Inner ideals in quadratic Jordan algebras

Abstract:

The inner ideals play a role in the theory of quadratic Jordan algebras analogous to that played by the one-sided ideals in the theory of associative algebras. In particular, the Jordan algebras with descending chain condition on inner ideals are intimately related to the Artinian associative algebras. In this paper we will completely characterize all inner ideals in the semisimple Jordan algebras with descending chain condition. It is well known that any left or right ideal $\mathfrak {B}$ in a semisimple Artinian $\mathfrak {A}$ is determined by an idempotent, $\mathfrak {B} = \mathfrak {A}f$ or $\mathfrak {B} = e\mathfrak {A}$. We show that any inner ideal in the quadratic Jordan algebra ${\mathfrak {A}^ + }$ has the form $\mathfrak {B} = e\mathfrak {A}f$, and if $\mathfrak {A}$ has involution $^\ast$ the inner ideals of the Jordan algebra $\mathfrak {H}(\mathfrak {A}, ^ \ast )$ of $^ \ast$-symmetric elements are “usually” of the form $\mathfrak {B} = {e^ \ast }\mathfrak {H}e$. We also characterize the inner ideals in the Jordan algebras $\mathfrak {J}(Q,c)$ or $\mathfrak {J}(N,c)$ determined by a quadratic or cubic form.