Representation Theory of Jordan Algebras

Abstract

Multiplication Representations in Classes of Algebras Defined by Identities.

In this chapter we develop the basic concepts of representation theory for an arbitrary class of algebras defined by identities. If f is an element of a free non-associative algebra over a field Φ then we say that an algebra r/I satisfies the identity f = 0 if f is mapped into 0 by every homomorphism of the free algebra into r. If S is a subset of a free non-associative algebra then we denote by C(S) the class of algebras satisfying every identity f = 0, f ϵ S. The representation theory for C(S) has as its starting point the notion of an S-bimodule for an r in the class C(S). This is a vector space m/I with bilinear compositions (a, u) → au, (a, u) → ua of ((r,m) into m such that the algebra ϵ = = r+m;, with multiplication (a₁ +u₁)(a₂ +u₂) = a₁ a₂ + a₁ u₂ + u₁ a₂, a_i υr, u_i υm is in the class C(S). We can derive the explicit conditions on au and ua for an S-bimodule of r from the set S of defining identities. Moreover, these conditions can be expressed as conditions on the linear transformations u →au, u →ua in m and this leads to the notion of an S-multiplication representation (S-birepresentation) of r in the associative algebra Horn I (m m). It is convenient to generalize this concept to that of an S-multiplication specialization in which Horn (m, m) is replaced by an arbitrary associative algebra with an identity element 1. This leads to the notion of a universal S-multiplication envelope for r in C(S). The determination of such envelopes is one of the basic problems of the representation theory since the S-bimodules and S-multiplication representations for can be identified with right modules and representations of the associative universal envelope.