Semidirect Products of Finite Group Schemes : Gabriel Quivers and Auslander-Reiten Components

Let G be a connected algebraic group over an algebraically closed field k. If char(k)=0, then there is a strong correspondence between representations of G and those of its Lie algebra \g. This changes dramatically in the situation char(k)=p>0 of positive characteristic. In this case, it turned out to be useful to approximate G by the ascending sequence (G_r)_{r\geq 1} consisting of its so-called Frobenius kernels. Each G_r is an infinitesimal group scheme, it is not uniquely determined by its group of k-rational points anymore; in fact, the latter are trivial. Its representation theory is equivalent to that of the dual Hopf algebra kG_r:=k[G_r]^* of its finite-dimensional coordinate ring k[G_r]. The Hopf algebra kG_1 is isomorphic to the restricted universal enveloping algebra \U_0(\g) of \g which shows that the representation theory of G_1 is equivalent to that of \g as a restricted Lie algebra. The representation theory of \g itself can be approximated by studying the family \{\Uchi(\g):\chi\in\g^*\} of its reduced enveloping algebras. Many results are known in case of reductive groups. In general, every group G is an extension of a reductive group H by a unipotent group U. If U is non-trivial, then the 'next best' case is that this extension splits, which in turn leads to a semidirect product G=U\rtimes H. Since the functor G\mapsto G_r is left exact, the rth Frobenius kernel of G is then the semidirect product U_r\rtimes H_r of the Frobenius kernels of U and H. It is exactly this point of view on which this thesis is build upon. Simple G_r-modules correspond to simple H_r-modules via the inflation functor mod(H_r)\to mod(G_r) defined by pullback along the projection G_r\to H_r and the principal indecomposable G_r-modules are induced by principal indecomposable H_r-modules. We will also establish a formula for the Gabriel quiver of the Hopf algebra kG_r and analyze the behaviour of the inflation functor in terms of Auslander-Reiten sequences. Furthermore, we will show that the stable Auslander-Reiten quiver of kG_r does not admit components of Euclidean type. We will formulate these results more general for finite group schemes and certain reduced enveloping algebras of restricted Lie algebras. As a major example, we will consider the Schrödinger group S:=H\rtimes SL(2), the semidirect product of the reductive group SL(2) with the Heisenberg group H\subseteq \SL(3), along with its quotient \Sbar\cong \G_a^2\rtimes SL(2) by the center. The group S has already been considered over the field of complex numbers and is related to physics. We will show that the Gabriel quivers of the Hopf algebras of the Frobenius kernels of S and \Sbar are all connected and take a closer look at reduced enveloping algebras U_\chi(\sbar) of the restricted Lie algebra \sbar=\Lie(\Sbar).



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