products in hochschild cohomology and grothendieck rings of group crossed products abstract we give a general construction of rings graded by the conjugacy classes of a finite group some examples of our construction are the hochschild cohomology ring of a finite group algebra the grothendieck ring of the drinfel d double of a group and the orbifold cohomology ring for a global quotient we generalize the first two examples by deriving product formulas for the hochschild cohomology ring of a group crossed product and for the grothendieck ring of an abelian extension of hopf algebras our results account for similarities in the product structures among these examples

clifford correspondence for finite dimensional hopf algebras abstract let b h be a crossed product algebra over an algebraically closed field with h a finite dimensional hopf algebra we give an explicit equivalence between the category of finite dimensional b h modules whose restriction to b is a direct sum of copies of a stable irreducible b module and the category of modules for a twisted product of h with the field this describes all finite dimensional irreducible b h modules containing a stable irreducible b submodule and thus generalizes the classical stable clifford correspondence for groups in case h is cocommutative we extend this correspondence to the nonstable case

localization and ideal theory in noetherian strongly group graded rings abstract we study primality hypercentrality simplicity and localization and the second layer condition in skew groups rings and group graded rings we give necessary and sufficient conditions for the skew group ring of a torsion free nilpotent group to be a simple ring and if the coefficient ring is commutative we give necessary and sufficient conditions for the skew group ring of an abelian group to be simple our method involves showing certain group graded rings are hypercentral our main results show that if g is a polycyclic by finite group and r is an artinian ring or a commutative noetherian ring then a strongly g graded ring with base ring r satisfies the second layer condition we discuss consequences of this for localization in such rings

heavily separable functors abstract prompted by an example related to the tensor algebra we introduce and investigate a stronger version of the notion of separable functor that we call heavily separable we test this notion on several functors traditionally connected to the study of separability

a noether skolem theorem for group graded rings abstract the classical noether skolem theorem gives conditions to determine if an automorphism or a derivation of a finite dimensional central simple algebra is inner the definition of inner has been extended to more general hopf algebra actions in this paper we study group gradings of central simple algebras and we give sufficient conditions for the grading to be inner this provides a partial answer to a question of j bergen and s montgomery

characterization of p schemes of prime cube order abstract let p be a prime let s be a non commutative p scheme of order p 3 and assume that s is not thin we prove that each closed subset of order p 2 of s is an elementary abelian p group we also construct non commutative schurian p schemes of order p 3

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

group graded rings and duality we give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend known results on skew group rings to corresponding results for large classes of group graded rings finally we modify the construction slightly to handle infinite groups introduction in the first section we give an alternate construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend results on skew group rings and their modules to corresponding results for large classes of groupgraded rings in the second section we modify the construction slightly to handle infinite groups and apply it to a theorem on prime and semiprime infinite crossed products i would like to thank my thesis advisor d s passman for his guidance and encouragement throughout the writing of this paper 1 finite group graded rings and the smash product let g be a multiplicative group an associative ring with identity is said to be g graded if 1 r e r x xec is a direct sum of additive subgroups r x with r x r y c r xy it follows that necessarily 1r e r 1 so that each r x is a unitary r l bimodule if r e r we write r x for the component of r in r x so that r exgcr x r is said to be strongly g graded if r x r y r xy for all x y e g throughout this section r is assumed to be g graded where g is a finite group with igi n let mc r denote the set of n x n matrices over r with the rows and columns indexed by the elements of g if a c mg r we write ax y for the entry in the x y position of a then if a 1 c mc r the matrix product a 8 is given by 2 x x y e xx zaz y z e c if u c g is any subset of g let r u ex eur x in particular r r g now suppose h c g is a subgroup of g we define r h c mc r by r h a e ms r ax y e r xhy l that is r h ex ygcr xhy l e x y where e x y is the matrix unit with 1r in the x y position and zeroes elsewhere note that r g mc r received by the editors september 27 1984 1980 mathematics subject classification primary 16a03 c 1985 american mathematical society 0002 9947 85 1 00 25 per page

generalized 2 cocycles of finite groups and maximal orders abstract letrbe a dedekind domain with quotient fieldkand letgbe a finite group let g g r 0 be a generalized 2 cocycle i e not necessarily taking its values in the units ofr and consider the generalized twisted group ringr g in this paper we derive criteria forr gto be a graded maximalr order resp a hereditaryr order or a maximalr order ink g

duality theorems for graded algebras and coalgebras the concepts of smash product and smash coproduct have been proved to be extremely useful tools in the theory of graded rings and graded coalgebras and more recently in the theory of action and coactions of a hopf algebra on an algebra or on a coalgebra once these concepts were defined a natural question appeared how can the smash product resp the smash coproduct be related to a matrix algebra resp to a matrix w x coalgebra the answer is given by the duality theorems proved in 3 for w x graded rings and in 5 for graded coalgebras various duality theorems w x w x for algebras have been proved during the last ten years cf 2 or 10 the purpose of this paper is to study duality for the smash product resp the smash coproduct associated to a g set and a g graded ring resp a g graded coalgebra the mentioned smash product and smash coproduct w x w x w x have been constructed in 10 and 6 it is shown by an example in 12 that for a g graded ring r and a left g set a we do not have a duality

noetherian semigroup algebras abstract it is shown that a semigroup s is finitely generated whenever the semigroup algebra k s is right noetherian and has finite gelfand kirillov dimension or s is a malcev nilpotent semigroup if furthermore s is a submonoid of a finitely generated nilpotent by finite group g then k s is right noetherian if and only if k s is left noetherian or equivalently s satisfies the ascending chain condition on right left ideals the latter condition is completely described in terms of the structure of s in case g is a nilpotent group the quotient group h of s contains a normal subgroup f such that h f is abelian by finite and f s finally also prime goldie contracted semigroup algebras are described

a maschke type theorem for doi hopf modules and applications abstract we prove a maschke type theorem for doi hopf modules a sufficient condition in order to have a maschke type property is that there exists a normalized integral map for the doi hopf datum in question the results are applied to graded modules and to yetter drinfel d modules as another application a characterization of cosemisimple coalgebras over a field is given

a duality theorem for crossed products of hopf algebras several variations have appeared recently see l 5 8 101 among the best are 2 2 1 3 2 2 but the former is restricted to the case of a trivial cocycle 0 and the latter to the case of finite dimensional h the main purpose of this paper is to prove a common generalization of these two theorems in section 2 we first recall preliminaries about crossed products then we introduce some new algebra constructions and we prove related useful results in particular we define an algebra a h op u that will replace a h u in the main theorem and we show a relationship between the two algebras in the new construction u is allowed to be more general for example we may take u h section 3 introduces a subalgebra h of h with a comodule structure co h h h they are our tools to formulate and handle certain awkward assumptions in the main theorem these can be regarded as finiteness conditions and they correspond to the u local finiteness and rl condition of 2 section 4 contains the duality theorem and its proof in section 5 we draw some corollaries in particular we show that 2 2 1 3 2 2 do follow from our theorem 153 0021 8693 92 3 00

clifford theory of simple modules clifford theory primarily is concerned with the representations of normal subgroups and quotient groups of finite groups the basic contribution being due to a h clifford 3 modern developments founded on the concepts of group graded rings and crossed products have shown close relations with commutative and non commutative galois theory this is most conspicuous in stable clifford theory of simple modules suppose n w h g is an extension of finite groups and v is a simple f n module where f is any field since passage from the inertia group to the whole group usually is easy to handle we may and do assume that v is g invariant or h invariant stable then the annihilator i ann v of v generates a homogeneous ideal z of f h and a f h z is a fully g graded algebra with simple l component r f n z general clifford theory is concerned with such g graded algebras a and their modules cf dade 7 lo assume the l component r a is simple artinian as above then the centre k z r has in a natural way the structure of a g field the main object of study is the endomorphism ring e end v of the a module induced from the simple r module v this is a fully g graded algebra as well generalizing wedderburn s theorem we show that a z m e as g graded algebras where d is the dimension of v over the division algebra d end v theorem 2 1 in particular k z d has the same g structure with respect to e and a and e are morita equivalent in some strong sense now assume that dim r d2m2 is finite and that the schur index m m v is relatively prime to the order of the automorphism group g c k of k induced by g then the graded structure of a e can be described by some distinguished cohomology class w v e h g k more precisely w oa v determines a crossed product sz of k with g

galois extensions for co frobenius hopf algebras abstract forhan infinite dimensional co frobenius hopf algebra over a fieldk andaanh comodule algebra the smash producta h ratis linked to the ring of coinvariantsacohby a morita context we use the morita setting to show that for co frobeniush equivalent conditions fora acohto be galois parallel the case ofhfinite dimensional

fixed rings and integrality abstract in this note we offer some variants of a theorem of pare and schelter on integrality in matrix rings we use these to deduce that a ring r is integral over the fixed subring r g under the action of a finite abelian group g provided g r r we also obtain an integrality version of the bergman isaacs theorem

cells in coxeter groups i abstract the purpose of this article is to shed new light on the combinatorial structure of kazhdan lusztig cells in infinite coxeter groups w our main focus is the set d of distinguished involutions in w which was introduced by lusztig in one of his first papers on cells in affine weyl groups we conjecture that the set d has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite coxeter groups moreover to each element of d we assign an explicitly defined set of equivalence relations on w that altogether conjecturally determine the partition of w into left right cells we are able to prove these conjectures only in a special case but even from these partial results we can deduce some interesting corollaries

crossed product algebras defined by separable extensions abstract we generalize the classical construction of crossed product algebras defined by finite galois field extensions to finite separable field extensions by studying properties of rings graded by groupoids we are able to calculate the jacobson radical of these algebras we use this to determine when the analogous construction of crossed product orders yield azumaya maximal or hereditary orders in a local situation thereby we generalize results by haile larson and sweedler

von neumann regularity of smash products associated with g set gradings abstract for a g graded ring r the problem of gr regularity i e von neumann regularity condition on homogeneous elements has been treated earlier see n st sescu and van oystaeyen 1982 11 once the concept of the smash product was introduced this problem has been resumed by several authors this paper is concerned with von neumann regularity of two smash products the smash product r g of the g graded ring r by the group g and the smash product r a of r by a finite left g set a the connections between the regularity of r a and the gr regularity of r are also investigated one consequence of our results is that the smash product r a is a von neumann regular ring if and only if the category g a r gr is regular in stenstrom s sense

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

the extended brauer quotient of n interior g algebras abstract in this note we generalize the extended brauer quotient defined by l puig and y zhou in 4 section 3 to the case of n interior g algebras where n is a normal subgroup of a finite group g we use this extended brauer quotient on permutation algebras to establish some correspondences that generalize known results that hold in the group algebra case

actions of divided power hopf algebras abstract we study prime ideals in enveloping algebra smash products and use a duality construction to obtain results on prime ideals in rings on which divided power hopf algebras act these actions correspond to higher derivations first we consider chains of prime ideals in an enveloping algebra smash product over an arbitrary ring where the lie algebra is assumed to be finite dimensional abelian over a field of positive characteristic we give a bound on the length of such a chain where the ideals all have the same intersection with the coefficient ring then using an explicit construction of a duality theorem of blattner and montgomery in this context we are able to apply results on enveloping algebra smash products to study the invariant ideals of prime ideals in a ring under a locally nilpotent divided power hopf algebra action

duality theorems for rings with actions or coactions graded for s a ring on which g acts as a group of automorphisms let a be s g the skew group ring then from our morita context we obtain g s g z mg s fi where mg is the ring of matrices over s with rows and columns indexed by g and with only finitely many nonzero entries note that if s is a field then mg s sn is dense in end s g with the finite discrete topology s j to obtain our second duality theorem namely that for

semisimple strongly graded rings is semisimple meaning in this paper semisimple artinian has been studied byseveral authors the most classical result is maschke s theorem for group rings for crossed products over elds there is a satisfactory answer given by aljadeffand robinson 3 another partial answer for skew group rings was given byalfaro et al 1 a reduction of the problem to crossed products over divisionrings was rst given by jespers and okninski 10 and a more constructiveversion was given by haefner and del rio 8 so in order to give a complete answer tothe problem there is still a gap between crossed products over division rings andcrossed products over elds the rst aim of the paper is to ll this gap showingthat the semisimplicity question for crossed products over division rings reducesto the same question for crossed products over elds in theorem 3 2 we makethis reductionand then in theorem3 3 we put togetherall the piecesof the puzzle a strongly

a schneider type theorem for hopf algebroids abstract comodule algebras of a hopf algebroid h with a bijective antipode i e algebra extensions b a by h are studied assuming that a lifted canonical map is a split epimorphism of modules of the non commutative base algebra of h relative injectivity of the h comodule algebra a is related to the galois property of the extension b a and also to the equivalence of the category of relative hopf modules to the category of b modules this extends a classical theorem by h j schneider on galois extensions by a hopf algebra our main tool is an observation that relative injectivity of a comodule algebra is equivalent to relative separability of a forgetful functor a notion introduced and analysed hereby

group graded algebras and almost polynomial growth let f be a field of characteristic 0 g a finite abelian group and a a g graded algebra we prove that a generates a variety of g graded algebras of almost polynomial growth if and only if a has the same graded identities as one of the following algebras 1 fcp the group algebra of a cyclic group of order p where p is a prime number and p g 2 ut2g f the algebra of 2 2 upper triangular matrices over f endowed with an elementary g grading 3 e the infinite dimensional grassmann algebra with trivial g grading 4 in case 2 g ez2 the grassmann algebra with canonical z2 grading

convergence in distribution for level continuous fuzzy random sets in this paper we give the relationships between tightness and convergence in distribution for level continuous fuzzy random sets to this end we first obtain some characterizations of compact subsets of the space of level continuous fuzzy sets in r p and establish the tightness for level continuous fuzzy random sets

the laws of large numbers for fuzzy random variables the new attempt of weak and strong law of large numbers for fuzzy random variables is discussed in this paper by proposing the convergence in probability and convergence with probability one for fuzzy random variables we first consider the limit properties of fuzzy numbers by invoking the hausdorff metric and then we extend it to the convergence in probability and convergence with probability one for fuzzy random variables we provide the notion of weak and strong convergence in probability and weak and strong convergence with probability one for fuzzy random variables finally we come up with the weak and strong law of large numbers for fuzzy random variables in weak and strong sense

graded semisimple algebras are symmetric abstract we study graded symmetric algebras which are the symmetric monoids in the monoidal category of vector spaces graded by a group we show that a finite dimensional graded semisimple algebra is graded symmetric the center of a symmetric algebra is not necessarily symmetric but we prove that the center of a finite dimensional graded division algebra is symmetric provided that the order of the grading group is not divisible by the characteristic of the base field

the central limit theorems for fuzzy random variables abstract the new concept of the central limit theorem for fuzzy random variables is discussed in this paper by proposing the convergence in distribution for fuzzy random variables we first consider the limit properties of fuzzy numbers by invoking the hausdorff metric and then we extend it to the weak and strong convergence of fuzzy distribution functions we provide a notion of fuzzy normal distribution then the central limit theorem for fuzzy random variables follows naturally

co frobenius hopf algebras integrals doi koppinen modules and injective objects abstract we investigate hopf algebras with non zero integral from a coalgebraic point of view categories of doi koppinen modules are studied in the special case where the defining coalgebra is left and right semiperfect and several pairs of adjoint functors are constructed as applications we give a very short proof for the uniqueness of the integrals and provide information about injective objects in the category of doi koppinen modules

covers of modules over a fixed ring abstract in this note we find conditions under which it is possible to prove the existence of relative injective covers of any module over the fixed ring r g by means of relative injective covers of modules over the base ring r the same problem is treated for flat covers

frobenius and separable functors for the category of entwined modules over cowreaths ii applications abstract let h be a quasi hopf algebra we apply results obtained in 8 to give necessary and sufficient conditions for the forgetful functor from doi hopf modules two sided hopf modules or yetter drinfeld modules over h to representations of the underlying algebra to be frobenius resp separable we show that in some situations these conditions reduce to the unimodularity and or co semisimplicity of the quasi hopf algebra h

epsilon strongly graded rings separability and semisimplicity abstract we introduce the class of epsilon strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products we determine precisely when an epsilon strongly graded ring is separable over its principal component thereby we simultaneously generalize a result for strongly group graded rings by n st sescu van den bergh and van oystaeyen and a result for unital partial crossed products by bagio lazzarin and paques we also show that the class of unital partial crossed products appears in the class of epsilon strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings thereby we obtain in the special case of unital partial crossed products a short proof of a general result by dokuchaev exel and simon concerning when graded rings can be presented as partial crossed products we also provide some interesting classes of examples of separable epsilon strongly graded rings with finite as well as infinite grading groups in particular we obtain an answer to a question raised by le bruyn van den bergh and van oystaeyen in 1988

advocating the use of imprecisely observed data in genetic fuzzy systems in our opinion and in accordance with current literature the precise contribution of genetic fuzzy systems to the corpus of the machine learning theory has not been clearly stated yet in particular we question the existence of a set of problems for which the use of fuzzy rules in combination with genetic algorithms produces more robust models or classifiers that are inherently better than those arising from the bayesian point of view we will show that this set of problems actually exists and comprises interval and fuzzy valued datasets but it is not being exploited current genetic fuzzy classifiers deal with crisp classification problems where the role of fuzzy sets is reduced to give a parametric definition of a set of discriminant functions with a convenient linguistic interpretation provided that the customary use of fuzzy sets in statistics is vague data we propose to test genetic fuzzy classifiers over imprecisely measured data and design experiments well suited to these problems the same can be said about genetic fuzzy models the use of a scalar fitness function assumes crisp data where fuzzy models a priori do not have advantages over statistical regression

two clifford s theorems for strongly group graded rings abstract let a be a strongly group graded ring in this paper we obtain the main results as following 1 in a a similar result to clifford s theorem of the decomposition of induced modules from normal subgroups it corrects the result of 4 2 frobenius reciprocity for a 3 we extend the theory of stable modules in two ways one of which is the localization theory of rings and modules in some special cases we can obtain the corresponding results of 1 2 3

clifford theory for association schemes abstract clifford theory of finite groups is generalized to association schemes it shows a relation between irreducible complex characters of a scheme and a strongly normal closed subset of the scheme the restriction of an irreducible character of a scheme to a strongly normal closed subset contains conjugate characters with same multiplicities moreover some strong relations are obtained

optimisation under hybrid uncertainty abstract fuzzy stochastic optimisation is emerging as a subfield of mathematical programming the disciplinary matrix of which consists of analysis of mathematical programs under fuzziness and randomness along with methods for solving them the primum movens of this paper is to describe a unifying methodological approach that is suitable for finding a satisfying solution of a mathematical program in the presence of fuzzy data and random variables properties of fuzzy random variables frvs serve as the backdrop to this approach which also lends itself better to handling mathematical programs with fuzzy random coefficients for the paper to be somewhat self contained the notion of frv is briefly discussed and a synopsis of fuzzy stochastic optimisation provided a systematically solved example aimed at illustrating the proposed approach is also included

infinite crossed products and group graded rings in this paper we precisely determine when a crossed product r g is semiprime or prime indeed we show that these conditions ultimately depend upon the analogous conditions for the crossed products r n of the finite subgroups n of g and upon the interrelationship between the normalizers of these subgroups and the ideal structure of r the proof offered here is combinatorial in nature using the a methods and is entirely self contained furthermore since the argument applies equally well to strongly g graded rings we have opted to work in this more general context let g be a multiplicative group and let r be a ring with 1 then a crossed product r g of g over r is an associative ring determined by g r and certain other parameters to be more precise for each x e g there exists an element je r g and every element a e r g is uniquely writable as a finite sum

light leaves and lusztig s conjecture we define a map f with domain a certain subset of the set of light leaves certain morphisms between soergel bimodules introduced by the author in an earlier paper and range the set of prime numbers using results of soergel we prove the following property of f if the image p f l of some light leaf l under f is bigger than the coxeter number of the corresponding weyl group then there is a counterexample to lusztig s conjecture in characteristic p we also introduce the double leaves basis which is an improvement of the light leaves basis that has already found interesting applications in particular it forms a cellular basis of soergel bimodules that allows us to produce an algorithm to find the bad primes for lusztig s conjecture

separable functors for the category of doi hopf modules applications abstract we prove a maschke type theorem for the category of doi hopf modules in fact we give necessary and sufficient conditions for the functor that forgets the c coaction to be separable this leads to a generalized notion of integrals our results can be applied to obtain maschke type theorems for yetter drinfel d modules long dimodules and modules graded by g sets existing maschke type theorems due to doi and the authors are recovered as special cases

separability and triangulated categories abstract we prove that the category of modules over a separable ring object in a tensor triangulated category admits a unique structure of triangulated category which is compatible with the original one this applies in particular to etale algebras more generally we do this for exact separable monads

monads and comonads on module categories let a be a ring and ma the category of right a modules it is well known in module theory that any a bimodule b is an a ring if and only if the functor ab ma ma is a monad or triple similarly an a bimodule c is an a coring provided the functor ac ma ma is a comonad or cotriple the related categories of modules or algebras of ab and comodules or coalgebras of ac are well studied in the literature on the other hand the right adjoint endofunctors homa b and homa c are a comonad and a monad respectively but the corresponding co module categories did not find much attention so far the category of homa b comodules is isomorphic to the category of b modules while the category of homa c modules called c contramodules by eilenberg and moore need not be equivalent to the category of c comodules the purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background this leads to a deeper understanding and characterisations of algebraic structures such as corings bialgebras and hopf algebras for example it turns out that the categories of c comodules and homa c modules are equivalent provided c is a coseparable coring furthermore we describe equivalences between categories of homa c modules and comodules over a coring d in terms of new galois properties of bicomodules finally we characterise hopf algebras h over a commutative ring r by properties of the functor homr h and the category of mixed homr h bimodules this generalises in particular the fact that a finite dimensional vector space h is a hopf algebra if and only if the dual space h is a hopf algebra

hypothesis testing for means in connection with fuzzy rating scale based data algorithms and applications abstract the fuzzy rating scale was introduced as a tool to measure intrinsically ill defined imprecisely valued attributes in a free way thus users do not have to choose a value from a class of prefixed ones like it happens when a fuzzy semantic representation of a linguistic term set is considered but just to draw the fuzzy number that better represents their valuation or measurement the freedom inherent to the fuzzy rating scale process allows users to collect data with a high level of richness accuracy expressiveness diversity and subjectivity what is especially valuable for statistical purposes this paper presents an inferential approach to analyze data obtained by using the fuzzy rating scale more concretely the paper is to be focused on testing different hypothesis about means on the basis of a sound methodology which has been stated during the last years all the procedures that have been developed to this aim will be presented in an algorithmic way adapted to the usual generic fuzzy rating scale based data and they will be illustrated by means of a real life example

a generalization of gabriel s galois covering functors and derived equivalences abstract let g be a group acting on a category c we give a definition for a functor f c c to be a g covering and three constructions of the orbit category c g which generalizes the notion of a galois covering of locally finite dimensional categories with group g whose action on c is free and locally bonded defined by gabriel here c g is defined for any category c and we do not require that the action of g is free or locally bounded we show that a g covering is a universal g invariant functor and is essentially given by the canonical functor c c g by using this we improve a covering technique for derived equivalences also we prove theorems describing the relationships between smash product construction and the orbit category construction by cibils and marcos 2006 without the assumption that the g action is free the orbit category construction by a cyclic group generated by an auto equivalence modulo natural isomorphisms e g the construction of cluster categories is justified by a notion of the colimit orbit category in addition we give a presentation of the orbit category of a category with a monoid action by a quiver with relations which enables us to calculate many examples

on the differential graded eilenberg moore construction abstract the eilenberg moore construction for modules over a differential graded monad is used to study a question of balmer regarding existence of an exact adjoint pair representing an exact monad a bousfield like localization for differential graded categories is realized as a special case of this construction using drinfeld quotients as applications we study some example coming from g equivariant triangulated categories and twisted derived categories

modules graded byg sets duality and finiteness conditions an efficient technique to uncover the relationship between the representations of a group g and the representations of a subgroup h is to consider gradings by grh the class of the left h cosets of g this l w x method of study was initiated by dade in 3 where he describes the structure of the simple objects in the category r gr of g graded r modules for a g graded ring r as a generalization of the classical clifford theory a systematic investigation of modules graded by g sets was started w x w x by nastasescu et al in 9 and then continued in other papers 1 8 5 categorical properties of the category g a r gr of a graded left r modules where r is a g graded ring and a a left g set were studied first and later a series of finiteness conditions for the objects of this category were considered one natural question to ask in this framework is the following if m is an object of g a r gr where a is a finite g set and if n is an r submodule of m then which finiteness conditions i e ascending chain condition descending chain condition existence of krull dimension on n

a duality theorem for vertex operator algebras abstract in this paper we prove that v h h m n v as s local vertex operator algebras where v is a simple vertex operator algebra and h is the group algebra of a finite subgroup of aut v and n dim h

clifford theory and galois theory i abstract we combine galois theory with crossed products by letting grading groups act on ground fields we show that clifford theory for suitable normal subgroups preserves such actions

graded equivalence theory with applications we develop the foundations for graded equivalence theory and apply them to investigate properties such as primeness finite representation type and vertex theory of graded rings the key fact that we prove is that for any two g graded rings r and s such that there is a category equivalence from gr r to gr s that commutes with suspensions then for any subgroup h of g the categories gr h g r and gr h g s of modules graded by the g set of right cosets are also equivalent

new bases of some hecke algebras via soergel bimodules abstract for extra large coxeter systems m s r 3 we construct a natural and explicit set of soergel bimodules d d w w w such that each d w contains as a direct summand or is equal to the indecomposable soergel bimodule b w when decategorified we prove that d gives rise to a set d w w w that is actually a basis of the hecke algebra this basis is close to the kazhdan lusztig basis and satisfies a positivity condition

covers and envelopes over gr gorenstein rings abstract in this paper we study the existence of gorenstein injective envelopes and gorenstein projective and flat covers in the category of graded modules and we relate them with the corresponding envelopes and covers in the category of modules

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space