INFINITESIMAL GROUPS OF TAME REPRESENTATION TYPE

ROLF FARNSTEINER

According to a fundamental theorem by Drozd the class of finite-dimensional algebras over an algebraically closed field k may be subdivided into the disjoint subclasses of representation-finite, tame, and wild algebras. A representation-infinite algebra is tame if in each dimension all but finitely many isoclasses of indecomposables occur in a finite number of one-parameter families. For certain types of algebras, such as hereditary algebras or group algebras of finite groups, these subclasses are well understood. For instance, a group algebra k[G] of a finite group G has finite representation type if and only if every Sylow subgroup for p := char(k) is cyclic, and tame if p = 2 and the Sylow-2-subgroups are dihedral, semidihedral, or generalized quaternion.

Infinitesimal groups give rise to cocommutative Hopf algebras that are complementary to modular group algebras. By general theory, a cocommutative Hopf algebra can be realized as the algebra H(G) of measures on a finite algebraic group G. Moreover, H(G) = H(G^{0})[G(k)] is a skew-group algebra over the Hopf algebra H(G^{0}) of the infinitesimal subgroup G^{0} ⊂ G. Infinitesimal groups of finite representation type are well understood [1, 4, 5], and the talk will be concerned with the following

Problem. Let G be an infinitesimal group. When does the principal block B_{0}(G) ⊂ H(G) have tame representation type?

This question has two interrelated aspects concerning the structure of G and the Morita type of B_{0}(G). The failure of the standard techniques from the modular representation theory of finite groups necessitates a different approach, which combines nullcones, schemes of tori, and methods from abstract representation theory (cf. [6, 7, 2, 3]). The objective of the talk is to explain how the interaction of these tools leads to a solution for infinitesimal groups of characteristic p ≥ 3.

References

[1] R. Farnsteiner, G. Röhrle and D. Voigt. Infinitesimal unipotent group schemes of complexity 1. Colloq. Math. 89 (2001), 179-192

[2] R. Farnsteiner and A. Skowroński. Classification of restricted Lie algebras with tame principal block. SFB Preprint No. 00-057, Universität Bielefeld (J. reine angew. Math. (to appear))

[3] . Infinitesimal groups with tame principal block. Manuscript in preparation.

[4] R. Farnsteiner and D. Voigt. Modules of solvable infinitesimal groups and the structure of representation-finite cocommutative Hopf algebras. Math. Proc. Camb. Phil. Soc. 127 (1999), 441-459

[5] . On cocommutative Hopf algebras of finite representation type. Adv. in Math. 155 (2000), 1-22

[6] . Schemes of tori and the structure of tame resstricted Lie algebras. J. London Math. Soc. 63 (2001), 553-570

[7] . On infinitesimal groups of tame representation type. SFB Preprint No. 99-106, Universität Bielefeld

Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany.

   E-mail address: rolf@mathematik.uni-bielefeld.de