restricted one dimensional central extensions of restricted simple lie algebras abstract we study the restricted one dimensional central extensions of an arbitrary finite dimensional restricted simple lie algebra for p 5 for h 2 g 0 we explicitly describe the cocycles spanning h 2 g and in the case h 2 g 0 we give a procedure to describe a basis for h 2 g

finite resolutions of modules for reductive algebraic groups of a suitable g module m by modules which are required to be direct sums of tensor products of exterior powers of the natural representation in particular in z 31 for partitions 1 and p resolutions are constructed for the modules l f and l f the schur functor corresponding to 1 and the skew schur functor corresponding to a p evaluated at f where f is a free module over a field or the integers the purpose of this paper is to characterise the gl modules which admit such a resolution as those modules which have a filtration in which each successive quotient has the form l f for some partition p by contrast with 2 3 we do not produce resolutions explicitly our methods are independent of those of akin and buchsbaum we give a new proof of their result on the existence of a resolution for l f and are algebraic group theoretic in nature we have therefore cast our main result the theorem of section 1 as a statement about resolutions for reductive algebraic groups over an algebraically closed field one may frequently wish to work over a more general coefficient ring see l 3 and so we treat in some detail chevalley group schemes and general linear group schemes over a principal ideal domain 2 6 and 2 7 giving the appropriate versions of our main result and relating it to the work of akin buchsbaum and weyman

cohomology for frobenius kernels of sl2 abstract let sl 2 r be the r th frobenius kernels of the group scheme sl 2 defined over an algebraically closed field of characteristic p 0 in this paper we give for r 1 a complete description of the cohomology groups for sl 2 r we also prove that the reduced cohomology ring h sl 2 r k red is cohen macaulay geometrically we show for each r 1 that the maximal ideal spectrum of the cohomology ring for sl 2 r is homeomorphic to the fiber product g b u r finally we adapt our calculations to obtain analogous results for the cohomology of higher frobenius lusztig kernels of quantized enveloping algebras of type sl 2

reduced standard modules and cohomology first cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts including wiles s famous paper 1995 internal to group theory 1 cohomology plays a role in the general theory of maximal subgroups of finite groups as developed by aschbacher and scott 1985 one can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible in this case r guralnick 1986 conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups this paper provides the first general positive results on this conjecture proving that the generic 1 cohomology h 1 gen g l lim h 1 g q l see cline parshall scott and van der kallen 1977 of a finite group g q of lie type with absolutely irreducible coefficients l in the defining characteristic of g is bounded by a constant depending only on the root system in all cases we are able to improve this result to a bound on h 1 g q l itself still depending only on the root system the generic h 1 result and related results for ext 1 emerge here as a consequence of a general study of interest in its own right of the homological properties of certain rational modules red red indexed by dominant weights for a reductive group g the modules red and red arise naturally from irreducible representations of the quantum enveloping algebra u of the same type as g at a pth root of unity where p 0 is the characteristic of the defining field for g finally we apply our ext 1 bounds results of bendel nakano and pillen 2006 as well as results of sin 1993 1992 1994 on the ree and suzuki groups to obtain the non generic bounds on h 1 g q l

cohomology of infinitesimal and discrete groups an infinitesimal algebraic group over a field k is a connected algebraic group scheme whose coordinate ring is a finite dimension al k algebra if one begins with a reductive algebraic group g defined over fp then one has a close relationship between the representation theory of g that of its infinitesimal subgroups g defined as the kernels of the frobenius endomorphisms rr g g and that of its finite chevalley subgroups g fq of fq rational points q pd a basic theorem of cline et al i 9 theorem 6 6 relates the cohomology of g in a rational representation to that of the finite groups g fq related work of cline et al 6 and of the authors 1 16 relates the cohomology of g to that of its infinitesimal subgroups a determination of the cohomology of the first infinitesimal subgroup g 1 of g with trivial coefficients has enabled the authors in 14 to make certain explicit cohomology calculations for the finite groups g fq with certain nontrivial coefficients in this paper we continue our study of infinitesimal subgroups of simple algebraic groups by proving various qualitative properties of their cohomology and by extending our computational knowledge in particular we provide a computation which applies in a range of degrees to chevalley groups over the prime field we also investigate the cohomology of infinitesimal subgroups of nilpotent groups enabling us for example to determine the cohomology of their lie algebras in certain cases we apply this lie algebra cohomology computation to determine the integral cohomology of various torsion free nilpotent groups modulo small prime torsion thereby extending a theorem of dwyer 12 in more detail the paper begins in sect 1 with a spectral sequence converging to the cohomology of the first infinitesimal subgroup g which was previously studied in 14 from this spectral sequence it easily follows that h g1 m is a noetherian h g1 k module for each finite dimensional rational gl module m we then apply the work of andersen and jantzen 1 on the gl cohomology of induced modules to obtain the analogous result for the second infinitesimal subgroup gz our analysis also permits a determination of the generic

skew modules for reductive groups skew modules for symmetric groups have been investigated by james and peel 13 farahat and peel 9 and clausen s more recently skew modules for general linear groups have been investigated by akin buchsbaum weyman 3 l 2 and kouwenhoven 161 our aim in this paper is to set up in a rather general context a theory of skew modules for reductive algebraic groups for a partition 1 of it james peel 13 obtain a filtration of the specht module sj viewed as a module for the young subgroup corresponding to a decomposition n a b of n the successive quotients in the filtration have the form s h sj p kronecker product where p runs over the partitions of a such that p is a skew partition and s denotes the skew specht module the analogue of this result is proved for general linear groups by akin buchsbaum and weyman 3 theorem 11 4 1 l and kouwenhoven 16 proposition ii 2 2a the analogue for reductive groups is more or less our definition of skew modules this filtration property is in fact a characterisation of skew modules see 1 3 4 and our definition for general linear groups therefore agrees with that of akin buchsbaum and weyman 3 and kouwenhoven 161 it becomes apparent rather quickly from our point of view that the skew modules have a particular kind of filtration see 2 1 1 which in the case of the general linear group is a filtration by schur modules this is a result of kouwenhoven 16 proposition 11 2 2b a for general linear groups kouwenhoven is able to give an explicit description of such a filtration one may then obtain as in kouwenhoven 16 proposition ii 2 2b b the result of james and peel 131 on the existence of specht series for skew modules for the symmetric groups by specializing to a certain weight space i e applying the schur functor as in ll chap 63 we are also able to prove see 2 3 3 the existence of a certain kind of resolution considered by akin and buchsbaum see 1 sects 83 for skew modules for general linear groups our main point of reference is the representation theory of reductive 465 0021 8693 88 3 00

extensions of simple modules for finite chevalley groups let g be an almost simple algebraic group over an algebraically closed field k of positive characteristic p we assume that g is defined and split over the prime field 5 for it 3 1 we denote by g n the finite group consisting of the points of g over the field with p elements our first result states that for a certain class of simple modules for g n the only extensions that can occur are those which are restrictions of g extensions see theorem 2 8 the full story about g extensions is not known yet except for a few rather small groups however much more is known about extensions for the algebraic group than about those for the finite group for instance the linkage principle l gives an upper bound for which g extensions may occur moreover a proof of the lusztig conjecture on the irreducible characters for g also implies strong statements about g extensions see 3 proposition 2 81 it is an open question whether all extensions between simple g n modules can be found via the extension theory for g we prove that in generic cases this is in fact so see theorem 3 2 moreover for the groups x 3 p and sp 4 p we are able to determine completely all extensions between simple modules from the corresponding known results for the algebraic groups x 3 k and sp 4 k our results extend previous results by cline parshall and scott s 91 who consider extensions of the trivial module by a minimal module and smith ls who considers extensions between certain minimal modules the method we use is a generalization of the one used by humphreys 131 who proves the existence of certain selfextensions for sp 4 p it should be pointed out that our methods do not work for p small on the other hand everything we do generalizes easily to the twisted chevalley groups see theorem 3 4 we demonstrate this by working out all extensions for su 3 p see section 4 3

elementary subalgebras of lie algebras abstract we initiate the investigation of the projective varieties e r g of elementary subalgebras of dimension r of a p restricted lie algebra g for various r 1 these varieties e r g are the natural ambient varieties for generalized support varieties for restricted representations of g we identify these varieties in special cases revealing their interesting and varied geometric structures we also introduce invariants for a finite dimensional u g module m the local r j radical rank and local r j socle rank functions which are lower upper semicontinuous on e r g examples are given of u g modules for which some of these rank functions are constant

batalin vilkovisky algebras and two dimensional topological field theories by a batalin vilkovisky algebra we mean a graded commutative algebraa together with an operator a a 1 such that 2 0 and a a is a graded derivation ofa for alla a in this article we show that there is a natural structure of a batalin vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions we make use of a technique from algebraic topology the theory of operads

higher genus surface operad detects infinite loop spaces abstract the operad studied in conformal field theory and introduced ten years ago by g segal s is built out of moduli spaces of riemann surfaces we show here that this operad which at first sight is a double loop space operad is indeed an infinite loop space operad this leads to a new proof of the fact that the classifying space of the stable mapping class group mathbb z times b gamma infty is an infinite loop space after plus construction t2 this new approach has various advantages in particular the infinite loop space structure is more explicid

on the cohomology of specht modules we investigate the cohomology of the specht module s for the symmetric group d we show if 0 i p 2 then hi d s is isomorphic to hs i b w0 where s d d 1 2 b is the borel subgroup of the algebraic group gld k and 1d is the weight of the determinant representation we obtain similar isomorphisms of ext di s 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 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

adjoint sensitivity analysis for nonlinear dynamic thermoelastic systems an adjoint approach is presented to formulate explicit sensitivities for a general functional associated with a nonlinear dynamic thermoelastic system uncoupled thermoelastic response and small deformation behavior are assumed the formulation uses the lagrange multiplier method to define an adjoint system the convolution operator to incorporate transient effects and domain parameterization to describe shape variations variations of the functional are expressed in explicit form with respect to perturbations of the design fields structural shape material properties applied loads prescribed boundary conditions and initial conditions the functional is defined in terms of the design fields and the implicit response fields displacement temperature strain temperature gradient stress heat flux vector reaction force and reaction surface flux in an example problem the finite element method is used to evaluate the real and adjoint responses and the shape sensitivities for a tank design problem the tank is modeled as a nonlinear quasistatic uncoupled thermoelastic system

second cohomology groups for frobenius kernels and related structures abstract 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

extensions of modules for algebraic groups let g be a connected semi simple and simply connected algebraic group over an algebraically closed field k of prime characteristic p in this paper we show how one can obtain the extensions of any two simple g modules if one knows the socles of all the g modules extg l l where g1 denotes the kernel of the frobenius endomorphism on g and l as well as l are simple g1 modules alternatively we prove how all such g extensions can be found once one knows the extensions of any two simple g modules belonging to a specified finite set of simple modules with small highest weight such information is available in certain low rank cases it should also be pointed out that it is possible from lusztig s conjecture on irreducible characters 11 to derive the dimensions of ext l1 l2 when l1 and l2 are simple with such small highest weights to obtain our results on exth we need to know that there exists no g1 extension of a simple module by itself this turns out to be somewhat

some simple modules for classical groups and p ranks of orthogonal and hermitian geometries abstract we determine the characters of the simple composition factors and the submodule lattices of certain weyl modules for classical groups the results have several applications the simple modules arise in the study of incidence systems in finite geometries and knowledge of their dimensions yields the p ranks of these incidence systems

central extensions and h1 l l of the graded lie algebras of cartan type let f be an algebraically closed field of characteristic p 0 l i rsl i a graded lie algebra of cartan type over f we give a description of the dual adjoint module l by means of the mixed product and the coinduced module such that the computation of h1 l l can be reduced to that of the first cohomology of l 0 then we determine the central extensions and h2 l f by computing h1 l l

localization for quantum groups at a root of unity let c be the field of complex numbers and fix q g c let g be a semi simple lie algebra over c and let g be the corresponding simply connected algebraic group let uq be the quantized enveloping algebra of q let oq be the algebra of quantized functions on g let oq b be the quotient hopf algebra of oq corresponding to a borel subgroup b of g in the paper bk we defined categories of equivariant quantum o modules and pg modules on the quantum flag variety of g we proved that the beilinson bernstein localization theorem holds at a generic q namely the global section functor gives an equivalence between categories of modules and vq iaod es on the quantum flag variety thus one can translate questions about the represen tation theory of quantum groups to the study of the noncommutative geometry of the quantum flag variety in this paper we prove that a derived version of this theorem holds at the root of unity case using this equivalence we get that one can understand the representation theory of quantum groups at roots of unity through the now commutative geometry of the springer fibers we now recall the main results in bk the constructions given there are crucial for the present paper and a fairly detailed survey of the material there is given in the next section we defined an equivariant sheaf of quasi coherent modules over the quantum flag variety to be a left oq mod e equipped with a right q b comod ile structure satisfying certain compatibility conditions such objects form a category denoted ai q gq it contains certain line bundles oq for in the weight lattice we proved that oq x is ample for 0 holds for every q this implies that the category m bq gq is a proj category in the sense of serre

cohomology of infinitesimal algebraic groups in this paper we study the cohomology h br k of a higher frobenius kernel br r 1 of a borel subgroup b of a reductive algebraic group g over an algebraically closed field k of a positive characteristic p for the first frobenius kernel b 1 the cohomology has been completely determined by h h andersen and j c jantzen 3 in case p h the coxeter number of g in low degrees the cohomology is independent of r 1 and their results allow us to compute the first different case h zp i br k for s lt i 3 6 in higher degrees however we have to resort to a different method although h b 2 k has already been computed for sl z also in 3 it is a considerably harder problem for sl 3 we apply algebraic steenrod operations and the transgression theorem to the lyndon hochschildserre or cartan eilenberg spectral sequence arising from a central series of the unipotent radical u of b to find h k for sl 3 5 1 the method we employ here was originally developed by m tezuka and n yagita 21 in order to investigate h u fq k q a power ofp in fact the results of sections 4 and 5 were first conceived from the corresponding results of h u fq k aside from the interests in relation to the cohomology of finite groups the work was also motivated by a close connection of the bl cohomology to g lusztig s conjectural character formula for the irreducible modules of the reductive group g which is described in section 2 hoping that the representation theory may shed more light on some topological problems we intend to take up the steenrod algebra and the a algebras in a subsequent paper

extensions for finite chevalley groups i let g be a connected semisimple algebraic group defined and split over the field fp with p elements and k be the algebraic closure of fp assume further that g is almost simple and simply connected and let g fq be the finite chevalley group consisting of fq rational points of g where q pr for a non negative integer r in this paper formulas are found relating extensions between simple kg fq modules and extensions over g considered as an algebraic group over k one of these formulas which only holds for primes p 3 h 1 where h is the coxeter number of g is then used to show the vanishing of self extensions between simple kg fq modules except for certain simple modules when r 1 and the underlying root system is of type a1 or cn

presenting the graded lie algebra associated to the nottingham group abstract the graded lie algebra l associated to the nottingham group is a loop algebra of the witt algebra w 1 the universal covering w 1 of w 1 has one dimensional centre so that the corresponding loop algebra m of w 1 has an infinite dimensional centre z m as m z m is isomorphic to l it follows from a result of b h neumann that l is not finitely presented however we are able to show that m itself is finitely presented we work more generally with the zassenhaus algebras w n in the group context examples of finitely presented groups whose centre is not finitely generated were given by v n remeslennikov and h abels

extensions for frobenius kernels 1 1 let g be a simple algebraic group over an algebraically closed field k of characteristic p 0 for r 1 let gr be the rth frobenius kernel of g it is well known that the representations for g1 are equivalent to the restricted representations for lie g historically the cohomology for frobenius kernels has been best understood for large primes friedlander and parshall fp first computed the cohomology ring h g1 k for p 3 h 1 where h is the coxeter number of the underlying root system they proved that the cohomology ring can be identified with the coordinate algebra of the nullcone andersen and jantzen aj later verified this fact for p h furthermore they generalized this calculation by looking at h g1 h0 where h0 indb for p h their results had some restrictions on the type of root system involved kumar lauritzen and thomsen klt removed the restrictions on the root systems through the use of frobenius splittings the cohomology ring h g1 k modulo nilpotents can be identified in general with the coordinate algebra of the restricted nullcone n1 x lie g x p 0 for good primes nakano parshall and vella npv proved that this variety is irreducible and can be identified with the closure of some richardson orbit recently carlson lin nakano and parshall clnp have given an explicit description of n1 these recent results provide some indication that one can systematically study extensions of frobenius kernels for small primes by using general formulas which exhibit generic behavior for large primes

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

vanishing of the batalin vilkovisky algebra structure for tcfts the purpose of this short note is to show that for topological conformal field theories in dimension two or closed string field theories the batalin vilkovisky algebra structure on the homology of the brst complex g z is trivial after a certain element of ghost number 0 is inverted the important point here is that we do not restrict our considerations to the tree level but incorporate the action of moduli spaces for higher genera

frobenius splitting of cotangent bundles of flag varieties let g be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic p 0 let u be the unipotent radical of a borel subgroup b g and u the lie algebra of u springer 16 has shown for good primes that there is a b equivariant isomorphism u u where b acts through conjugation on u and through the adjoint action on u for g sln one has the well known equivariant isomorphism a 7 a i between unipotent and nilpotent upper triangular matrices let p be a good prime for g then there is an isomorphism of homogeneous bundles x g b u g b u where the latter can be identified with the cotangent bundle t g b of g b motivated in part by 12 we establish a link between the g invariant form on the steinberg module st h0 g b p 1 cf 1 8 and frobenius splittings 15 of the cotangent bundle of g b the representation h0 g b 2 p 1 is a quotient of the space of functions h x ox on x here h0 g b m denotes the g module induced from the b module m and half the sum of the roots r opposite to the roots

extensions for finite chevalley groups ii let g be a semisimple simply connected algebraic group defined and split over the field fp with p elements let g fq be the finite chevalley group consisting of the fq rational points of g where q p r and let g r be the rth frobenius kernel the purpose of this paper is to relate extensions between modules in mod g f q and mod g r with extensions between modules in mod g among the results obtained are the following for r 2 and p 3 h 1 the g f q extensions between two simple g f q modules are isomorphic to the g extensions between two simple p r restricted g modules with suitably twisted highest weights for p 3 h 1 we provide a complete characterization of h 1 g f q h0 where h 0 ind g b a and a is p r restricted furthermore for p 3 h 1 necessary and sufficient bounds on the size of the highest weight of a g module v are given to insure that the restriction map h 1 g v h 1 g f q v is an isomorphism finally it is shown that the extensions between two simple p r restricted g modules coincide in all three categories provided the highest weights are close together

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

second cohomology groups for algebraic groups and their frobenius kernels let g be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic p 0 let t be a maximal split torus in g b t be a borel subgroup of g and u its unipotent radical let f g g be the frobenius morphism for r 1 define the frobenius kernel gr to be the kernel of f iterated with itself r times define ur respectively br to be the kernel of the frobenius map restricted to u respectively b let x t be the integral weight lattice and x t be the dominant integral weights the computations of particular importance are h2 u1 k h2 br for x t h2 gr h0 for x t and h2 b for x t the above cohomology groups for the case when the field has characteristic 2 are computed in this paper these computations complete the picture started by bendel nakano and pillen 2007 5 for p 3 furthermore the computations show h2 gr h0 has a good filtration

algebraic group actions in the cohomology theory of lie algebras of cartan type abstract given a lie algebra g of cartan type we construct an infinite dimensional cocommutative hopf algebrad g u g which is the analog of the distribution algebra of a connected reductive algebraic group we show that the simpleu g modules lift to ad g u g structure this additional structure is used to formulate and prove relative projectivity theorems for lie algebras of cartan type the support varieties of certaind g u g modules are computed by using these algebraic group techniques

quantum flag varieties equivariant quantum d modules and localization of quantum groups abstract let o q g be the algebra of quantized functions on an algebraic group g and o q b its quotient algebra corresponding to a borel subgroup b of g we define the category of sheaves on the quantum flag variety of g to be the o q b equivariant o q g modules and prove that this is a proj category we construct a category of equivariant quantum d modules on this quantized flag variety and prove the beilinson bernstein s localization theorem for this category in the case when q is transcendental

on the cohomology of algebraic and related finite groups in the above theorem h gl m denotes the rational cohomology of the algebraic group gl with coefficients in m the rational gl module obtained from the adjoint module m m through twisting by the r th power of the frobenius endomorphism on gl the algebra structure on the cohomology is induced from the associative algebra structure on m on the other hand we provide the following systematic determination of certain rational cohomology groups h g v r where v r is the rational gmodule obtained from the rational g module v by applying the r th frobenius twist see 1 2 this is a somewhat sharpened version of theorem 4 of 36

sheaf cohomology on gb and tensor products of weyl modulus let k be an algebraically closed field it is assumed that char k p 0 unless otherwise indicated g is a simply connected semisimple algebraic group over k t is a fixed maximal torus and b a bore1 subgroup of g containing t let qi g t be the root system of g d the set of all simple roots and and the sets of all positive and negative roots relative to d respectively we assume that the set a is chosen such that b corresponds to 0 let x be the weight lattice of sp which may be considered as the character group of t or b therefore if 1 e x by abuse of notation we also denote by l the one dimensional tor b module determined by a let x be the set of all dominant weights and w the fundamental weight corresponding to a simple root a e a we denote by 6 the half sum of all positive roots and we know that 6 is the sum of all fundamental weights let w be the weyl group of 0 the reflection with respect to a root a is denoted by s the longest element of w by w there are two ways of w acting on x i the ordinary action if 1 e x m e w the image of i under w is denoted by 2 ii dot action w a 5 2 6 s in particular let a w a the dual root system of is denoted by and the dual root of a e by a i e u 2a a u where is the inner product invariant under the ordinary action of w of the euclidean space spanned by for 1 e x a e let a a 1 a all modules of an algebraic group we consider in this paper are assumed to be rational in particular for an infinite dimensional module the term rational module means this module is a direct limit of finite dimensional rational modules let h be a closed subgroup of g containing t such as g b or t m an h module we define for l e x

invariant forms on irreducible modules of simple algebraic groups let g be a simple linear algebraic group over an algebraically closed field k of characteristic p 0p 0 and let v be an irreducible rational g module with highest weight when v is self dual a basic question to ask is whether v has a non degenerate g invariant alternating bilinear form or a non degenerate g invariant quadratic form if p 2p 2 the answer is well known and easily described in terms of in the case where p 2p 2 we know that if v is self dual it always has a non degenerate g invariant alternating bilinear form however determining when v has a non degenerate g invariant quadratic form is a classical problem that still remains open we solve the problem in the case where g is of classical type and is a fundamental highest weight i i and in the case where g is of type alal and r s r s for 1 r s l1 r s l we also give a solution in some specific cases when g is of exceptional type as an application of our results we refine seitz s 1987 description of maximal subgroups of simple algebraic groups of classical type one consequence of this is the following result if x y sl v x y sl v are simple algebraic groups and v xv x is irreducible then one of the following holds 1 v yv y is not self dual 2 both or neither of the modules v yv y and v xv x have a non degenerate invariant quadratic form 3 p 2p 2 x so v x so v and y sp v y sp v

a note on decomposition numbers for reductive algebraic groups in a recent paper 7 james proves two theorems on the decomposition numbers for the general linear groups and working within the context of green s monograph 5 derives properties of the decomposition numbers for the symmetric groups this note grew out of an attempt to formulate for arbitrary reductive groups these theorems of james we prove a natural result on decomposition numbers for reductive groups from which james s theorems are deduced the result the theorem of section 2 is the analogue of jantzen s satz 1 18 of 8 on decomposition numbers for representations of complex semisimple lie algebras also proved independently by deodhar very little direct use of 5 is made though some of the ideas are cribbed from that source as one may see by comparing the properties of the functor fa in section 2 with green s treatment see section 6 2 of s of ese theory s a ring e an idempotent

on ad semisimple lie algebras the question whether any ad semisimple lie algebra is commutative is related to the existence of simple semiabelian lie algebras these algebras determine the brauer group of a field and are in addition to that significant for the study of lie algebras all whose proper subalgebras are nilpotent in this paper we consider a class of lie algebras related to several topics of lie algebra theory and field theory we begin with the study of finite dimensional lie algebras g having the property that the adjoint representation sends every element of g to a semisimple derivation we call such a lie algebra ad semisimple our study of these may be regarded as a complement to the classical investigation of ad nilpotent lie algebras due to engel however the results of the first section show that in contrast with engels theory the theory of ad semisimple lie algebras is highly sensitive to the choice of the base field this feature becomes more evident if we confine our studies to the subclass of simple semiabelian lie algebras sections 3 5 are devoted to a brief discussion of some aspects of the structure of this interesting class a more thorough investigation can be found in 15 1 theorem 5 2 and section 6 are closely related to a paper by towers 11 i after discussing the relationship between simple semiabelian lie algebras and restricted lie algebras having a nonsingular p map i e a p map whose only zero is 0 we illustrate in the final section some aspects of the linkage between the brauer group of the underlying field and the isomorphism classes of simple semiabelian lie algebras the brauer group of the field admitting no simple semiabelian lie algebras is trivial it is conceivable that the brauer group determines the isomorphism classes of simple semiabelian lie algebras the cases of ir and 6 for example indicate the existence of a more general yet discerning relationship

new perspectives on the brst algebraic structure of string theory motivated by the descent equation in string theory we give a new interpretation for the action of the symmetry charges on the brst cohomology in terms of what we callthe gerstenhaber bracket this bracket is compatible with the graded commutative product in cohomology and hence gives rise to a new class of examples of what mathematicians call agerstenhaber algebra the latter structure was first discussed in the context of hochschild cohomology theory 11 off shell in the chiral brst complex all the identities of a gerstenhaber algebra hold up to homotopy applying our theory to thec 1 model we give a precise conceptual description of the brst gerstenhaber algebra of this model we are led to a direct connection between the bracket structure here and the anti bracket formalism in bv theory 29 we then discuss the bracket in string backgrounds with both the left and the right movers we suggest that the homotopy lie algebra arising from our gerstenhaber bracket is closely related to the hla recently constructed by witten zwiebach finally we show that our constructions generalize to any topological conformal field theory

reduced basis technique for evaluating the sensitivity coefficients of the nonlinear tire response an efficient reduced basis technique is presented for calculating the sensitivity of nonlinear tire response to variations in the desin variables the tire is discretized by using three field mixed finite element models the vector of structural response and its first and second order sensitivity coefficients derivatives with respect to design variables are each expressed as linear combinations of a small number of basis or global approximation vectors the bubnov galerkin technique is then used to approximate each of the finite element equations governing the response and the sensitivity coefficients by a small number of algebraic equations in the amplitudes of these vectors

the frobenius morphism in invariant theory abstract let r be the homogeneous coordinate ring of the grassmannian g gr 2 n defined over an algebraically closed field of characteristic p 0 in this paper we give a completely characteristic free description of the decomposition of r considered as a graded r p module into indecomposables frobenius summands as a corollary we obtain a similar decomposition for the frobenius pushforward of the structure sheaf of g and we obtain in particular that this pushforward is almost never a tilting bundle on the other hand we show that r provides a noncommutative resolution for r p when p n 2 generalizing a result known to be true for toric varieties in both the invariant theory and the geometric setting we observe that if the characteristic is not too small the frobenius summands do not depend on the characteristic in a suitable sense in the geometric setting this is an explicit version of a general result by bezrukavnikov and mirkovic on frobenius decompositions for partial flag varieties we are hopeful that it is an instance of a more general p uniformity principle

cohomology of classical algebraic groups from the functorial viewpoint abstract we prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups this result was previously known only for general linear groups we give several applications to the study of classical algebraic groups such as a cohomological stabilization property the injectivity of external cup products and the existence of hopf algebra structures on the stable cohomology of a classical algebraic group with coefficients in a hopf algebra our result also opens the way to new explicit cohomology computations we give an example inspired by recent computations of djament and vespa

on making large nonlinear problems small abstract a number of aspects of reduction methods for solution of large scale nonlinear problems are discussed including a selection of basis vectors for steady state problems b identification and determination of bifurcation and limit points and tracing post limit point and post bifurcation point paths using reduction methods c application of reduction methods to nonlinear problems with prescribed nonzero values of the field variable and d use of reduction methods in conjunction with multifield mixed finite element models four numerical examples are presented to demonstrate the effectiveness of using reduction methods for the solution of nonlinear thermal and structural problems also a number of research areas which have high potential for application of reduction methods are identified

good filtrations of symmetric algebras and strong f regularity of invariant subrings in this paper we prove that a linear action of a reductive group on a polynomial ring with good filtrations over a field of characteristic p 0 yields a strongly f regular in particular cohen macaulay invariant subring the strongly f regular property of some known examples of invariant subrings such as the coordinate rings of schubert varieties in grassmannians are recovered a similar result over a field of characteristic zero is also proved

strata hasse invariants hecke algebras and galois representations we construct group theoretical generalizations of the hasse invariant on strata closures of the stacks mathop g hbox mathtt zip nolimits mu restricting to zip data of hodge type we obtain a group theoretical hasse invariant on every ekedahl oort stratum closure of a general hodge type shimura variety a key tool is the construction of a stack of zip flags mathop g hbox mathtt zipflag nolimits mu fibered in flag varieties over mathop g hbox mathtt zip nolimits mu it provides a simultaneous generalization of the classical case homogeneous complex manifolds studied by griffiths schmid and the flag space for siegel varieties studied by ekedahl van der geer four applications are obtained 1 pseudo representations are attached to the coherent cohomology of hodge type shimura varieties modulo a prime power 2 galois representations are associated to many automorphic representations with non degenerate limit of discrete series archimedean component 3 it is shown that all ekedahl oort strata in the minimal compactification of a hodge type shimura variety are affine thereby proving a conjecture of oort 4 part of serre s letter to tate on mod p modular forms is generalized to general hodge type shimura varieties

differentiating the weyl generic dimension formula with applications to support varieties abstract the authors compute the support varieties of all irreducible modules for the small quantum group u g where g is a finite dimensional simple complex lie algebra and 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

design sensitivity analysis and optimization of nonlinear structuralresponse using incremental procedure this paper describes procedures for design sensitivity analysis of nonlinear structural response using the incremental solution scheme the total and updated lagrangian approaches are discussed sensitivity analyses for the stress strain and displacement constraints are developed in the optimization process the buckling load constraint is also imposed numerical aspects of design sensitivity analysis are discussed three numerical examples are solved to illustrate the design sensitivity analysis and optimization process the numerical procedures are stable and work well they can be applied to more complex structures to optimize them

extensions for finite chevalley groups iii rational and generic cohomology abstract 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 f q are established which generalize earlier work of hiller in addition to stability ranges for generic cohomology which improve on seminal work of cline parshall scott and van der kallen

complexes and vanishing of cohomology for group schemes abstract completely reducible homology complexes were originally introduced in the context of finite groups and used to study the question of vanishing of cohomology in this paper we study these complexes and the vanishing of cohomology for arbitrary finite dimensional cocommutative hopf algebras applications are later provided for infinitesimal group schemes of reductive and solvable algebraic groups

support varieties of line bundle cohomology groups for sl3 k abstract let g sl 3 k where k is a field of characteristic p 0 and let x t be any weight with corresponding line bundle l on g b in this paper we compute the support varieties for all modules of the form h i h i g b l over the first frobenius kernel g 1 the calculation involves certain recursive character formulas given by donkin cf 5 which can be used to compute the characters of the line bundle cohomology groups in the case where is a p regular weight and m h i 0 for some i these formulas are used to show that any pth root of unity is not a root of the generic dimension of m to handle the case where is not p regular we employ techniques similar to those used by drupieski nakano and parshall cf 7 to show that the module h i is not projective over g 1 whenever it is nonzero and lies outside of the steinberg block

on conjugating representations and adjoint representations of semisimple groups where ea is a direct sum of irreducible g modules of highest weight 2 here we consider the conjugating representation in arbitrary characteristic and prove the appropriate version of richardson s theorem under small and almost certainly unnecessary characteristic root system restrictions one can no longer decompose a g into homogeneous components but instead we prove the existence of an ascending

derivations and central extensions of finitely generated graded lie algebras abstract 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

infinitesimal 1 parameter subgroups and cohomology this is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p 0 whereas sfb is concerned with detection of cohomology classes the present paper introduces the graded algebra k vr g of functions on the scheme of infinitesimal 1 parameter subgroups of height r on an affine group scheme g and demonstrates that this algebra is essentially a retract of h g k provided that g is an infinitesimal group scheme of height kr this work is a continuation of f s in which the existence of certain universal extension classes was established thereby enabling the proof of finite generation of h g k for any finite group scheme g over k the role of the scheme of infinitesimal 1 parameter subgroups of g was foreshadowed in f p where h g 1 k was shown to be isomorphic to the coordinate algebra of the scheme of p nilpotent elements of g lie g for g a smooth reductive group g 1 the first frobenius kernel of g and p char k sufficiently large indeed p nilpotent elements of g correspond precisely to infinitesimal 1 parameter subgroups of g 1 much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1 parameter subgroups in 1 we construct the affine scheme vr g of homomorphisms from ga r to g which we call the scheme of infinitesimal 1 parameter subgroups of height r in g in the special case r 1 this is the scheme of p nilpotent elements of the p restricted lie algebra lie g for various classical groups g vr g is the scheme of r tuples of p nilpotent pairwise commuting elements of lie g more generally an embedding g c gln determines a closed embedding of vr g into the scheme of r tuples of p nilpotent pairwise commuting elements of gl1 lie gln the relationship between k vr g the coordinate algebra of vr g and h g k is introduced in theorem 1 14 a natural homomorphism of graded k algebras

the associative forms of the graded cartan type lie algebras this paper determines the cartan type lie algebras that possess a nonsingular associative form o introduction the structure of cartan type lie algebras has been extensively studied for several years derivation algebras and isomorphism classes of cartan subalgebras are well known and it is the objective of the present paper to determine those graded cartan type lie algebras which possess a nonsingular associative form in contrast with trace forms general nonsingular associative forms do not play an eminent role in the classification theory of simple lie algebras of prime characteristic their significance primarily rests on cohomology theory nonsingular associative forms allow the study of central extensions of lie algebras by means of derivations we will employ a slight generalization of this known interrelation in a forthcoming paper the introductory remarks of the first section are followed by a brief investigation of cartan subalgebras of graded lie algebras our results generalize those of m frank cf 8 3 establishes general properties of associative forms of graded simple lie algebras and provides the basic tools for the study of the graded cartan type lie algebras the author would like to express his gratitude to professor helmut strade for many helpful suggestions as well as to the referee for proposing proposition 3 4 and the resulting simplifications in 4 1 graded representations of graded lie algebras in the sequel l ik r l r k 1 denotes a graded lie algebra over an arbitrary field f a representation t l gl v is said to be graded if v 13 vj and t x vj c vl j for every x e li and s 1 we consider the subalgebras l eik l li and l d3 li using the p b w theorem it can be readily verified that we have u l u l u lo u l u l u lo u l for the respective universal enveloping algebras received by the editors may 17 1984 and in revised form april 10 1985 1980 mathematics subject classification primary 17b20 t1986 american mathematical society 0002 9947 86 1 00 25 per page

vertex operator representations for tkk algebras abstract it is known that up to isomorphism there are only one lattice s and one non lattice semi lattice s in r 2 in s tan tkk algebras and vertex operator representations j algebra 211 1999 298 342 we obtained a vertex operator representation for the baby tkk algebra g t s the goal of the present paper is to study the tkk algebra g t s which includes the baby tkk algebra as a subalgebra and its vertex operator representations as a by product we also obtain two representations for the baby tkk algebra g t s one of the representations recovers the construction given in s tan tkk algebras and vertex operator representations j algebra 211 1999 298 342

variational method for design sensitivity analysis in nonlinear structural mechanics this paper describes a unified variational theory of design sensitivity analysis of linear and nonlinear structures for shape nonshape and material selection problems the concepts of reference volume and adjoint structure are used to develop the unified viewpoint a general formula for design sensitivity analysis is derived several analytical linear and nonlinear examples are used to interpret various terms of the formula and demonstrate its use such analytical examples give insight into the application of the general formula to more complex problems that must be treated numerically adjoint state fields are also interpreted as sensitivities for the functional related to the primary structure

limits of weight spaces lusztig s q analogs and fiberings of adjoint orbits let g be a connected complex semisimple algebraic group and t a maximal torus inside a borel subgroup b with g t and b their lie algebras let v be a representation in the category a for g the t decomposition v met va of v into a direct sum of finite dimensional weight spaces is central in the representation theory of g in this paper we introduce on weight spaces a new structure the principal filtration je v where e is a principal nilpotent in g chosen to be compatible with t for example e can be the sum of the simple root vectors relative to t b this filtration is constructed in a very simple way by taking j v to be the space of vectors annihilated by the p 1 th power of e for p 0 our approach is motivated by kostant s fundamental work ki k2 on actions of the principal tds three dimensional subalgebra and coordinate rings of regular adjoint orbits in theorem 3 4 we give a new description in terms of the dimension jumps of the principal filtration of the weight space of lusztig s l q analog mm q of dominant weight multiplicity in a finite dimensional irreducible g representation v the polynomial m q was defined algebraically as an alternating sum over the weyl group through a q analog of kostant s weight multiplicity formula we prove that m q is equal to the jump polynomial

modular representation theory of lie algebras on etudie les representations non necessairement restreintes de l algebre universelle enveloppante u 7b g pour une algebre de lie restreinte 7b g

an adjoint variable method for sensitivity analysis of non linear elastic systems summary an adjoint variable method for design sensitivity analysis of non linear elastic systems is presented the method uses domain parameterization and a mutual form of the hu washizu energy principle and extends results reported in a recent work for linear elastic systems to non linear elasticity non linearities due to finite deformations and non linear hyperelastic constitutive models are considered in contrast to other methods for non linear sensitivity analysis the present formulation can be applied with force displacement or mixed approximate solution methods the mutual energy expression used in the adjoint sensitivity derivation is developed from a non linear extension of the hu washizu energy functional and yields a linear governing equation for the adjoint system this has important ramifications for the computational cost of a sensitivity analyses of non linear systems excluding the cost of determining the response of the system the cost of a sensitivity analysis for a non linear system is essentially the same as that for a linear system finite element implementation of the resulting sensitivity expressions is discussed and two numerical examples are presented the first example involves large deformations of a mooney rivlin body while the second involves design sensitivity analysis for mixed solution methods

support varieties for modules over chevalley groups and classical lie algebras let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 g 1 be the first frobenius kernel and g f p be the corresponding finite chevalley group let m be a rational g module in this paper we relate the support variety of m over the first frobenius kernel with the support variety of m over the group algebra kg 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

a note on the adjoint action of a semisimple group on the coordinate algebra of an infinitesimal subgroup abstract let g be a connected semisimple simply connected algebraic group over an algebraically closed field k of positive characteristic the action of g on itself by conjugation induces on the coordinate algebra k g and hence on the coordinate algebra k g r of the infinitesimal subgroup g r the structure of a rational g module in the recent paper 3 by bendel nakano pillen and sobaje the authors ask whether the tensor product of k g r with the rth steinberg module st r has a good filtration here we show that this is the case provided that the characteristic of k is good for g we do so by relating k g r to a certain quotient of the coordinate algebra k g of the lie algebra g of g and using a koszul resolution we note that the argument may also be applied to certain polynomial modules occurring in the context of doty s conjecture 17 4 2 11

absolutely indecomposable representations and kac moody lie algebras a conjecture of kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated kac moody lie algebra in this paper we prove these conjectures for indivisible dimension vectors