Graded representations of graded Lie algebras and generalized representations of Jordan algebras

Abstract

We introduce and discuss a connection between representations of a certain class of graded Lie algebras and representations of Jordan algebras. This connection is stimulating in both directions. On the one hand it allows to produce an unified point of view on ordinary and Jacobson representations of Jordan algebras and formulate a notion of a generalized representation of a Jordan algebra, which includes ordinary and Jacobson representations as very special cases. The classification of irreducible generalized representations of simple Jordan algebras is given. On the other hand we prove that there are no infinite dimensional irreducible finitely graded representations of graded semisimple Lie algebras and classify the finite dimensional representations of this kind. The theorem about nonexistence of infinite dimensional irreducible finitely graded representations of graded semisimple Lie algebras has in fact as motivation the theorem about the absence of infinite dimensional irreducible representations of the semisimple finite dimensional Jordan algebra A (what is under considered connection can be formulated as the absence of 2-graded irreducible infinite dimensional representations of the 3-graded Lie algebra L(A) = U-1 circle plus U-0 circle plus U-1). (Less)