Odd-quadratic Lie superalgebras

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Section: Doing Easy Calculations This Condition Is Equivalent Tomentioning

confidence: 95%

Quadratic Malcev Superalgebras with Reductive Even Part

Albuquerque

Barreiro

Benayadi

2010

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“…Recall that a quadratic Lie color algebra (see [1,18]) is Lie color algebra (g, [·, ·]) together with a ε-symmetric, nondegenerate, invariant bilinear form B : g × g −→ R, such that for any x, y, z ∈ g,…”

Section: Is a Complex Consisting Of The Following Datamentioning

confidence: 99%

Derivation Simple Color Algebras and Semisimple Lie Color Algebras

Zhang

Zhou

2009

“…Recall that in [9], classical Lie superalgebras don't have simultaneously quadratic and odd-quadratic structure. Later, in [1], it has been proved that any perfect Lie superalgebra with even part is a reductive Lie algebra does not possess simultaneously quadratic and odd-quadratic structure. More generally, we can see in the same way of the previous proposition , that any non abelian Lie superalgebra does not possess simultaneously a quadratic and an odd-quadratic structures.…”

Section: Introductionmentioning

confidence: 99%

Associative Superalgebras with Homogeneous Symmetric Structures

Ayadi

Benayadi

2012

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A homogeneous symmetric structure on an associative superalgebra A is a non-degenerate, supersymmetric, homogeneous (i.e. even or odd) and associative bilinear form on A. In this paper, we show that any associative superalgebra with non null product can not admit simultaneously even-symmetric and odd-symmetric structure. We prove that all simple associative superalgebras admit either even-symmetric or odd-symmetric structure and we give explicitly, in every case, the homogeneous symmetric structures. We introduce some notions of generalized double extensions in order to give inductive descriptions of even-symmetric associative superalgebras and odd-symmetric associative superalgebras. We obtain also an other interesting description of odd-symmetric associative superalgebras whose even parts are semi-simple bimodules without using the notions of double extensions.