on global deformation quantization in the algebraic case abstract we give a proof of yekutieli s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems instead we use an argument inspired by algebraic de rham cohomology

on the interpretation and identification of dynamic takagi sugeno fuzzy models dynamic takagi sugeno fuzzy models are not always easy to interpret in particular when they are identified from experimental data it is shown that there exists a close relationship between dynamic takagi sugeno fuzzy models and dynamic linearization when using affine local model structures which suggests that a solution to the multiobjective identification problem exists however it is also shown that the affine local model structure is a highly sensitive parametrization when applied in transient operating regimes due to the multiobjective nature of the identification problem studied here special considerations must be made during model structure selection experiment design and identification in order to meet both objectives some guidelines for experiment design are suggested and some robust nonlinear identification algorithms are studied these include constrained and regularized identification and locally weighted identification their usefulness in the present context is illustrated by examples

scalar extensions of derived categories and non fourier mukai functors abstract orlov s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a fourier mukai functor this result has been extended by lunts and orlov to include functors from perfect complexes to quasi coherent complexes in this paper we show that the latter extension is false without the full faithfulness hypothesis our results are based on the properties of scalar extensions of derived categories whose investigation was started by pawel sosna and the first author

deformation theory of abelian categories in this paper we develop the basic infinitesimal deformation theory of abelian categories this theory yields a natural generalization of the well known deformation theory of algebras developed by gerstenhaber as part of our deformation theory we define a notion of flatness for abelian categories we show that various basic properties are preserved under flat deformations and we construct several equivalences between deformation problems

hochschild cohomology of abelian categories and ringed spaces this paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors we show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of hochschild cohomology for abelian categories we then show that this hochschild cohomology coincides with the one defined by gerstenhaber schack and swan in the case of module categories over diagrams and schemes and also with the hochschild cohomology for exact categories introduced recently by keller in addition we show in complete generality that hochschild cohomology satisfies a mayer vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf

dg quotients of dg categories abstract keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with verdier s notion of quotient of a triangulated category modulo a triangulated subcategory this work is an attempt to further develop his theory

the first obstructions to enhancing a triangulated category in this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement we show that these obstructions do not vanish in an example of triangulated category without models we also obtain cohomological characterizations of pre triangulated dg a infinity and spectral categories

a two parameter quantum deformation of the unit disc we define and study a two parameter deformation of the unit disc this deformation is described in terms of a family of type i c algebras c q we study the representation theory for c q and construct an action of the quantized universal enveloping algebra uq sl 2 on c q

the cohomology of presheaves of algebras i presheaves over a partially ordered set to each presheaf over a poset of associative algebras a we associate an algebra a we define a full exact embedding of the category of presheaf a bimodules in that of a bimodules we show that this embedding preserves neither enough relative injectives nor enough relative projectives but nonetheless preserves relative yoneda cohomology the cohomology isomorphism links the deformations of manifolds algebraic presheaves and algebras it also implies that the cohomology of any triangulable space is isomorphic to the hochschild cohomology of an associative algebra the latter isomorphism preserves all known cohomology operations we conclude the paper by exhibiting for each associative algebra and triangulable space a product which is again an associative algebra 0 introduction presheaves are rather familiar objects and arise in many situations some of which are discussed below our primary purpose in this paper is to prove a theorem which reduces questions about the cohomology of presheaves of rings to questions about the classical cohomology of rings specifically fix a commutative ring k and a partially ordered set poset jy let a be a presheaf of associative k algebras over j hereinafter a diagram then a has naturally associated with it an abelian category of bimodules and a relative yoneda cohomology bifunctor ext a all the foregoing formalism is introduced in 1 we define for each such diagram a a single k algebra a and a full exact embedding m m of the category of a bimodules in the category of a bimodules 2 the embedding induces a natural transformation of yoneda cohomology bifunctors w exta a ext l 2 our central result is the special cohomology comparison theorem cct namely for an arbitrary diagram a over an arbitrary poset w exta exta h is an isomorphism the proof occupies 3 4 the scheme of the proof will be given shortly first we describe some of the history and utility of the cct in gs1 we proved a primitive version of the cct applicable to arbitrary diagrams defined over arbitrary finite posets and to a narrow class of diagrams defined over a limited class of infinite posets the proofs given there for the finite received by the editors march 6 1987 1980 mathematics subject classification 1985 revision primary 16a58 16a61 16a62 18g10 18g25 secondary 18e25 32g05 55n25 55u10

moduli of non commutative polarized schemes we construct using geometric invariant theory a quasi projective deligne mumford stack of stable graded algebras we also construct a derived enhancement which classifies twisted bundles of stable graded a infty algebras the tangent complex of the derived scheme is given by graded hochschild cohomology which we relate to ordinary hochschild cohomology we obtain a version of hilbert stability for non commutative projective schemes

a quantum deformation of invariants of higher binary forms abstract we use the theory of the quantum group u q gl 2 r to develop a quantum theory of invariants and show a decomposition of invariants into a gordan capelli series higher binary forms are introduced on the basis of braided algebras we define quantised invariants and give basic examples we show that the symbolic method of clebsch and gordan works also in the quantised case we discuss the deformed discriminant of the quadratic and the cubic form the deformed invariants i 1 i 2 of the quartic form and further invariants without a classical analog

on deformations of triangulated models abstract this paper is the first part of a project aimed at understanding deformations of triangulated categories and more precisely their dg and a models and applying the resulting theory to the models occurring in the homological mirror symmetry setup in this first paper we focus on models of derived and related categories based upon the classical construction of twisted objects over a dg or a algebra for a hochschild 2 cocycle on such a model we describe a corresponding curvature compensating deformation which can be entirely understood within the framework of twisted objects we unravel the construction in the specific cases of derived a and abelian categories homotopy categories and categories of graded free qdg modules we identify a purity condition on our models which ensures that the structure of the model is preserved under deformation this condition is typically fulfilled for homotopy categories but not for unbounded derived categories

algebroid prestacks and deformations of ringed spaces for a ringed space x o we show that the deformations of the abelian category mod o of sheaves of o modules lowen and van den bergh 2006 are obtained from algebroid prestacks as introduced by kontsevich in case x is a quasi compact separated scheme the same is true for qch o the category of quasi coherent sheaves on x it follows in particular that there is a deformation equivalence between mod o and qch o

deformations of algebroid stacks abstract in this paper we consider deformations of an algebroid stack on an etale groupoid we construct a differential graded lie algebra dgla which controls this deformation theory in the case when the algebroid is a twisted form of functions we show that this dgla is quasiisomorphic to the twist of the dgla of hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe

quantum theory of the real and the complexified projective line quantum deformations of sets of points of the real and the complexified projective line are constructed these deformations depend on the deformation parameter q and certain further parameters ij the deformations for which the subspace of polynomials of degree three has a basis of ordered monomials are selected we show that the corresponding algebras of three points have polynomiality invariant elements which turn out to be cross ratios in the classical limit are defined for the special case ij 1 a quantum cross ratio with properties similar to the classical case is presented as an application a quantum version of the real euclidean distance is given

an example of a non fourier mukai functor between derived categories of coherent sheaves orlov s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a fourier mukai functor in this paper we show that this result is false without the fully faithfulness hypothesis we also show that our functor does not lift to the homotopy category of spectral categories if the ground field is mathbb q

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation