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• W788045324 abstract "In 1990 George Lusztig introduced a quantum analogue of restricted enveloping algebras of Lie algebras, the small quantum groups. In his PhD thesis Christopher M. Drupieski generalized these to analogues of Frobenius kernels, the Frobenius-Lusztig kernels. Theydepend on three parameters, the characteristic p of the field, the order ell of a root of unity and the height r. The case r = 0 gives the small quantum groups defined by George Lusztig. Our thesis concerns basic classification questions from classical representation theory of these algebras. The determination of the representation type of an algebra answers the question if it is (at least theoretically) possible to classify the indecomposable modules of a given algebra up to isomorphism. If the small quantum group is not attached to an sl2, we show that all blocks different from the simple Steinberg block are of wild representation type. Thesl2-case was known before: the small half-quantum groups are tame, their half-quantum groups are representation-finite. The higher Frobenius-Lusztig kernels are all wild with the exception of explicitly known SL(2)-blocks in height one and the simple Steinberg blocks. For this result we have to assume certain niteness assumptions on cohomology which are known in the height zero case, but not proven in general. The Borel and nilpotent partsof the higher Frobenius-Lusztig kernels are wild.Auslander-Reiten theory is a tool to understand the module category of a finite dimensional algebra by describing the irreducible homomorphisms. The isomorphism classes of indecomposable modules and the irreducible maps are arranged in an oriented graph, theAuslander-Reiten quiver. We are interested in the shape of the connected components of this quiver. In the tame and representation-finite cases, the Auslander-Reiten quiver of the Frobenius-Lusztig kernels is completely understood. For the wild cases we assumeG is not SL(2). It was known before that the connected components of the Auslander-Reiten quiver of a Frobenius-Lusztig kernel are attached to a (finite or innite) Dynkin diagramor a Euclidean diagram. For the Frobenius-Lusztig kernels we show that there are no components attached to finite Dynkin or Euclidean diagrams. In the height zero case we show that the components containing the restriction of a module for the infinite dimensionalquantum group U_zeta(g) are attached to A_infty. For the small half-quantum groups we show that the components containing compatibly graded modules, i.e. modules for u_zeta(b)U_zeta^0(g) are also attached to A_infty." @default.
• W788045324 created "2016-06-24" @default.
• W788045324 creator A5074552281 @default.
• W788045324 date "2012-05-21" @default.
• W788045324 modified "2023-09-26" @default.
• W788045324 title "Representation type and Auslander-Reiten theory of Frobenius-Lusztig kernels" @default.
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