We propose a new algebraic structure, called a weak Jordan algebra, which we define outside of characteristics 2,3. Any Jordan algebra is a weak Jordan algebra, and the converse holds in characteristics different from 5. However, in characteristic 5 there are many examples of simple weak Jordan algebras which are not Jordan, and not even power associative – they are only power associative up to degree 5 (note that by a theorem of Albert, an algebra in characteristic 0 or ≥ 7 which is power associative up to degree 5 and even 4 is power associative in all degrees). These algebras correspond (via a version of the Kantor-Koehler-Tits construction) to Lie algebras in the Fibonacci tensor category Fib in characteristic 5, which can be obtained from Lie algebras in characteristic 5 with a derivation d such that d5 = 0 by the procedure of semisimplification. This allows one to view the notion of a weak Jordan algebra as an example from a new subject that may be called ‘Lie theory in tensor categories’.
This is joint work with A. Kannan and V. Ostrik.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
