nLab Jordan superalgebra



Super-Algebra and Super-Geometry



The notion of Jordan superalgebra is the analog in superalgebra/supergeometry of that of Jordan algebra:

A Jordan superalgebra is a supercommutative superalgebra with underlying /2\mathbb{Z}/2-graded algebra J=J 0J 1J = J_0 \oplus J_1, where:


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}.


  • 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)

  • M. L. Racine, E. I. Zel’manov, An octonionic construction of the Kac superalgebra K 10K_10, Proc. Amer. Math. Soc. 143 (2015), 1075-1083, (pdf).

Last revised on February 16, 2023 at 10:42:11. See the history of this page for a list of all contributions to it.