Search

In this paper the authors introduce an analogue of the nilpotent cone, N {\mathcal N} , for a classical Lie superalgebra, g {\mathfrak g} , that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, g = g 0 ¯ ⊕ g 1 ¯ {\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1} with Lie G 0 ¯ = g 0 ¯ \text {Lie }G_{\bar 0}={\mathfrak g}_{\bar 0} , it is shown that there are finitely many G 0 ¯ G_{\bar 0} -orbits on N {\mathcal N} . Later the authors prove that the Duflo-Serganova commuting variety, X {\mathcal X} , is contained in N {\mathcal N} for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper.

The problem of whether the cohomological support map of a finite dimensional Hopf algebra has the tensor product property has attracted a lot of attention following the earlier developments on representations of finite group schemes. Many authors have focussed on concrete situations where positive and negative results have been obtained by direct arguments. In this paper we demonstrate that it is natural to study questions involving the tensor product property in the broader setting of a monoidal triangulated category. We give an intrinsic characterization by proving that the tensor product property for the universal support datum is equivalent to complete primeness of the categorical spectrum. From these results one obtains information for other support data, including the cohomological one. Two theorems are proved giving compete primeness and non-complete primeness in certain general settings. As an illustration of the methods, we give a proof of a recent conjecture of Negron and Pevtsova on the tensor product property for the cohomological support maps for the small quantum Borel algebras for all complex simple Lie algebras., 20 pages

In this paper the authors investigate the q-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type Α. The authors present a coordinate algebra type construction that allows us to realize these q-Schur algebras as the duals of the dth graded components of certain graded coalgebras. Under suitable conditions an isomorphism theorem is proved that demonstrates that the representation theory reduces to the q-Schur algebra of type Α. This enables the authors to address the questions of cellularity, quasi-hereditariness and representation type of these algebras. Later it is shown that these algebras realize the 1-faithful quasi hereditary covers of the Hecke algebras of type Β. As a further consequence, the authors demonstrate that these algebras are Morita equivalent to the category 𝒪 for rational Cherednik algebras for the Weyl group of type Β. In particular, we have introduced a Schur-type functor that identifies the type Β Knizhnik–Zamolodchikov functor.

In this paper the authors study the behavior of the sheaf cohomology functors $R^{\bullet}\text{ind}_{B}^{G}(-)$ where $G$ is an algebraic group scheme corresponding to a simple classical Lie superalgebra and $B$ is a BBW parabolic subgroup as defined by D. Grantcharov, N. Grantcharov, Nakano and Wu. We provide a systematic treatment that allows us to study the behavior of these cohomology groups $R^{\bullet}\text{ind}_{B}^{G}L_{\mathfrak f}(\lambda)$ where $L_{\mathfrak f}(\lambda)$ is an irreducible representation for the detecting subalgebra ${\mathfrak f}$. In particular, we prove an analog of Kempf's vanishing theorem and the Bott-Borel-Weil theorem for large weights.

In this paper the authors produce a projective indecomposable module for the Frobenius kernel of a simple algebraic group in characteristic p that is not the restriction of an indecomposable tilting module. This yields a counterexample to Donkin’s longstanding Tilting Module Conjecture. The authors also produce a Weyl module that does not admit a p-Weyl filtration. This answers an old question of Jantzen, and also provides a counterexample to the ( p , r ) {(p,r)} -Filtration Conjecture.

Let ${\mathcal H}_{q}(d)$ be the Iwahori-Hecke algebra for the symmetric group, where $q$ is a primitive $l$th root of unity. In this paper we develop a theory of support varieties which detects natural homological properties such as the complexity of modules. The theory the authors develop has a canonical description in an affine space where computations are tractable. The ideas involve the interplay with the computation of the cohomology ring due to Benson, Erdmann and Mikaelian, the theory of vertices due to Dipper and Du, and branching results for cohomology by Hemmer and Nakano. Calculations of support varieties and vertices are presented for permutation, Young and classes of Specht modules. Furthermore, a discussion of how the authors' results can be extended to other Hecke algebras for other classical groups is presented at the end of the paper.

In the representation theory of split reductive algebraic groups, it is well known that every Weyl module with minuscule highest weight is irreducible over every field. Also, the adjoint representation of E 8 is irreducible over every field. In this paper, we prove a converse to these statements, as conjectured by Gross: if a Weyl module is irreducible over every field, it must be either one of these, or trivially constructed from one of these. We also prove a related result on non-degeneracy of the reduced Killing form.

We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$\Delta$Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M$\Delta$C, ${\bf K}$, and then to associate to ${\bf K}$ a topological space--the Balmer spectrum $\operatorname{Spc} {\bf K}$. We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that $\operatorname{Spc} {\bf K}$ is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M$\Delta$C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of ${\bf K}$, which in turn can be applied to classify the thick two-sided ideals and $\operatorname{Spc} {\bf K}$. As a special case, our approach can be applied to the stable module categories of arbitrary finite dimensional Hopf algebras that are not necessarily cocommutative (or quasitriangular). We illustrate the general theorems with classifications of the Balmer spectra and thick two-sided/right ideals for the stable module categories of all small quantum groups for Borel subalgebras, and classifications of the Balmer spectra and thick two-sided ideals of Hopf algebras studied by Benson and Witherspoon., Comment: 34 pages

In this paper the authors introduce a class of parabolic subalgebras for classical simple Lie superalgebras associated to the detecting subalgebras introduced by Boe, Kujawa and Nakano. These parabolic subalgebras are shown to have good cohomological properties governed by the Bott-Borel-Weil theorem involving the zero component of the Lie superalgebra in conjunction with the odd roots. These results are later used to verify an open conjecture given by Boe, Kujawa and Nakano pertaining to the equality of various support varieties.

Let $G$ be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic $p>0$. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of Donkin: one on tilting modules and the lifting of projective modules for Frobenius kernels of $G$ and another on the existence of certain filtrations of $G$-modules. A key question related to these conjectures is whether the tensor product of the $r$th Steinberg module with a simple module with $p^{r}$th restricted highest weight admits a good filtration. In this paper we verify this statement when (i) $p\geq 2h-4$ ($h$ is the Coxeter number), (ii) for all rank two groups, (iii) for $p\geq 3$ when the simple module corresponds to a fundamental weight and (iv) for a number of cases when the rank is less than or equal to five.

The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $E_{8}$ or its highest weight is minuscule. In this paper, we prove an analogous criteria for irreducibility of Weyl modules over the quantum group $U_{\zeta}({\mathfrak g})$ where ${\mathfrak g}$ is a complex simple Lie algebra and $\zeta$ ranges over roots of unity.

In this paper we investigate Donkin's ( p , r ) -Filtration Conjecture, and present two proofs of the “if” direction of the statement when p ≥ 2 h − 2 . One proof involves the investigation of when the tensor product between the Steinberg module and a simple module has a good filtration. One of our main results shows that this holds under suitable requirements on the highest weight of the simple module. The second proof involves recasting Donkin's Conjecture in terms of the identifications of projective indecomposable G r -modules with certain tilting G -modules, and establishing necessary cohomological criteria for the ( p , r ) -filtration conjecture to hold.

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ :G → G be a strict endomorphism (i.e., the subgroup G(σ) of σ-fixed points is finite). Also, let Gσ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the dimension of the degree m cohomology group Hm(G(σ),L) for any irreducible kG(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. These bounds are actually established for the degree m extension groups \( Ext^{m}_{G(\sigma )}(L,L^{\prime })\) between irreducible kG(σ)-modules \(L,L^{\prime }\), with a similar result holding for Gσ. In these Extm results, the bounds also depend on the highest weight associated to L, but are, nevertheless, independent of the characteristic p.

Let $G$ be a connected reductive algebraic group and $B$ be a Borel subgroup defined over an algebraically closed field of characteristic $p>0$. In this paper, the authors study the existence of generic $G$-cohomology and its stability with rational $G$-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of $G$ and $B$-cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of $G({\mathbb F}_{q})$ are established which generalizes earlier work of Hiller, in addition to stability ranges for generic cohomology which improves on seminal work of Cline, Parshall, Scott and van der Kallen., Corrected proof of Theorem 5.1.1, added Remark 5.2.3

In this paper the authors investigate the structure of the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose E 2-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen–Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifying as a ring with the E 2-term. We present criteria for the collapsing of this spectral sequence and provide some examples where the ring isomorphism fails. Furthermore, we show that there are instances when the spectral sequence does not collapse and yields cohomology rings which are not Cohen-Macaulay.

In this paper we provide a complete classification of the representation type for the blocks for the Hecke algebra of type A A , stated in terms of combinatorical data. The computation of the complexity of Young modules is a key component in the proof of this classification result.

Let $G$ be an affine algebraic group scheme over an algebraically closed field $k$ of characteristic $p>0$, and let $G_r$ denote the $r$-th Frobenius kernel of $G$. Motivated by recent work of Friedlander, the authors investigate the class of mock injective $G$-modules, which are defined to be those rational $G$-modules that are injective on restriction to $G_r$ for all $r\geq 1$. In this paper the authors provide necessary and sufficient conditions for the existence of non-injective mock injective $G$-modules, thereby answering a question raised by Friedlander. Furthermore, the authors investigate the existence of non-injective mock injectives with simple socles. Interesting cases are discovered that show that this can occur for reductive groups, but will not occur for their Borel subgroups.

Let ${\Xmathfrak g}={\Xmathfrak g}_{\zerox }\oplus {\Xmathfrak g}_{\onex }$ be a classical Lie superalgebra and let ℱ be the category of finite-dimensional ${\Xmathfrak g}$-supermodules which are completely reducible over the reductive Lie algebra ${\Xmathfrak g}_{\zerox }$. In [B. D. Boe, J. R. Kujawa and D. K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), 696–724], we demonstrated that for any module M in ℱ the rate of growth of the minimal projective resolution (i.e. the complexity of M) is bounded by the dimension of ${\Xmathfrak g}_{\onex }$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\Xmathfrak {gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.

The authors compute the support varieties of all irreducible modules for the small quantum group u ζ ( g ) , where g is a finite-dimensional simple complex Lie algebra, and ζ is a primitive l-th root of unity with l larger than the Coxeter number of g . The calculation employs the prior calculations and techniques of Ostrik and of Nakano, Parshall, and Vella, as well as deep results involving the validity of the Lusztig character formula for quantum groups and the positivity of parabolic Kazhdan–Lusztig polynomials for the affine Weyl group. Analogous support variety calculations are provided for the first Frobenius kernel G 1 of a reductive algebraic group scheme G defined over the prime field F p .

The support varieties for the induced modules or Weyl modules for a reductive algebraic group G were computed over the first Frobenius kernel G1 by Nakano, Parshall and Vella. A natural generalization of this computation is the calculation of the support varieties of Demazure modules over the first Frobenius kernel, B1, of the Borel subgroup B. In this paper we initiate the study of such computations. We complete the entire picture for reductive groups with underlying root systems A1 and A2. Moreover, we give complete answers for Demazure modules corresponding to a particular (standard) element in the Weyl group, and provide results relating support varieties between different Demazure modules which depend on the Bruhat order.

We prove the analog of Kostant's Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostant's cohomology formula holds for quantum groups at a generic parameter $q$, recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra $\mathfrak{g} = \mathfrak{sl}(n)$. We also show that Kostant's formula holds when $q$ is specialized to an $\ell$-th root of unity for odd $\ell \ge h-1$ (where $h$ is the Coxeter number of $\mathfrak{g}$) when the highest weight of the coefficient module lies in the lowest alcove. This can be regarded as an extension of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.

Let g be the finite-dimensional Witt algebra W(1,1) over an algebraically closed field of characteristic p>3. It is well known that all simple W(1,1)-modules are finite-dimensional. Each simple module admits a character χ∈g∗. Given χ∈g∗ one can form the (finite-dimensional) reduced enveloping algebra u(g,χ). The simple modules for u(g,χ) are precisely those simple W(1,1)-modules admitting the character χ. In this paper the authors compute Ext1 between pairs of simple modules for u(g,χ).

In this paper we determine the group of endotrivial modules for certain symmetric and alternating groups in characteristic p. If p = 2, then the group is generated by the class of Ωn(k) except in a few low degrees. If p > 2, then the group is only determined for degrees less than p2. In these cases we show that there are several Young modules which are endotrivial.

We investigate blocks of the Category O for the Virasoro algebra over C. We demonstrate that the blocks have Kazhdan‐Lusztig theories and that the truncated blocks give rise to interesting Koszul algebras. The simple modules have BGG resolutions, and from this we compute the extensions between Verma modules and simple modules, and between pairs of simple modules.

Let G be a connected reductive algebraic group with a Frobenius morphism F: G → G defined over a finite field □ pr . The main result of the paper is to prove that any rational G-module M which is projective when restricted to the Frobenius kernel G r = Ker(F) is also projective over the split and twisted finite Chevalley groups. In 1987, Parshall conjectured this statement for r = 1 in the split case. The authors verified this in 1999 with the possible exclusion of primes 2 and 3 in non-simply laced cases. The converse of the main result is also discussed for split groups in this paper.

Let G G be a connected reductive algebraic group over an algebraically closed field of characteristic p > 0 p>0 , G 1 G_{1} be the first Frobenius kernel, and G ( F p ) G({\mathbb F}_{p}) be the corresponding finite Chevalley group. Let M M be a rational G G -module. In this paper we relate the support variety of M M over the first Frobenius kernel with the support variety of M M over the group algebra k G ( F p ) kG({\mathbb F}_{p}) . This provides an answer to a question of Parshall. Applications of our new techniques are presented, which allow us to extend results of Alperin-Mason and Janiszczak-Jantzen, and to calculate the dimensions of support varieties for finite Chevalley groups.

Let G be a reductive algebraic group scheme defined over Fp and G1 be the first Frobenius kernel. For any dominant weight λ, one can construct the Weyl module V(λ). When p is a good prime for G, the G1-support variety of V(λ) was computed by Nakano, Parshall and Vella in [D.K. Nakano, B.J. Parshall, D.C. Vella, Support varieties for algebraic groups, J. Reine Angew. Math. 547 (2002) 15–49]. We complete this calculation by computing the G1-supports of the Weyl modules over fields of bad characteristic.

Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p . Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B -modules. We also consider the standard induced modules obtained by inducing a simple B -module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie( U ) with coefficients in k .

The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the questions of whether a given complex representation is symplectic or orthogonal have been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from z hold by using similar proofs. This paper considers analogues of these results for simple, induced, and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

The complexity of a module is an important homological invariant that measures the polynomial rate of growth of its minimal projective resolution. For the symmetric group $\Sigma_n$, the Lie module $\mathsf{Lie}(n)$ has attracted a great deal of interest in recent years. We prove here that the complexity of $\mathsf{Lie}(n)$ in characteristic $p$ is $t$ where $p^t$ is the largest power of $p$ dividing $n$, thus proving a conjecture of Erdmann, Lim and Tan. The proof uses work of Arone and Kankaanrinta which describes the homology $\operatorname{H}_\bullet(\Sigma_n, \mathsf{Lie}(n))$ and earlier work of Hemmer and Nakano on complexity for modules over $\Sigma_n$ that involves restriction to Young subgroups.

Let G be a finite group such that \(\mathop {{\text {SL}}}\nolimits (n,q)\subseteq G \subseteq \mathop {{\text {GL}}}\nolimits (n,q)\) and Z be a central subgroup of G. In this paper we determine the group T(G / Z) consisting of the equivalence classes of endotrivial k(G / Z)-modules where k is an algebraically closed field of characteristic p such that p does not divide q. The results in this paper complete the classification of endotrivial modules for all finite groups of (untwisted) Lie type A, initiated earlier by the authors.

We investigate the cohomology of the Specht module Sλ for the symmetric group Σd. We show if 0⩽i⩽p−2, then Hi(Σd,Sλ) is isomorphic to Hs+i(B,w0⋅λ′−δ) where s=d(d−1)2, B is the Borel subgroup of the algebraic group GLd(k) and δ=(1d) is the weight of the determinant representation. We obtain similar isomorphisms of ExtΣdi(Sλ,M) with B-cohomology, which in turn yield isomorphisms of cohomology for Borel subgroups of GLn(k) for varying n⩾d. In the case i=0, and the case i=1 for certain λ, we apply our result and known symmetric group results of James and Erdmann to obtain new information about B-cohomology. Finally we show that Specht module cohomology is closely related to cohomology for the Frobenius kernel B1 for small primes.

Let G be a reductive algebraic group over an algebraically closed field k of characteristic p > 0 with g = Lie ( G ) . In this paper the authors investigate the following problem. Given a nilpotent orbit O in the restricted nullcone N 1 ( g ) , construct a finite-dimensional (tilting) G-module such that the support variety of M, V g ( M ) , is the closure of O .

Suppose that $$G$$ is a finite group such that $$\mathrm{SL }(n,q)\subseteq G \subseteq \mathrm{GL }(n,q)$$ , and that $$Z$$ is a central subgroup of $$G$$ . Let $$T(G/Z)$$ be the abelian group of equivalence classes of endotrivial $$k(G/Z)$$ -modules, where $$k$$ is an algebraically closed field of characteristic $$p$$ not dividing $$q$$ . We show that the torsion free rank of $$T(G/Z)$$ is at most one, and we determine $$T(G/Z)$$ in the case that the Sylow $$p$$ -subgroup of $$G$$ is abelian and nontrivial. The proofs for the torsion subgroup of $$T(G/Z)$$ use the theory of Young modules for $$\mathrm{GL }(n,q)$$ and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.

In this paper the authors investigate the representation type of the blocks of the relative (parabolic) category O S for complex semisimple Lie algebras. A complete classification of the blocks corresponding to regular weights is given. The main results of the paper provide a classification of the blocks in the “mixed” case when the simple roots corresponding to the singular set and S do not meet.

Let Hq (d) be the Iwahori–Hecke algebra of the symmetric group, where q is a primitive lth root of unity. Using results from the cohomology of quantum groups and recent results about the Schur functor and adjoint Schur functor, it is proved that, contrary to expectations, for l 4 the multiplicities in a Specht or dual Specht module filtration of an Hq (d)-module are well defined. A cohomological criterion is given for when an Hq (d)-module has such a filtration. Finally, these results are used to give a new construction of Young modules that is analogous to the Donkin–Ringel construction of tilting modules. As a corollary, certain decomposition numbers can be equated with extensions between Specht modules. Setting q = 1, results are obtained for the symmetric group in characteristic p 5. These results are false in general for p = 2 or 3.

The nucleus was a concept first developed in the cohomology theory for finite groups. In this paper the authors investigate the nucleus for restricted Lie algebras. The nucleus is explicitly described for several important classes of Lie algebras.

1.1. In the late 1970s, Alperin [A] defined an invariant called the complexity of a module as a way to relate the modules with the complexes and resolutions that they admit. Several years later, Carlson [Ca1,Ca2] defined affine algebraic varieties corresponding to modules over group algebras. These varieties are subvarieties of the spectrum of the cohomology ring which was earlier described by Quillen [Q]. They are known in present day language as support varieties. It was discovered early on that the complexity of a module is equal to the dimension of the support variety of the module. Geometric methods involving support varieties have played a fundamental role in understanding the interplay between the modular representation theory and cohomology for finite groups. Despite substantial progress in this direction, there have been few explicit computations of support varieties for important classes of modules over certain groups. The goal of this paper is to introduce methods and techniques for computing support varieties for modules over the symmetric group Σd . In the process, we will provide explicit computations of support varieties for certain classes

LetGGbe a semisimple simply connected algebraic group defined and split over the fieldFp{\mathbb {F}}_pwithppelements, letG(Fq)G(\mathbb {F}_{q})be the finite Chevalley group consisting of theFq{\mathbb {F}}_{q}-rational points ofGGwhereq=prq = p^r, and letGrG_{r}be therrth Frobenius kernel. The purpose of this paper is to relate extensions between modules inMod(G(Fq))\text {Mod}(G(\mathbb {F}_{q}))andMod(Gr)\text {Mod}(G_{r})with extensions between modules inMod(G)\text {Mod}(G). Among the results obtained are the following: forr>2r >2andp≥3(h−1)p\geq 3(h-1), theG(Fq)G(\mathbb {F}_{q})-extensions between two simpleG(Fq)G(\mathbb {F}_{q})-modules are isomorphic to theGG-extensions between two simpleprp^r-restrictedGG-modules with suitably “twisted" highest weights. Forp≥3(h−1)p \geq 3(h-1), we provide a complete characterization ofH1(G(Fq),H0(λ))\text {H}^{1}(G(\mathbb {F}_{q}),H^{0}(\lambda ))whereH0(λ)=indBG λH^{0}(\lambda )=\text {ind}_{B}^{G}\ \lambdaandλ\lambdaisprp^r-restricted. Furthermore, forp≥3(h−1)p \geq 3(h-1), necessary and sufficient bounds on the size of the highest weight of aGG-moduleVVare given to insure that the restriction mapH1⁡(G,V)→H1⁡(G(Fq),V)\operatorname {H}^{1}(G,V)\rightarrow \operatorname {H}^{1}(G(\mathbb {F}_{q}),V)is an isomorphism. Finally, it is shown that the extensions between two simpleprp^r-restrictedGG-modules coincide in all three categories provided the highest weights are “close" together.

Tensor triangular geometry as introduced by Balmer is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category. In this paper we provide a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the Balmer spectrum. For a classical Lie superalgebra ${\mathfrak g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$, we construct a Zariski space from a detecting subalgebra of ${\mathfrak g}$ and demonstrate that this topological space governs the tensor triangular geometry for the category of finite dimensional ${\mathfrak g}$-modules which are semisimple over ${\mathfrak g}_{\bar{0}}$., to appear in Advances in Mathematics

Let k be an algebraically closed field of characteristic p > 0, and let G be a simple simply-connected algebraic group over k. In this paper we investigate situations where the dimension of a rational cohomology group for G can be bounded by a constant times the dimension of the coefficient module. As an application, effective bounds on the first cohomology of the symmetric group are obtained. We also show how, for finite Chevalley groups, our methods permit significant improvements over previous estimates for the dimensions of second cohomology groups.

In this paper the authors explore relationships between the cohomology of the general linear group and the symmetric group. Stability results are given which show that the cohomology of these groups agree in a certain range of degrees.

Let G be a semisimple simply connected algebraic group defined and split over the field F p with p elements, G( F q ) be the finite Chevalley group consisting of the F q -rational points of G where q=pr, and Gr be the rth Frobenius kernel of G. This paper investigates relationships between the extension theories of G, G( F q ) , and Gr over the algebraic closure of F p . First, some qualitative results relating extensions over G( F q ) and Gr are presented. Then certain extensions over G( F q ) and Gr are explicitly identified in terms of extensions over G.