A Hom-version of the affinizations of Balinskii–Novikov and Novikov superalgebras

Zhang, Runxuan; Hou, Dongping; Bai, Chengming

doi:10.1063/1.3546025

Cited by 22 publications

(22 citation statements)

References 15 publications

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“…The following result, given in [24], extends Yau' construction of Hom-Novikov algebras from Novikov algebras with an algebra endomorphism [21]. Then (A, • α , α) is a Hom-Novikov superalgebra with…”

Section: )mentioning

confidence: 69%

“…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.…”

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confidence: 99%

“…For detailed discussions we refer the reader to the literatures (e.g. [1,7,21,23,24] and references therein).…”

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confidence: 99%

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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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Section: )mentioning

confidence: 69%

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confidence: 99%

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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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“…Hom-left-symmetric algebras, or hom-pre-Lie algebras were first introduced in [9], and then further studied in [16] and [18].…”

Section: The Construction Of Strict Hom-lie 2-algebras From Hom-left-mentioning

confidence: 99%

Hom-Lie 2-algebras

Sheng

Chen

2013

Journal of Algebra

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In this paper, we introduce the notions of hom-Lie 2-algebras, which is the categorification of hom-Lie algebras, $HL_\infty$-algebras, which is the hom-analogue of $L_\infty$-algebras, and crossed modules of hom-Lie algebras. We prove that the category of hom-Lie 2-algebras and the category of 2-term $HL_\infty$-algebras are equivalent. We give a detailed study on skeletal hom-Lie 2-algebras. In particular, we construct the hom-analogues of the string Lie 2-algebras associated to any semisimple involutive hom-Lie algebras. We also proved that there is a one-to-one correspondence between strict hom-Lie 2-algebras and crossed modules of hom-Lie algebras. We give the construction of strict hom-Lie 2-algebras from hom-left-symmetric algebras and symplectic hom-Lie algebras.Comment: 21 pages, Journal of Algebra, 376 (2013) 174-19

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“…The representation theory and the cohomology theory of Hom-Lie superalgebras were studied in [17]. In [18], some infinite-dimensional Hom-Lie superalgebras were constructed, induced by affinizations of the Hom-Balinskii-Novikov superalgebras and Hom-Novikov superalgebras. Furthermore, the central extensions of the infinite-dimensional Hom-Lie superalgebras were studied.…”

Section: Introductionmentioning

confidence: 99%