Restricted representations of classical Lie algebras of types 𝐴₂ and 𝐵₂

Abstract

The dimensions of the finite dimensional irreducible restricted modules for a Lie algebra of classical type have never been determined. C. W. Curtis ([5], [6]) has given sufficient, but not necessary, conditions that the dimension be given by Weyl's formula. In this paper, we give results which determine the dimensions of a certain class of finite dimensional irreducible restricted modules for a simple algebra of type A 2 or B 2 over a field of characteristic p>3. Counterexamples to a conjecture of N. Jacobson regarding the complete reducibility of certain modules for an algebra of classical type are given. Our method involves rather lengthy, though elementary, calculations (given in detail in [3]), and depends on a character formula for algebras of types A 2 or B 2 over the complex field, which is due to J. P. Antoine ([l], [2]). R. Steinberg has mentioned that the results for A 2 were obtained by Mark ([9]).