Rolf Farnsteiner

Algebraic Groups

Algebraic groups are analogues of Lie groups, where the role of differential manifolds and differentiable maps is taken by algebraic varieties and their morphisms. Accordingly, many notions and techniques of this theory have their origin in its classical precursor, and in early articles algebraic groups were referred to as "algebraic Lie groups" (J. Dieudonné: Sur les groupes de Lie algébriques sur un corps de charactéristique p>0, Rend. Circ. Mat. Palermo 2 (1952), 380-402).

The theory of affine algebraic groups (linear algebraic groups) is mainly based on the work by Borel and Chevalley in the 50's and 60's of the last century. Tits, Steinberg and others contributed further results, and the theory became a field in its own right. A new powerful point of view was introduced by Grothendieck's functorial approach, in which the underlying varieties were replaced by schemes.

In the classical theory the local structure of Lie groups is investigated by means of their Lie algebras. For fields of characteristic zero, the correspondence between algebraic groups and Lie algebras is still an effective tool. In the modular case, however, the Lie algebra Lie(G) of an algebraic group G no longer encapsulates the structural features of G. One is thus led to "approximate" G by an ascending sequence (Gr)r>0 of normal subgroups, the so-called Frobenius kernels of G, with the first Frobenius kernel G1 corresponding to Lie(G). The structure and representation theory of the infinitesimal subgroups  (Gr)r>0 has been an active field of research, particularly in the case where G is a smooth reductive group.


   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