Hom Gel'fand-Dorfman bialgebras and Hom-Lie conformal algebras

Yuan, Lamei

doi:10.1063/1.4870870

Cited by 16 publications

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“…In particular, we introduce quadratic Hom-Lie conformal superalgebras and establish equivalence of quadratic Hom-Lie conformal superalgebras and super Hom-Gel'fand-Dorfman bialgebras. This generalizes the equivalent theorem in the ungraded case obtained in [23], and in classical Lie conformal superalgebra case given in [4,13]. Section 5 is devoted to central extensions of Hom-Lie conformal superalgebras.…”

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“…Finally, we characterize one-dimensional central extensions of quadratic Hom-Lie conformal superalgebras by using certain bilinear forms of super Hom-Gel'fand-Dorfman bialgebras.In Section 2 of this paper, we summarize the main Hom-type (super)algebras and recall the notions of super Gel'fand-Dorfman bialgebra and Lie conformal superalgebra. In Section 3, we introduce super Hom-Gel'fand-Dorfman bialgebras and provide five different ways for constructing super Hom-Gel'fand-Dorfman bialgebras by extending some constructions of Hom-Gel'fand-Dorfman bialgebras given in [23] and Hom-Novikov algebras given in [21]. We also construct affinization of super Hom-Gel'fand-Dorfman bialgebras, which leads us to a class of infinite-dimensional Hom-Lie superalgebras.…”

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Hom-Gel’fand–Dorfman super-bialgebras and Hom-Lie conformal superalgebras

Yuan

Chen

2016

Acta. Math. Sin.-English Ser.

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The purpose of this paper is to introduce and study super Hom-Gel'fand-Dorfman bialgebras and Hom-Lie conformal superalgebras. In this paper, we provide different ways for constructing super Hom-Gel'fand-Dorfman bialgebras and obtain some infinite-dimensional Hom-Lie superalgebras from affinization of super Hom-Gel'fand-Dorfman bialgebras. Also, we give a general construction of Hom-Lie conformal superalgebras from Hom-Lie superalgebras and establish equivalence of quadratic Hom-Lie conformal superalgebras and super Hom-Gel'fand-Dorfman bialgebras. Finally, we characterize one-dimensional central extensions of quadratic Hom-Lie conformal superalgebras by using certain bilinear forms of super Hom-Gel'fand-Dorfman bialgebras.In Section 2 of this paper, we summarize the main Hom-type (super)algebras and recall the notions of super Gel'fand-Dorfman bialgebra and Lie conformal superalgebra. In Section 3, we introduce super Hom-Gel'fand-Dorfman bialgebras and provide five different ways for constructing super Hom-Gel'fand-Dorfman bialgebras by extending some constructions of Hom-Gel'fand-Dorfman bialgebras given in [23] and Hom-Novikov algebras given in [21]. We also construct affinization of super Hom-Gel'fand-Dorfman bialgebras, which leads us to a class of infinite-dimensional Hom-Lie superalgebras. In Section 4, we introduce the notion of Hom-Lie conformal superalgebra, which is a superanalogue of Hom-Lie conformal algebras introduced in [23] as well as a Hom-version of Lie conformal superalgebras. In particular, we introduce quadratic Hom-Lie conformal superalgebras and establish equivalence of quadratic Hom-Lie conformal superalgebras and super Hom-Gel'fand-Dorfman bialgebras. This generalizes the equivalent theorem in the ungraded case obtained in [23], and in classical Lie conformal superalgebra case given in [4,13]. Section 5 is devoted to central extensions of Hom-Lie conformal superalgebras. We characterize one-dimensional central extensions of quadratic Hom-Lie conformal superalgebras by using certain bilinear forms of super Hom-Gel'fand-Dorfman bialgebras, generalizing a construction due to Hong [6]. We should point out that all these results could naturally be extended to more general graded cases.Throughout this paper, all linear spaces and tensor products are over the complex field C. In addition to the standard notation Z, we use Z + to denote the set of nonnegative integers. §2. PreliminariesLet us begin with some definitions, including Hom-Lie superalgebra, Hom-associative algebra, Hom-Novikov superalgebra, super Gel'fand-Dorfman bialgebra, Hom-Gel'fand-Dorfman bialgebra, Hom-Lie conformal algebra and Lie conformal superalgebra. For detailed discussions we refer the reader to the literatures (e.g. [1,7,21,23,24] and references therein).Let V be a superspace that is a Z 2 -graded linear space with a direct sum V = V 0 ⊕ V 1 . The elements of V j , j = {0, 1}, are said to be homogenous and of parity j. The parity of a homogeneous element x is denoted by |x|. Throughout what follows, if |x| occurs in an...

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confidence: 58%

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confidence: 99%

Hom-Gel’fand–Dorfman super-bialgebras and Hom-Lie conformal superalgebras

Yuan

Chen

2016

Acta. Math. Sin.-English Ser.

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“…In this section we recall some basic definitions and results related to our paper from [10] and [13].…”

Section: Preliminariesmentioning

confidence: 99%

“…Later, Liberati in [10] introduced a conformal analog of Lie bialgebras including the conformal classical Yang-Baxter equation, the conformal Manin triples and conformal Drinfeld's double. As a generalization of [1], Hong and Li introduced the definition of left-symmetric conformal algebra in [8] and developed a conformal theory of left-symmetric bialgebras in [7].Recently, the Hom-Lie conformal algebra was introduced and studied in [13], where it was proved that a Hom-Lie conformal algebra is equivalent to a Hom-Gel'fand-Dorfman bialgebra. From then on, similar generalizations of certain algebraic structures became a very popular subject.…”

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The construction of Hom-left-symmetric conformal bialgebras

Guo

Zhang

Wang

2018

Linear and Multilinear Algebra

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In this paper, we first introduce the notion of Hom-left-symmetric conformal bialgebras and show some nontrivial examples. Also, we present construction methods of matched pairs of Hom-Lie conformal algebras and Homleft-symmetric conformal algebras. Finally, we prove that a finite Hom-leftsymmetric conformal bialgebra is free as a C[∂]-module is equivalent to a Homparakähler Lie conformal algebra. In particular, we investigate the coboundary Hom-left-symmetric conformal bialgebras.left-invariant affine structures on Lie groups in [3]. Furthermore, a theory of left-symmetric bialgebras was developed by Bai in [1], he proved that a left-symmetric bialgebra is equivalent to a parakahler Lie algebra which is the Lie algebra of a Lie group G with a G-invariant parakähler structure, studied coboundary left-symmetric bialgebras and obtained an analog of the classical Yang-Baxter equation and called it S-equation.Lie conformal algebras were introduced by Kac in [9], he gave an axiomatic description of the singular part of the operator product expansion of chiral fields in conformal field theory. It is an useful tool to study vertex algebras and has many applications in the theory of Lie algebras. Moreover, Lie conformal algebras have close connections to Hamiltonian formalism in the theory of nonlinear evolution equations. Lie conformal algebras were widely studied in the following aspects: the structure theory [6], representation theory [4], [5] and cohomology theory [2] of finite Lie conformal algebras. Later, Liberati in [10] introduced a conformal analog of Lie bialgebras including the conformal classical Yang-Baxter equation, the conformal Manin triples and conformal Drinfeld's double. As a generalization of [1], Hong and Li introduced the definition of left-symmetric conformal algebra in [8] and developed a conformal theory of left-symmetric bialgebras in [7].Recently, the Hom-Lie conformal algebra was introduced and studied in [13], where it was proved that a Hom-Lie conformal algebra is equivalent to a Hom-Gel'fand-Dorfman bialgebra. From then on, similar generalizations of certain algebraic structures became a very popular subject. In [11], Sheng and Bai introduced a new definition of Hom-Lie algebras and studied their properties. In [12], Sun and Li proved that a Hom-leftsymmetric bialgebra is equivalent to a Hom-parakahler Lie algebra, extending the result given in [1]. Recently, Zhao, Yuan and Chen developed the cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, they introduced the notion of derivations of multiplicative Hom-Lie conformal algebras and study their properties in [14].Motivated by these results, this paper is organized as follows. In Section 2, we introduce the definition of a parakähler Hom-Lie conformal algebra, and study the parakähler Hom-Lie conformal algebra in terms of Hom-left-symmetric conformal algebras. In Section 3, we introduce the matched pairs of Hom-Lie conformal algebras and Hom-...

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“…Hom-Lie conformal algebras were introduced and studied in [8]. Lately, similar generalizations of certain algebraic structures became a very popular subject.…”

Section: Introductionmentioning

confidence: 99%

Deformations and generalized derivations of Hom-Lie conformal algebras

2017

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The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α k -derivations of multiplicative Hom-Lie conformal algebras and study their properties.

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Hom Gel'fand-Dorfman bialgebras and Hom-Lie conformal algebras

Cited by 16 publications

References 35 publications

Hom-Gel’fand–Dorfman super-bialgebras and Hom-Lie conformal superalgebras

Hom-Gel’fand–Dorfman super-bialgebras and Hom-Lie conformal superalgebras

The construction of Hom-left-symmetric conformal bialgebras

Deformations and generalized derivations of Hom-Lie conformal algebras

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