In this paper we show that for an n-Filippov algebra g, the tensor power g ^n-1 is endowed with a structure of anti-symmetric co-representation over the Leibniz algebra g ^n-1. This co-representation is used to define...In this paper we show that for an n-Filippov algebra g, the tensor power g ^n-1 is endowed with a structure of anti-symmetric co-representation over the Leibniz algebra g ^n-1. This co-representation is used to define some relative theories for Leibniz n-algebras with n 〉 2 and obtain exact sequences relating them. As a result, we construct a spectral sequence for the Leibniz homology of Filippov algebras.展开更多
We study the Leibniz n-algebra U_(n)(L),whose multiplication is defined via the bracket of a Leibniz algebra L as[x1,…,xn]=[x1,[…,[xn−2,[xn−1,xn]]…]].We show that U_(n)(L)is simple if and only if L is a simple Lie ...We study the Leibniz n-algebra U_(n)(L),whose multiplication is defined via the bracket of a Leibniz algebra L as[x1,…,xn]=[x1,[…,[xn−2,[xn−1,xn]]…]].We show that U_(n)(L)is simple if and only if L is a simple Lie algebra.An analog of Levi's theorem for Leibniz algebras in U_(n)(Lb)is established and it is proven that the Leibniz n-kernel of U_(n)(L)for any semisimple Leibniz algebra L is the n-algebra U_(n)(L).展开更多
Compactness of subspaces of a Z<sub>2</sub>-graded vector space is introduced and used to study simple Leibniz superalgebras. We introduce left and right super-invariance of bilinear forms over superalgebr...Compactness of subspaces of a Z<sub>2</sub>-graded vector space is introduced and used to study simple Leibniz superalgebras. We introduce left and right super-invariance of bilinear forms over superalgebras. Pseudo-quadratic Leibniz superalgebras are Leibniz superalgebras endowed with a non degenerate, supersymmetric and super-invariant bilinear form. In this paper, we show that every nondegenerate, supersymmetric and super-invariant bilinear form over a Leibniz superalgebra induce a Lie superalgebra over the underlying vector space. Then by using double extension extended to Leibniz superalgebras, we study pseudo-quadratic Leibniz superalgebras and the induced Lie superalgebras. In particular, we generalize some results on Leibniz algebras to Leibniz superalgebras.展开更多
文摘In this paper we show that for an n-Filippov algebra g, the tensor power g ^n-1 is endowed with a structure of anti-symmetric co-representation over the Leibniz algebra g ^n-1. This co-representation is used to define some relative theories for Leibniz n-algebras with n 〉 2 and obtain exact sequences relating them. As a result, we construct a spectral sequence for the Leibniz homology of Filippov algebras.
基金The first author was supported by the National Science Foundation(grant number 1658672),USA.
文摘We study the Leibniz n-algebra U_(n)(L),whose multiplication is defined via the bracket of a Leibniz algebra L as[x1,…,xn]=[x1,[…,[xn−2,[xn−1,xn]]…]].We show that U_(n)(L)is simple if and only if L is a simple Lie algebra.An analog of Levi's theorem for Leibniz algebras in U_(n)(Lb)is established and it is proven that the Leibniz n-kernel of U_(n)(L)for any semisimple Leibniz algebra L is the n-algebra U_(n)(L).
文摘Compactness of subspaces of a Z<sub>2</sub>-graded vector space is introduced and used to study simple Leibniz superalgebras. We introduce left and right super-invariance of bilinear forms over superalgebras. Pseudo-quadratic Leibniz superalgebras are Leibniz superalgebras endowed with a non degenerate, supersymmetric and super-invariant bilinear form. In this paper, we show that every nondegenerate, supersymmetric and super-invariant bilinear form over a Leibniz superalgebra induce a Lie superalgebra over the underlying vector space. Then by using double extension extended to Leibniz superalgebras, we study pseudo-quadratic Leibniz superalgebras and the induced Lie superalgebras. In particular, we generalize some results on Leibniz algebras to Leibniz superalgebras.
