Group-Graded Algebras, Extensions of Infinitesimal Groups, and Applications

Farnsteiner, Rolf

doi:10.1007/s00031-009-9049-x

Cited by 8 publications

(6 citation statements)

References 35 publications

(18 reference statements)

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“…Hence

\overset{W}{true}

is indecomposable and reducible. The structure of such modules was determined in (see [, § 4.1] for a more recent account): they have Loewy length 2 with isotypic socle and head. Thus the head of

\overset{W}{true}

consists of trivial modules.…”

Section: Unique Embeddings Of Nilpotent Elements Into Sl2‐subalgebrasmentioning

confidence: 99%

The Jacobson-Morozov theorem and complete reducibility of Lie subalgebras

Stewart

Thomas

2017

Proc. London Math. Soc.

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In this paper we determine the precise extent to which the classical sl2-theory of complex semisimple finite-dimensional Lie algebras due to Jacobson-Morozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to Springer-Steinberg and Carter/Spaltenstein. Our main advance arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple algebraic groups over algebraically closed fields k are G-completely reducible, a notion essentially due to Serre. For example, if G is exceptional and char k = p 5, we classify the triples (h, g, p) such that there exists a non-G-completely reducible subalgebra of g = Lie(G) isomorphic to h. We do this also under the restriction that h be a p-subalgebra of g. We find that the notion of subalgebras being G-completely reducible effectively characterises when it is possible to find bijections between the conjugacy classes of sl2-subalgebras and nilpotent orbits and it is this which allows us to prove our main theorems.For absolute completeness, we also show that there is essentially only one occasion in which a nilpotent element cannot be extended to an sl2-triple when p 3: this happens for the exceptional orbit in G2 when p = 3.

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“…Hence

\overset{W}{true}

is indecomposable and reducible. The structure of such modules was determined in (see [, § 4.1] for a more recent account): they have Loewy length 2 with isotypic socle and head. Thus the head of

\overset{W}{true}

consists of trivial modules.…”

Section: Unique Embeddings Of Nilpotent Elements Into Sl2‐subalgebrasmentioning

confidence: 99%

The Jacobson-Morozov theorem and complete reducibility of Lie subalgebras

Stewart

Thomas

2017

Proc. London Math. Soc.

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“…Therefore we obtain new realizations of those modules and are able to show when they have an SL(2) 1 T r -module structure. These are exactly those modules which were missing in [11]. We then turn to a different topic in section 4.…”

Section: Introductionmentioning

confidence: 98%

“…A foundation for this is Premets work ( [22]) on the representation theory of the restricted Lie algebra sl (2). Farnsteiner started in [11] to extend these results to the infinitesimal case, the group schemes SL(2) 1 T r for r ≥ 1. These results will be summarized in the first section.…”

Section: Introductionmentioning

confidence: 99%

“…The indecomposable representations of the restricted Lie algebra sl(2) were classified by Premet in [22]. In [11,4.1] Farnsteiner incorporated these results into the Auslander-Reiten theory of this algebra. Let T ⊆ SL(2) be the standard torus of diagonal matrices.…”

Section: Introductionmentioning

confidence: 99%

“…These modules (and certain twists of them) belong to the exceptional tubes. There was no realization given in [11] of the modules belonging to homogeneous tubes. But the following was shown: Lemma 1.2.…”

Section: Introductionmentioning

confidence: 99%

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Classification of Indecomposable Modules for Finite Group Schemes of Domestic Representation Type

Kirchhoff

2017

Transformation Groups

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We investigate the representation theory of domestic group schemes G over an algebraically closed field of characteristic p > 2. We present results about filtrations of induced modules, actions on support varieties, Clifford theory for certain group schemes and applications of Clifford theory for strongly group graded algebras to the structure of Auslander-Reiten quivers. The combination of these results leads to the classification of modules belonging to the principal blocks of the group algebra kG.

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Support varieties, AR-components, and good filtrations

Farnsteiner

Röhrle²

2009

Math. Z.

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Abstract. Let G be a reductive group, defined over the Galois field Fp with p being good for G. Using support varieties and covering techniques based on GrT -modules, we determine the position of simple modules and baby Verma modules within the stable Auslander-Reiten quiver Γs(Gr) of the r-th Frobenius kernel of G. In particular, we show that the almost split sequences terminating in these modules usually have an indecomposable middle term.Concerning support varieties, we introduce a reduction technique leading to isomorphismsfor baby Verma modules of certain highest weights λ, µ ∈ X(T ), which are related by the notion of depth. IntroductionIn the representation theory of finite-dimensional self-injective algebras, the stable AuslanderReiten quiver has proven to be an important homological invariant, which has been studied for group algebras of finite groups, reduced enveloping algebras of restricted Lie algebras and distribution algebras of infinitesimal group schemes. In the classical context of Frobenius kernels of reductive groups, the relevant algebras are usually of wild representation type, rendering a classification of their indecomposable modules a hopeless task. This fact notwithstanding, one does have a fairly good understanding of the connected components of the corresponding stable Auslander-Reiten quiver. It is therefore of interest to relate this information to certain classes of indecomposable modules, such as simple modules, Weyl modules or Verma modules, and to identify the position of the latter within the AR-quiver. The main problem in this undertaking is the lack of a suitable presentation of the underlying algebras, as required by the techniques of abstract representation theory. On the other hand, the theory of rank varieties and support varieties has seen considerable progress over the last years, so that one can hope to exploit these tools in the aforementioned context.In continuation of work begun in [22], we study in this article those Auslander-Reiten components of the algebras Dist(G r ) that contain simple modules or baby Verma modules. The underlying algebra Dist(G r ) consists of the distributions of the r-th Frobenius kernel of the smooth reductive group scheme G, defined over an algebraically closed field of positive characteristic p. The case r = 1, which pertains to restricted enveloping algebras of reductive Lie algebras, was settled in [22] by means of a detailed analysis of nilpotent orbits in rank varieties, leading to the consideration of groups of types SL(2) 1 ×SL(2) 1 , SL(3) 1 and SO(5) 1 . For r > 1, rank varieties are less tractable and the structure of the cohomology rings defining support varieties is more complicated. We address these problems by passage to G r T -modules, as described below. With regard to Auslander-Reiten theory, our main results can roughly be summarized as follows:Theorem. Let G be a smooth reductive group scheme, defined over F p . Given a character λ ∈ X(T ) and r ≥ 1, the following statements hold:

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Group-Graded Algebras, Extensions of Infinitesimal Groups, and Applications

Cited by 8 publications

References 35 publications

The Jacobson-Morozov theorem and complete reducibility of Lie subalgebras

The Jacobson-Morozov theorem and complete reducibility of Lie subalgebras

Classification of Indecomposable Modules for Finite Group Schemes of Domestic Representation Type

Support varieties, AR-components, and good filtrations

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