projectivity and freeness over comodule algebras let h be a hopf algebra and a an h simple right h comodule algebra it is shown that under certain hypotheses every h a hopf module is either projective or free as an a module and a is either a quasi frobenius or a semisimple ring as an application it is proved that every weakly finite in particular every finite dimensional hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras and the latter are frobenius algebras similar results are obtained for h simple h module algebras

another characterization of parabolic subgroups let g be a reductive algebraic group defined over an algebraically closed field k let h be a closed connected subgroup of g containing a maximal torus t of g in 13 it was shown at least in characteristic zero that the parabolic subgroups of g can be characterized among all such subgroups h by a certain finiteness property of the induction functor iz and its derived functors lk g this theme is continued in the present paper where it is shown that the parabolic subgroups can be characterized by yet another familiar property of the induction functor at least in characteristic zero we also obtain several results which are independent of the characteristic by added hypotheses on h or by using the restriction functor

deformations of lie algebra representations let g be a reductive algebraic group defined over an algebraically closed field k of positive characteristic p in this paper we continue our previous investigation 7 of the representation theory of the lie algebra f of g a key ingredient in our earlier work involved the consideration of a family a x s of finite dimensional algebras parameterized by the set 0 qs 1homka1g y b k of characters on a certain central subalgebra 0 6 of the universal enveloping algebra u 6 of g each irreducible module is naturally an irreducible module for exactly one such ax with the special case x 0 corresponding to restricted modules we introduce a so called deformation algebra s4 intermediate between the universal enveloping algebra u and certain of the algebras ax here sl is a quotient of u q such that each ax containing the restricted enveloping algebra v a of a fixed borel subalgebra 2a is in turn a quotient of s4 given a module mx for such an a we introduce the important notion of a deformation family of mx in definition 1 3 the hypothesis that the behavior of mx is related to that of the corresponding modules for near by algebras at motivates the present study a basic result 7 theorem 3 2 reduces the representation theory of ax for x 0 to that of an associated a for a subalgebra of to which the restriction t of x is nilpotent in the sense of 121 thus a study of the representation theory of the algebras ax for x a nilpotent character subsumes the classical problem of determining the irreducible representations of the lie algebra each g conjugacy class in the nullcone x of nilpotent elements of 0 y f defines a family of isomorphic algebras ax presumably the restricted enveloping algebra v ao corresponding to the conjugacy class o is the most complicated algebra of this kind its representation theory has been extensively investigated in recent years because of its intimate connections

a mackey imprimitivity theory for algebraic groups let g be an affine algebraic group over an algebraically closed field k and let h be a closed subgroup of g if v is a rational h module a comodule for the coordinate ring of h there is a now well known notion of an induced module vi g for g defined as the space morph g v of all h equivariant morphisms from g to a finite dimensional subspace of v with obvious g action the question arises given a rational g module m how can one recognize m as an induced module vig for a finite group g the answer is the mackey imprimitivity theorem the module m is induced if and only if it is a direct sum of subspaces permuted transitively by g with h the stabilizer of one of these subspaces called v one uses this result for example in proving the famous mackey decomposition theorem which describes the restriction of any induced module to a second subgroup l as a direct sum of suitable induced modules given the imprimitivity theorem the proof is just a matter of grouping the summands permuted by g into their orbits under l in the case of algebraic groups the situation is quite different for some subgroups h all g modules are induced this occurs for example if g is connected and k g h k e g if h is parabolic also if g is a connected unipotent group then a rational g module m is induced from some proper subgroup if and only if its endomorphism ring contains a two dimensional submodule e for the conjugation action of g with e k 1 and such that k l is precisely the subspace annihilated by the action of the lie algebra of g on e 27 cf also 5 5 below the latter is an application of a general criterion in case g h is affine a rational g module m is induced if and only if there is an action of the coordinate ring a of g h compatible with the action of g on both a and m note that this generalizes the imprimitivity theorem in the case of finite groups since then a has a k basis of g h orthogonal idempotents permuted transitively by g

coring stabilizers for a hopf algebra coaction abstract the coring stabilizer stab p is introduced for any prime ideal p of a right h comodule algebra a such that the factor ring a p is either right or left goldie this notion is used to obtain hopf algebraic analogs of two category equivalences associated with a homogeneous space the category of linearized quasicoherent sheaves on a noncommutative homogeneous space is interpreted as a suitable quotient category of the category of hopf modules birational invariance of such quotient categories is proved it is shown that for a birational h coequivariant extension b of a properly defined subsets of prime ideals of a and b correspond to each other bijectively

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

adaptive sample size and importance sampling in estimation based local search for the probabilistic traveling salesman problem the probabilistic traveling salesman problem is a paradigmatic example of a stochastic combinatorial optimization problem for this problem recently an estimation based local search algorithm using delta evaluation has been proposed in this paper we adopt two well known variance reduction procedures in the estimation based local search algorithm the first is an adaptive sampling procedure that selects the appropriate size of the sample to be used in monte carlo evaluation the second is a procedure that adopts importance sampling to reduce the variance involved in the cost estimation we investigate several possible strategies for applying these procedures to the given problem and we identify the most effective one experimental results show that a particular heuristic customization of the two procedures increases significantly the effectiveness of the estimation based local search

local search for the probabilistic traveling salesman problem correction to the 2 p opt and 1 shift algorithms abstract the probabilistic traveling salesman problem concerns the best way to visit a set of customers located in some metric space where each customer requires a visit only with some known probability a solution to this problem is an a priori tour which visits all customers and the objective is to minimize the expected length of the a priori tour over all customer subsets assuming that customers in any given subset must be visited in the same order as they appear in the a priori tour this problem belongs to the class of stochastic vehicle routing problems a class which has received increasing attention in recent years and which is of major importance in real world applications several heuristics have been proposed and tested for the probabilistic traveling salesman problem many of which are a straightforward adaptation of heuristics for the classical traveling salesman problem in particular two local search algorithms 2 p opt and 1 shift were introduced by bertsimas in a previous report we have shown that the expressions for the cost evaluation of 2 p opt and 1 shift moves as proposed by bertsimas are not correct in this paper we derive the correct versions of these expressions and we show that the local search algorithms based on these expressions perform significantly better than those exploiting the incorrect expressions

representations of hopf algebras arising fromlie algebras of cartan type c w x 1 1 this is a continuation of 13 in an effort to use algebraic group actions to study the representations of restricted lie algebras of cartan type in this paper we study the representation theory of certain finite diw x mensional hopf algebras that arise from the construction in 13 the representation theory of these hopf algebras closely resembles the representation theory of frobenius kernels of reductive algebraic groups in positive characteristics let g be a connected reductive algebraic group scheme over an algebraically closed field k of characteristic p 0 it is well known that if f g a g is the r th iteration of the frobenius map and g s ker f r r there is a chain of normal subgroup schemes g eg e eg in earlier 1 2 terminology this is equivalent to the existence of a chain of finite dimen sional cocommutative hopf algebras dist g dist g 1 2 dist g where dist g is known as the r th hyperalgebra of the distribur tion algebra dist g of g let g be the lie algebra of g and u g be its restricted universal enveloping algebra there exists an isomorphism of

extension of the 2 p opt and 1 shift algorithms to the heterogeneous probabilistic traveling salesman problem abstract the probabilistic traveling salesman problem is a well known problem that is quite challenging to solve it involves finding the tour with the lowest expected cost for customers that will require a visit with a given probability there are several proposed algorithms for the homogeneous version of the problem where all customers have identical probability of being realized from the literature the most successful approaches involve local search procedures with the most famous being the 2 p opt and 1 shift procedures proposed by bertsimas d j bertsimas l howell further results on the probabilistic traveling salesman problem european journal of operational research 65 1 1993 68 95 recently however evidence has emerged that indicates the equations offered for these procedures are not correct and even when corrected the translation to the heterogeneous version of the problem is not simple in this paper we extend the analysis and correction to the heterogeneous case we derive new expressions for computing the cost of 2 p opt and 1 shift local search moves and we show that the neighborhood of a solution may be explored in o n2 time the same as for the homogeneous case instead of o n3 as first reported in the literature

existence of multiple solutions to nonlinear elliptic equations of nondivergence form abstract in this paper by means of a priori estimates and topological degree theory we prove the existence of two or more solutions to nonlinear elliptic equations of nondivergence form

a cohomological characterization of parabolic subgroups of reductive algebraic groups let g be an algebraic group defined over an algebraically closed field k and rat g the category of rational g modules for any closed subgroup h of g and any rational h module v let v ig denote the rational g module induced from i the induction functor ig rat h rat g is left exact and we denote its derived functors by l j for n o 1 2 of course these derived functors are 0 for n 2 1 whenever 1 g is exact and by a result of cline parshall and scott 3 this happens precisely when g h is an afftne variety so the case of an afftne quotient is characterized by the vanishing of the higher derived functors in a like vein in this paper we will attempt to find a property of these functors which characterizes the case of a projective quotient since by definition g h is projective iff h p is a parabolic subgroup of g we are in effect looking for a cohomological characterization of parabolic subgroups of g the property of the functors l j we consider is a certain finiteness condition explained below the author was originally led to consider this question in 14 where induction from certain nonparabolic subgroups is studied and is shown to have this finiteness property it is well known that for every rational h module v there is an induced bundle l on g h with rank equal to dim i such that l j v identifies naturally with the sheaf cohomology group h g h l see 4 for details thus we may apply standard results from algebraic geometry to obtain some basic facts for example lg g 0 for n dim g h and serre s theorem gives the easy half of the result of cline parshall and scott men

on injective modules for infinitesimal algebraic groups ii let g be a semisimple algebraic group scheme defined and split over the prime subfield of an algebraically closed field k of characteristic p 0 if f g g is the frobenius morphism and t is a maximal split torus in g then tg is the inverse image of t under f thus tg is a non reduced subgroup scheme of g let b be a bore1 subgroup of gz containing 7 whose opposite b determines the positive system of roots in the root system of g relative to t let tb b xc tg b n tg the scheme theoretic intersection of b with 7 g it is well known j2 sect 3 1 that injective tg mods as a factor of q p coincides with the multiplicity of l p as a composition factor in lnd i at the same time if i is generic relative to an alcove in the sense defined in cps 1 appendix a the multiplicities of l p in tndyi i coincide with the multiplicities inde i l p where l f denotes the rational irreducible g module of high weight i relative to b thus in the generic case one can translate the problem of computing multiplicities of irreducibles in indg l to 2 question about the structure of rational injectivc tg modules we begin here a study of the structure of rational injectivc 7 modules from the point of view of the parabolic subgroup schemes of 7 g in this we proceed by analogy with the early work h of harish chandra sp of springer and with the more recent work rs of ronan and smith if p is a parabolic subgroup scheme of g containing t let tp p xc tg p n tg then tp has a levi factorization tp tl x lip where c denotes the unipotcnt radical of tp and tl its levi factor

models of quasiprojective homogeneous spaces for hopf algebras given an affine group scheme g of finite type over a field k a homogeneous space for g is a scheme x over k containing a rational point x such that g operates transitively on x assuming that g operates on the right we may identify x with the quotient k g of g by the left action of the stabilizer k of x in g the representation theoretic significance ofk g is that the induction functor indk from k modules to g modules factors as a category equivalence

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

on the generic structure of cohomology modules for semisimple algebraic groups let c be a connected semisimple algebraic group over a field of positive characteristic denote by b a borel subgroup our main result says that generically the cohomology modules for line bundles on g b have simple socles and simple heads and wc identify the corresponding highest weights as one of the conse quences we discover a certain symmetry among extensions of simple modules for g in this paper we continue the study of the 1nodule structure of the cohomology groups h x h g b t x for g a sesimple algebraic group over a field of positive characteristic p b a borel subgroup of g and 2 x the line bundle on g b induced by the character x of b for earlier work on various aspects of this question see e g 1 4 7 9 11 13 here we will consider the case where x is generic i e lies far away from the walls of the weyl chamber containing it see 2 1 for the precise condition in this case h x is nonzero in exactly one degree our main result theorem 2 2 says that this nonzero cohomology module has simple socle and simple head for x dominant it is a classical result that h x has simple socle the first nondominant case was treated in 2 where we proved that h1 x also always has a simple socle it should likewise be mentioned that j c jantzen has proved that weyl modules which may be identified with hn k n dimg b in the generic case have simple socles 12 6 3 and that j e humphreys in 10 has formulated conjectures which suggest that for x generic h x has the same module structure as a weyl module recently doty and sullivan have announced results which include special cases of our result see their forthcoming paper 8 to formulate some of the consequences of our main theorem we need a little more notation we set w equal to the weyl group of g relative to a maximal torus t in b for w e w we denote by l w the length of w and if x is a generic dominant weight then we let l xw be the simple g module which by our theorem occurs as

simulated annealing for discrete optimization with estimation we extend the basic convergence results for the simulated annealing sa algorithm to a stochastic optimization problem where the objective function is stochastic and can be evaluated only through monte carlo simulation hence disturbed with random error this extension is important when either the objective function cannot be evaluated exactly or such an evaluation is computationally expensive we present a modified sa algorithm and show that under suitable conditions on the random error the modified sa algorithm converges in probability to a global optimizer computational results and comparisons with other approaches are given to demonstrate the performance of the proposed modified sa algorithm

a priori optimization consider a complete graph g v e in which each node is present with probability p we are interested in solving combinatorial optimization problems on subsets of nodes which are present with a certain probability we introduce the idea of a priori optimization as a strategy competitive to the strategy of reoptimization under which the combinatorial optimization problem is solved optimally for every instance we consider four problems the traveling salesman problem tsp the minimum spanning tree vehicle routing and traveling salesman facility location we discuss the applicability of a priori optimization strategies in several areas and show that if the nodes are randomly distributed in the plane the a priori and reoptimization strategies are very close in terms of performance we characterize the complexity of a priori optimization and address the question of approximating the optimal a priori solutions with polynomial time heuristics with provable worst case guarantees finally we use the tsp as an example to find practical solutions based on ideas of local optimality