arXiv subjects

Run-Xuan Zhang

Publications and source records attributed to Run-Xuan Zhang.

3 recordsLinked to original sources

arXiv2014

Classification of three dimensional complex omega-Lie algebras

A complex $ω$-Lie algebra is a vector space $L$ over the complex field, equipped with a skew symmetric bracket $[-,-]$ and a bilinear form $ω$ such that $$[[x,y],z]+[[y,z],x]+ [[z,x],y]=ω(x,y)z+ω(y,z)x+ω(z,x)y$$ for all $x,y,z\in L$. The notion of $ω$-Lie algebras, as a generalization of Lie algebras, was introduced in Nurowski \cite{Nur2007}. Fundamental results about finite-dimensional $ω$-Lie algebras were developed by Zusmanovich \cite{Zus2010}. In \cite{Nur2007}, all three dimensional non-Lie real $ω$-Lie algebras were classified. The purpose of this note is to provide an approach to classify all three dimensional non-Lie complex $ω$-Lie algebras. Our method also gives a new proof of the classification in Nurowski \cite{Nur2007}.

math.RA↗

arXiv2013

Cohomologies and Deformations of Generalized Left-symmetric Algebras

The purpose of this paper is to develop a cohomology and deformation theories for generalized left-symmetric algebras.We introduce the notions of generalized left-symmetric cohomology and deformation. We also generalize a theorem of Dzhumadil'daev on connections between the right-symmetric cohomology and Chevalley-Eilenberg cohomology. As an application, we obtain a factorization theorem in left-symmetric superalgebras cohomology. Finally, we obtain all complex simple left-symmetric superalgebras of dimension 3 by the infinitesimal deformations of a given left-symmetric superalgebras.

math.RA↗

arXiv2013

Left-symmetric Structures on Complex Simple Lie Superalgebras

A well-known fact is that there does not exist any compatible left-symmetric structures on a finite-dimensional complex semisimple Lie algebra (see \cite{Chu1974}). This result is not valid in semisimple Lie superalgebra case. In this paper, we study the compatible Left-symmetric superalgebra (LSSA for short) structures on complex simple Lie superalgebras. We prove that there is not any compatible LSSA structure on a finite-dimensional complex simple Lie superalgebra except for the classical simple Lie superalgebra $A(m,n)(m\neq n)$ and Cartan simple Lie superalgebra $W(n)(n\geq 3)$. We also classify all compatible LSSAs with a right-identity on A(0,1).

math.RA↗