# Universal Enveloping Algebras

Universal enveloping algebras are the Lie theoretic analogues of group algebras. Nowadays enveloping algebras and their deformations are studied in a number of interrelated fields such as quantum groups, Lie superalgebras or restricted Lie algebras. Certain structural features of these algebras account for various phenomena in Lie theory that remain elusive if one confines one's attention to the category of Lie algebras. For instance, the dichotomy between the classical and the modular representation theory of Lie algebras is best understood in the context of enveloping algebras.

Finite dimensional semisimple complex Lie algebras and their representations are well-understood: According to Weyl's Theorem these representations decompose into direct sums of irreducible subrepresentations. The irreducible constituents in turn can be parametrized by "highest weights". If the highest weight of an irreducible module is known, then Weyl's Dimension Formula gives its dimension. The modern proofs of these results employ structural features of the universal enveloping algebras of the underlying Lie algebras.

Let L be a finite dimensional Lie algebra with universal enveloping algebra U(L). Every (irreducible) L-module M is also an (irreducible) module for the enveloping algebra U(L). If we are working over an algebraically closed field, then the center Z(L) of U(L) operates on every finite dimensional irreducible L-module via scalars. In the aforementioned classical context, these scalars separate the irreducible modules, so that each finite dimensional irreducible module is uniquely determined by its central character. This fact depends on Harish-Chandra's Theorem, which relates Z(L) to polynomial invariants of the Weyl group of L.

If we work over fields of positive characteristic (the modular case), the
situation changes completely. Among other factors, the difference resides in the fact
that the
center of the enveloping algebra U(L) tends to be bigger. Given an n-dimensional Lie
algebra L one can always find a subalgebra O(L) of Z(L) such that O(L) is a
polynomial ring in n variables, and U(L) is a free O(L)-module of finite rank.
One notable consequence is the finite dimensionality of irreducible
L-modules, which markedly contrasts with the classical situation. For
algebraically closed fields the irreducible L-modules may thus again be subdivided
according to their characters on O(L). However, this time different irreducibles
may give rise to the same character. In fact, the irreducible modules belonging
to the same character are modules over a finite dimensional associative algebra,
the so-called *reduced enveloping algebra*.
One thus obtains an algebraic family of finite dimensional algebras that is
parametrized by the maximal spectrum of O(L).

**Publications**

1. *Extension functors of modular Lie algebras.*

Math. Ann. **288** (1990),
713-730

2. *Cohomology groups of reduced enveloping algebras.*

Math. Z. **206**
(1991), 103-117

3. *Shapiro's Lemma and its consequences
in the cohomology theory of modular Lie algebras.*

Math. Z. **206** (1991),
153-168 (with Helmut Strade)

4. *On the theory of
Frobenius extensions and its application to Lie superalgebras.*

Trans. Amer. Math. Soc. **335**
(1993), 407-424 (with Allen Bell)

5. *Representations
of blocks associated to induced modules of restricted Lie algebras.*

Math. Nachr. **179**
(1996), 57-88

6. *Classification of restricted Lie
algebras with tame principal block.*

J. reine angew. Math. **546**
(2002), 1-45 (with Andrzej Skowronski)

* *7.* Block
representation type of Frobenius kernels of smooth groups. *

J. reine angew. Math. **586 **(2005), 45-69

8. *Auslander-Reiten components for G _{1}T-modules.
*

J. Algebra Appl.

**4**(2005), 739-759

9. *Tameness and complexity of finite group schemes*.

Bull. London Math. Soc. **39 (**2007),
63-70

10. *Galois actions and blocks of tame infinitesimal group
schemes*.

Trans. Amer. Math. Soc. 359 (2007),
5867-5898 (with Andrzej Skowronski)