Abstract
In this article, we prove that if B is a simple binary-Lie superalgebra whose even part is isomorphic to sl2(F) and whose odd part is a completely reducible binary-Lie-module over the even part, then B is a Lie superalgebra. We introduce also a binary-Lie module over which is sl2(F) not completely reducible.
Highlights
All algebras mentioned in this article are algebras over a fixed arbitrary field of characteristic zero.A superalgebra is a 2 -graded algebra, i.e., an algebra = A A0 ⊕ A1 such that AiA j ⊆ Ai+ j for every{ } i, j ∈ 0,1 = 2
We say that an anti-commutative superalgebra B is a binary-Lie superalgebra, if it satisfies φs (a,b, c, d ) = 0 for every a,b, c, d homogeneous in B, where φs is the super-analog of the function φ, i.e
We notice that not every binary Lie module over sl2 ( ) is completely reducible as we can see in the following example: Example 1
Summary
All algebras mentioned in this article are algebras over a fixed arbitrary field of characteristic zero. (2015) On Simple Completely Reducible Binary-Lie Superalgebras over sl ( ). We say that an anti-commutative superalgebra B is a binary-Lie superalgebra, if it satisfies φs (a,b, c, d ) = 0 for every a,b, c, d homogeneous in B , where φs is the super-analog of the function φ , i.e. A subset J of a superalgebra A is said to be a super-ideal of A , if and only if J is an ideal of A and. Our aim is to characterize binary-Lie superalgebras whose even part is isomorphic to sl ( ) and whose odd part is a completely reducible module over the even part. Let = B B0 ⊕ B1 be a simple binary-Lie superalgebra, such that B0 is and B1 is a completely reducible module over B0.
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