hilbert schemes and noncommutative deformations of type a kleinian singularities abstract let h k be a symplectic reflection algebra corresponding to a cyclic subgroup sl 2 c of order n and u k e h k e the spherical subalgebra of h k we show that for suitable k there is a filtered z n 1 algebra r such that 1 there is an equivalence of categories r qgr u k mod 2 there is an equivalence of categories gr r qgr coh hilb c 2 here coh hilb c 2 is the category of coherent sheaves on the hilbert scheme and for a graded algebra r we write r qgr for the quotient category of finitely generated graded r modules modulo torsion

quantization and dynamisation of trace poisson brackets the quantization problem for the trace bracket algebra derived from double poisson brackets is discussed we obtain a generalization of the boundary ybe or so called abcd algebra for the quantization of quadratic trace brackets a dynamical deformation is proposed on the lines of gervais neveu felder dynamical quantum algebras

module structures on the k theory of graded rings abstract let r be a commutative ring a a0 a1 a graded r algebra and a the graded ideal a1 a2 then ki a ki a0 ki a a we show that the groups ki a a are naturally modules over the ring w r of witt vectors they also have a natural filtration whose associated graded groups are r modules when r contains a field of characteristic zero ki a a is an r module and the filtration is by r submodules

poisson gerstenhaber brackets in representation algebras abstract we introduce cyclic bilinear forms on coalgebras and use them to generalize van den bergh s poisson brackets in representation algebras

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

reflection functors and symplectic reflection algebras for wreath products we construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver these functors give equivalences of categories associated to generic parameters which are in the same orbit under the weyl group action we give applications to the representation theory of symplectic reflection algebras of wreath product groups

a class of algebras similar to the enveloping algebra of sl 2 fix f e c x define r c a b h subject to the relations ha ah a hb bh b ab ba f h we study these algebras for different f and in particular show how they are similar to and different from u sl 2 the enveloping algebra of sl 2 c there is a notion of highest weight modules and a category a for such r for each n 0 if f x x 1 n l xn 1 then r has precisely n simple modules in each finite dimension and every finite dimensional r module is semisimple

modified double poisson brackets abstract we propose a non skew symmetric generalization of the original definition of double poisson bracket by m van den bergh it allows one to explicitly construct a more general class of h 0 poisson structures on finitely generated associative algebras we show that modified double poisson brackets inherit certain major properties of the double poisson brackets

sheaves on fibered threefolds and quiver sheaves this paper classifies a class of holomorphic d branes closely related to framed torsion free sheaves on threefolds fibered in resolved ade surfaces over a general curve c in terms of representations with relations of a twisted kronheimer nakajima type quiver in the category coh c of coherent sheaves on c for the local calabi yau case c cong bf a 1 and special choice of framing one recovers the n 1 ade quiver studied by cachazo katz vafa

double poisson algebras in this paper we develop poisson geometry for non commutative algebras this generalizes the bi symplectic geometry which was recently and independently introduced by crawley boevey etingof and ginzburg our quasi poisson brackets induce classical quasi poisson brackets on representation spaces as an application we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by crawley boevey and shaw carry a natural poisson structure

canonical singularities of orders over surfaces we define and study canonical singularities of orders over surfaces these are non commutative analogues of kleinian singularities that arise naturally in the minimal model program for orders over surfaces d chan and c ingalls invent math 161 2005 427 452 we classify canonical singularities of orders using their minimal resolutions which we define we describe them explicitly as invariant rings for the action of a finite group on a full matrix algebra over a regular local ring we also prove that they are gorenstein describe their auslander reiten quivers and note a simple version of the mckay correspondence

noncommutative resolutions of ade fibered calabi yau threefolds in this paper we construct noncommutative resolutions of a certain class of calabi yau threefolds studied by cachazo et al geometric transitions and n 1 quiver theories http arxiv org abs hep th 0108120v2 2001 the threefolds under consideration are fibered over a complex plane with the fibers being deformed kleinian singularities the construction is in terms of a noncommutative algebra introduced by ginzburg calabi yau algebras http arxiv org abs math 0612139v2 2006 which we call the n 1 ade quiver algebra

poisson deformations of symplectic quotient singularities abstract we establish a connection between smooth symplectic resolutions and symplectic deformations of a possibly singular affine poisson variety in particular let v be a finite dimensional complex symplectic vector space and g sp v a finite subgroup our main result says that the so called calogero moser deformation of the orbifold v g is in an appropriate sense a versal poisson deformation that enables us to determine the algebra structure on the cohomology h x c of any smooth symplectic resolution x v g multiplicative mckay correspondence we prove further that if g gl h is an irreducible weyl group and v h h then no smooth symplectic resolution of v g exists unless g is of types a b c

noncommutative extensions of elliptic integrable euler arnold tops and painleve vi equation in this paper we suggest generalizations of elliptic integrable tops to matrix valued variables our consideration is based on the r matrix description which provides lax pairs in terms of quantum and classical r matrices first we prove that for relativistic and non relativistic tops such lax pairs with spectral parameters follow from the associative yang baxter equation and its degenerations then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction we describe a matrix version of the reduced elliptic top and verify that the latter constraints are fulfilled in this case the construction of matrix extensions is naturally generalized to the monodromy preserving equation in this way we get matrix extensions of the painleve vi equation and its multidimensional analogues written in the form of non autonomous elliptic tops finally it is mentioned that the matrix valued variables can be replaced by elements of noncommutative associative algebra at the end of the paper we also describe special elliptic gaudin models which can be considered as matrix extensions of the reduced elliptic top

calabi yau algebras and the shifted noncommutative symplectic structure abstract in this paper we show that for a koszul calabi yau algebra there is a shifted bi symplectic structure in the sense of crawley boevey etingof ginzburg 15 on the cobar construction of its co unitalized koszul dual coalgebra and hence its dg representation schemes in the sense of berest khachatryan ramadoss 3 have a shifted symplectic structure in the sense of pantev toen vaquie vezzosi 28

preprojective algebra structure on skew group algebras abstract we give a class of finite subgroups g s l n k for which the skew group algebra k x 1 x n g does not admit the grading structure of a higher preprojective algebra namely we prove that if a finite group g s l n k is conjugate to a subgroup of s l n 1 k s l n 2 k for some n 1 n 2 1 then the skew group algebra k x 1 x n g is not morita equivalent to a higher preprojective algebra this is related to the preprojective algebra structure on the tensor product of two koszul bimodule calabi yau algebras we prove that such an algebra cannot be endowed with a grading structure as required for a higher preprojective algebra moreover we construct explicitly the bound quiver of the higher preprojective algebra over a finite dimensional koszul algebra of finite global dimension we show in addition that preprojective algebras over higher representation infinite koszul algebras are derivation quotient algebras whose relations are given by a superpotential

the combinatorics of c fixed points in generalized calogero moser spaces and hilbert schemes abstract in this paper we study the combinatorial consequences of the relationship between rational cherednik algebras of type g l 1 n cyclic quiver varieties and hilbert schemes we classify and explicitly construct c fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles furthermore we give a combinatorial description of the bijections between c fixed points induced by the etingof ginzburg isomorphism and nakajima reflection functors we apply our results to obtain a new proof as well as a generalization of the q hook formula

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

isomorphism problems of noncommutative deformations of type d kleinian singularities we construct all possible noncommutative deformations of a kleinian singularity c 2 gamma of type d n in terms of generators and relations and solve the isomorphism problem for the associative algebras thus constructed we prove that in our parametrization all isomorphisms arise from the action of the normalizer n sl 2 gamma on c gamma we deduce that the moduli space of isomorphism classes of noncommutative deformations in type d n is isomorphic to a vector space of dimension 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

quantized symplectic oscillator algebras of rank one a quantized symplectic oscillator algebra of rank 1 is a pbw deformation of the smash product of the quantum plane with uq sl2 we study its representation theory and in particular its category o

on the connected part of the covariant tatep divisible group and the function of the family of hyperelliptic curvesy 2 1 x n modulo various primes

tempered distributions on heisenberg groups whose convolution with schwartz class functions is schwartz class abstract let n be a nilpotent lie group and q a tempered distribution on n we say that q is a left j multiplier if convolution on the left by q takes schwartz class functions to schwartz class functions there is a similar definition for right j multipliers we show that if is an irreducible unitary representation of n then one can define q h h whenever q is a left j multiplier the main results of the paper characterize left j multipliers q on heisenberg groups in terms of the transform operators q and show how this characterization can be used to find fundamental solutions of some left invariant differential operators there is also an example of a left j multiplier which is not a right j multiplier

generalized double affine hecke algebras of rank 1 and quantized del pezzo surfaces abstract let d be a simply laced dynkin diagram of rank r whose affinization has the shape of a star i e d 4 e 6 e 7 e 8 to such a diagram one can attach a group g whose generators correspond to the legs of the affinization have orders equal to the leg lengths plus 1 and the product of the generators is 1 the group g is then a 2 dimensional crystallographic group g z l z 2 where l is 2 3 4 and 6 respectively in this paper we define a flat deformation h t q of the group algebra c g of this group by replacing the relations saying that the generators have prescribed orders by their deformations saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots which are deformation parameters the algebra h t q for d 4 is the cherednik algebra of type c c 1 which was studied by noumi sahi and stokman and controls askey wilson polynomials we prove that h t q is the universal deformation of the twisted group algebra of g and that this deformation is compatible with certain filtrations on c g we also show that if q is a root of unity then for generic t the algebra h t q is an azumaya algebra and its center is the function algebra on an affine del pezzo surface for generic q the spherical subalgebra e h t q e provides a quantization of such surfaces we also discuss connections of h t q with preprojective algebras and painleve vi

noncommutative deformations of type a kleinian singularities abstract the type a kleinian singularities are the surfaces of the form c 2 g where g is a finite cyclic subgroup of sl2 c the standard noncommutative analog is the fixed ring of the weyl algebra a1 c under the induced action of g these fixed rings are however only part of a large family of algebras whose associated graded ring is the coordinate ring of such a singularity the structure of these algebras is studied with reference to the structure of the associated singularity and to the structure of primitive factor rings of u s l 2 c

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

simple ambiskew polynomial rings abstract we determine simplicity criteria in characteristics 0 and p for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring one obstruction to simplicity is the possible existence of a canonical normal element z in the case where this element exists we give simplicity criteria for the rings obtained by inverting z and the rings obtained by factoring out the ideal generated by z the results are illustrated by numerous examples including higher quantized weyl algebras and generalizations of some low dimensional symplectic reflection algebras

reflection functors for quiver varieties and weyl group actions we introduce reflectionfunctors on quiver varieties they are hyper kahler isometries between quiver varieties with different parameters related by elements in the weyl group the definition is motivated by the origial reflection functor given by bernstein gelfand ponomarev 1 but they behave much nicely they are isomorphisms and satisfy the weyl group relations as an application we define weyl group representations of homology groups of quiver varieties they are analogues of slodowy s construction of springer representations of the weyl group

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

quantized multiplicative quiver varieties abstract beginning with the data of a quiver q and its dimension vector d we construct an algebra d q d q mat d q which is a flat q deformation of the algebra of differential operators on the affine space mat d q the algebra d q is equivariant for an action by a product of quantum general linear groups acting by conjugation at each vertex we construct a quantum moment map for this action and subsequently define the hamiltonian reduction a d q of d q with moment parameter we show that a d q is a flat formal deformation of lusztig s quiver varieties and their multiplicative counterparts for all dimension vectors satisfying a flatness condition of crawley boevey indeed the product on a d q yields a fedosov quantization the of symplectic structure on multiplicative quiver varieties as an application we give a description of the category of representations of the spherical double affine hecke algebra of type a n 1 and its generalization constructed by etingof oblomkov and rains in terms of a quotient of the category of equivariant d q modules by a serre subcategory of aspherical modules

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

the construction of ale spaces as hyper kahler quotients on decrit la construction d une famille particuliere de 4 varietes hyper kahler les espaces asymptotiquement localement euclidiens ale on decrit une 4 variete de riemann avec juste une extremite qui a l infini ressemble au quotient r 4 de l espace euclidien r 4 par un groupe fini d identifications

quantum groups and deformations of preprojective algebras abstract let be a quiver containing finite cyclic blocks and be the deformed preprojective algebra of it is proved that a suitable quotient uq of for an appropriate vector is a finite dimensional quantum group some properties of uq are also obtained an isomorphism as hopf algebras is given between uq and a restricted quantum group u q sl 2

on functions on graphs and representations of a certain class of algebras abstract we study representations of certain algebras which can be described in terms of graphs and positive functions weights on the set of their vertices continuing an earlier investigation of the case where the graph is a dynkin graph or an extended dynkin graph with a weight of a special kind for the cases where the graph is one of the extended dynkin graphs d 4 e 6 e 7 or e 8 we prove that all irreducible representations of the corresponding algebras are finite dimensional in the case of a graph which properly contains an extended dynkin graph we study the evolution of weights under the action of the coxeter functors in particular we show that there exist two linearly independent p invariant weights we also prove that there exists a weight which makes the corresponding algebra to have an infinite dimensional irreducible representation

special transverse slices and their enveloping algebras let g be a simple simply connected algebraic group over image image lie g image image the nilpotent cone of image and e h f an imageimage2 triple in image let s e ker ad f the special transverse slice to the adjoint orbit of e and s0 s image image the coordinate ring image s0 is naturally graded see slodowy simple singularities and simple algebraic groups lecture notes in mathematics vol 815 springer verlag berlin heidelberg new york 1980 let z image be the centre of the enveloping algebra u image and z image image an algebra homomorphism identify image with image via a killing isomorphism and let denote the linear function on image corresponding to e following kawanaka generalized gelfand graev representations and ennola duality in algebraic groups and related topics advanced studies in pure mathematics vol 6 pp 175 206 north holland amsterdam new york oxford 1985 moeglin c r acad sci paris ser i 303 no 17 1986 845 848 and premet invent math 121 1995 79 117 we attach to a nilpotent subalgebra image subset ofimage of dimension dim 2 and a 1 dimensional image module image let imageh height 14 width 14 denote the algebra opposite to endimage u image multiply sign in circleu image image and imageh height 14 width 14 imageh height 14 width 14 multiply sign in circlez image image it is proved in the paper that the algebra imageh height 14 width 14 has a natural filtration such that gr imageh height 14 width 14 the associated graded algebra is isomorphic to image s0 this construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of image

k theory of endomorphism rings and of rings of invariants abstract this paper gives the following description ofk0of the endomorphism ring of a finitely generated projective module t heorem let t be a ring and p a finitely generated projective t module let i be the trace ideal of p then k0 end pt is isomorphic to a subgroup of k0 t i if further the natural map k1 t k1 t i is surjective then k0 end pt is isomorphic to the subgroup of k0 t generated by the direct summands of pn for n n as a corollary we can determinek0of the ring of invariants for many free linear actions in particular the following result is proved t heorem let v be a fixed point free linear representation of a finite group g over a field k of characteristic zero and let s v be the symmetric algebra of v let k be any finite dimensional k vector space then k0 s v g ks k s v g ks k similar results are given for suitable noncommutative versions ofs v

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

quantization of minimal resolutions of kleinian singularities in this paper we prove an analogue of a recent result of gordon and stafford that relates the representation theory of certain noncommutative deformations of the coordinate ring of the nth symmetric power of c 2 with the geometry of the hilbert scheme of n points in c 2 through the formalism of z algebras our work produces for every regular noncommutative deformation o of a kleinian singularity x c 2 as defined by crawley boevey and holland a filtered z algebra which is morita equivalent to o s uch that the associated graded z algebra is morita equivalent to the minimal resolution of x the construction uses the description of the algebras o as quantum hamiltonian reductions due to holland and a git

non commutative integrability of the grassmann pentagram map abstract the pentagram map is a discrete dynamical system first introduced by schwartz in 1992 it was proved to be integrable by schwartz ovsienko and tabachnikov in 2010 gekhtman shapiro and vainshtein studied poisson geometry associated to certain networks embedded in a disc or annulus and its relation to cluster algebras later gekhtman et al and tabachnikov reinterpreted the pentagram map in terms of these networks and used the associated poisson structures to give a new proof of integrability in 2011 mari beffa and felipe introduced a generalization of the pentagram map to certain grassmannians and proved it had a lax representation we reinterpret this grassmann pentagram map in terms of noncommutative algebra in particular the double brackets of van den bergh and generalize the approach of gekhtman et al to establish a noncommutative version of integrability

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

central extensions of preprojective algebras the quantum heisenberg algebra and 2 dimensional complex reflection groups preprojective algebras of quivers were introduced in 1979 by gelfand and ponomarev gp because for quivers of finite ade type they are models for indecomposable representations they contain each indecomposable exactly once twenty years later these algebras and their deformed versions introduced in cbh for arbitrary quivers became a subject of intense interest since their representation varieties called quiver varieties played an important role in geometric representation theory ironically it is exactly for quivers of finite ade type that preprojective algebras fail to have good properties they are not koszul and their deformed versions are not flat

kp hierarchy for the cyclic quiver we introduce a generalisation of the kp hierarchy closely related to the cyclic quiver and the cherednik algebra hk zm this hierarchy depends on m parameters one of which can be eliminated with the usual kp hierarchy corresponding to the m 1 case generalising the result of wilson invent math 133 1 1 41 1998 we show that our hierarchy admits solutions parameterised by suitable quiver varieties the pole dynamics for these solutions is shown to be governed by the classical calogero moser system for the wreath product zm sn and its new spin version these results are further extended to the case of the multi component hierarchy