CLASSIFICATION OF POINTED WEAK TORSION-FREE REPRESENTATIONS FOR CLASSICAL LIE-ALGEBRAS

Let g(A) be the modular classical Lie algebra over an algebraically cl osed field k of characteristic p > 0, defined by a Cartan matrix A and Serre's relations and generators E(i), H-i, F-i. A representation V o f g(A) is called weak torsion free if the generators E(i), F-i act inj ectively on V and is called pointed if V is irreducible and has a one dimensional weight space. In this paper assuming p > 3 and A not equal A(ip-1), we classify for all indecomposable Cartan matrices A the poi nted (weak) torsion free representations of g(A) up to isomorphism. It turns out to be the only g(A) for A of type A(l)(p + l + 1) and C-l, admit pointed (weak) torsion free representations. Explicit constructi ons of these representations are given by specifying actions of genera tors E(i), F-i, H-i on a chosen basis. Also all these pointed weak tor sion free representations are realized by differential operators throu gh the modular Weyl algebras. (C) 1995 Academic Press, Inc.