integrable modules for affine lie superalgebras irreducible nonzero level modules with finite dimensional weight spaces are discussed for nontwisted affine lie superalgebras a complete classification of such modules is obtained for superalgebras of type a m n and c n using mathieu s classification of cuspidal modules over simple lie algebras in other cases the classification problem is reduced to the classification of cuspidal modules over finite dimensional cuspidal lie superalgebras described by dimitrov mathieu and penkov based on these results a complete classification of irreducible integrable in the sense of kac and wakimoto modules is obtained by showing that any such module is of highest weight in which case the problem was solved by kac and wakimoto

generalized verma modules over sln 2 induced from u hn free sln 1 modules abstract a class of generalized verma modules over sl n 2 are constructed from sl n 1 modules which are u h n free modules of rank 1 the necessary and sufficient conditions for these sl n 2 modules to be simple are determined this leads to a class of new simple sl n 2 modules

coherent families of weight modules of lie superalgebras and an explicit description of the simple admissible sl n 1 1 modules abstract let g be a classical lie superalgebra of type i we introduce coherent families of weight g modules with bounded weight multiplicities and establish a correspondence between cuspidal and highest weight submodules of these families by extending mathieu s work ann inst fourier 50 2000 537 this enables us to reduce the description of the g 0 module structure of arbitrary simple weight g modules with bounded weight multiplicities to the g 0 module structure of highest weight modules with bounded weight multiplicities we then construct tensor coherent families for g sl n 1 1 which yield an explicit description of the g 0 structure of an arbitrary simple weight module with bounded weight multiplicities in particular we show that for g sl n 1 1 the maximal length of an indecomposable g 0 submodule of a simple weight module with bounded multiplicities equals 5

on some natural complete operators abstract an operator is a mapping from integer functions to integer functions it is known from a result by baker gill and solovay 1975 that there exists an operator computable in polynomial time by a nondeterministic oracle turing machine called an np operator but not computable in polynomial time by any deterministic oracle turing machine we investigate several natural operators which share similar properties we use the concept of completeness to give a precise classification of the complexity of these operators for example the question of finding maximum values of polynomial time computable functions can be formulated as an operator complete for the class of np operators

oracle dependent properties of the lattice of np sets abstract we consider under the assumption p np questions concerning the structure of the lattice of np sets together with the sublattice p we show that two questions which are slightly more complex than the known splitting properties of this lattice cannot be settled by arguments which relativize the two questions which we consider are whether every infinite np set contains an infinite p subset and whether there exists an np simple set we construct several oracles all of which make p np and which in addition make the above mentioned statements either true or false in particular we give a positive answer to the question raised by bennett and gill 1981 whether an oracle b exists making p b np b and such that every infinite set in np b has an infinite subset in p b the constructions of the oracles are finite injury priority arguments

singular gelfand tsetlin modules of gl n abstract the classical gelfand tsetlin formulas provide a basis in terms of tableaux and an explicit action of the generators of gl n for every irreducible finite dimensional gl n module these formulas can be used to define a gl n module structure on some infinite dimensional modules the so called generic gelfand tsetlin modules the generic gelfand tsetlin modules are convenient to work with since for every generic tableau there exists a unique irreducible generic gelfand tsetlin module containing this tableau as a basis element in this paper we initiate the systematic study of a large class of non generic gelfand tsetlin modules the class of 1 singular gelfand tsetlin modules an explicit tableaux realization and the action of gl n on these modules is provided using a new construction which we call derivative tableaux our construction of 1 singular modules provides a large family of new irreducible gelfand tsetlin modules of gl n and is a part of the classification of all such irreducible modules for n 3

on the existence of mathfrak g mathfrak k modules of finite type let g be a reductive lie algebra over an algebraically closed field of characteristic zero and k be a subalgebra reductive in g we prove that g admits an irreducible g k module m which has finite k multiplicities and which is not a g k 0 module for any proper inclusion of reductive subalgebras k k 0 g if and only if k contains its centralizer in g the main point of the proof is a geometric construction of g k modules which is analogous to cohomological induction for g gl n we show that whenever k contains its centralizer there is an irreducible g k module m of finite type over k such that k coincides with the subalgebra of all g 2 g which act locally finitely on m finally for a root subalgebra k gl n we describe all possibilities for the subalgebra l k of all elements acting locally finitely on some m math subject classification 2000 primary 17b10 secondary 22e46

cuspidal representations of sl n 1 abstract in this paper we study the subcategory of cuspidal modules of the category of weight modules over the lie algebra sl n 1 our main result is a complete classification and an explicit description of indecomposable cuspidal modules

generalized verma modules induced from sl 2 c and associated verma modules abstract with each generalized verma module induced from a well embedded parabolic subalgebra of a lie algebra with triangular decomposition we associate a verma module over the same algebra in a natural way in the case when the semisimple part of the levi factor of the parabolic subalgebra is isomorphic to sl 2 c and the generalized verma module is induced from an infinite dimensional simple module we prove that the associated verma module is simple if and only if the original generalized verma module is simple

new singular gelfand tsetlin mathfrak gl n modules of index 2 singular gelfand tsetlin modules of index 2 are modules whose tableaux bases may have singular pairs but no singular triples of entries on each row in this paper we construct singular gelfand tsetlin modules for arbitrary singular character of index 2 explicit bases of derivative tableaux and the action of the generators of mathfrak gl n are given for these modules our construction leads to new families of irreducible gelfand tsetlin modules and also provides tableaux bases for some simple verma modules

root fernando kac subalgebras of finite type abstract let g be a finite dimensional lie algebra and m be a g module the fernando kac subalgebra of g associated to m is the subset g m g of all elements g g which act locally finitely on m a subalgebra l g for which there exists an irreducible module m with g m l is called a fernando kac subalgebra of g a fernando kac subalgebra of g is of finite type if in addition m can be chosen to have finite jordan holder l multiplicities under the assumption that g is simple i penkov has conjectured an explicit combinatorial criterion describing all fernando kac subalgebras of finite type which contain a cartan subalgebra in the present paper we prove this conjecture for g e 8

on the structure of weight modules given any simple lie superalgebra g we investigate the structure of an arbitrary simple weight 0 module we introduce two invariants of simple weight modules the shadow and the small weyl group generalizing results of fernando and futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a levi subsuperalgebra then we classify the cuspidal levi subsuperalgebras of all simple classical lie superalgebras and of the lie superalgebra w n most of them are simply levi subalgebras of go in which case the classification of all finite cuspidal representations has recently been carried out by one of us mathieu our results reduce the classification of the finite simple weight modules over all classical simple lie superalgebras to classifying the finite cuspidal modules over certain lie superalgebras which we list explicitly

complete reducibility of torsion free cn modules of finite degree abstract we show that every torsion free weight module with finite dimensional weight spaces over a symplectic complex lie algebra which is different from sp 2 c is completely reducible

extensions of the conformal representations for orthogonal lie algebras abstract the conformal transformations with respect to the metric defining o n c give rise to an inhomogeneous polynomial representation of o n 2 c using shen s technique of mixed product we generalize the above representation to an inhomogeneous representation of o n 2 c on the tensor space of any finite dimensional irreducible o n c module with the polynomial space where a hidden central transformation is involved moreover we find a condition on the constant value taken by the central transformation such that the generalized representation is irreducible in our approach pieri s formulas invariant operators and the idea of kostant s characteristic identities play key roles the result could be useful in understanding higher dimensional conformal field theory with the constant value taken by the central transformation as the central charge our representations virtually provide natural extensions of the conformal transformations on a riemannian manifold to its vector bundles

sparse sets in np p exptime versus nexptime this paper investigates the structural properties of sets in np p and shows that the computational difficulty of lower density sets in np depends explicitly on the relations between higher deterministic and nondeterministic time bounded complexity classes the paper exploits the recently discovered upward separation method which shows for example that there exist sparse sets in np p if and only if exptime nexptime in addition the paper uses relativization techniques to determine logical possibilities limitations of these proof techniques and exhibits one of the first natural structural differences between relativized np and conp

simple sl_ n 1 module structures on u h abstract we study the category m consisting of u sl n 1 modules whose restriction to u h is free of rank 1 in particular we classify isomorphism classes of objects in m and determine their submodule structure this leads to new sl n 1 modules for n 1 we also find the central characters and derive an explicit formula for taking tensor product with a simple finite dimensional module

supports of weight modules over witt algebras as the first step towards a classification of simple weight modules with finite dimensional weight spaces over witt algebras w n we explicitly describe the supports of such modules we also obtain

eigenvalue coincidences and k orbits i abstract we study the variety g l consisting of matrices x gl n c such that x and its n 1 by n 1 cutoff x n 1 share exactly l eigenvalues counted with multiplicity we determine the irreducible components of g l by using the orbits of gl n 1 c on the flag variety b of gl n c more precisely let b b be a borel subalgebra such that the orbit gl n 1 c b in b has codimension l then we show that the set y b ad g x x b g l g gl n 1 c is an irreducible component of g l and every irreducible component of g l is of the form y b where b lies in a gl n 1 c orbit of codimension l an important ingredient in our proof is the flatness of a variant of a morphism considered by kostant and wallach and we prove this flatness assertion using an analogue of the steinberg variety

irreducible completely pointed modules of quantum groups of type a abstract we give a classification of all irreducible completely pointed u q sl n 1 modules over a characteristic zero field in which q is not a root of unity this generalizes the classification result of benkart britten and lemire in the non quantum case we also show that any infinite dimensional irreducible completely pointed u q sl n 1 can be obtained from some irreducible completely pointed module over the quantized weyl algebra a n 1 q

sparse sets in np p relativizations we construct an oracle relative to which bf p ne bf np and there are no sparse sets in bf np bf p the well known construction of baker gill and solovay siam j comput 4 1975 pp 431 442 gives an oracle relative to which there is a sparse set in bf np bf p together these results show that simple modifications of conventional proof techniques cannot establish whether or not sparse sets exist in bf np bf p even if one assumes bf p ne bf np

finite weight modules over twisted affine lie superalgebras abstract this work provides the first step toward the classification of irreducible finite weight modules over twisted affine lie superalgebras we divide the class of such modules into two subclasses called hybrid and tight we reduce the classification of hybrid irreducible finite weight modules to the classification of cuspidal modules of finite dimensional cuspidal lie superalgebras which is discussed in a work of dimitrov mathieu and penkov

a koszul category of representations of finitary lie algebras abstract we find for each simple finitary lie algebra g a category t g of integrable modules in which the tensor product of copies of the natural and conatural modules is injective the objects in t g can be defined as the finite length absolute weight modules where by absolute weight module we mean a module which is a weight module for every splitting cartan subalgebra of g the category t g is koszul in the sense that it is antiequivalent to the category of locally unitary finite dimensional modules over a certain direct limit of finite dimensional koszul algebras we describe these finite dimensional algebras explicitly we also prove an equivalence of the categories t o and t sp corresponding respectively to the orthogonal and symplectic finitary lie algebras o sp

nonlinear mean filters in image processing the use of nonlinear means in image processing is introduced the properties of these means in the presence of different types of noise are investigated it is shown that nonlinear filters based on these means behave well for both additive and impulse noise their performance in the presence of signal dependent noise is satisfactory they preserve the edges better than linear filters and they reject the noise better than median filters

a combinatorial description of blocks in o p λ associated with sl 2 induction abstract we study the category o p where is an admissible category of dense weight sl 2 modules we give a combinatorial description of projectively stratified algebras arising from o p and prove a double centralizer property moreover we determine the characters of tilting modules in o p and prove that the finite dimensional algebra associated with the principal block of our o p is its own ringel dual

determination of means by invariance abstract if m is a mean on r n and m f x 1 f x 2 f x n f m x 1 x 2 x n then we say that m is invariant under f the problem is to find a class of functions that by invariance determines a mean uniquely we focus on the geometric mean which can be transformed to obtain results for other means

reductions among polynomial isomorphism types abstract a set a is polynomial many one reducible to a set b a is karp reducible to b if there is a polynomially computable function f such that for all x x a iff f x b arbitrary sets a and b are of the same polynomial many one degree if each is polynomial many one reducible to the other a and b are polynomially isomorphic if the function f can be taken one to one onto and polynomially invertible in classical recursive function theory all many one complete sets are recursively isomorphic berman and hartmanis have observed that all known np complete sets are polynomially isomorphic berman and hartmanis have observed that all known np complete sets are polynomially isomorphic and have conjectured that all np complete sets complete under karp reducibility are isomorphic in this paper we show that not just the complete degree but every polynomial many one degree consists either of a single isomorphism type or else contains infinitely many isomorphism types densely ordered under one one size increasing polynomially invertible reductions and also contains infinitely many isomorphisms types which are incomparable under one one invertible reductions in fact we show that every countable partial ordering can be embedded in any such many one degree we also exhibit polynomial degrees which have infinitely many isomorphism types no examples are known of degrees consisting of a single isomorphism type

galois orders in skew monoid rings abstract we introduce a new class of noncommutative rings galois orders realized as certain subrings of invariants in skew semigroup rings and develop their structure theory the class of galois orders generalizes classical orders in noncommutative rings and contains many important examples such as the generalized weyl algebras the universal enveloping algebra of the general linear lie algebra associated yangians and finite w algebras

classification of irreducible nonzero level modules with finite dimensional weight spaces for affine lie algebras abstract we classify the irreducible weight affine lie algebra modules with finite dimensional weight spaces on which the central element acts nontrivially in particular we show that any such module is a quotient of a generalized verma module the classification of such irreducible modules is reduced to the classification of irreducible torsion free modules for algebras of type a n and c n which is due to o mathieu forthcoming

on modules of bounded multiplicities for the symplectic algebras simple infinite dimensional highest weight modules having bounded weight multipicities are classified as submodules of a tensor product also it is shown that a simple torsion free module of finite degree tensored with a finite dimensional module is completely reducible

lie algebra modules with finite dimensional weight spaces i let g denote a reductive lie algebra over an algebraically closed field of characteristic zero and let 4 denote a cartan subalgebra of g in this paper we study finitely generated g modules that decompose into direct sums of finite dimensional 4 weight spaces we show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules those we call torsion free modules we also show that if g is a simple lie algebra that admits a torsion free module then g is of type a or c

modular representations of the special linear groups with small weight multiplicities abstract we classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities for the natural embeddings of the classical groups inductive systems of representations with totally bounded weight multiplicities are classified an analogue of the steinberg tensor product theorem for arbitrary indecomposable inductive systems for such embeddings is proved

faces and maximizer subsets of highest weight modules abstract in this paper we study general highest weight modules v over a complex finite dimensional semisimple lie algebra g we present three formulas for the set of weights of a large family of modules v which include but are not restricted to all simple modules and all parabolic verma modules these formulas are direct and do not involve cancellations and were not previously known in the literature our results extend the notion of the weyl polytope to general highest weight g modules v we also show that for all simple modules the convex hull of the weights is a w j invariant polyhedron for some parabolic subgroup w j we compute its vertices faces and symmetries more generally we also do this for all parabolic verma modules and for all modules v with highest weight not on a simple root hyperplane to show our results we extend the notion of convexity to arbitrary additive subgroups a r of coefficients our techniques enable us to completely classify weak a faces of the support sets wt v in the process extending classical results of satake borel tits vinberg and casselman as well as modern variants by chari dolbin ridenour and cellini marietti to general highest weight modules

downward translations of equality in this paper we construct oracles relative to which dtime t n equals ntime t n and dtime t n does not equal ntime t n for t n sufficiently smaller than t n a stronger result than this is also obtained though for fewer t n expressedin two parts for t n 2n there is an oracle relative to which dtime t n equals ntime t n and ntime 2n contains a set not in dtime t n for any t n growing more slowly than t n for t n 22n 0 1 there is an oracle relative to which dtime t n equals ntime t n and ntime log t n contains a set not in dtime t n for t n growing more slowly than t n these results expand on those obtained by dekhtyar 1976 book et al 1982 and allender 1989

axiomatic characterization of nonlinear homomorphic means one important problem in mathematics and statistics is to formulate a mean from a set of data various definitions of mean have been studied to represent in some sense an average of data the applications of these means involve all the fields of science and technology with nonlinear filtering being a primary example in the area of image processing 1 3 the goal of this work is to axiomatically investigate the rationale for a large class of means which is referred to as quasi arithmetic mean 4 nonlinear homomorphic mean or nonlinear homomorphic filter 3

cuspidal sln modules and deformations of certain brauer tree algebras we show that the algebras describing blocks of the category of cuspidal weight resp generalized weight sln modules are one parameter resp multi parameter deformations of certain brauer tree algebras we explicitly determine these deformations both graded and ungraded the algebras we deform also appear as special centralizer subalgebras of temperley lieb algebras or as generalized khovanov algebras they show up in the context of highest weight representations of the virasoro algebra in the context of rational representations of the general linear group and schur algebras and in the study of the milnor fiber of kleinian singularities

relativizing time space and time space consider the classes of formal languages specified by nondeterministic acceptors that operate simultaneously within time bounds from a set mathcal t and space bounds from a set mathcal s how large must the time bounds be in order to obtain all of the languages specified by nondeterministic acceptors that operate within space bounds from mathcal s how large must the space bounds be in order to obtain all of the languages specified by nondeterministic acceptors that operate with time bounds from mathcal t the first question is shown to be equivalent with appropriate restrictions on mathcal s and mathcal t to the question of whether it matters if the time bounds apply to all of the steps or only to the steps which query the oracle the second question is shown to be equivalent to the question of whether it matters if the space bounds apply to all of the configurations or only to the configurations in which the oracle is queried these results generalize a more specific result 6

large annihilator category o for sl o sp abstract we construct a new analogue of the bgg category o for the infinite dimensional lie algebras g sl o sp a main difference with the categories studied in 9 and 2 is that all objects of our category satisfy the large annihilator condition introduced in 5 despite the fact that the splitting borel subalgebras b of g are not conjugate one can eliminate the dependency on the choice of b and introduce a universal highest weight category ola of g modules the letters la coming from large annihilator the subcategory of integrable objects in ola is precisely the category t g studied in 5 we investigate the structure of ola and in particular compute the multiplicities of simple objects in standard objects and the multiplicities of standard objects in indecomposable injectives we also complete the annihilators in u g of simple objects of ola

a classification of simple lie modules having a 1 dimensional weight space let l denote a simple lie algebra over the complex numbers in this paper we classify and construct all simple l modules which may be infinite dimensional but have at least one 1 dimensional weight space this completes the study begun earlier by the authors for the case of l a the approach presented here relies heavily on the results of suren fernando whose dissertation dealt with simple weight modules and their weight systems 0 introduction let l be a finite dimensional simple lie algebra over the complex field c having a cartan subalgebra h and denote by c l the centralizer of h in the universal enveloping algebra u of l if x h c is a weight function of a 1 dimensional weight space mx in a simple l module m then q c l c defined by q c v cv for v e mx and c e c l is an algebra homomorphism called a mass function of m clearly x restricted to h is equal to x conversely given any algebra homomorphism q c l c one can construct a unique simple l module which admits x as a mass function 4 10 in 4 the authors determined all algebra homomorphisms q c l c for the simple lie algebras l of type an and using these classified all pointed an modules where we call a module pointed if it is simple and has at least one 1 dimensional weight space in this paper we complete the classification of all pointed l modules for arbitrary simple lie algebras the collection of all pointed l modules clearly includes the highest weight l modules and is included in the collection of all harish chandra l modules relative to the cartan subalgebra h i e simple l modules having a weight space decomposition and finite dimensional weight spaces relative to h the latter inclusion is strict since there exist examples of harish chandra a2 modules in which every weight space is two dimensional in the special case of a1 modules every harish chandra a1 module is pointed our approach to this problem makes heavy use of the results from fernando s thesis 8 in particular from fernando s results we see that in place of determining all algebra homomorphisms q c l c it suffices to find only those algebra homomorphisms x which are associated with so called pointed torsion free l received by the editors may 23 1984 this paper was presented at the 1984 canadian mathematical society s summer seminar on lie algebras and related topics held at the university of windsor 1980 mathematics subject classification primary 17b10

irreducible representations of _ with a 1 dimensional weight space in this paper we classify all irreducible linear representations of the simple lie algebra an which admit a one dimensional weight space with respect to some cartan subalgebra h of a we first show that the problem is equivalent to determining all algebra homomorphisms from the centralizer of the cartan subalgebra h in the universal enveloping algebra of an to the base field we construct all such algebra homomorphisms and provide conditions under which two such algebra homomorphisms provide inequivalent irreducible representations of an

weight modules for current algebras for any finite dimensional simple lie algebra g and commutative associative algebra s of finite type we give a complete classification of the simple weight modules of g s with bounded weight multiplicities the classification of simple weight modules is widely seen as a difficult problem in a spectacular tour de force olivier mathieu gave a complete description of such modules for finite dimensional simple lie algebras ma the crucial part of mathieu s classifi cation was understanding the simple admissible highest weight modules in the present paper we consider this question in the context of generalised current algebras l g k s where g is a finite dimensional simple lie algebra over k and s is a commutative associative and unital k algebra which is of finite type these algebras include the finite dimensional simple lie algebras g and the loop algebras g c c t t 1 of affine kac moody theory our methods work not only for the classification of simple admissible highest weight modules but also describe the simple objects in the much larger category of admissible modules weight modules whose weight multiplicities are uniformly bounded we hope that this paper will contribute to solving the harder problem of classifying all simple weight modules for these algebras the category of admissible modules includes the finite dimensional simple modules all of which are known to be evaluation modules see the introductions of l1 cfk or the more general results in nss lp l2 for details what is surprising is that this larger category of generally infinite dimensional modules can also be described in terms of evaluations we now summarize the content of this paper funding from the natural sciences and engineering research council of canada is gratefully ac knowledged

z graded oscillator representations of symplectic lie algebras abstract in this paper we study two parameter z graded oscillator representations of symplectic lie algebras obtained from the natural oscillator representation by partially swapping differential operators and multiplication operators we prove that the homogeneous subspaces with nonzero degree or unequal parameters are infinite dimensional irreducible weight submodules with finite dimensional weight subspaces when the two parameters are equal the homogeneous subspace of degree zero is exactly a direct sum of two infinite dimensional irreducible weight submodules with finite dimensional weight subspaces in generic case these irreducible modules are non unitary modules without highest weight vectors our results are extensions of howe s oscillator construction of infinite dimensional multiplicity free irreducible representations for special linear lie algebras