A bilinear form f on a nonassociative triple system T is said to be invariant if and only if f( abc ,d) = f(a, dcb ) = f(c, bad ) for all a,b,c,d ∈ T . (T ,f) is called a pseudo-metric triple system if f is non-degen...A bilinear form f on a nonassociative triple system T is said to be invariant if and only if f( abc ,d) = f(a, dcb ) = f(c, bad ) for all a,b,c,d ∈ T . (T ,f) is called a pseudo-metric triple system if f is non-degenerate and invariant. A decomposition theory for triple systems and pseudo-metric triple systems is established. Moreover, the ?nite-dimensional metric Lie triple systems are characterized in terms of the structure of the non-degenerate, invariant and symmetric bilinear forms on them.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A bilinear form f on a nonassociative triple system T is said to be invariant if and only if f( abc ,d) = f(a, dcb ) = f(c, bad ) for all a,b,c,d ∈ T . (T ,f) is called a pseudo-metric triple system if f is non-degenerate and invariant. A decomposition theory for triple systems and pseudo-metric triple systems is established. Moreover, the ?nite-dimensional metric Lie triple systems are characterized in terms of the structure of the non-degenerate, invariant and symmetric bilinear forms on them.
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.
