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Let $(\mathfrak{g},[p])$ be a finite dimensional restricted Lie algebra over a perfect field $\mathbbm{k}$ of characteristic $p\!\ge \!3$. By combining methods from recent work of Benson-Carlson \cite{BC20} with those of \cite{CF21,Fa17} we obtain a description of the endotrivial $(\mathfrak{g},[p])$-modules in case $\mathfrak{g}$ is supersolvable., Comment: final version, to appear in J. Algebra

Let be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

Given a finite group scheme $\cG$ over an algebraically closed field $k$ of characteristic $\Char(k)=p>0$, we introduce new invariants for a $\cG$-module $M$ by associating certain morphisms $\deg^j_M : U_M \lra \Gr_d(M) \ \ (1\!\le\!j\!\le\! p\!-\!1)$ to $M$ that take values in Grassmannians of $M$. These maps are studied for two classes of finite algebraic groups, infinitesimal group schemes and elementary abelian group schemes. The maps associated to the so-called modules of constant $j$-rank have a well-defined degree ranging between $0$ and $j\rk^j(M)$, where $\rk^j(M)$ is the generic $j$-rank of $M$. The extreme values are attained when the module $M$ has the equal images property or the equal kernels property. We establish a formula linking the $j$-degrees of $M$ and its dual $M^\ast$. For a self-dual module $M$ of constant Jordan type this provides information concerning the indecomposable constituents of the pull-back $\alpha^\ast(M)$ of $M$ along a $p$-point $\alpha : k[X]/(X^p) \lra k\cG$.

Let U be the unipotent radical of a Borel subgroup of a connected reductive algebraic group G, which is defined over an algebraically closed field k. In this paper, we extend work by Goodwin-R\"ohrle concerning the commuting variety of Lie(U) for char(k)=0 to fields, whose characteristic is good for G, Comment: Final version

Let $(\mathfrak{g},[p])$ be a restricted Lie algebra over an algebraically closed field $k$ of characteristic $p\!\ge \!3$. Motivated by the behavior of geometric invariants of the so-called $(\mathfrak{g},[p])$-modules of constant $j$-rank ($j \in \{1,\ldots,p\!-\!1\}$), we study the projective variety $\mathbb{E}(2,\mathfrak{g})$ of two-dimensional elementary abelian subalgebras. If $p\!\ge\!5$, then the topological space $\mathbb{E}(2,\mathfrak{g}/C(\mathfrak{g}))$, associated to the factor algebra of $\mathfrak{g}$ by its center $C(\mathfrak{g})$, is shown to be connected. We give applications concerning categories of $(\mathfrak{g},[p])$-modules of constant $j$-rank and certain invariants, called $j$-degrees., Comment: Fixed a few typos in the last version and added a missing argument in the proof of the published version

In this article we study the interplay between algebro-geometric notions related to $\pi$-points and structural features of the stable Auslander-Reiten quiver of a finite group scheme. We show that $\pi$-points give rise to a number of new invariants of the AR-quiver on one hand, and exploit combinatorial properties of AR-components to obtain information on $\pi$-points on the other. Special attention is given to components containing Carlson modules, constantly supported modules, and endo-trivial modules.

Using the variety of infinitesimal one-parameter subgroups introduced in [55, 56] by Suslin-Friedlander-Bendel, we define a numerical invariant for representations of an infinitesimal group scheme G. For an indecomposable G-module M of complexity 1, this number, which may also interpreted as the height of a “vertex” UM ⊆ G, is related to the period of M . In the context of the Frobenius category of GrT -modules associated to a smooth reductive group G and a maximal torus T ⊆ G, our methods give control over the behavior of the Heller operator of such modules, as well as precise values for the periodicity of their restrictions to Gr. Applications include the structure of stable Auslander-Reiten components of GrT -modules as well as the distribution of baby Verma modules.

In their recent article [5], the authors endow every finite-dimensional simple complex Lie algebra g with a coalgebra structure such that the composition μ◦δ of the two structure maps δ : g −→ g⊗Cg and μ : g⊗C g −→ g coincides with the identity. Moreover, the dual algebra (g∗, δ∗) associated to the Lie coalgebra is isomorphic to (g, μ). The coalgebra map δ is given explicitly for sl(n), those for the other types are obtained via embeddings g ↪→ sl(n). The purpose of the present short note is to elicit the conceptual sources of [5], starting from the observation that the coalgebra maps defined in [5] are in fact homomorphisms of g-modules. For Lie algebras affording non-degenerate symmetric associative forms, such coalgebra maps naturally arise by dualizing the multiplication. This immediately implies the abovementioned duality, and the formulae displayed in [5, §4] can also be subsumed under our general approach. By demanding that δ be g-linear, we depart from the usual compatibility condition of a Lie bialgebra, which requires δ to be a derivation. In that case, μ ◦ δ is a derivation of g. For fields of characteristic zero, only nilpotent Lie algebras afford invertible derivations (cf. [7]), so that non-zero Lie bialgebras over such fields never satisfy μ ◦ δ = idg. Let k be a field. Given a finite-dimensional k-vector space V , we consider the k-linear maps

Let G be a reductive group, defined over the Galois field $${\mathbb{F}_p}$$ with p being good for G. Using support varieties and covering techniques based on G r T-modules, we determine the position of simple modules and baby Verma modules within the stable Auslander–Reiten quiver Γ s (G r ) of the rth 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 isomorphisms $$\mathcal{V}_{G_r}(Z_r(\lambda)) \cong \mathcal{V}_{G_{r-d}}(Z_{r-d}(\mu))$$ for baby Verma modules of certain highest weights $${\lambda, \mu \in X(T)}$$ , which are related by the notion of depth.

Using results on algebras that are graded by p-groups, we study representations of infinitesimal groups $ \mathcal{G} $ that possess a normal subgroup $ \mathcal{N}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \triangleleft } \mathcal{G} $ with a diagonalizable factor group $ \mathcal{G} \mathord{\left/ {\vphantom {\mathcal{G} \mathcal{N}}} \right. \kern-\nulldelimiterspace} \mathcal{N} $ . When combined with rank varieties, Auslander-Reiten theory and Premet's work on SL(2)1-modules, these techniques lead to the determination of the indecomposable modules of the infinitesimal groups of domestic representation type.

Given an infinitesimal group G \mathcal {G} , that is defined over an algebra- ically closed field of characteristic p ≥ 3 p \ge 3 , we determine the block structure of the algebra of measures H ( G ) H(\mathcal {G}) in case its principal block B 0 ( G ) \mathcal {B}_0(\mathcal {G}) is tame and the height of the factor group G / M ( G ) \mathcal {G}/\mathcal {M}(\mathcal {G}) of G \mathcal {G} by its multiplicative center M ( G ) \mathcal {M}(\mathcal {G}) is at least two. Our results yield a complete description of the stable Auslander-Reiten quiver of H ( G ) H(\mathcal {G}) along with a criterion for the domesticity of H ( G ) H(\mathcal {G}) .

Using a representation-theoretic interpretation of support varieties due to Friedlander- Pevtsova (18), we show that the complexity of tame blocks of finite group schemes is bounded by 2. In this context, our result salvages a theorem by Rickard (23), whose proof is flawed.

Given an algebraically closed field k of characteristic p≥3, we classify the finite algebraic k-groups whose algebras of measures afford a principal block of tame representation type. The structure of such a group $\mathcal{G}$ is largely determined by a linearly reductive subgroup scheme $\hat{\mathcal{G}}$ of SL(2), with the McKay quiver of $\hat{\mathcal{G}}$ relative to its standard module being the Gabriel quiver of the principal block $\mathcal{B}_0(\mathcal{G})$ . The graphs underlying these quivers are extended Dynkin diagrams of type $\tilde{A}, \tilde{D}$ or $\tilde{E}$ , and the tame blocks are Morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type.

In this paper we study the Auslander–Reiten quiver of the highest weight category of finite-dimensional G1T-modules, associated to a smooth reductive algebraic group G. By relating properties of stable Auslander–Reiten components to those of their rank varieties we show that there are at most three isomorphism types of these components. For the Frobenius kernels G1Tr the maximal ranks of tubes are determined.

This paper investigates varieties of tori and Cartan subalgebras of a finite-dimensional restricted Lie algebra ( g , [ p ] ) (\mathfrak {g},[p]) , defined over an algebraically closed field k k of positive characteristic p p . We begin by showing that schemes of tori may be used as a tool to retrieve results by A. Premet on regular Cartan subalgebras. Moreover, they give rise to principal fibre bundles, whose structure groups coincide with the Weyl groups in case g = Lie ⁡ ( G ) \mathfrak {g}= \operatorname {Lie}(\mathcal {G}) is the Lie algebra of a smooth group G \mathcal {G} . For solvable Lie algebras, varieties of tori are full affine spaces, while simple Lie algebras of classical or Cartan type cannot have varieties of this type. In the final sections the quasi-projective variety of Cartan subalgebras of minimal dimension rk ( g ) \textrm {rk}(\mathfrak {g}) is shown to be irreducible of dimension dim k ⁡ g − rk ( g ) \dim _k\mathfrak {g}-\textrm {rk}(\mathfrak {g}) , with Premet’s regular Cartan subalgebras belonging to the regular locus.

In this note we give a description of those infinitesimal group schemes of characteristic 2 whose Hopf algebras have finite representation type.

Much of the recent progress in the representation theory of infinitesimal group schemes rests on the application of algebro-geometric techniques related to the notion of cohomological support varieties (cf. [6, 8–10]). The noncohomological characterization of these varieties via the so-called rank varieties (see [21, 22]) involves schemes of additive subgroups that are the infinitesimal counterparts of the elementary abelian groups. In this note we introduce another geometric tool by considering schemes of tori of restricted Lie algebras. Our interest in these derives from the study of infinitesimal groups of tame representation type, whose determination [12] necessitates the results to be presented in §4 and §5 as well as techniques from abstract representation theory.In contrast to the classical case of complex Lie algebras, the information on the structure of a restricted Lie algebra that can be extracted from its root systems is highly sensitive to the choice of the underlying maximal torus. Schemes of tori obviate this defect by allowing us to study algebraic families of root spaces. Accordingly, these schemes may also shed new light on various aspects of the structure theory of restricted Lie algebras. We intend to pursue these questions in a forthcoming paper [13], and focus here on first applications within representation theory.

We classify the uniserial infinitesimal unipotent commutative groups of finite representation type over algebraically closed fields. As an application we provide detailed information on the structure of those infinitesimal groups whose distribution algebras have a representation-finite principal block.

Let (L,[p]) a finite dimensional nilpotent restricted Lie algebra of characteristic \(p \geq 3, \chi \in L^*\) a linear form. In this paper we study the representation theory of the reduced universal enveloping algebra \(u(L,\chi )\). It is shown that \(u(L,\chi )\) does not admit blocks of tame representation type. As an application, we prove that the nonregular AR-components of \(u(L,\chi )\) are of types \(\Bbb Z [A_\infty ]\) or \(\Bbb Z [A_n]/(\tau )\).

Let (L, [p]) be a finite dimensional restricted Lie algebra over an algebraically closed field F of characteristic p ≥ 3, X ∈ L* a linear form. In this article we study the Auslander-Reiten quivers of certain blocks of the reduced enveloping algebra u(L,x). In particular, it is shown that the enveloping algebras of supersolvable Lie algebras do not possess AR-components of Euclidean type.

This article extends techniques concerning Frobenius extensions in order to study enveloping algebras of restricted Lie superalgebras. In particular, we determine those restricted Lie superalgebras whose enveloping algebras are of finite representation type. In contrast to the non-graded case these algebras are not necessarily serial.

Let (L, [p]) be a finite dimensional restricted Lie algebra over an algebraically closed field F of characteristic p ≧ 3, and x e L* be a linear form. In this article we investigate the structure and representation theory of those blocks of the reduced enveloping algebra u(L,x) that are associated to irreducible modules of the form u(L, x) ⊗ u(K,x|k)Fλ.

Let k be an algebraically closed field. Given an extension A : B of finite-dimensional k-algebras, we establish criteria ensuring that the representation-theoretic notion of polynomial growth is preserved under ascent and descent. These results are then used to show that principal blocks of finite group schemes of odd characteristic are of polynomial growth if and only if they are Morita equivalent to trivial extensions of radical square zero tame hereditary algebras. In that case, the blocks are of domestic representation type and the underlying group schemes are closely related to binary polyhedral group schemes.

By using an approach to the theory of Frobenius extensions that emphasizes notions related to associative forms, we obtain results concerning the trace and corestriction mappings and transitivity. These are employed to show that the extension of enveloping algebras determined by a subalgebra of a Lie superalgebra is a Frobenius extension, and to study certain questions in representation theory.

Let (g,[p]) be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p>0. The scheme of tori of maximal dimension of g gives rise to a finite group S(g) that coincides with the Weyl group of g in case g is a Lie algebra of classical type. In this paper, we compute the group S(g) for Lie algebras of Cartan type and provide applications concerning weight space decompositions, the existence of generic tori and polynomial invariants., 32 pages. Minor updates according to referee comments. Accepted for publication in Forum Math

In this article we are primarily concerned with the relativized cohomology theory of ring extensions, as initiated by G. Hochschild [ 141. Our main objective is to lay the foundation for a general vanishing theory of relativized extension functors. The significance of such a theory rests on the unification it provides for the various classical results from the cohomology theories of groups, associative algebras, and Lie algebras. The approach chosen here heavily emphasizes locally nilpotent operators as an underlying principle for vanishing and isomorphism theorems. Although in algebraic homology theory these concepts have appeared only recently (cf. [5]), they occur implicitly in singular homology theory of topological spaces. The conventional proof of the excision theorem, for instance, resorts, by means of barycentric subdivision, to the construction of a locally nilpotent operator that is chain homotopic to the identity. In Section 1 we study the question to what an extent properties of operators are inherited by resolutions. It is shown that resolutions defined by injective envelopes and projective covers are suitable tools for ordinary homology. In the framework of relative homological algebra of a given ring extension R:S the availability of standard resolutions allows the application of more computational techniques. We show that the commutator algebra of the centralizer C,(S) of S in R operates on these resolutions in a natural fashion. This result in conjunction with a certain homotopy is subsequently utilized to show that the associated action on the spaces Ext;,, S) (M, N) is trivial. The succeeding section combines these techniques with basic facts concerning locally finite operators to establish various vanishing theorems for extension functors. The third section deals with the study of the cohomology theory of Frobenius extensions. We address questions emerging from recent developments in the cohomology theories of finite dimensional modular [9] and finite dimensional restricted Lie algebras [7-93. Our results are closely

Letbe an infinitesimal group scheme, defined over an algebraically closed field of characteristic p. We employ rank varieties of-modules to study the stable Auslander-Reiten quiver of the distribution algebra of. As in case of finite groups, the tree classes of the AR-components are finite or infinite Dynkin diagrams, or Euclidean diagrams. We classify the components of finite and Euclidean type in caseis supersolvable or a Frobenius kernel of a smooth, reductive group.

In this paper we indicate how algebro-geometric techniques can be employed in order to obtain information concerning the representation finite reduced enveloping algebras of restricted Lie algebras.

It is well-known that the classical vanishing results of the cohomology theory of Lie algebras depend on the characteristic of the underlying base field. The theorems of Cartan and Zassenhaus, for instance, entail that non-modular simple Lie algebras do not admit non-trivial central extensions. In contrast, early results by Block [3] prove that this conclusion loses its validity if the underlying base field has positive characteristic.Central extensions of a given Lie algebra L, or equivalently its second cohomology group H(L, F), can be conveniently described by means of derivations φ:L → L*.

This paper establishes sufficient conditions for the vanishing of the homology and cohomology groups of an associative algebra with coefficients in a two-sided module. 0. Introduction. Algebraic homology and cohomology theory may be considered an extension of ordinary representation theory. Certain problems of the latter discipline initially motivated the definition of low-dimensional cohomology groups which subsequently was generalized to arbitrary dimension (cf. [2, 8]). The theory of derived functors set forth by Cartan and Eilenberg in [2] provided the appropriate unifying concept for the various ramifications of cohomology theory and also layed the foundation for the study of homology groups. Among the most notable applications of cohomology theory are Weyl's Theorem for finite-dimensional semisimple nonmodular Lie algebras, the theorem concerning the complete reducibility of finite-dimensional modules of compact topological groups as well as Maschke's theorem. The proofs of these classical results are primarily based on certain averaging techniques which produce central elements whose action forces the vanishing of the pertinent homology and cohomology groups. This paper is concerned with various generalizations of the above-mentioned classical vanishing theorems. It is conceivable that our new results may be applicable in some nonclassical contexts, notably in the cohomology theory of modular and Zgraded Lie algebras of arbitrary dimension. In §§3 and 4 we present, in accordance with the philosophy of the theory of algebraic complexes, a treatment of the most general case, namely the homology and cohomology of associative algebras. The third section establishes conditions for the vanishing of cohomology groups of an associative algebra A in terms of the action of the enveloping algebra Ae on the A-bimodule M. By emphasizing the role of central elements of the augmentation ideal, this approach naturally extends previous results (cf. [6, Theorem 1.2; 7, Proposition C 6]). The methods of §4 sharply contrast those employed in §3. The algebra A is considered a Lie algebra which operates on Hn(A, M) in a natural fashion. In particular, we shall demonstrate that the results of [6] retain their validity if one utilizes elements that act locally nilpotently on A via their adjoint representation. Incidentally, the above mentioned Lie action appears to be useful in the study of restricted cohomology groups which hitherto were not amenable to methods from ordinary Lie cohomology theory. Received by the editors February 2, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 18H15. (r)1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

(1985). Conditions for the commutativity of restricted lie algebras. Communications in Algebra: Vol. 13, No. 7, pp. 1475-1489.

Some general results concerning derivations of finitely generated Lie algebras are established. These are employed in order to determine the derivations and central extensions of Kac-Moody Lie algebras.

Lie algebras which are isomorphic to central quotients of quaternion division algebras are investigated.

Restricted Lie algebras ( L , [ p ] ) (L,[p]) having the property that some power of the p p -mapping [ p ] [p] is semilinear are investigated.

This paper presents a sufficient condition for the vanishing of the cohomology groups of an associative algebra. 0. Introduction. In this paper, we shall present a vanishing theorem for the cohomology groups (cf. [6]) of an associative F-algebra. The theory of derived functors, set forth by Cartan and Eilenberg in [2], links the cohomology groups of a Lie algebra L to the cohomology groups of its universal enveloping algebra. We shall use this fact in order to interpret our general result within the framework of Lie algebra cohomology. Since its initiation by the Whitehead Lemmas [8, 9], this branch of cohomology theory has enjoyed considerable attention. However, the customary methods exclusively employed the description of Lie algebra cohomology as given in [3]. This had the effect that the proofs of vanishing theorems necessitated additional assumptions concerning the behavior of suitably chosen elements [3, 5]. Our present result does not require such considerations and generalizes the results alluded to above. The author wishes to acknowledge his indebtedness to G. P. Hochschild. Aside from providing kind advice concerning the exposition of this paper, he has also contributed to the final formulation of Proposition 1.1, which amounts to a clarification and simplification of the author's original proof of the vanishing theorem. 1. The main result. Throughout this section, A is assumed to be an associative algebra with unity over a field F. Let M be a two sided A-module, i.e. an F-vector space together with two module operations (left and right) such that a.(m.b)=(a.m).b Va,bEAVmcM. We denote by Cn (A, M) the F-space of (n F)-linear mappings from the n-fold Cartesian product of A into M (n > 1). We put C0(A, M) = M and Cn (A, M) = 0 for n 0. Then P2 = 0 and Hn(A, M) :=ker6 0 Cn(A,M)/im6nCn-l (A, M) for n > 0 is called the nth cohomology group of A with coefficients in M (cf. [6]). Received by the editors January 22, 1986. 1980 Mathematics Subject Clasification (1985 Reision). Primary 16A61, 17B56. (?1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page