Finite-dimensional representations of PI algebras

Abstract

Shirshov’s theorem [6, 4.2.81 says that if R =k(xi, . . . . x,} is a PI algebra over the field k of degree d and every monomial of degree <d in the x’s is algebraic over k, then R is finite-dimensional over k. The truly remarkable aspect of this result is the requirement of algebraicity on just a finite number of elements albeit chosen from a generating set. Another result which has the same flavor is a theorem of Kostant [2, 8.11 which gives a criterion for a left ideal I in the enveloping algebra U of a finitedimensional Lie algebra L over a field k of characteristic zero to be of finite codimension: I is of finite codimension in U if and only if every element of a Lie generating set of L is “algebraic” over I; i.e., for each element u EL there is 0 #g(t) ~k[t] such that g(u) E I. The proof of Kostant’s result relies on some deep results. This paper arose out of an attempt to simplify the proof of Kostant’s results and to generalize appropriately his result to other classes of rings. Kostant’s theorem fails in characteristic p, as Makar-Limanov showed us, and care is needed even for an appropriate generalization to finitely generated (afline) PI algebras. The analog of Kostant’s result for affine PI algebras is given in our Theorem 5; unfortunately, we must require that every element of an afline PI algebra be algebraic over a left ideal to force finite codimension. An easy example is given which shows that even requiring all monomials of some generating set to be algebraic over a left ideal fails to ensure finite codimen-