Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 1, pp. 203-221 (2000)

The Root Space Decomposition of the Quadratic Lie Superalgebras

Saïd Benayadi

Université de Metz, Département de Mathématiques, CNRS. UPRES- A - 7035, Ile du Saulcy, 57 045 Metz cedex 1, France. e-mail:}

Abstract: \font\frak=eufm10 \def\f#1{\hbox{\frak#1}} A quadratic Lie superalgebra is a Lie superalgebra ${\f g}={\f g}_{\bar 0}\oplus{\f g}_{\bar 1}$ with a non-degenerate, supersymmetric, even and ${\f g}$-invariant bilinear form $B$, $B$ is called an invariant scalar product of ${\f g}$. In this paper, we study properties of the decomposition of a quadratic Lie superalgebra ${\f g}={\f g}_{\bar 0}\oplus{\f g}_{\bar 1}$ relative to a fixed Cartan subalgebra of ${\f g}_{\bar 0}$. Finally, we give two characterizations of the basic classical Lie superalgebras among the quadratic Lie superalgebras.

Keywords: Classical Simple Lie Superalgebras, Quadratic Lie Superalgebras, Reductive Lie algebras, Roots of Lie superalgebras.

Classification (MSC2000): 17A60, 17A70, 17B05, 17B20, 17B70

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