Homology, Homotopy and Applications, vol.4(2), 2002, pp.141–163 NONSPLIT EXTENSIONS OF MODULAR LIE ALGEBRAS OF RANK 2

Second cohomology groups of irreducible representations of classical Lie algebras A2, B2 andG2 over an algebraically closed field of characteristic p> h are calculated. Here h is the Cox-eter number. To Jan–Erik Roos on his sixty–fifth birthday 1. Formulation of the main result Levi-Mal’cev theorem has cohomological origin. It states that any finite-dimen-sional extension of a finite-dimensional simple Lie algebra over a field of charac-teristic 0 is split. In case of characteristic p> 0 any Lie algebra has at least one nonsplit extension and the number of irreducible modules with such a property is finite [8]. For example, the 3-dimensional simple Lie algebra A1 = sl2 has ex-actly one irreducible module, namely the (p − 1)-dimensional module Vp−2, with H2(A1, Vp−2) 6 = 0 [7]. The aim of our paper is to calculate second cohomology groups with coefficient