Nodal Non-Commutative Jordan Algebras

A finite dimensional power-associative algebra 𝒰 with a unity element 1 over a field J is called a nodal algebra by Schafer (7) if every element of 𝒰 has the form α1 + z where α is in J, z is nilpotent, and if 𝒰 does not have the form 𝒰 = ℐ1 + n with n a nil subalgebra of 𝒰 . An algebra SI is called a non-commutative Jordan algebra if 𝒰 is flexible and 𝒰 + is a Jordan algebra. Some examples of nodal non-commutative Jordan algebras were given in (5) and it was proved in (6) that if 𝒰 is a simple nodal noncommutative Jordan algebra of characteristic not 2, then 𝒰 + is associative. In this paper we describe all simple nodal non-commutative Jordan algebras of characteristic not 2.

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