Editing Jordan algebra (section)

===Jordan superalgebras===

Jordan [[superalgebra]]s were introduced by Kac, Kantor and Kaplansky; these are <math>\mathbb{Z}/2</math>-graded algebras <math>J_0 \oplus J_1</math> where <math>J_0</math> is a Jordan algebra and <math>J_1</math> has a "Lie-like" product with values in <math>J_0</math>.<ref>{{harvnb|McCrimmon|2004|pp=9–10}}</ref>

Any <math>\mathbb{Z}/2</math>-graded associative algebra <math>A_0 \oplus A_1</math> becomes a Jordan superalgebra with respect to the graded Jordan brace

:<math>\{x_i,y_j\} = x_i y_j + (-1)^{ij} y_j x_i \ . </math>

Jordan simple superalgebras over an algebraically closed field of characteristic 0 were classified by {{harvtxt|Kac|1977}}.  They include several families and some exceptional algebras, notably <math>K_3</math> and <math>K_{10}</math>.