nLab
Jordan superalgebra

Idea

A Jordan superalgebra is the analog of a Jordan algebra in superalgebra/supergeometry. A Jordan superalgebra $J$ is a $\mathbb{Z}_2$-graded algebra $J = J_0 \oplus J_1$, where $J_0$ is a Jordan algebra and $J_1$ a $J_0$-bimodule with a “Lie-like” product into $J_0$.

Elements of $J$ are supercommutative, that is, $a \cdot b = (-1)^{|a|\cdot|b|} b \cdot a$, and satisfy the super Jordan identity.

Classification

Simple Jordan superalgebras over an algebraically closed field of characteristic 0 were classified by Kac (Kac 77). The only exceptional finite-dimensional example is the 10-dimensional Jordan superalgebra $K_{10}$.

References

• Victor Kac, Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras, Comm. Algebra 5(1977), 1375–1400, (doi: 10.1080/00927877708822224)

• Isaiah Kantor, Connections between Poisson brackets and Jordan and Lie superalgebras, Lie theory, differential equations and representation theory, Montreal (1989), 213–225.

• Irving Kaplansky, Graded Jordan algebras I (preprint).

• Consuelo Martínez, Simplicity of Jordan Superalgebras and Relations with Lie Structures, Irish Math. Soc. Bulletin 50(2003), 97–116 (pdf)

• M. E. Martin, Classification of three-dimensional Jordan superalgebras (arXiv:1708.01963)