Abstract :
Let image(A) be the modular classical Lie algebra over an algebraically closed field k of characteristic p > 0, defined by a Cartan matrix A and Serre′s relations and generators Ei, Hi, Fi. A representation V of image(A) is called weak torsion free if the generators Ei, Fi act injectively 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 ≠ Aip − 1, we classify for all indecomposable Cartan matrices A the pointed (weak) torsion free representations of image(A) up to isomorphism. It turns out to be the only image(A) for A of type Al (p[formula]l + 1) and Cl, admit pointed (weak) torsion free representations. Explicit constructions of these representations are given by specifying actions of generators Ei, Fi, Hi on a chosen basis. Also all these pointed weak torsion free representations are realized by differential operators through the modular Weyl algebras.
