cofree quiver settings abstract we give a complete classification of all quivers q and dimension vectors for which the representation space rep q is cofree as a representation of its natural symmetry group gl

middle convolution and harnad duality dettweiler reiter s additive description of katz s middle convolution can be interpreted in terms of the harnad duality using this interpretation together with mumford s geometric invariant theory we generalize the additive middle convolution to have a multi parameter and act on systems of linear ordinary differential equations with irregular singularities we show that the generalized operation holds basic properties of the original one and additionally show that katz s algorithm involving our generalization works well in generic case

some algorithms for semi invariants of quivers abstract we present some theorems and algorithms for calculating perpendicular categories and locally semi simple decompositions we implemented a computer program tetiva based on these algorithms and we offer this program for everybody s use

a revision of the lyapunov exponent in 1d quadratic maps abstract only misiurewicz points strictly have a lyapunov exponent positive measure in one dimensional 1d quadratic maps hence we represent here the lyapunov exponent of all the misiurewicz points which were found and inventoried by us in a former work as a result of this representation an infinity of alignments of misiurewicz points are obtained and two types of jumps in the values of are found

qurves and quivers abstract in this paper we associate to a k qurve a formerly known as a quasi free algebra j cuntz d quillen algebra extensions and nonsingularity j amer math soc 8 1995 251 289 or formally smooth algebra m kontsevich a rosenberg noncommutative smooth spaces math ag 9812158 1998 the one quiver q 1 a and dimension vector 1 a this pair contains enough information to reconstruct for all n n the gl n etale local structure of the representation scheme rep n a in an appendix we indicate how one might extend this to qurves over non algebraically closed fields further we classify all finitely generated groups g such that the group algebra kg is a k qurve if char k 0 these are exactly the virtually free groups we determine the one quiver setting in this case and indicate how it can be used to study the finite dimensional representations of virtually free groups as this approach also applies to fundamental algebras of graphs of separable k algebras we state the results in this more general setting

on the equations and classification of toric quiver varieties toric quiver varieties moduli spaces of quiver representations are studied given a quiver and a weight there is an associated quasi projective toric variety together with a canonical embedding into projective space it is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3 in each fixed dimension d up to isomorphism there are only finitely many d dimensional toric quiver varieties a procedure for their classification is outlined

normality of marsden weinstein reductions for representations of quivers we prove that the marsden weinstein reductions for the moment map associated to representations of a quiver are normal varieties we give an application to conjugacy classes of matrices

poisson structures on moduli spaces of representations abstract we show that a poisson structure can be induced on the affine moduli space of semisimple representations of an associative algebra from a suitable lie algebra structure on the zeroth hochschild homology of the algebra in particular this applies to necklace lie algebra for path algebras of doubled quivers and preprojective algebras we call such structures h 0 poisson structures and show that they are well behaved for azumaya algebras and under morita equivalence

asymptotic expansions for trace functionals abstract we obtain taylor approximations for functionals v tr f h 0 v defined on the bounded self adjoint operators where h 0 is a self adjoint operator with compact resolvent and f is a sufficiently nice scalar function relaxing assumptions on the operators made in 17 and derive estimates and representations for the remainders of these approximations

quiver varieties and lusztig s algebra abstract we study preprojective algebras of graphs and their relationship to module categories over representations of quantum sl 2 as an application ade quiver varieties of nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by lusztig

universal scaling law for the largest lyapunov exponent in coupled map lattices coupled map lattices cmls 1 as discrete analogs tocoupled oscillators and partial differential equations havein recent years become the model of choice for developingintuitions and concepts in the study of spatiotemporalchaotic systems among the variety of quantities appliedto characterize the dynamics of a cml the spectrum oflyapunov exponents stands out for its ease of numericalevaluation and for the wealth of information that derivesfrom it for example from pesin s formula 2 thesum of all positive lyapunov exponents provides anestimate of the kolmogorov sinai entropy meanwhile one obtains the information dimension of the chaoticattractor by employing the kaplan yorke conjecture 3 moreover the largest lyapunov exponent lle denotedhenceforth by l quanti es the sensitive dependence oninitial conditions a hallmark of chaos 4 by yieldingthe rate of exponential divergence of two typical nearbytrajectories in this letter we focus on the scaling behavior of lnear certain limiting situations consider a diffusivelycoupled map lattice of the form

classes of degeneracy loci for quivers the thom polynomial point of view the chern classes for degeneracy loci of quivers are natural generalizations of the thom porteous giambelli formula suppose that e f are vector bundles over a manifold m and that s e f is a vector bundle homomorphism the question is which cohomology class is defined by the set6k s m consisting of points m where the linear map s m has corank k the answer due to i porteous is a determinant in terms of chern classes of the bundles e f we can generalize the question by giving more bundles over m and bundle maps among them the situation can be conveniently coded by an oriented graph called a quiver assigning vertices for bundles and arrows for maps we give a new method for calculating chern class formulae for degeneracy loci of quivers we show that for representation finite quivers this is a special case of the problem of calculating thom polynomials for group actions this allows us to apply a method for calculating thom polynomials developed by the authors the method reducing the calculations to solving a system of linear equations is quite different from the method of a buch and w fulton developed for calculating chern class formulae for degeneracy loci of a n quivers and it is more general can be applied to an dn e6 e7 and e8 quivers we provide sample calculations for a 3 and d4 quivers

on the littlewood richardson polynomials we prove the equivalence of several descriptions of generators of rings of semiinvariants of quivers due to domokos and zubkov schofield and van den bergh and our earlier work we also show that the dimensions of semi invariants of weights n depend polynomially on n 2002 elsevier science usa all rights reserved

fibers of generic brauer severi schemes abstract in this note we investigate the fibers of van den bergh s brauer severi scheme bs m n v m n of the trace ring of m generic n n matrices over the variety of matrix invariants if v m n is a point of representation type m 1 n 1 m r n r then the reduced variety of 1 has m i m i irreducible components each of dimension equation

the flat locus of brauer severi fibrations of smooth orders given a cayley hamilton smooth order a in a central simple algebra we determine the flat locus of the brauer severi fibration of a moreover we give a classification of all reduced central singularities where the flat locus differs from the azumaya locus and show that the fibers over the flat non azumaya points near these central singularities can be described as fibered products of graphs of projection maps this generalizes an old result of artin on the fibers of the brauer severi fibration of a maximal order over a ramified point finally we show these fibers are also toric quiver varieties and use this fact to compute their cohomology

a proof of the popov conjecture for quivers abstract let q be a quiver with dimension vector we show that if the space of isomorphism classes of semisimple representations iss q of q of dimension vector is not smooth then the quotient map rep q iss q is not equidimensional in other words we prove the popov conjecture for the natural action of the linear reductive group gl on the space rep q of dimensional representations of the quiver q

local structure of schelter procesi smooth orders in this paper we give the etale local classification of schelterprocesi smooth orders in central simple algebras in particular we prove that if is a central simple k algebra of dimension n2 where k is a field of trancendence degree d then there are only finitely many etale local classes of smooth orders in this result is a non commutative generalization of the fact that a smooth variety is analytically a manifold and so has only one type of etale local behaviour

noncommutative geometry and quiver algebras abstract we develop a new framework for noncommutative differential geometry based on double derivations this leads to the notion of moment map and of hamiltonian reduction in noncommutative symplectic geometry for any smooth associative algebra b we define its noncommutative cotangent bundle t b which is a basic example of noncommutative symplectic manifold applying hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras b that includes preprojective algebras associated with quivers our formalism of noncommutative hamiltonian reduction provides the space with a lie algebra structure analogous to the poisson bracket on the zero fiber of the moment map in the special case where is the preprojective algebra associated with a quiver of non dynkin type we give a complete description of the gerstenhaber algebra structure on the hochschild cohomology of in terms of the lie algebra

on quiver grassmannians and orbit closures for representation finite algebras we show that auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective injective its endomorphism ring is called the projective quotient algebra for any representation finite algebra we use the projective quotient algebra to construct desingularizations of quiver grassmannians orbit closures in representation varieties and their desingularizations this generalizes results of cerulli irelli feigin and reineke

a formal inverse to the cayley hamilton theorem abstract we prove that a q algebra r with formal trace can be realized as n n matrices if and only if it satisfies the cayley hamilton identity of degree n

necklace lie algebras and noncommutative symplectic geometry recently v ginzburg proved that calogero phase space is a coadjoint orbit for some infinite dimensional lie algebra coming from noncommutative symplectic geometry 12 in this note we generalize his argument to specific quotient varieties of representations of deformed preprojective algebras this result was also obtained independently by v ginzburg 13 using results of w crawley boevey and m holland 10 8 and 9 we give a combinatorial description of all the relevant couples alpha lambda which are coadjoint orbits we give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness

invariant theory of canonical algebras abstract based on the first fundamental theorem of classical invariant theory we present a reduction technique for computing relative invariants for quivers with relations this is applied to the invariant theory of canonical algebras and yields an explicit construction of the moduli spaces together with the quotient morphisms from the corresponding representation spaces for families of modules with a fixed dimension vector belonging to the central sincere separating subcategory by means of a tilting process we extend these results to the invariant theory of concealed canonical algebras thus covering the cases of tame hereditary tame concealed and tubular algebras respectively our approach yields in particular a uniform treatment to an essential part of the invariant theory of extended dynkin quivers a topic popular over the years but stretches far beyond since also concealed canonical algebras of tubular or wild representation type are covered

the tb theorem on non homogeneous spaces 0 introduction main objects and results 3 0 1 main results 4 0 2 an application of t1 heorem electric intensity capacity 7 0 3 how to interpret calderon zygmund operator t 9 0 3 1 bilinear form is defined on lipschitz functions 10 0 3 2 bilinear form is defined for smooth functions 11 0 3 3 apriori boundedness 12 0 4 plan of the paper 15

scaling behavior of nonhyperbolic coupled map lattices coupled map lattices of nonhyperbolic local maps arise naturally in many physical situations described by discretized reaction diffusion equations or discretized scalar field theories as a prototype for these types of lattice dynamical systems we study diffusively coupled tchebyscheff maps of nth order which exhibit strongest possible chaotic behavior for small coupling constants a we prove that the expectations of arbitrary observables scale with sqrt of a in the low coupling limit contrasting the hyperbolic case which is known to scale with a moreover we prove that there are log periodic oscillations of period ln n2 modulating the sqrt of a dependence of a given expectation value we develop a general 1st order perturbation theory to analytically calculate the invariant one point density show that the density exhibits log periodic oscillations in phase space and obtain excellent agreement with numerical results

an analytic estimate of the maximum lyapunov exponent in products of tridiagonal random matrices in this work we present a theoretical and numerical study of the behaviour of the maximum lyapunov exponent in products of random tridiagonal matrices in the limit of small coupling and small fluctuations such a problem is directly motivated by the investigation of coupled map lattices in a regime where the chaotic properties are quite robust and yet a complete understanding has still not been reached we derive some approximate analytic expressions by introducing a suitable continuous time formulation of the evolution equation as a first result we show that the perturbation of the lyapunov exponent due to the coupling depends only on a single scaling parameter which in the case of strictly positive multipliers is the ratio of the coupling strength with the variance of local multipliers an explicit expression for the lyapunov exponent is obtained by mapping the original problem onto a chain of nonlinear langevin equations which are eventually reduced to a single stochastic equation the probability distribution of this dynamical equation provides an excellent description for the behaviour of the lyapunov exponent finally multipliers with random signs are also considered finding that the lyapunov exponent again depends on a single scaling parameter which however has a different expression

quiver representations with an irreducible null cone abstract let d be a prehomogeneous dimension vector for a quiver q there is an action of the product gl d of linear groups on the vector space rep q d of representations of q with dimension vector d and there is a representation t with a dense gl d orbit in rep q d we give a construction for a dense subset f q d of the variety z q d of common zeros of all semi invariants in k rep q d of positive degree and we show that this set is stable for big dimension vectors i e f q n d x t n 1 x f q d moreover we show that the existence of a dense orbit in z q d depends on a quiver q such that the category of representations of q is equivalent to the right perpendicular category t

smooth quiver representation spaces abstract in this paper we classify all quivers and corresponding dimension vectors having a smooth space of semisimple representation classes the result is that these quiver settings can be reduced via some specific reduction steps to 3 simple types

characterization and semiadditivity of the c 1 harmonic capacity the c 1 harmonic capacity c plays a central role in problems of approximation by harmonic functions in the c 1 norm in n 1 in this paper we prove the comparability between the capacity c and its positive version c as a corollary we deduce the semiadditivity of c this capacity can be considered as a generalization in n 1 of the continuous analytic capacity in c moreover we also show that the so called inner boundary conjecture fails for dimensions n 1 unlike in the case n 1

stratifications of marsden weinstein reductions for representations of quivers and deformations of symplectic quotient singularities we investigate the poisson geometry of the marsden weinstein reductions of the moment map associated to the cotangent bundle of the space of representations of a quiver we show that the stratification by representation type equals the stratification by symplectic leaves the deformed symplectic quotient singularities spectra of centres of symplectic reflection algebras associated to a wreath product are shown to be isomorphic to reductions of a certain quiver this establishes a method to calculate when these deformations are smooth furthermore the isomorphism identifies symplectic leaves and so one can give a description of their symplectic leaves in terms of roots of the quiver

perpendicular categories null cones and dense orbits abstract let q be a quiver k an algebraically closed field and d a prehomogeneous dimension vector for q with canonical decomposition d i 1 r i dim t i we denote by z q d the set of common zeros of all semi invariants in k rep q d of positive degree we study the relations between the existence of a dense gl d orbit in the null cone z q d and the position of the representations t 1 t r in the auslander reiten quiver associated to q

multiplicative preprojective algebras middle convolution and the deligne simpson problem abstract we introduce a family of algebras which are multiplicative analogues of preprojective algebras and their deformations as introduced by m p holland and the first author we show that these algebras provide a natural setting for the middle convolution operation introduced by n m katz in his book rigid local systems and put in an algebraic setting by m dettweiler and s reiter and h volklein we prove a homological formula relating the dimensions of hom and ext spaces study varieties of representations of multiplicative preprojective algebras and use these results to study simple representations we apply this work to the deligne simpson problem obtaining a sufficient and conjecturally necessary condition for the existence of an irreducible solution to the equation a 1 a 2 a k 1 with the a i in prescribed conjugacy classes in gl n c

finitely presented simple modules over leavitt path algebras abstract let e be an arbitrary graph and k be any field we construct various classes of non isomorphic simple modules over the leavitt path algebra l k e induced by vertices which are infinite emitters closed paths which are exclusive cycles and paths which are infinite and call these simple modules chen modules it is shown that every primitive ideal of l k e can be realized as the annihilator ideal of some chen module our main result establishes the equivalence between a graph theoretic condition and various conditions concerning the structure of simple modules over l k e

littlewood paley theory and the t 1 theorem with non doubling measures let be a radon measure on rd which may be non doubling the only condition that must satisfy is b x r crn for all x rd r 0 and for some fixed 0 n d in this paper littlewood paley theory for functions in lp is developed one of the main difficulties to be solved is the construction of reasonable approximations of the identity in order to obtain a calderon type reproducing formula moreover it is shown that the t 1 theorem for n dimensional calderon zygmund operators without doubling assumptions can be proved using the littlewood paley type decomposition that is obtained for functions in l2 as in the classical case of homogeneous spaces

invariants of quivers under the action of classical groups abstract we consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical linear groups instead of a product of the general linear groups and by considering the dual action of groups on vertex vector spaces together with the usual action a generating system for the corresponding algebra of invariants is found in particular semi invariants of supermixed representations of quivers are established as a consequence a generating system for the algebra of so n invariants of several matrices is constructed over a field of characteristic different from 2 the proof uses the reduction to semi invariants of mixed representations and the decomposition formula that generalizes amitsur s formula for the determinant

symmetric quiver settings with a regular ring of invariants in this paper we classify all the symmetric quivers and corresponding dimension vectors having a smooth space of semisimple representation classes the result we obtain is that such quivers can be decomposed as a connected sum of a few number of basic quivers

on the symplectic structure of instanton moduli spaces abstract we study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of su 2 instantons on both commutative and non commutative r 4 we identify global darboux coordinates and quadratic hamiltonians on classical phase spaces for which these quiver varieties are natural completions we also show that the group of non commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non commutative instantons this paper should be viewed as a step towards extending known results for calogero moser spaces to the instanton moduli spaces

the semiadditivity of continuous analytic capacity and the inner boundary conjecture let a e be the continuous analytic capacity of a compact set e c c in this paper we obtain a characterization of a in terms of curvature of measures with zero linear density and we deduce that a is countably semiadditive this result has important consequences for the theory of uniform rational approximation on compact sets in particular it implies the so called inner boundary conjecture 1 introduction the continuous analytic capacity a was introduced by erokhin and vitushkin see vi in the 1950s in order to study problems of uniform rational approximation in compact subsets of the complex plane up to now the geometric properties of the capacity a have not been well understood it has been an open question if a is semiadditive as a set function that is to say if

isomorphisms of quantizations via quantization of resolutions abstract in this paper we prove the existence of isomorphisms between certain non commutative algebras that are interesting from the representation theoretic perspective and arise as quantizations of certain poisson algebras we show that quantizations of kleinian singularities obtained by three different constructions are isomorphic to each other the constructions are via symplectic reflection algebras quantum hamiltonian reduction and w algebras next we prove that parabolic w algebras in type a are isomorphic to quantum hamiltonian reductions associated to quivers of type a finally we show that the symplectic reflection algebras for wreath products of the symmetric group and a kleinian group are isomorphic to certain quantum hamiltonian reductions our results involving w algebras are new while for those dealing with symplectic reflection algebras we just find new proofs a key ingredient in our proofs is the study of quantizations of symplectic resolutions of appropriate poisson varieties

generating graded central simple algebras abstract we solve the generator problem for z graded central simple algebras applications are given to automorphisms of trace rings of generic matrices and to periodic fat point modules

riesz transforms and harmonic lip1 capacity in cantor sets we estimate the l norm of the s dimensional riesz transforms on some cantor sets in rd to this end we show that the riesz transforms truncated at different scales behave in quasiorthogonal way as an application we obtain some precise numerical estimates for the lipschitz harmonic capacity of these sets

noncommutative smoothness and coadjoint orbits abstract in math ag 0010030 math z to appear r bocklandt and the author proved that certain quotient varieties of representations of deformed preprojective algebras are coadjoint orbits for the necklace lie algebra n q of the corresponding quiver q a conjectural ring theoretical explanation of these results was given in terms of noncommutative smoothness in the sense of c procesi j algebra 107 1987 63 74 in this paper we prove these conjectures the main tool in the proof is the etale local description due to w crawley boevey math ag 0105247 along the way we determine the smooth locus of the marsden weinstein reductions for quiver representations

formal structures and representation spaces abstract m kapranov introduced and studied in math ag 9802041 the noncommutative formal structure of a smooth affine variety in this note we show that his construction is a particular case of micro localization and extend the construction functorially to representation schemes of affine algebras we describe explicitly the formal completions in the case of path algebras of quivers and initiate the study of their finite dimensional representations

minimal generating set for semi invariants of quivers of dimension two abstract a minimal by inclusion generating set for the algebra of semi invariants of a quiver of dimension 2 2 is established over an infinite field of arbitrary characteristic the mentioned generating set consists of the determinants of generic matrices and the traces of tree paths of pairwise different multidegrees where in the case of characteristic different from two we take only admissible paths as a consequence we describe relations modulo decomposable semi invariants

generating invariant rings of quivers in arbitrary characteristic abstract it is well known that the ring of polynomial invariants of a reductive group is finitely generated however it is difficult to give strong upper bounds on the degrees of the generators especially over fields of positive characteristic in this paper we make use of the theory of good filtrations along with recent results on the null cone to provide polynomial bounds for matrix semi invariants in arbitrary characteristic and consequently for matrix invariants our results generalize to invariants and semi invariants of quivers

solvable model for spatiotemporal chaos we show that the dynamical behavior of a coupled map lattice where the individual maps are bernoulli shift maps can be solved analytically for integer couplings we calculate the invariant density of the system and show that it displays a nontrivial spatial behavior we also introduce and calculate a generalized spatiotemporal correlation function

hopf algebras with trace and representations we study the restriction of representations of cayley hamilton algebras to subalgebras this theory is applied to determine tensor products and branching rules for representations of quantum groups at roots of 1

quivers geometric invariant theory and moduli of linear dynamical systems we use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems a general approach to this problem is presented and applied to two well known cases we show how both lomadze s and helmke s compactification arises naturally as a geometric invariant theory quotient both moduli spaces are proven to be smooth projective manifolds furthermore a description of lomadze s compactification as a quot scheme is given whereas helmke s compactification is shown to be an algebraic grassmann bundle over a quot scheme this gives an algebro geometric description of both compactifications as an application we determine the cohomology ring of helmke s compactification and prove that the two compactifications are not isomorphic when the number of outputs is positive