On Bernstein Algebras of n-th Order

Abstract

Let (A,w) be a finite dimensional n-th order Bernstein algebra over an infinite field K (charK ≠ 2). If e ∈ A is a nontrivial idempotent then A = K _e ⊕ U _e ⊕ V _e where ${U_e} = \left\{ {x \in Kerw/ex = \frac{1}{2}x} \right\}$ and V _e = {x ∈ Kerw/R ⁿ _e x = 0}. In this paper we show the following results:(1) The dimension of U _e does not depend on idempotent e,(2) if K = ℝ and dimU _e = r then there is an r-parametric family of idempotents of the A, (3) if K = ℝ then dim _R V _e ≥ n.

Partially supported by D.G.A. P. CB-6/91.