weyl groups of fine gradings on matrix algebras octonions and the albert algebra given a grading a g gag on a nonassociative algebra a by an abelian group g we have two subgroups of aut a the automorphisms that stabilize each component ag as a subspace and the automorphisms that permute the components by the weyl group of we mean the quotient of the latter subgroup by the former in the case of a cartan decomposition of a semisimple complex lie algebra this is the automorphism group of the root system i e the so called extended weyl group a grading is called fine if it cannot be refined we compute the weyl groups of all fine gradings on matrix algebras octonions and the albert algebra over an algebraically closed field of characteristic different from 2 in the case of the albert algebra

graded morita theory for infinite groups abstract having shown that not all graded rings are graded equivalent via classical morita theory to a skew group ring we extend classical morita theory which is based on rings with identity to a generalized graded morita theory for rings with local units this enables us to give necessary and sufficient conditions for graded morita equivalence between two rings graded by a group we show that the strongly graded property is a graded morita invariant and we show that a graded ring is graded morita equivalent to a skew group ring namely r g g if and only if it is strongly graded these results extend the classical cohen montgomery duality theory to rings graded by infinite groups the fundamental tools include rings with local units and the results of boisen

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

the brauer splitting theorem and projective representations of finite groups over rings in the theory of representations of finite groups the projective representations arise naturally when one studies the relations between the representations of the group and representations of certain subgroups the role played by the group rings in representation theory is taken by the twisted groups rings when one considers projective representations for representations over algebraically closed fields the theory of schur multipliers provides a very satisfactory tool that may be used to lift projective representations of a finite group to usual representations of a finite central extension of that group furthermore there exists a satisfactory theory of projective characters block theory for projective representations etc for which we may refer to karpilovsky s recent book 3 since for finite groups the twisted group rings defined over commutative rings are azumaya algebras when a mild condition holds it is natural to ask for an equivalent of the brauer splitting theorem for group rings in this paper we derive this splitting theorem for twisted group rings over commutative rings without nontrivial idempotents this is obtained by introducing a suitable extension of the schur multiplier theorem in such a way that this result may be applied to a single representation and over arbitrary fields or rings of the forementioned type along the way we get involved in some cocycle manipulations that will lead to the determination of the center of a twisted group ring in terms of ray classes the results derived in this paper should be of some use in the study of properties and

group gradings on associative algebras abstract let r g g rg be a g graded ring we describe all types of gradings on r if g is torsion free and r is artinian semisimple if r is a matrix algebra over an algebraically closed field f then we give a description of all g gradings on r provided that g is an abelian group in the case of an abelian group g we also classify all finite dimensional graded simple algebras and finite dimensional graded division algebras over an algebraically closed field of characteristic zero

representability and specht problem for g graded algebras abstract let w be an associative pi algebra over a field f of characteristic zero graded by a finite group g let id g w denote the t ideal of g graded identities of w we prove 1 g graded pi equivalence there exists a field extension k of f and a finite dimensional z 2 z g graded algebra a over k such that id g w id g a where a is the grassmann envelope of a 2 g graded specht problem the t ideal id g w is finitely generated as a t ideal 3 g graded pi equivalence for affine algebras let w be a g graded affine algebra over f then there exists a field extension k of f and a finite dimensional algebra a over k such that id g w id g a

graded identities of matrix algebras and the universal graded algebra in the last decade group gradings and graded identities of finite dimensional central simple algebras have been an active area of research we refer the reader to bahturin et al 6 and 7 there are two basic kinds of group grading elementary and fine it was proved by bahturin and zaicev 7 that any group grading of mn c is given by a certain composition of an elementary grading and a fine grading in this paper we are concerned with fine gradings on mn c and their corresponding graded identities let r be a simple algebra finite dimensional over its center k and g a finite group we say that r is fine graded by g if r g grg is a grading and dimk rg 1 thus any component is either 0 or isomorphic to k as a k vector space it is easy to show that supp r the subset of elements of g for which rg is not 0 is a subgroup of g moreover

grusin operators riesz transforms and nilpotent lie groups we establish that the riesz transforms of all orders corresponding to the grusin operator h n nabla x 2 x 2n nabla y 2 and the first order operators nabla x x nu nabla y where x in mathbf r n y in mathbf r m n in mathbf n and nu in 1 ldots n n are bounded on l p mathbf r n m for all p in langle 1 infty rangle and are also weak type 1 1 moreover the transforms of order less than or equal to n 1 corresponding to h n and the operators nabla x x n nabla y are bounded on l p mathbf r n m for all p in langle 1 infty rangle but if n is odd all transforms of order n 2 are bounded if and only if p in langle 1 n rangle the proofs are based on the observation that the nabla x x nu nabla y generate a finite dimensional nilpotent lie algebra the corresponding connected simply connected nilpotent lie group is isometrically represented on the spaces l p mathbf r n m and h n is the corresponding sublaplacian

graded artin algebras a graded artin algebra is a graded ring which neglecting the grading is an artin algebra although this definition has the merit of brevity it does not expose some of the basic properties of graded artin algebras for instance that there is a bound on absolute values of degrees of homogeneous elements indeed a different definition is given in section 1 and the discrepancy is rectified by theory 1 4 we admit that our chief interest in graded artin algebras lies in applications to the representation theory of artin algebras see 5 1 we feel nonetheless that graded artin algebras are interesting objects of study in their own right and some of our techniques especially in sections 5 and 6 are equally applicable to arbitrary graded rings thus this paper unlike its sequel 51 is designed to be read by a general ring theoretical audience it is in addition self contained from the standpoint of knowledge concerning graded rings required the major concern of the paper is the existence of gradings on given modules over a graded artin algebra we use the term gradable when a gradation exists our transcendent result along these lines is that direct summands of finitely generated gradable modules are gradable it follows immediately see section 3 that simple modules and projective modules over a graded artin algebra are gradable also inasmuch as we do not restrict ourselves to positively graded rings endomorphism rings of graded finitely generated modules over graded artin algebras are graded artin algebras in a similar vein duality and horn

analysis of degenerate elliptic operators of grusin type we analyse degenerate second order elliptic operators h in divergence form on l 2 r n r m we assume the coefficients are real symmetric and a 1 h h a 2 h for some a 1 a 2 0 where h delta nabla x 1 cdot c delta 1 delta 1 x 1 nabla x 1 c delta 2 delta 2 x 1 nabla x 2 2 here x 1 r n x 2 r m and c delta i delta i are positive measurable functions such that c delta i delta i x behaves like x delta i as x 0 and x delta i as x to infty with delta 1 delta 1 in 0 1 rangle and delta 2 delta 2 geq 0 our principal results state that the submarkovian semigroup s t e th is conservative and its kernel k t satisfies bounds 0 leq k t x y leq a b x t 1 2 b y t 1 2 1 2 where b x r denotes the volume of the ball b x r centred at x with radius r measured with respect to the riemannian distance associated with h the proofs depend on detailed subelliptic estimations on h a precise characterization of the riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation we discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive nor continuous

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

on the structure of graded commutative algebras abstract consider a a commutative algebra graded by means of an abelian group g we show that under the hypothesis a 0 g a g a g where g g a g 0 the algebra a is of the form a j i j with any i j a well described graded ideal of a satisfying i j i k 0 if j k under certain conditions the simplicity of a is characterized and it is shown that a is the direct sum of the family of its minimal graded ideals each one being a simple graded commutative algebra

actions of picard groups on graded rings throughout this paper g denotes a group with identity e if r is a g graded ring we write r g grg and we refer to re as the coefficient ring we denote the category of graded left r modules i e those left r modules m with a g grading m g gmg such that rhmg mhg by r gr the aim of this paper is to present a method to reduce the study of one strongly graded ring r to the study of another strongly graded ring r that is more tractable this reduction process has two aspects reducing the coefficient ring or reducing the grading since we apply this method in our study of module theoretic properties such as semisimplicity for graded rings we require that this reduction process preserve the category of modules r mod and the category of graded modules r gr this leads us to the notion of graded equivalence in this paper a morita context is the usual tuple a b p q where we assume and are isomorphisms a g graded context is a morita context r r p q where r and r are g graded rings rpr and r qr are graded bimodules i e they have a grading which makes them into both graded r modules and graded r modules on the corresponding side and and are graded bimodule homomorphisms see 14 for details about graded modules and homomorphisms given a functor f r gr r gr we say that f is a graded equivalence of categories if any of the following equivalent conditions hold see for example 8

graded embeddings of finite dimensional simple graded algebras abstract let a b be finite dimensional g graded algebras over an algebraically closed field k with char k 0 where g is an abelian group and let id g a be the set of graded identities of a resp id g b we show that if a b are g simple then there is a graded embedding a b if and only if id g b id g a we also give a weaker generalization for the case where a is g semisimple and b is arbitrary

the spectral parabolic equation and three dimensional backscattering problems involving dependence of the acoustic parameters in depth and one of the horizontal cartesian coordinates are solved using the cross range wave number spectrum and the parabolic equation pe method the self starter and the two way pe are used to solve the two dimensional wave equation that is obtained after transforming to wave number space although the spectral pe solution breaks down for energy that propagates nearly parallel to the cross range axis it is accurate in other directions for both outgoing and backscattered energy the spectral pe is practical for solving large scale reverberation problems in three dimensions

identities and isomorphisms of graded simple algebras abstract let g be an arbitrary abelian group and let a and b be two finite dimensional g graded simple algebras over an algebraically closed field f such that the orders of all finite subgroups of g are invertible in f we prove that a and b are isomorphic if and only if they satisfy the same g graded identities we also describe all isomorphism classes of finite dimensional g graded simple algebras

on the graded identities for elementary gradings in matrix algebras over infinite fields we find a basis for the g graded identities of the n n matrix algebra mn k over an infinite field k of characteristic p 0 with an elementary grading such that the neutral component corresponds to the diagonal of mn k

hopf cocycle deformations and invariant theory for a given finite dimensional hopf algebra h we describe the set of all equivalence classes of cocycle deformations of h as an affine variety using methods of geometric invariant theory we show how our results specialize to the universal coefficients theorem in the case of a group algebra and we also give examples from other families of hopf algebras including dual group algebras and bosonizations of nichols algebras in particular we use the methods developed here to classify the cocycle deformations of a dual pointed hopf algebra associated to the symmetric group on three letters we also give an example of a cocycle deformation over a dual group algebra which has only rational invariants but which is not definable over the rational field this differs from the case of group algebras in which every 2 cocycle is equivalent to one which is definable by its invariants

three dimensional propagation modeling in shallow water much interest exists in problems of shallow water propagation where three dimensional spatially dependent waveguide effects are sometimes very large in these cases the three dimensional dependence of the acoustic field can be accounted for in a variety of ways other than solving the often intractable fully coupled problem there exist several pseudo three dimensional models that are based on the parabolic equation method e g n 2d pe j s perkins and r n bear j acoust soc am 72 515 522 1982 the adiabatic pe m d collins j acoust soc am 93 2269 2278 1993 the coupled mode pe a t abawi et al j acoust soc am 102 233 238 1997 and the spectral pe g j orris and m d collins j acoust soc am 96 3499 3503 1994 yet because these models are extensions of efficient two dimensional models they have explicit and implicit limitations on their applicability some of the advantages and limitations of these pseudo three dimensional models will be demonstrated through their application to data collected during the swellex 96 and rds 1 matched field experiments

on groups of central type a finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements equivalently the degree of this character is the square root of the index of the center in 1964 iwahori and matsumoto 15 conjectured that a group of central type must be solvable the paper of liebler and yellen 16 aims to prove this but there is a gap in their proof as explained below nevertheless they do correctly obtain substantial information about a minimal counterexample to the conjecture the aim of this paper is to fill the gap in the liebler yellen proof using as they do the classification of finite simple groups and to provide further information about which solvable groups can have central type we show for instance that every normal subgroup of the central factor group of a solvable group of central type has the property that its system normalizers have square index it is interesting to contrast this restrictive condition with the result of gagola that every solvable group is embeddable in the central factor group of a group of central type see 6 theorem 1 2 the arguments in 16 show that a minimal counterexample g to the iwahori matsumoto conjecture must have the following structure if k z is any minimal normal subgroup of g z where z z g the center then k is abelian k z is a 2 group s cg k is the maximal solvable normal subgroup of g and any minimal normal subgroup his of g s is the direct product of a number l of copies of gl 3 2 indeed the liebler yellen methods can be used to show that the sylow 2 subgroup of k considered as an h s module is a product o f copies of a gl 3 2 module of dimension 4 this module having a unique minimal submodule which is trivial and an irreducible factor module of dimension 3 the minimal submodules are amalgamated in the product

graphs with relations coverings and group graded algebras the paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs to include gradings of all basic finite dimensional algebras over an algebraically closed field a theory of coverings of graphs with relations is introduced the object of this paper is to relate group gradings on algebras to coverings of a graph which is associated to the algebra the linking of the theories allows one to relate purely algebraic questions to questions in algebraic topology group theory or combinatorics in the representation theory of artin algebras the association to each algebra of a finite directed graph called the quiver of the algebra has been a useful tool the reason that the quiver of such an algebra is of interest is that there is a natural definition of representations of the quiver so that the category of representations of the quiver satisfying certain relations is equivalent to the category of finitely generated modules over the algebra 1 gives a slight extension of these concepts to finitely generated algebras the main emphasis of the paper is to show that the theory of coverings of graphs with relations introduced by c riedtmann 9 and expanded by p gabriel 2 and the theory of group graded algebras are essentially the same although the original connection between coverings and gradings was inspired by the similarity of results for z graded artin algebras 3 4 and p gabriel s announced results 2 the context of this paper is more general and deals with all finitely generated algebras over a field we associate to each such algebra a finite directed graph which we still call a quiver of the algebra we show that for each regular covering t of a quiver t0 of an algebra a with certain prescribed restrictions we get a g grading of the algebra a where g is the automorphism group of the covering y over ro conversely given a certain type of g grading of a where g is a group we construct a regular covering y of the quiver t0 of the algebra such that g is isomorphic to the automorphism group of t over ro furthermore if y is a regular covering of t0 with automorphism group g we show that the category of representations of y satisfying a certain set of relations is equivalent to the category of finite dimensional graded g modules we

verbally prime t ideals and graded division algebras abstract let f be an algebraically closed field of characteristic zero and let g be a finite group we consider graded verbally prime t ideals in the free g graded algebra it turns out that equivalent definitions in the ordinary case i e ungraded extend to nonequivalent definitions in the graded case namely verbally prime g graded t ideals and strongly verbally prime t ideals at first following kemer s ideas we classify g graded verbally prime t ideals the main bulk of the paper is devoted to the stronger notion we classify g graded strongly verbally prime t ideals which are t ideal of affine g graded algebras or equivalently g graded t ideals that contain a capelli polynomial it turns out that these are precisely the t ideal of g graded identities of finite dimensional g graded central over f i e z a e f which admit a g graded division algebra twisted form over a field k which contains f or equivalently over a field k which contains enough roots of unity e g a primitive n root of unity where n o r d g

a three dimensional propagation algorithm using finite azimuthal aperture the use of three dimensional 3 d propagation models is becoming increasingly common in current underwater acoustics applications such models typically treat the propagation as a function of range depth and bearing consistent with previous multibearing two dimensional n 2 d models however to obtain accurate 3 d solutions at long ranges many bearings must be computed in order to maintain the necessary cross range resolution between bearings in this paper a 3 d parabolic equation pe model is developed with a marching algorithm based on the split step fourier technique in both depth and azimuthal bearing the algorithm includes a scheme to compute the solution for only a finite azimuthal aperture rather than the full 360 of bearing the success of this algorithm allows for increased cross range resolution at long range without increasing the number of bearings needed in the calculation results of this technique are compared to results from a suggested benchmark case

four dimensional yetter drinfeld module algebras over h4 abstract in this paper we classify four dimensional yetter drinfeld module algebras over sweedler s 4 dimensional hopf algebra h 4 all h 4 azumaya algebras of dimension 4 have been given in particular all yetter drinfeld h 4 module algebra structures on the matrix algebra m 2 k are classified

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

representation theory of graded artin algebras having initiated the study of graded artin algebras a in 91 here we initiate the study of their representation theory our point of view is to study graded i modules which we believe to be somewhat more tractable than ungraded ones in order to obtain information about all i modules this point of view leads to the introduction in section 1 of the full subcategory mod of mod i consisting of the gradable objects of mod i that is the finitely generated i modules which support a gradation in these terms one of the major results of the paper asserts that mod has finite representation type if and only if mod i has finite representation type thus in section 3 we introduce a number g g 4 designed to measure the size of mod i if g co we show that i has infinite representation type if g co we show that there is a graded artin algebra q of a certain specified form such that i has infinite representation type precisely when 0 has infinite representation type the artin algebra fl has in particular desirable diagrammatic properties and these we will exploit elsewhere we speculate that when g is finite every finitely generated i module is gradable now in case n has finite representation type it is obvious that g is finite we show indeed that for every gradation of a given artin algebra of finite representation type every module is gradable this result and the others cited are chiefly consequences of a result proved in section 4 if a component of the auslander reiten graph of a graded artin algebra contains a gradable module then the component

projective schur algebras of nilpotent type are brauer equivalent to radical algebras let k be a field and g a finite group for any element h g k k denotes the units of k here as a trivial g module we can form the twisted group algebra kg it is isomorphic to the group algebra as a vector space and its multiplication satisfies the rule xu yu xyf u where f g g k is a 2 cocycle representing it is easily checked that the 2 cocycle condition satisfied by f insures the associativity of the multiplication in kg a projective schur algebra over k is by definition a k central simple algebra which is a homomorphic image of a twisted group algebra kg for some finite group g and some h g k equivalently a k central simple algebra a is a projective schur algebra over k if it contains a subgroup of the units of a which is finite modulo k and spans a as k vector space we often use the notation k clearly a projective schur algebra over k determines an element in the brauer group br k of the field k the subgroup in br k generated by and in fact consisting of such elements is called the projective schur group of k and denoted by ps k the notions of projective schur algebra and projective schur group were introduced in 6 by lorenz and opolka this construction can be viewed as the projective analogue of schur algebras and the schur group denoted by s k of the field k see 10 it is natural to restrict attention to special types of groups g our interest here is in k central simple algebras that are homomorphic images of twisted group algebras kg with g nilpotent resp abelian see 6 we call these algebras projective schur algebras of nilpotent type resp abelian type clearly as above these algebras represent classes in br k and we denote by pnil k resp pab k the subgroup they generate in 6 it is shown that if k is a number field and n is the number of roots of unity in k then pab k nbr k where nb denotes the n torsion subgroup of b more generally since a symbol algebra of degree n over k is a homomorphic

on weak equivalences of gradings abstract when one studies the structure e g graded ideals graded subspaces radicals or graded polynomial identities of graded algebras the grading group itself does not play an important role but can be replaced by any other group that realizes the same grading here we come to the notion of weak equivalence of gradings two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component the following question arises naturally when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group it turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading namely a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group the same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset we show that if n 349 then on the full matrix algebra m n f there exists an elementary group grading that is not weakly equivalent to any grading by a finite semi group and if n 3 then any elementary grading on m n f is weakly equivalent to an elementary grading by a finite group

the center of the generic g crossed product abstract let g be a finite group and let f be a field of characteristic zero in this paper we construct a generic g crossed product over f using generic graded matrices the center of this generic g crossed product denoted by f g is then the invariant field of a suitable g action on a field of rational functions in several indeterminates the main goal of this paper is to study the extensions f g f given that f contains enough roots of unity and determine how close they are to being purely transcendental in particular we show that f g f is a stably rational extension for g c 2 c 2 n where n is odd and for g n 2 m e 1 1 where g c d n 2 m 1 furthermore we prove that if h k are groups of coprime orders then f h k is stably rationally equivalent to the fraction field of f h f k

group gradings on full matrix rings we study g gradings of the matrix ring mn k k a field and give a complete description of the gradings where all the elements ei j are homogeneous called good gradings among these we determine the ones that are strong gradings or crossed products if g is a finite cyclic group and k contains a primitive g th root of 1 we show how all g gradings of mn k can be produced in particular we give a precise description of all c2 gradings of m2 k and show that for algebraically closed k any such grading is isomorphic to one of the two good gradings

gradings on the algebra of upper triangular matrices and their graded identities abstract let k be an infinite field and let ut n k denote the algebra of n n upper triangular matrices over k we describe all elementary gradings on this algebra further we describe the generators of the ideals of graded polynomial identities of ut n k and we produce linear bases of the corresponding relatively free graded algebras we prove that one can distinguish the elementary gradings by their graded identities we describe bases of the graded polynomial identities in several typical cases although in these cases we consider elementary gradings by cyclic groups the same methods serve for elementary gradings by any finite group

pauli gradings on lie superalgebras and graded codimension growth abstract we introduce grading on certain finite dimensional simple lie superalgebras of type p t by elementary abelian 2 group this grading gives rise to pauli matrices and is a far generalization of z 2 z 2 grading on lie algebra of 2 2 traceless matrices we use this grading for studying numerical invariants of polynomial identities of lie superalgebras in particular we compute graded pi exponent corresponding to pauli grading

graded polynomial identities of matrices we consider g graded polynomial identities of the p p matrix algebra mp k over a field k of characteristic 0 graded by an arbitrary group g we find relations between the g graded identities of the g graded algebra mp k and the g h graded identities of the tensor product of mp k and the h graded algebra mq k with a fine h grading we also find a basis of the g graded identities of mp k with an elementary grading such that the identity component coincides with the diagonal of mp k

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

the graded grothendieck group and the classification of leavitt path algebras this paper is an attempt to show that parallel to elliott s classification of af c algebras by means of k theory the graded k0 group classifies leavitt path algebras completely in this direction we prove this claim at two extremes namely for the class of acyclic graphs graphs with no cycles and multi headed comets or rose graphs graphs in which each head is connected to a cycle or to a collection of loops or a mixture of these graphs i e polycephaly graphs

partial representations of hopf algebras in this work the notion of a partial representation of a hopf algebra is introduced and its relationship with partial actions of hopf algebras is explored given a hopf algebra h one can associate to it a hopf algebroid hpar which has the universal property that each partial representation of h can be factorized by an algebra morphism from hpar we define also the category of partial modules over a hopf algebra h which is the category of modules over its associated hopf algebroid hpar the hopf algebroid structure of hpar enables us to enhance the category of partial h modules with a monoidal structure and such that the algebra objects in this category are the symmetric partial actions some examples of categories of partial h modules are explored in particular we can describe fully the category of partially z2 graded modules