arXiv 1409.43422014-09math.RTmath-phmath.DGmath.MP

Commutative $n$-ary superalgebras with an invariant skew-symmetric form

Elizaveta Vishnyakova

备注：: 27 pages
发表信息：: J.Geom.Phys. 98 (2015) 340-354
DOI：: 10.1016/j.geomphys.2015.08.015

英文摘要：

We study $n$-ary commutative superalgebras and $L_{\infty}$-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their $n$-ary generalizations, commutative associative and Jordan algebras with an invariant form. We give a classification of anti-commutative $m$-dimensional $(m-3)$-ary algebras with an invariant form, and a classification of real simple $m$-dimensional Lie $(m-3)$-algebras with a positive definite invariant form up to isometry. Furthermore, we develop the Hodge Theory for $L_{\infty}$-algebras with a symmetric invariant form, and we describe quasi-Frobenius structures on skew-symmetric $n$-ary algebras.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

On the deformations and derivations of $n$-ary multiplicative Hom-Nambu-Lie superalgebras

In this paper, we introduce the relevant concepts of $n$-ary multiplicative Hom-Nambu-Lie superalgebras and construct three classes of $n$-ary multiplicative Hom-Nambu-Lie superalgebras. As a generalization of the notion of derivations for $n$-ary multiplicative Hom-Nambu-Lie algebras, we discuss the derivations of $n$-ary multiplicative Hom-Nambu-Lie superalgebras. In addition, the theory of one parameter formal deformation of $n$-ary multiplicative Hom-Nambu-Lie superalgebras is developed by choosing a suitable cohomology.

2014-01-03math.RAmath.RT

On the cohomology and extensions of $n$-ary multiplicative Hom-Nambu-Lie superalgebras

In this paper, we discuss the representations of $n$-ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notion of representations for $n$-ary multiplicative Hom-Nambu-Lie algebras. We also give the cohomology of an $n$-ary multiplicative Hom-Nambu-Lie superalgebra and obtain a relation between extensions of an $n$-ary multiplicative Hom-Nambu-Lie superalgebra $\mathfrak{b}$ by an abelian one $\mathfrak{a}$ and $Z^1(\mathfrak{b}, \mathfrak{a})_{\bar{0}}$. We also introduce the notion of $T^*$-extensions of $n$-ary multiplicative Hom-Nambu-Lie superalgebras and prove that …

2014-01-02math.RT

Cohomology of Hom-Leibniz and $n$-ary Hom-Nambu-Lie superalgebras

The aim of this paper is to study the cohomology of Hom-Leibniz superalgebras. We construct the $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type and provide as application the computations of the derivations and second cohomology group. Moreover, we extend to graded case the Takhtajan's construction of a cohomology of $n$-ary Hom-Nambu-Lie algebras starting from cohomology of Hom-Leibniz algebras.

2014-06-14math.RAmath.RT

李超代数的Howe对偶

Howe Duality for Lie Superalgebras

研究了R意义下一般线性李超代数的对偶对。豪给出了对称和斜对称代数（在超意义下）在对偶对作用下的显式无乘分解，并给出了该对称代数的每个同构子空间中最高权向量的显式计算公式.本文给出了$gl(m|n)$的自然表示的对称平方的对称和反对对称代数的不可约$gl(m|n)$ -模的显式无乘法分解。在前一种情况下，我们发现，以及明确的公式为最高权向量。我们的工作统一和推广了对称和反对称模型中的经典结果，并得到了一些应用。

2000-08-13math.RT

辛费米子顶点算子超代数的Z_2-orbifold模型

A Z_2-orbifold model of the symplectic fermionic vertex operator superalgebra

构造了一个与具有非退化反称双线性型的有限维向量空间相关联的无理C_2-余有限顶点算子代数.对它的不可约模的等价类进行了分类，并确定了它的自同构群。

2006-05-24math.QAmath.RT

A Note on Skew Characters of Symmetric Groups

In previous work Regev used part of the representation theory of Lie superalgebras to compute the values of a character of the symmetric group whose decomposition into irreducible constituents is described by semistandard $(k,\ell)$-tableaux. In this short note we give a new proof of Regev's result using skew characters.

2016-03-12math.RT

更多相关论文收起更多论文

李超代数$\hat{\frak{gl}}_{\infty|\infty}$的有理半标准表和character公式

Rational semistandard tableaux and character formula for the Lie superalgebra $\hat{\frak{gl}}_{\infty|\infty}$

本文给出了作用于无限维Fock空间$n$水平$\frak{F}^n$上的Howe对偶对$(\hat{\frak{gl}}_{\infty|\infty},\frak{gl}_n)$的一种新的组合解释.在$\frak{F}^n$中出现的$\hat{\frak{gl}}_{\infty|\infty}$的准有限不可约最高权表示的特征是根据某些倾斜形状的bitableaux来实现的。研究了这类表的一般组合学，包括Robinson-Schensted-Knuth对应和Littlewood-Richardson规则，以及它与$\frak{gl}_n$的有理半标准表的对偶关系.这一结果也解释了包括$\frak{gl}_n$在内的其他Howe对偶对。

2006-04-29math.RT

Derived supersymmetries of determinantal varieties

We show that the linear strands of the Tor of determinantal varieties in spaces of symmetric and skew-symmetric matrices are irreducible representations for the periplectic (strange) Lie superalgebra. The structure of these linear strands is explicitly known, so this gives an explicit realization of some representations of the periplectic Lie superalgebra. This complements results of Pragacz and Weyman, who showed an analogous statement for the generic determinantal varieties and the general linear Lie superalgebra. We also give a simpler proof of their result. Via Koszul duality, this is an o…

2012-07-13math.ACmath.RT

Noetherianity of some degree two twisted skew-commutative algebras

A major open problem in the theory of twisted commutative algebras (tca's) is proving noetherianity of finitely generated tca's. For bounded tca's this is easy, in the unbounded case, noetherianity is only known for Sym(Sym^2(C^\infty)) and Sym(\wedge^2(C^\infty)). In this paper, we establish noetherianity for the skew-commutative versions of these two algebras, namely \wedge(Sym^2(C^\infty)) and \wedge(\wedge^2(C^\infty)). The result depends on work of Serganova on the representation theory of the infinite periplectic Lie superalgebra, and has found application in the work of Miller-Wilson on…

2020-11-11math.ACmath.RT