On structure and TKK algebras for Jordan superalgebras

ABSTRACT

We compare a number of different definitions of structure algebras and TKK constructions for Jordan (super)algebras appearing in the literature. We demonstrate that, for unital superalgebras, all the definitions of the structure algebra and the TKK constructions reduce to one of two cases. Moreover, one can be obtained as the Lie superalgebra of superderivations of the other. We also show that, for non-unital superalgebras, more definitions become nonequivalent. As an application, we obtain the corresponding Lie superalgebras for all simple finite dimensional Jordan superalgebras over an algebraically closed field of characteristic zero.

KEYWORDS:

• Lie superalgebras
• Jordan superalgebras
• TKK-construction

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B60
• 17B70
• 17C50

Acknowledgment

SB is a PhD Fellow of the Research Foundation - Flanders (FWO). KC is supported by Australian Research Council Discover-Project Grant DP140103239 and a postdoctoral fellowship of the Research Foundation - Flanders (FWO). The authors thank Hendrik De Bie, Tom De Medts, and Erhard Neher for helpful discussion and comments.