On structure and TKK algebras for Jordan superalgebras

TL;DR

This paper compares various definitions of structure and TKK algebras for Jordan superalgebras, revealing their relationships and differences, especially between unital and non-unital cases, and classifies the associated Lie superalgebras for simple finite-dimensional cases.

Contribution

It clarifies the relationships between different TKK and structure algebra definitions for Jordan superalgebras and classifies the resulting Lie superalgebras for simple finite-dimensional cases.

Findings

All definitions coincide for unital superalgebras, with one being the Lie superalgebra of superderivations of the other.

For non-unital superalgebras, the definitions are generally non-equivalent.

Explicit classification of Lie superalgebras for all simple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic zero.

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 fall apart into 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 non-equivalent. As an application, we obtain the corresponding Lie superalgebras for all simple finite dimensional Jordan superalgebras over an algebraically closed field of characteristic zero.