designing structured tight frames via an alternating projection method tight frames also known as general welch bound equality sequences generalize orthonormal systems numerous applications including communications coding and sparse approximation require finite dimensional tight frames that possess additional structural properties this paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems ieps which includes the frame design problem to apply this method one needs only to solve a matrix nearness problem that arises naturally from the design specifications therefore it is the fast and easy to develop versions of the algorithm that target new design problems alternating projection will often succeed even if algebraic constructions are unavailable to demonstrate that alternating projection is an effective tool for frame design the paper studies some important structural properties in detail first it addresses the most basic design problem constructing tight frames with prescribed vector norms then it discusses equiangular tight frames which are natural dictionaries for sparse approximation finally it examines tight frames whose individual vectors have low peak to average power ratio par which is a valuable property for code division multiple access cdma applications numerical experiments show that the proposed algorithm succeeds in each of these three cases the appendices investigate the convergence properties of the algorithm

strongly graded rings and almost split sequences let a ogsg a be an algebra over a field k strongly graded by a finite group g with identity e for example a crossed product r g in this paper we will study connections between mod a and mod a where mod a mod a denotes the category of finitely presented ac modules a modules it follows from 2 that there is a ring a morita equivalent to a and an action of g on a such that the skew group ring a g and a are morita equivalent from this it follows that if we know that a property holds for mod a if and only if it holds for mod a g then holds for mod a if and only if it holds for mod a in the case of both strongly graded rings and skew group rings we have natural ring maps a a and a a g which induce natural functors between the module categories by restriction and tensor product to be able to transfer properties of these functors from skew group rings to strongly graded rings we prove that there are equivalences i mod a modaandj moda moda g suchthatj f f iandi h2 h j where f f and h h are induction and restriction for the rings a and a a and a g j respectively some equivalences occurring in 1 2 51 turn out to work as an application we get the following if a is an artin algebra k is algebraically closed and the order of g is invertible in k then the natural functors f and h map an almost split sequence to a direct sum of almost split sequences an interesting special case of this is k n k g where n is a normal subgroup of a finite group g and the order of g n is invertible in k throughout this paper k will denote a field let a be a graded ring then mod a gr mod a and mod a gr mod a will denote the category of graded a modules and the category of finitely presented graded a modules if two rings a and b are morita equivalent we will write a b if g is a group then igi will denote the order of g

skew group algebras of deformed preprojective algebras abstract suppose that q is a finite quiver and g aut q is a finite group k is an algebraic closed field whose characteristic does not divide the order of g for any algebra k q i where i is an arbitrary ideal of path algebra kq we give all the indecomposable g modules from indecomposable modules when g is abelian in particular we apply this result to the deformed preprojective algebra q and get a reflection functor for the module category of q g furthermore we construct a new quiver q g and prove that q g is morita equivalent to q g for some

realizing stable categories as derived categories abstract in this paper we discuss a relationship between representation theory of graded self injective algebras and that of algebras of finite global dimension for a positively graded self injective algebra a such that a 0 has finite global dimension we construct two types of triangle equivalences first we show that there exists a triangle equivalence between the stable category of z graded a modules and the derived category of a certain algebra of finite global dimension secondly we show that if a has gorenstein parameter l then there exists a triangle equivalence between the stable category of z l z graded a modules and a derived orbit category of which is a triangulated hull of the orbit category of the derived category

configurations of trivial extensions of dynkin type a n for a given trivial extensionof cartan class dn we find a combinatorial algorithm giving the configuration of zdn associated to this algorithm is stated in the theorem 6 8 and illustrated in the example 6 3 in order to do that we develop an inductive embedding technique for the auslander reiten quiver of a trivial extensionof type dn this technique uses the ordinary quiver q which is the union of oriented cycles and a well known set of relationssuch thatis the quotient path algebra kq see 1 5 and 6 giving a trivial exten sion of type dn we get a trivial extensionof type dn m 1 by inserting a cycle with m vertices in one of the so called in sertion vertices of q in section 3 we describe the embedding of into this will allow us to deduce properties offrom and since q has less cycles that q we will use this embed ding to reduce the study ofto the minimal trivial extensions which were done in section 2

representations of skew group algebras induced from isomorphically invariant modules over path algebras abstract suppose that q is a connected quiver without oriented cycles and is an automorphism of q let k be an algebraically closed field whose characteristic does not divide the order of the cyclic group the aim of this paper is to investigate the relationship between indecomposable kq modules and indecomposable k q k modules it has been shown by hubery that any k q k module is an isomorphically invariant kq module i e ii module in this paper we call it equivalent kq module and conversely any equivalent kq module induces a k q k module in this paper the authors prove that a k q k module is indecomposable if and only if it is an indecomposable equivalent kq module namely a method is given in order to induce all indecomposable k q k modules from all indecomposable equivalent kq modules the number of non isomorphic indecomposable k q k modules induced from the same indecomposable equivalent kq module is given in particular the authors give the relationship between indecomposable k q k modules and indecomposable kq modules in the cases of indecomposable simple projective and injective modules

complexity periodicity and one parameter subgroups using the variety of infinitesimal one parameter subgroups introduced in 55 56 by suslin friedlander bendel we define a numerical invariant for representations of an infinitesimal group scheme g for an indecomposable g module m of complexity 1 this number which may also interpreted as the height of a vertex um g is related to the period of m in the context of the frobenius category of grt modules associated to a smooth reductive group g and a maximal torus t g our methods give control over the behavior of the heller operator of such modules as well as precise values for the periodicity of their restrictions to gr applications include the structure of stable auslander reiten components of grt modules as well as the distribution of baby verma modules

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

the left and right parts of the module category of crossed product group algebras let g be a finite group k be a field with characteristic not dividing the order of g a nd b e at wistedkg module algebra such that g or g for short exists as a crossed product algebra in this paper we study the relationship between the class l g or r g which consists of all finitely generated indecomposable g modules whose predecessors or successors have projective dimension or injective dimension at most one and the class l or r then we study those properties which rely on the left or right part of that are preserved under the extension of crossed product group algebra g

quotients of cluster categories higher cluster categories were recently introduced as a generalization of cluster categories this paper shows that in dynkin types a and d half of all higher cluster categories can be obtained as quotients of cluster categories the other half are quotients of 2 cluster categories the lowest type of higher cluster categories hence in dynkin types a and d all higher cluster phenomena are implicit in cluster categories and 2 cluster categories in contrast the same is not true in dynkin type e

maps between representations of clans abstract we give a description of the homomorphisms between representations of a clan see crawley boevey j london math soc 40 1989 9 30 more or less as it was done by krause j algebra 137 1991 186 194 for string algebras as a consequence of our findings we can establish the irreducibility of all maps which are expected to be so this allows us after defining a natural exact structure on the category of representations to calculate in many situations auslander reiten sequences like in the string algebra case

tame algebras and tits quadratic forms abstract we solve a long standing problem concerning the connection between the tameness of simply connected algebras and the weak nonnegativity of the associated tits integral quadratic forms and derive some consequences

a characterization of selfinjective algebras of strictly canonical type abstract we prove that the class of selfinjective algebras of strictly canonical type investigated in kwiecien and skowronski 2009 27 kwiecien and skowronski 2009 28 coincides with the class of selfinjective algebras having triangular galois coverings with infinite cyclic group and the auslander reiten quiver with quasi tubes maximally saturated by simple and projective modules satisfying natural conditions

local representation theory of transporter categories abstract we attempt to generalize the p modular representation theory of finite groups to finite transporter categories which are regarded as generalized groups we shall carry on our tasks through modules of transporter category algebras a type of gorenstein skew group algebras the kan extensions upgrading the induction and co induction are our main tools to establish connections between representations of a transporter category and of its transporter subcategories some important constructions and theorems in local representation theory of finite groups 1 are generalized

sensing matrix optimization for block sparse decoding recent work has demonstrated that using a carefully designed sensing matrix rather than a random one can improve the performance of compressed sensing in particular a well designed sensing matrix can reduce the coherence between the atoms of the equivalent dictionary and as a consequence reduce the reconstruction error in some applications the signals of interest can be well approximated by a union of a small number of subspaces e g face recognition and motion segmentation this implies the existence of a dictionary which leads to block sparse representations in this work we propose a framework for sensing matrix design that improves the ability of block sparse approximation techniques to reconstruct and classify signals this method is based on minimizing a weighted sum of the interblock coherence and the subblock coherence of the equivalent dictionary our experiments show that the proposed algorithm significantly improves signal recovery and classification ability of the block omp algorithm compared to sensing matrix optimization methods that do not employ block structure

the existence of covering functors for orders and consequences for stable components with oriented cycles the starting point for the covering theory for algebras of finite type over algebraically closed fields was riedtmann s paper algebren darstellungskocher uberlagerungen und zurtick 14 1 her main result from today s point of view is the existence of covering functors i e the existence of a bijection between the morphisms between indecomposable modules and purely combinatorial defined vector spaces generated by paths in a translation quiver cf 1 1 141 this bijection has recently turned out to be more and more important in the representation theory of algebras it was the first step leading to the use of auslander reiten quivers as a tool for the classification of algebras of linite type 15 81 the main purpose of this paper is to initiate an analog for classical orders over complete dedekind domains we shall formulate our results and proofs only for orders but it is always obvious that everything carries immediately over to the algebra situation let r be a complete dedekind domain with quotient field k and let n be an r order in a separable k algebra a 161 n is a subring of a with the same identity and moreover a is a full r sublattice which means that i is an r lattice and contains a k basis of a we denote by r the auslander reiten quiver of i i e vertices of r are the isomorphism classes of indecomposable n lattices and two vertices are joined by an arrow provided there exists an irreducible i map between the corresponding a lattices moreover we denote by f the universal cover of i 8 in particular the vertices of r are homotopy classes of walks in i with covering morphism f f p r 8 for each arrow x y in f we define as the space of irreducible maps irr fx fy 4 171 from fx to fy moreover for indecomposable i lattices iv n let r m n c hom m n be the ith functorial radical i e r m n ri n m where ri n is the ith power of the radical of the presentable fun or n hom n 21 292 0021 8693 85 3 00

the cohomology group associated with jacobi cusp forms over cayley numbers we consider the cohomology group associated with jacobi cusp forms over cayley numbers as an example of the general theory of the cohomology groups associated with cusp forms developed by eichler and shimura the theory establishes isomorphisms between the cohomology groups and the vector spaces of vector valued cusp forms and then expresses their common dimensions in terms of certain geometric invariants in the corresponding quotient spaces this is an analogue of the riemann roch theorem applied to the cases of vector valued modular forms

singular blocks of restricted sl3 abstract we compute generators and relations for the basic algebra of a non semisimple singular block of the restricted enveloping algebra of sl 3 over an algebraically closed field of characteristic p 3 working directly with the basic algebra we compute its centre and the internal degree zero part of its first hochschild cohomology and show its verma modules are koszul

a near optimal sampling strategy for sparse recovery of polynomial chaos expansions abstract compressive sampling has become a widely used approach to construct polynomial chaos surrogates when the number of available simulation samples is limited originally these expensive simulation samples would be obtained at random locations in the parameter space it was later shown that the choice of sample locations could significantly impact the accuracy of resulting surrogates this motivated new sampling strategies or design of experiment approaches such as coherence optimal sampling which aim at improving the coherence property in this paper we propose a sampling strategy that can identify near optimal sample locations that lead to improvement in local coherence property and also enhancement of cross correlation properties of measurement matrices we provide theoretical motivations for the proposed sampling strategy along with several numerical examples that show that our near optimal sampling strategy produces substantially more accurate results compared to other sampling strategies

on the degree of irreducible morphisms abstract we study the degree of irreducible morphisms in generalized standard convex components of the auslander reiten quiver of an artin algebra with the property that paths with the same origin and end vertices have equal length we call the components with this last property components with length in particular we give two criteria to determine wether the degree of such an irreducible morphism f is finite or infinite one of them is given in terms of the compositions of f with non zero maps between modules in the component the other states that the left degree of an irreducible map f is finite if and only if ker f belongs to the component we apply our results to irreducible morphisms over artin algebras of finite representation type and over tame hereditary algebras

on projection matrix optimization for compressive sensing systems this paper considers the problem of designing the projection matrix phi for a compressive sensing cs system in which the dictionary psi is assumed to be given the optimal projection matrix design is formulated in terms of finding those phi such that the frobenius norm of the difference between the gram matrix of the equivalent dictionary phi psi and the identity matrix is minimized a class of the solutions is derived in a closed form which is a generalization of the existing results more interestingly it is revealed that this solution set is characterized by an arbitrary orthonormal matrix this freedom is then used to further enhance the performance of the cs system by minimizing the coherence between the atoms of the equivalent dictionary an alternating minimization based algorithm is proposed for solving the corresponding minimization problem experiments are carried out and simulations show that the projection matrix obtained by the proposed approach significantly improves the signal recovery accuracy of the cs system and outperforms those by existing algorithms

superspecies and their representations abstract superspecies are introduced to provide a construction of all finite dimensional superalgebras all acyclic superspecies are classified according to their graded representation types

almost laura algebras abstract in this paper we propose a generalization for the class of laura algebras called almost laura we show that this new class of algebras retains most of the essential features of laura algebras especially concerning the important role played by the non semiregular components in their auslander reiten quivers also we study more intensively the left supported almost laura algebras showing that these are characterized by the presence of a generalized standard convex and faithful component finally we prove that almost laura algebras behave well with respect to full subcategories split by nilpotent extensions and skew group algebras

stable categories of graded maximal cohen macaulay modules over noncommutative quotient singularities abstract tilting objects play a key role in the study of triangulated categories a famous result due to iyama and takahashi asserts that the stable categories of graded maximal cohen macaulay modules over quotient singularities have tilting objects this paper proves a noncommutative generalization of iyama and takahashi s theorem using noncommutative algebraic geometry namely if s is a noetherian as regular koszul algebra and g is a finite group acting on s such that s g is a gorenstein isolated singularity then the stable category cm z s g of graded maximal cohen macaulay modules has a tilting object in particular the category cm z s g is triangle equivalent to the derived category of a finite dimensional algebra

the theory of jacobi forms over the cayley numbers m eie and a kriegabstract as a generalization of the classical theory of jacobi forms we discussjacobi forms on xc8 which are related with integral cayley numbers usingthe selberg trace formula we give a simple explicit formula for the dimensionof the space of jacobi forms the orthogonal complement of the space of cuspforms is shown to be spanned by certain types of eisenstein series

skew group algebras in the representation theory of artin algebras then the corresponding crossed product algebra i g or i x g for short has an elements cgieg gi a li i addition is componentwise and multiplication is given by g a g and g g2 y g g2 m in this paper we assume that the values of y lie in the center z a of i hence the action of g on i is given by a group homomorphism g p aut i and 3 can be left out in the special case that y is the trivial map we write 1g instead of i g and the elements as c c li gi ag is then called a skew group ring there is a lot of literature on skew group algebras and crossed product algebras and on the relationship with the ring ac whose elements are those elements of a left fixed by g much work has been done on which properties of li are inherited by n g or ac some of the work on the relationship between these rings has its roots in trying to develop a galois theory for noncommutative rings we refer to 3 7 13 15 19 21 23 25 27 281 and their references in this paper we study these constructions when i is an artin algebra and g a finite group of order n such that n is invertible in 1 under these assumptions the construction preserves central properties of interest in the

projection design for statistical compressive sensing a tight frame based approach in this paper we develop a framework to design sensing matrices for compressive sensing applications that lead to good mean squared error mse performance subject to sensing cost constraints by capitalizing on the mse of the oracle estimator whose performance has been shown to act as a benchmark to the performance of standard sparse recovery algorithms we use the fact that a parseval tight frame is the closest design in the frobenius norm sense to the solution of a convex relaxation of the optimization problem that relates to the minimization of the mse of the oracleestimator with respect to the equivalent sensing matrix subject to sensing energy constraints based on this result we then propose two sensing matrix designs that exhibit two key properties the designs are closed form rather than iterative the designs exhibit superior performance in relation to other designs in the literature which is revealed by our numerical investigation in various scenarios with different sparse recovery algorithms including basis pursuit de noise bpdn the dantzig selector and orthogonal matching pursuit omp

semisimple extensions and elements of trace 1 let h be a hopf algebra which is frobenius over a commutative ring k let a be an h module algebra a h its ring of invariants under the h action and a h the associated semidirect or smash product this setup has been shown by the authors and s montgomery cf cfm to give rise to a morita context ah a a a h with structure maps a aha a h and a a ah thecomodule theoretic concept of h galois extensions was shown cfm under these circumstance to be equivalent to surjectivity of 1 in this paper we start out by showing that another important comodule theoretic concept that of a total integral dl can be translated to a module theoretic concept as well we show that existence of a total integral is equivalent to the surjectivity of which we also termed suggestively existence of an element c e a of trace 1 furthermore if this element c centralizes ah then ah is an ah bimodule direct summand of a we apply these to the morita context associated with the h module algebra a h to prove the main theorem of this paper theorem 1 8 this theorem deals with semisimple extensions sometimes coined maschke type theorems or better yet with separable extensions we show that when a ah is h galois

low dimensional orders of finite representation type in this paper we study noncommutative surface singularities arising from orders the singularities we study are mild in the sense that they have finite representation type or equivalently are log terminal in the sense of the mori minimal model program for orders chan and ingalls in invent math 161 2 427 452 2005 these were classified independently by artin in terms of ramification data and reiten van den bergh in terms of their ar quivers the first main goal of this paper is to connect these two classifications by going through the finite subgroups g subset mathrm gl 2 explicitly computing h 2 g k and then matching these up with artin s list of ramification data and reiten van den bergh s ar quivers this provides a semi independent proof of their classifications and extends the study of canonical orders in chan et al proc lond math soc 3 98 1 83 115 2009 to the case of log terminal orders a secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows let b k zeta llbracket x y rrbracket be the skew power series ring where zeta is a root of unity or more generally a terminal order over a complete local ring we consider rings of the form a b f where f in z b which we interpret to be the ring of functions on a noncommutative plane curve we classify those noncommutative plane curves which are of finite representation type and compute their ar quivers

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

support varieties ar components and good filtrations let g be a reductive group defined over the galois field mathbb f p with p being good for g using support varieties and covering techniques based on grt modules we determine the position of simple modules and baby verma modules within the stable auslander reiten quiver s gr of the rth frobenius kernel of g in particular we show that the almost split sequences terminating in these modules usually have an indecomposable middle term concerning support varieties we introduce a reduction technique leading to isomorphisms mathcal v g r z r lambda cong mathcal v g r d z r d mu for baby verma modules of certain highest weights lambda mu in x t which are related by the notion of depth

polyhedral groups mckay quivers and the finite algebraic groups with tame principal blocks given an algebraically closed field k of characteristic p 3 we classify the finite algebraic k groups whose algebras of measures afford a principal block of tame representation type the structure of such a group mathcal g is largely determined by a linearly reductive subgroup scheme hat mathcal g of sl 2 with the mckay quiver of hat mathcal g relative to its standard module being the gabriel quiver of the principal block mathcal b 0 mathcal g the graphs underlying these quivers are extended dynkin diagrams of type tilde a tilde d or tilde e and the tame blocks are morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type

perpendicular categories with applications to representations and sheaves this paper is concerned with the omnipresence of the formation of the subcategories right left perpendicular to a subcategory of objects in an abelian category we encounter these subcategories in various contexts the formation of quotient categories with respect to localizing subcategories cf section 2 the deletion of vertices and shrinking of arrows see 37 in the representation theory of finite dimensional algebras cf section 5 the comparison of the representation theories of different extended dynkin quivers cf section 10 the theory of tilting cf sections 4 and 6 the study of homological epimorphisms of rings cf section 4 the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme cf section 7 and 21 the study of maximal cohen macaulay modules over surface singularities cf sections the comparison of weighted projective lines for different weight sequences cf section 9 the formation of affine and local algebras attached to path algebras of extended dynkin quivers canonical algebras and weighted projective lines cf section 11 and 21 and the concept of universal localization in 40 formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization this applies in particular to applications in the domain of finite dimensional algebras and their representations several applications of the methods presented in this paper are already in existence partly published or appearing in print in the near future see for instance 40 39 4 45 26 49 46 and have show the versatility of the notion of a perpendicular category it seems that right perpendicular categories first appeared as the subcategories of so called closed objects in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing serre subcategory see 18 47 34 another natural occurrence is encountered in commutative algebra forming the possibly infinitely generated modules of depth 2 cf section 7 the concept and some of the central applications were first presented in a talk given by the first author at the honnef meeting in january 1985 we also note that the perfectly matching nomination perpendicular category was coined by a schofield who discovered independently the usefulness of this concept in dealing with hereditary algebras see 39 cf also section 7 the authors further acknowledge the support of the deutsche forschungsgemeinschaft spp darstellungstheorie von endlichen gruppen und endlichdimensionalen algebren throughout this paper rings are associative with unit and modules are unitary right modules mod r respectively mod r denotes the category of all respectively all finitely presented right r modules

representations of quivers over the algebra of dual numbers abstract the representations of a quiver q over a field k the kq modules where kq is the path algebra of q over k have been studied for a long time and one knows quite well the structure of the module category mod k q it seems to be worthwhile to consider also representations of q over arbitrary finite dimensional k algebras a here we draw the attention to the case when a k is the algebra of dual numbers the factor algebra of the polynomial ring k t in one variable t modulo the ideal generated by t 2 thus to the modules where k q k q t t 2 the algebra is a 1 gorenstein algebra thus the torsionless modules are known to be of special interest as the gorenstein projective or maximal cohen macaulay modules they form a frobenius category l thus the corresponding stable category l is a triangulated category as we will see the category l is the category of perfect differential kq modules and l is the corresponding homotopy category the category l is triangle equivalent to the orbit category of the derived category d b mod k q modulo the shift and the homology functor h mod mod k q yields a bijection between the indecomposables in l and those in mod k q our main interest lies in the inverse it is given by the minimal l approximation also we will determine the kernel of the restriction of the functor h to l and describe the auslander reiten quivers of l and l

the cartan map on categories of graded modules this paper presents a technique based on elementary k theory it relies on the fact that k for a category of graded modules can be viewed as a z t t l module for some variable t in this context we will deduce a great deal of information from an explicit computation of the coordinates of the matrix of the cartan homomorphism in section 1 we begin by recalling some definitions and results that we will use throughout this paper from these we then describe our basic machinery section 2 will provide some applications of this machine we note in particular the following uses nakayama 131 proposed that if a is a finite dimensional algebra of infinite dominant dimension then a is self injective auslander and reiten 4 noticed that the following stronger statement would imply the truth of nakayama s conjecture given a finite dimensional algebra a with minimal injective resolution 0 a e e se every indecomposable injective module is isomorphic to a summand of some ei here we prove this generalized nakayama conjecture for graded algebras another application of this technique shows that the determinant of the cartan map is one for graded algebras of finite global dimension this in turn is used to show that if b is any algebra not necessarily graded with only finitely many nonisomorphic indecomposable modules m 1 m then the matrix whose ijth coordinate is dim hom mi mj has determinant one finally we give an application to the rationality of power series let f be a field and let a be a finite dimensional f algebra we will assume everywhere that a is basic that is for every simple a module s end s f this assumption is primarily to simplify notation and the arguments given could be modified to handle algebras that are not basic we will use the notation mod a for the category of finitely generated left a modules let g be a full subcategory of mod 4 and let x 55 be the free 390 0021 8693 83 3 00

skew group algebras of path algebras and preprojective algebras abstract we compute explicitly up to morita equivalence the skew group algebra of a finite group acting on the path algebra of a quiver and the skew group algebra of a finite group acting on a preprojective algebra these results generalize previous results of reiten and riedtmann 1985 rr for a cyclic group acting on the path algebra of a quiver and of reiten and van den bergh 1989 rv proposition 2 13 for a finite subgroup of sl c x c y acting on c x y

weighted projective lines of tubular type and equivariantization abstract we prove that the categories of coherent sheaves over weighted projective lines of tubular type are explicitly related to each other via the equivariantization with respect to certain cyclic group actions

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

minimal model of ginzburg algebras abstract we compute the minimal model for ginzburg algebras associated to acyclic quivers q in particular we prove that there is a natural grading on the ginzburg algebra making it formal and quasi isomorphic to the preprojective algebra in non dynkin type and in dynkin type is quasi isomorphic to a twisted polynomial algebra over the preprojective with a unique higher a composition to prove these results we construct and study the minimal model of an a envelope of the derived category d b q whose higher compositions encode the triangulated structure of d b q

rank varieties and projectivity for a class of local algebras we consider algebras open image in new window where k is an algebraically closed fields to any finite dimensional module for this algebra we associate a rank variety when char k 2 we recover carlson s rank variety the main result states that a module is projective if and only if its rank variety vanishes this has applications to other algebras including tensor products of certain brauer tree algebras and certain parabolic hecke algebras in addition the result has implications for the graph structure of the stable auslander reiten quiver mathematics subject classification 2000 16g10 18g05 20c20

diagram automorphisms of quiver varieties abstract we show that the fixed point subvariety of a nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the split quotient quiver introduced by reiten and riedtmann as a special case quiver varieties of type d arise as the connected components of fixed point subvarieties of diagram involutions of quiver varieties of type a in the case where the quiver varieties of type a correspond to small self dual representations we show that the diagram involutions coincide with classical involutions of two row slodowy varieties it follows that certain quiver varieties of type d are isomorphic to slodowy varieties for orthogonal or symplectic lie algebras

homological domino effects and the first finitistic dimension conjecture counterexamples to the first finitistic dimension conjecture are constructed in fact for any fieldk and any integerm 2 there exist finite dimensionalk algebras such that the little finitistic dimension of ism while the big finitistic dimension ism 1 our examples are monomial relation algebras within this class of algebras the big and little finitistic dimensions cannot differ by more than 1 the analysis of the examples is based on a sharp picture of arbitrary second syzygies over monomial relation algebras

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

wild two point algebras abstract the representation type of two point algebras is determined it is also shown that a two point algebra is wild if and only if it is controlled wild moreover it turns out that a minimal wild two point algebra is strongly minimal wild if and only if it is not strictly wild

constructing coherently g invariant modules abstract let g be a reductive group acting on a path algebra kq as automorphisms we assume that g admits a graded polynomial representation theory and the action is polynomial we describe the quiver q g of the smash product algebra k q k m g where m g is the associated algebraic monoid of g we use q g representations to construct coherently g invariant modules of q as an application we construct algebraic semi invariants on the quiver representation spaces from those g invariant modules

highest weight theory for finite dimensional graded algebras with triangular decomposition abstract we show that the category of graded modules over a finite dimensional graded algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category when the algebra is self injective we show furthermore that this highest weight category has tilting modules in the sense of ringel this provides a new perspective on the representation theory of such algebras and leads to several new structures attached to them there are a wide variety of examples in algebraic lie theory to which this applies restricted enveloping algebras lusztig s small quantum groups hyperalgebras finite quantum groups and restricted rational cherednik algebras

cartan leray spectral sequence for galois coverings of linear categories abstract we provide a cartan leray type spectral sequence for the hochschild mitchell co homology of a galois covering of linear categories we infer results relating the galois group and hochschild cohomology in degree one

on generalized standard auslander reiten components having only finitely many non directing modules abstract let be a connected component of the auslander reiten quiver mod a of an artin algebra we show that is quasi directed if and only if the paths in ind a having end points in satisfy certain conditions moreover we describe the shapes of such components and identify the classes of algebras to which they are related

skew group algebras of jacobian algebras for a quiver with potential q w with an action of a finite cyclic group g we study the skew group algebra lambda g of the jacobian algebra lambda p q w by a result of reiten and riedtmann

graded selfinjective and koszul algebras abstract in this paper we continue our work on koszul algebras initiated in earlier studies the consideration about the existence of almost split sequences for koszul modules appeared in our early work and only a partial answer is known koszul duality relates finite dimensional algebras of infinite global dimension with infinite dimensional algebras such that the injective dimension of the graded simples is finite two questions naturally arise the existence of almost split sequences for infinite dimensional graded algebras and the relations between the global dimension of a graded algebra and the injective dimension of the graded simples in the first part of the paper we deal with the first matter in the second we consider the relations between the graded global dimension and the maximum of the injective dimensions of the graded simples proving that in the noetherian case they coincide in the third part we consider finite dimensional graded selfinjective quiver algebras and prove that for such algebras all indecomposable projective modules have the same lowey length in the forth and last part of the paper we specialize to selfinjective koszul algebras and characterize their yoneda algebras proving they constitute non commutative versions of the commutative regular algebras

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

noetherian hereditary abelian categories satisfying serre duality norwegian univ sci technol dept math sci n 7491 trondheim norway limburgs univ ctr dept wni b 3590 diepenbeek belgium reiten i norwegian univ sci technol dept math sci n 7491 trondheim norway