nLab
Jordan superalgebra

Contents

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

  • 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)

Last revised on July 8, 2020 at 03:11:44. See the history of this page for a list of all contributions to it.