nLab
Jordan superalgebra

Contents

Idea

A Jordan superalgebra is the analog of a Jordan algebra in superalgebra/supergeometry. A Jordan superalgebra JJ is a 2\mathbb{Z}_2-graded algebra J=J 0J 1J = J_0 \oplus J_1, where J 0J_0 is a Jordan algebra and J 1J_1 a J 0J_0-bimodule with a “Lie-like” product into J 0J_0.

Elements of JJ are supercommutative, that is, ab=(1) |a||b|baa \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 10K_{10}.

References