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

geometric characterizations of the representation type of hereditary algebras and of canonical algebras abstract we show that a finite connected quiver q with no oriented cycles is tame if and only if for each dimension vector d and each integral weight of q the moduli space m q d s s of semi stable d dimensional representations of q is just a projective space in order to prove this we show that the tame quivers are precisely those whose weight spaces of semi invariants satisfy a certain log concavity property furthermore we characterize the tame quivers as being those quivers q with the property that for each schur root d of q the field of rational invariants k rep q d gl d is isomorphic to k or k t next we extend this latter description to canonical algebras more precisely we show that a canonical algebra is tame if and only if for each generic root d of and each indecomposable irreducible component c of rep d the field of rational invariants k c gl d is isomorphic to k or k t along the way we establish a general reduction technique for studying fields of rational invariants on schur irreducible components of representation varieties

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

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

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

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

local random quantum circuits are approximate polynomial designs we prove that local random quantum circuits acting on n qubits composed of o t10n2 many nearest neighbor two qubit gates form an approximate unitary t design previously it was unknown whether random quantum circuits were a t design for any t 3 the proof is based on an interplay of techniques from quantum many body theory representation theory and the theory of markov chains in particular we employ a result of nachtergaele for lower bounding the spectral gap of frustration free quantum local hamiltonians a quasi orthogonality property of permutation matrices a result of oliveira which extends to the unitary group the path coupling method for bounding the mixing time of random walks and a result of bourgain and gamburd showing that dense subgroups of the special unitary group composed of elements with algebraic entries are copy tensor product expanders we also consider pseudo randomness properties of local random quantum circuits of small depth and prove that circuits of depth o t10n constitute a quantum t copy tensor product expander the proof also rests on techniques from quantum many body theory in particular on the detectability lemma of aharonov arad landau and vazirani we give applications of the results to cryptography equilibration of closed quantum dynamics and the generation of topological order in particular we show the following pseudo randomness property of generic quantum circuits almost every circuit u of size o nk on n qubits cannot be distinguished from a haar uniform unitary by circuits of size o n k 9 11 that are given oracle access to u

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

explicit constructions of unitary transformations between equivalent irreducible representations irreducible representations irreps of a finite group g are equivalent if there exists a similarity transformation between them in this paper we describe an explicit algorithm for constructing this transformation between a pair of equivalent irreps assuming that we are given an algorithm for computing the matrix elements of these irreps along the way we derive a generalization of the classical orthogonality relations for matrix elements of irreps of finite groups we give an explicit form of such unitary matrices for the important case of conjugated young yamanouchi representations when our group g is the symmetric group s n

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

subsystem eth motivated by the qualitative picture of canonical typicality we propose a refined formulation of the eigenstate thermalization hypothesis eth for chaotic quantum systems the new formulation which we refer to as subsystem eth is in terms of the reduced density matrix of subsystems this strong form of eth outlines the set of observables defined within the subsystem for which it guarantees eigenstate thermalization we discuss the limits when the size of the subsystem is small or comparable to its complement in the latter case we outline the way to calculate the leading volume proportional contribution to the von neumann and renyi entanglment entropies finally we provide numerical evidence for the proposal in the case of a one dimensional ising spin chain

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

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

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

concentration of haar measures with an application to random matrices abstract we show that the mixing times of random walks on compact groups can be used to obtain concentration inequalities for the respective haar measures as an application we derive a concentration inequality for the empirical distribution of eigenvalues of sums of random hermitian matrices with possible applications in free probability the advantage over existing techniques is that the new method can deal with functions that are non lipschitz or even discontinuous with respect to the usual metrics

approach to thermal equilibrium of macroscopic quantum systems we consider an isolated macroscopic quantum system let mathcal h be a microcanonical energy shell i e a subspace of the system s hilbert space spanned by the finitely many energy eigenstates with energies between e and e ensuremath delta e the thermal equilibrium macrostate at energy e corresponds to a subspace mathcal h eq of mathcal h such that text dim text mathcal h eq text dim text mathcal h is close to 1 we say that a system with state vector ensuremath psi mathcal h is in thermal equilibrium if ensuremath psi is close to mathcal h eq we show that for typical hamiltonians with given eigenvalues all initial state vectors ensuremath psi 0 evolve in such a way that ensuremath psi t is in thermal equilibrium for most times t this result is closely related to von neumann s quantum ergodic theorem of 1929

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

optimal filtrations on representations of finite dimensional algebras we present a representation theoretic description of the non empty strata in the hesselink stratification of the nullcone of representations of quivers we use this stratification to define optimal filtrations on representations of finite dimensional algebras as an application we investigate the isomorphism problem for uniserial representations 1 optimal filtrations 1 1 let a be a finite dimensional algebra over an algebraically closed field k in this paper we want to parameterize isomorphism classes of finite dimensional amodules having a specific jordan holder sequence in particular we want to relate the recent results of k bongartz and b huisgen zimmermann 3 4 1 on uniserial modules to the hesselink stratification of nullcones by morita theory we may reduce to the case that a is a basic algebra that is all simple a modules are one dimensional in this case we can write a as the quotient of the path algebra of a quiver and relate finite dimensional a modules to representations of this quiver 1 2 a quiver q is a directed graph on a finite set of vertices vl vn let aij be the number of directed arrows from vi to vj or loops if v vj the euler form of q is the bilinear form x z x zn z determined by the matrix x xij i j c mn z with entries xij 6ij aij clearly x encodes the structure of the directed graph q a representation v of a quiver q of dimension vector a al an e nn assigns to every arrow vi vj in q a matrix v q e majixi k the set of all a dimensional representations form an affine space

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

optimal stopping time problem for stochastic navier stokes equations and infinite dimensional variational inequalities we prove the existence of a solution for the obstacle problem associated with the kolmogorov operator corresponding to the stopping time problem for stochastic navier stokes equations in 2 d 2005 elsevier ltd all rights reserved

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

deformations of polyhedra and polygons by the unitary group we introduce the set of framed convex polyhedra with n faces as the symplectic quotient c2n su 2 a framed polyhedron is then parametrized by n spinors living in c2 satisfying suitable closure constraints and defines a usual convex polyhedron plus extra u 1 phases attached to each face we show that there is a natural action of the unitary group u n on this phase space which changes the shape of faces and allows to map any framed polyhedron onto any other with the same total boundary area this identifies the space of framed polyhedra to the grassmannian space u n su 2 u n 2 we show how to write averages of geometrical observables polynomials in the faces area and the angles between them over the ensemble of polyhedra distributed uniformly with respect to the haar measure on u n as polynomial integrals over the unitary group and we provide a few methods to compute these integrals systematically we also use the itzykson zuber formula from matrix models as the generating function

the cut off phenomenon for random reflections ii complex and quaternionic cases in this paper we generalize the random reflections problem ono n considered in an earlier paper to the complex and quaternionic cases we give precise estimates on the speed of convergence to stationarity for specific examples of random walks onu n andsp n for which the one step distribution is a certain probability measure concentrated on reflections our results show that in both cases the so called cut off phenomenon occurs atk0 1 2n logn

concentration and convergence rates for spectral measures of random matrices the topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest including in particular matrices drawn from haar measure on the classical lie groups random compressions of random hermitian matrices and the so called random sum of two independent random matrices in each case we estimate the expected wasserstein distance from the empirical spectral measure to a deterministic reference measure and prove a concentration result for that distance as a consequence we obtain almost sure convergence of the empirical spectral measures in all cases

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

approach to typicality in many body quantum systems the recent discovery that for large hilbert spaces almost all that is typical hamiltonians have eigenstates that place small subsystems in thermal equilibrium has shed much light on the origins of irreversibility and thermalization here we give numerical evidence that many body lattice systems generically approach typicality as the number of subsystems is increased and thus provide further support for the eigenstate thermalization hypothesis our results indicate that the deviation of many body systems from typicality decreases exponentially with the number of systems further by averaging over a number of randomly selected nearest neighbor interactions we obtain a powerlaw for the atypicality as a function of the hilbert space dimension distinct from the power law possessed by random hamiltonians

local random quantum circuits are approximate polynomial designs numerical results we numerically investigate the statement that local random quantum circuits acting on n qubits composed of polynomially many nearest neighbor two qubit gates form an approximate unitary poly n design brandao et al 2012 arxiv 1208 0692 using a group theory formalism spectral gaps that give a ratio of convergence to a given t design are evaluated for a different number of qubits n up to 20 and degrees t t 2 3 4 and 5 improving previously known results for n 2 in the case of t 2 and 3 their values lead to the conclusion that the previously used lower bound that bounds spectral gaps values may give very little information about the real situation and in most cases only tells us that a gap is closed we compare our results to another lower bounding technique again showing that its results may not be tight

numerical evidence for approximate consistency and markovianity of some quantum histories in a class of finite closed spin systems closed quantum systems obey the schrodinger equation whereas nonequilibrium behavior of many systems is routinely described in terms of classical markovian stochastic processes evidently there are fundamental differences between those two types of behavior we discuss the conditions under which the unitary dynamics may be mapped onto pertinent classical stochastic processes this is first principally addressed based on the notions of consistency and markovianity numerical data are presented that show that the above conditions are to good approximation fulfilled for heisenberg type spin models comprising 12 20 spins the accuracy to which these conditions are met increases with system size

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

on the explicit constructions of certain unitary t designs unitary t designs are good finite subsets of the unitary group ug d that approximate the whole unitary group ug d well unitary t designs have been applied in randomized benchmarking tomography quantum cryptography and many other areas of quantum information science if a unitary t design itself is a group then it is called a unitary t group although it is known that unitary t designs in ug d exist for any t and d the unitary t groups do not exist for t geq 4 if d geq 3 as it is shown by guralnick tiep 2005 and bannai navarro rizo tiep bnrt 2018 explicit constructions of exact unitary t designs in ug d are not easy in general in particular explicit constructions of unitary 4 designs in ug 4 have been an open problem in quantum information theory we prove that some exact unitary t 1 designs in the unitary group ug d are constructed from unitary t groups in ug d that satisfy certain specific conditions based on this result we specifically construct exact unitary 3 designs in ug 3 from the unitary 2 group sl 3 2 in ug 3 and also unitary 4 designs in ug 4 from the unitary 3 group sp 4 3 in ug 4 numerically we also discuss some related problems and a few applications to physics problems

from quantum dynamics to the canonical distribution general picture and a rigorous example derivation of the canonical or boltzmann distribution based only on quantum dynamics is discussed consider a closed system which consists of a mutually interacting subsystem and a heat bath and assume that the whole system is initially in a pure state which can be far from equilibrium with small energy fluctuation under the hypothesis of equal weights for eigenstates we derive the canonical distribution in the sense that at sufficiently large and typical time the instantaneous quantum mechanical expectation value of an arbitrary operator of the subsystem is almost equal to the desired canonical expectation value we present a class of examples in which the above derivation can be rigorously established without any unproven hypotheses

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

normal typicality and von neumann s quantum ergodic theorem we discuss the content and significance of john von neumann s quantum ergodic theorem qet of 1929 a strong result arising from the mere mathematical structure of quantum mechanics the qet is a precise formulation of what we call normal typicality i e the statement that for typical large systems every initial wave function 0 from an energy shell is normal it evolves in such a way that t t is for most t macroscopically equivalent to the micro canonical density matrix the qet has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s however we point out that this criticism does not apply to the actual qet a correct statement of which does not appear in these papers but to a different indeed weaker statement furthermore we formulate a stronger statement of normal typicality based on the observation that the bound on the deviations from the average specified by von neumann is unnecessarily coarse and a much tighter and more relevant bound actually follows from his proof

impulse control of stochastic navier stokes equations optimal control theory of 2uid dynamics has numerous applications such as aero hydrodynamic control combustion control tokomak magnetic fusion as well as ocean and atmospheric prediction during the past decade several fundamental advances have been made by a number of researchers as documented in sritharan 21 22 in this paper we develop a new direction to this subject namely we mathematically formulate and resolve impulse and stopping time problems impulse control of navier stokes equations has signi7cance beyond control theory in fact in optimal weather prediction the task of updating the initial data optimally at strategic times can be reformulated precisely as an impulse control problem for the primitive cloud equations which consist of the navier stokes equation coupled with temperature and species evolution equations cf 11 see 1 9 15 for the study of optimal stopping problem alone it is possible to impose regularity assumptions on the stopping cost however in our case optimal stopping problems are used as intermediate steps to treat the impulse control problem through an iteration process this dictates that we must work with stopping costs which have only continuity property optimal stopping and impulse control problems are very well known particularly for di usion processes e g see the books of bensoussan and lions 3 4 for degenerate

convergence rate for spectral distribution of addition of random matrices abstract let a and b be two n by n deterministic hermitian matrices and let u be an n by n haar distributed unitary matrix it is well known that the spectral distribution of the sum h a u b u converges weakly to the free additive convolution of the spectral distributions of a and b as n tends to infinity we establish the optimal convergence rate 1 n in the bulk of the spectrum

viscosity solutions of dynamic programming equations for the optimal control of the two dimensional navier stokes equations in this paper we study the infinite dimensional hamilton jacobi equation associated with the optimal feedback control of viscous hydrodynamics we resolve the global unique solvability problem of this equation by showing that the value function is the unique viscosity solution

convergence to equilibrium under a random hamiltonian we analyze equilibration times of subsystems of a larger system under a random total hamiltonian in which the basis of the hamiltonian is drawn from the haar measure we obtain that the time of equilibration is of the order of the inverse of the arithmetic average of the bohr frequencies to compute the average over a random basis we compute the inverse of a matrix of overlaps of operators which permute four systems we first obtain results on such a matrix for a representation of an arbitrary finite group and then apply it to the particular representation of the permutation group under consideration

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

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

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

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

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

a concentration inequality and a local law for the sum of two random matrices let h n a n u n b n u n ast where an and bn are two n by n hermitian matrices and un is a haar distributed random unitary matrix and let mu h n mu a n mu b n be empirical measures of eigenvalues of matrices hn an and bn respectively then it is known see pastur and vasilchuk in commun math phys 214 249 286 2000 that for large n the measure mu h n is close to the free convolution of measures mu a n and mu b n where the free convolution is a non linear operation on probability measures the large deviations of the cumulative distribution function of mu h n from its expectation have been studied by chatterjee j funct anal 245 379 389 2007 in this paper we improve chatterjee s concentration inequality and show that it holds with the rate which is quadratic in n in addition we prove a local law for eigenvalues of h n n by showing that the normalized number of eigenvalues in an interval approaches the density of the free convolution of a and b provided that the interval has width log n 1 2

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

statistical description of small quantum systems beyond the weak coupling limit an explicit expression is derived for the statistical description of small quantum systems which are relatively weakly and directly coupled to only small parts of their environments the derived expression has a canonical form but is given by a renormalized self hamiltonian of the studied system which appropriately takes into account the influence of the system environment interaction in the case that the system has a narrow spectrum and the environment is sufficiently large the modification to the self hamiltonian usually has a mean field feature given by an environmental average of the interaction hamiltonian in other cases the modification may be beyond the mean field approximation

variational inequalities and optimization for markov processes associated with semi dirichlet forms we consider an optimal stopping problem and an impulsive control for markov processes associated with semi dirichlet forms we show that the value functions are the maximum solutions of certain variational inequalities examples both in finite dimensions and infinite dimensions are given

stein s method for concentration inequalities we introduce a version of stein s method for proving concentration and moment inequalities in problems with dependence simple illustrative examples from combinatorics physics and mathematical statistics are provided

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

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