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Jordan supersystems related to Lie superalgebras. (English) Zbl 1441.17020

In this paper, the authors extend the construction of Jordan algebras and subquotients associated to Lie algebras (see A. Fernández López et al. [J. Algebra 308, No. 1, 164–177 (2007; Zbl 1113.17015); Int. Math. Res. Not. 2007, No. 16, rnm051, 34 p. (2007; Zbl 1145.17008)]) to Lie superalgebras.
Any ad-nilpotent element of index less than or equal to three of the even part \(L_0\) of a Lie superalgebra \(L=L_0\oplus L_1\) gives rise to a Jordan superalgebra by using the Grassmann envelope. For ad-nilpotent elements of index less than or equal to four in \(L_1\), the construction is slightly modified and not a Jordan superalgebra but a Jordan superpair is obtained.
The authors also extend to the supersetting the subquotients associated to abelian inner ideals of Lie superalgebras, and show that they are Jordan superpairs. These subquotients coincide with the Jordan superalgebras/superpairs attached to ad-nilpotent elements when considering the appropriate abelian inner ideals.

MSC:

17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17C50 Jordan structures associated with other structures
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