Sheaves Of Algebras Research Articles

Deformations of sheaves of algebras

A construction of the tangent dg Lie algebra of a sheaf of operad algebras on a site is presented. The requirements on the site are very mild; the requirements on the algebra are more substantial. A few applications including the description of deformations of a scheme and equivariant deformations are considered. The construction is based upon a model structure on the category of presheaves which should be of an independent interest.

Deformations of quasicoherent sheaves of algebras

Gerstenhaber and Schack [NATO Adv. Sci. Inst. Ser. C, Vol. 247, 1986] developed a deformation theory of presheaves of algebras on small categories. We translate their cohomological description to sheaf cohomology. More precisely, we describe the deformation space of (admissible) quasicoherent sheaves of algebras on a quasiprojective scheme X in terms of sheaf cohomology on X and X × X . These results are applied to the study of deformations of the sheaf D X of differential operators on X . In particular, in case X is a flag variety we show that any deformation of D X , which is induced by a deformation of O X , must be trivial. This result is used in [Lunts, Rosenberg, manuscript], where we study the localization construction for quantum groups.

Finitary Čech-de Rham Cohomology

The present paper continues (Mallios & Raptis, International Journal of Theoretical Physics, 2001, 40, 1885) and studies the curved finitary spacetime sheaves of incidence algebras presented therein from a Cech cohomological perspective. In particular, we entertain the possibility of constructing a nontrivial de Rham complex on these finite dimensional algebra sheaves along the lines of the first author's axiomatic approach to differential geometry via the theory of vector and algebra sheaves (Mallios, Geometry of Vector Sheaves: An Axiomtic Approach to Differential Geometry, Vols. 1–2, Kluwer, Dordrecht, 1998a; Mathematica Japonica (International Plaza), 1998b, 48, 93). The upshot of this study is that important “classical” differential geometric constructions and results usually thought of as being intimately associated with C∞-smooth manifolds carry through, virtually unaltered, to the finitary-algebraic regime with the help of some quite universal, because abstract, ideas taken mainly from sheaf-cohomology as developed in Mallios (1998a,b). At the end of the paper, and due to the fact that the incidence algebras involved have been interpreted as quantum causal sets (Raptis, International Journal of Theoretical Physics, 2000, 39, 1233; Mallios & Raptis, 2001), we discuss how these ideas may be used in certain aspects of current research on discrete Lorentzian quantum gravity.

Gerbes of chiral differential operators

In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by A.Vaintrob and two of the authors. Hopefully our result clarifies to some extent the constructions of the above work.

Topological algebra sheaves and applications

Topological algebra sheaves and applications

Chiral de Rham Complex

The aim of this note is to define certain sheaves of vertex algebras on smooth manifolds. For each smooth complex algebraic (or analytic) manifold $X$, we construct a sheaf $\Omega^{ch}_X$, called the {\bf chiral de Rham complex} of $X$. It is a sheaf of vertex algebras in the Zarisky (or classical) topology, It comes equipped with a $\BZ$-grading by {\it fermionic charge}, and the {\it chiral de Rham differential} $d_{DR}^{ch}$, which is an endomorphism of degree 1 such that $(d_{DR}^{ch})^2=0$. One has a canonical embedding of the usual de Rham complex $(\Omega_X, d_{DR})\hra (\Omega_X^{ch}, d_{DR}^{ch})$ which is a quasiisomorphism. If $X$ is Calabi-Yau then this sheaf admits an N=2 supersymmetry. For some $X$ (for example, for curves or for the flag spaces $G/B$), one can construct also a purely even analogue of this sheaf, a {\it chiral structure sheaf} $\CO^{ch}_X$. For the projective line, the space of global sections of the last sheaf is the irreducible vacuum $\hsl(2)$-module on the critical level.

Vertex Algebras and Mirror Symmetry

Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in toric varieties. We establish the relation between these vertex algebras for mirror Calabi-Yau manifolds. This should eventually allow us to rewrite the whole story of toric Mirror Symmetry in the language of sheaves of vertex algebras. Our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra, as well as some basic results of the theory of vertex algebras. Ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties. This paper could also be of interest to physicists, because it contains explicit descriptions of A and B models of Calabi-Yau hypersurfaces and complete intersection in terms of free bosons and fermions.

On the Quantization of Poisson Brackets

In this paper we introduce two classes of Poisson brackets on algebras (or on sheaves of algebras). We call them locally free and nonsingular Poisson brackets. Using the Fedosov's method we prove that any locally free nonsingular Poisson bracket can be quantized. In particular, it follows from this that all Poisson brackets on an arbitrary field of characteristic zero can be quantized. The well-known theorem about the quantization of nondegenerate Poisson brackets on smooth manifolds follows from the main result of this paper as well.

Cohen-Macaulay Section Rings

In this paper, we study the section rings of sheaves of Cohen-Macaulay algebras (over a field F F ) on a ranked poset. A necessary and sufficient condition for these rings to be Cohen-Macaulay will be given. This is a further generalization of a result of Yuzvinsky, which generalizes Reisner’s theorem concerning Stanley-Reisner rings.

On topological algebra sheaves

AbstractGiven two topological algebra sheaves, we seek that their tensor product be an (algebra) sheaf of the same type. We further study the latter sheaf in connection with the set of morphisms which are defined on it. As an application, we finally consider fundamental notions and results related to algebras of holomorphic functions in the framework of topological algebra sheaves.

Sheaves of Einstein algebras

To include all types of singularities into a geometrically tractable theoretical scheme we change from Einstein algebras, an algebraic generalization of general relativity, to sheaves of Einstein algebras. The theory of such spaces, called Einstein structured spaces, is developed. Both quasiregular and curvature singularities are studied in some detail. Examples of the closed Friedmann world model and the Schwarzschild spacetime show that Schmidt'sb-boundary is a useful theoretical tool when considered in the category of structured spaces.

Einstein algebras and general relativity

A purely algebraic structure called an Einstein algebra is defined in such a way that every spacetime satisfying Einstein's equations is an Einstein algebra but not vice versa. The Gelfand representation of Einstein algebras is defined, and two of its subrepresentations are discussed. One of them is equivalent to the global formulation of the standard theory of general relativity; the other one leads to a more general theory of gravitation which, in particular, includes so-called regular singularities. In order to include other types of singularities one must change to sheaves of Einstein algebras. They are defined and briefly discussed. As a test of the proposed method, the sheaf of Einstein algebras corresponding to the spacetime of a straight cosmic string with quasiregular singularity is constructed.

Algebraic Deformations and Bicohomology

AbstractRecently we have introduced an enriched cohomology theory for categories that are tripleable (algebraic) over a category of modules. The cohomology admits a circle product, related to the obstruction problem for algebraic deformations, making the total complex a graded ring. We here offer similar constructions in two other situations - coalgebraic and bialgebraic categories. Examples include categories of bialgebras, sheaves of modules, and sheaves of algebras over a sheaf of rings.

Sheaves of Noncommutative Algebras and the Beilinson-Bernstein Equivalence of Categories

Let $X$ be an irreducible algebraic variety defined over a field $k$, let $\mathcal {R}$ be a sheaf of (noncommutative) noetherian $k$-algebras on $X$ containing the sheaf of regular functions $\mathcal {O}$ and let $R$ be the ring of global sections. We show that under quite reasonable abstract hypotheses (concerning the existence of a faithfully flat overring of $R$ obtained from the local sections of $\mathcal {R}$) there is an equivalence between the category of $R$-modules and the category of sheaves of $\mathcal {R}$-modules which are quasicoherent as $\mathcal {O}$-modules. This shows that the equivalence of categories established by Beilinson and Bernstein as the first step in their proof of the KazhdanLusztig conjectures (where $R$ is a primitive factor ring of the enveloping algebra of a complex semisimple Lie algebra, and $\mathcal {R}$ is a sheaf of twisted differential operators on a generalised flag variety) is valid for more fundamental reasons than is apparent from their work.

Sheaves of noncommutative algebras and the Beilinson-Bernstein equivalence of categories

Let $X$ be an irreducible algebraic variety defined over a field $k$, let $\mathcal {R}$ be a sheaf of (noncommutative) noetherian $k$-algebras on $X$ containing the sheaf of regular functions $\mathcal {O}$ and let $R$ be the ring of global sections. We show that under quite reasonable abstract hypotheses (concerning the existence of a faithfully flat overring of $R$ obtained from the local sections of $\mathcal {R}$) there is an equivalence between the category of $R$-modules and the category of sheaves of $\mathcal {R}$-modules which are quasicoherent as $\mathcal {O}$-modules. This shows that the equivalence of categories established by Beilinson and Bernstein as the first step in their proof of the KazhdanLusztig conjectures (where $R$ is a primitive factor ring of the enveloping algebra of a complex semisimple Lie algebra, and $\mathcal {R}$ is a sheaf of twisted differential operators on a generalised flag variety) is valid for more fundamental reasons than is apparent from their work.

Equational compactness of sheaves of algebras on a Noetherian locale

Equational compactness of sheaves of algebras on a Noetherian locale

Recovering a frame from its sheaves of algebras

Let ▪ be a frame, α an element of ▪ and T a finitary algebraic theory. In this paper we compare the category ▪ of sheaves of T-algebras on ▪ with the category Sh(α↓, T) of sheaves of T-algebras on α↓ (where α↓ is the initial segment {β|β<α}). This comparison suggests the definition of formal initial segments of the category ▪. For a large class of theories to be called ‘integral’ (examples of which are sets, monoids, groups, rings, modules on a integral domain, boolean algebras,hellip;) the formal initial segments of ▪ coincide with the usual initial segments of ▪: this means that we are able to recover ▪ axiomatically from ▪. When ▪ is the initial frame {0, 1}, the frame of formal initial segments of ▪ is the frame of open subsets of a compact space Spec T, called the spectrum of the theory T. When T is the theory of modules on some ring R, we recover in this way various well known notions of spectra and their corresponding sheaf-representation of the ring.

Infinitesimal automorphisms of $\textit{g}$-structures and certain intransitive infinite Lie algebra sheaves

Infinitesimal automorphisms of $\textit{g}$-structures and certain intransitive infinite Lie algebra sheaves

Coherent algebraic sheaves in real algebraic geometry

In the following paper we give examples of vector bundlesF→X defined on a regular, compact, connected affine varietyX such thatF has no algebraic structure.

Local properties of intransitive infinite Lie algebra sheaves

Local properties of intransitive infinite Lie algebra sheaves