Rolf Farnsteiner

Cohomology of Lie Algebras

The cohomology theory of Lie algebras was initiated by C. Chevalley and S. Eilenberg in their article "Cohomology theory of Lie groups and Lie algebras", Trans. Amer. Math. Soc. 63 (1948), 85-124. Their main motivation was the computation of the real cohomology of the underlying topological space of a compact connected Lie group in terms of its associated Lie algebra. From the very beginning it was realized, however, that cohomology groups could also be used to provide elegant algebraic proofs for structural results on complex Lie algebras such as Whitehead's Lemmas, Weyl's Theorem and the Theorem of Levi-Malcev. Since then many more applications have emerged, and the theory nowadays is a discipline in its own right.

For restricted Lie algebras of characteristic p, a second theory is available, which is more suitable for the study of the "restricted" modules of such an algebra. These so-called restricted cohomology groups were introduced by G. P. Hochschild (Cohomology of restricted Lie algebras, Amer. J. Math. 76 (1954), 555-580), who established interpretations for groups of low degree as well as a connection with the Chevalley-Eilenberg cohomology. Being extension groups of Frobenius algebras, restricted cohomology groups behave much like their precursors for finite groups. In particular, E. M. Friedlander and B. J. Parshall proved an analogue of the Evens-Venkov Theorem, and thus laid the foundation for support varieties of restricted Lie algebras. 

 

Publications

1.  Central extensions and invariant forms of graded Lie algebras.
        Algebras, Groups, Geom. 3 (1986), 431-455

   2.  Dual space derivations and H2(L,F) of modular Lie algebras.
        Can. J. Math. 39 (1987), 1078-1106

   3.  On the cohomology of associative algebras and Lie algebras.
        Proc. Amer. Math. Soc. 99 (1987), 414-420

   4.  Cohomology groups of infinite dimensional algebras.
        Math. Z. 199 (1988), 407-423

   5.  Derivations and central extensions of finitely generated graded Lie algebras.
        J. Algebra 118 (1988), 33-45

   6.  Extension functors of modular Lie algebras.
        Math. Ann. 288 (1990), 713-730

   7.  Cohomology groups of reduced enveloping algebras.
        Math. Z. 206 (1991), 103-117

   8.  Shapiro's Lemma and its consequences in the cohomology theory of modular Lie algebras.
        Math. Z. 206 (1991), 153-168  (with Helmut Strade)