Rolf Farnsteiner

Modular Lie Algebras

The theory of Lie groups and Lie algebras was initiated by the Norwegian mathematician Marius Sophus Lie (1842-1899), who studied systems of differential equations by means of their transformation groups. In this context Lie algebras occurred as "infinitesimal transformations" and were called "infinitesimal groups" until H. Weyl introduced the current terminology in 1934. The general structure theory complex Lie algebras was developed by E. Cartan, F. Engel, and W. Killing between 1888 and 1894. Many of the classical results, such as H. Weyl's theorem on the complete reducibility of representations of complex semi-simple Lie algebras, were based on analytic techniques. Later, purely algebraic proofs were found. In his paper "Rational methods in the theory of Lie algebras", Ann. of Math. 36 (1936), 875-881, N. Jacobson showed that most of the results for real and complex Lie algebras retain their validity for arbitrary fields of characteristic 0. For fields of positive characteristic, however, new phenomena arise. Not only do the classical methods no longer work, also most of the aforementioned fundamental results cannot be transferred to this "modular" setting.

The above facts notwithstanding, there are interesting classes of modular Lie algebras that behave better than others. Roughly speaking, many features from the classical structure and representation theory can be salvaged as long as the Lie algebra to be studied comes from a smooth algebraic group scheme. Such Lie algebras have an additional structure, a so-called p-mapping, which, along with the adjoint representation of the underlying algebraic group, affords a good structure theory. In fact, the representation theory of reductive Lie algebras (i.e., those associated to reductive algebraic groups) is currently a rather active field of research.

E. Witt's 1937 discovery of a non-classical simple Lie algebra, that is, a Lie algebra not associated to a smooth algebraic group scheme, is generally considered to be the starting point of the theory of modular Lie algebras. Subsequently, many more simple Lie algebras not arising as analogues of complex simple Lie algebras were found, and in 1966 A. Kostrikin and I. Shafarevic enunciated their famous conjecture asserting that any restricted simple Lie algebra is either classical or of Cartan type. In the following two decades the structure theory of abstract restricted Lie algebras enjoyed considerable attention and the work of many mathematicians culminated in the affirmative solution of the Kostrikin-Shafarevic conjecture by R. Block and R. Wilson in 1988.

Much of the current work on modular Lie algebras focuses on support varieties and the investigation of irreducible representations of reductive Lie algebras.

Publications

1.  Modules of solvable infinitesimal groups and the structure of representation-finite cocommutative Hopf algebras.
        Math. Proc. Cambridge Philos. Soc. 127 (1999), 441-459   (with Detlef Voigt)

   2.  On cocommutative Hopf algebras of finite representation type.
        Adv. in Math. 155 (2000), 1-22   (with Detlef Voigt)

   3.  Auslander-Reiten components for Lie algebras of reductive groups.
        Adv. in Math. 155 (2000), 49-83  

   4.  On the Auslander-Reiten quiver of an infinitesimal group.
        Nagoya Math. J. 160 (2000), 102-121  

   5.  Infinitesimal unipotent group schemes of complexity 1.
        Colloq. Math. 89  (2001),  179-192   (with Gerhard Röhrle and Detlef Voigt)

   6.  Representations of infinitesimal groups of characteristic 2.
        Arch. Math.78 (2002),  184-188   (with Detlef Voigt)

   7.  On infinitesimal groups of tame representation type.
        Math. Z. 244 (2003), 479-513   (with Detlef Voigt)

   8.  Block representation type of Frobenius kernels of smooth groups.
        J. reine angew. Math. 586 (2005), 45-69

   9.  Auslander-Reiten components for G1T-modules.
        J. Algebra Appl. 4 (2005), 739-759

  10.  The tame infinitesimal groups of odd characteristic. (with Andrzej Skowronski)
         Adv. in Math. 205 (2006), 229-274  

 11.  Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks.
        Invent. math. 166 (2006), 27-94

 12.  Galois actions and blocks of tame infinitesimal group schemes.
        Trans. Amer. Math. Soc. 359 (2007), 5867-5898