Cohomology of restricted Lie algebras

In this dissertation, we investigate the cohomology theory of restricted Lie algebras. Motivations for the definition of a restricted Lie algebra are given and the theory of ordinary Lie algebra cohomology is briefly reviewed, including a discussion on algebraic interpretations of the low dimensional cohomology spaces of ordinary Lie algebras. The general Cartan-Elienberg construction of the standard cochain complex is given for ordinary Lie algebras. The representation theory of restricted Lie algebras is reviewed including a description of the restricted universal enveloping algebra Ures.( g ) of a restricted Lie algebra. In the case of an abelian restricted Lie algebra, we construct an augmented complex of free U res.( g ) modules that is exact in dimensions less than p and hence define the cohomology theory of these algebras in dimension less than p. Explicit formulas for the dimensions of the cochain spaces are given in the abelian case. In particular we show that the dimension of Ck( g ) is the same as that of the symmetric algebra Sk( g ). In the non-abelian case, we explicitly construct a cochain complex {Ck( g ; M), deltak} for any coefficient module M for k ≤ 3 and give explicit formulas for the coboundary operators in these dimensions. It is shown that classical and restricted cohomology do not differ at all in dimension zero and that the restricted cohomology space H 1( g ; M) is canonically injected into the classical cohomology H1cl. ( g ; M). A canonical map H2( g ; M) → H2cl. ( g ; M) is constructed and the kernel is investigated for specific coefficient modules. The corresponding notions of the usual algebraic interpretations of ordinary low dimensional cohomology are defined and we show that our restricted cohomology spaces encode this information as well. The dissertation concludes with some remarks on multiplicative structures in our complex as well as directions for further research.