie bialgebras and construct solutions of the classical Hom-Yang-Baxter equations in some special Hom-Lie algebras using Hom-$\mathcal~O$-operators.
31 citations
Posted Content•
TL;DR: In this paper, the derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.
Abstract: Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. In this paper, we study cohomology and representations of Bihom-Lie algebras. In particular, derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.
30 citations
TL;DR: In this article, the notion of hom-Batalin-Vilkovisky algebras and strong differential hom-Gerstenhaber algesbras are defined and a homology associated to a hom-Poisson manifold is defined.
26 citations
TL;DR: In this article, the authors modify the definition of a Hom-Lie algebroid introduced by Laurent-Gengoux and Teles and give its equivalent dual description, which is a natural generalization of a purely homogeneous bialgebra and a Lie algebroid.
25 citations
TL;DR: In this article, the notion of hom-big brackets, which is a gene-ralization of Kosmann-Schwarzbach's big brackets, was introduced and used to describe hom-Lie bialgebras and hom-Nijenhuis operators.
Abstract: In this paper, we introduce the notion of hom-big brackets, which is a gene- ralization of Kosmann-Schwarzbach's big brackets. We show that it gives rise to a graded hom-Lie algebra. Thus, it is a useful tool to study hom-structures. In particular, we use it to describe hom-Lie bialgebras and hom-Nijenhuis operators.
22 citations