Algebraic Bundles Research Articles

On the foundations of nonlinear generalized functions I and II

Part 1. On the Foundations of Nonlinear Generalized Functions I: Introduction Notation and terminology Scheme of construction Calculus C- and J-formalism Calculus on $U_\varepsilon(\Omega)$ Construction of a diffeomorphism invariant Colombeau algebra Sheaf properties Separating the basic definition from testing Characterization results Differential equations Part 2. On the Foundations of Nonlinear Generalized Functions II: Introduction to Part 2 A simple condition equivalent to negligibility Some more calculus Non-injectivity of the canonical homomorphism from ${\mathcal G}^d(\Omega)$ into ${\mathcal G}^e(\Omega)$ Classification of smooth Colombeau algebras between ${\mathcal G}^d(\Omega)$ and ${\mathcal G}^e(\Omega)$ The algebra ${\mathcal G}^2$ classification results Concluding remarks Acknowledgments Bibliography.

Linear bundles of Lie algebras and their applications

We consider the algebraic properties of families of Lie brackets, also called linear bundles of Lie algebras, and especially ones that can be defined over the classical algebras. Some applications to the theory of the integrable systems are given as such bundles naturally define compatible Poisson–Lie (Kirillov) tensors.

THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS

Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aω of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that . It is proved that the set of all C*-algebras of sections of locally trivial C*-algebra bundles over S2 with fibres Aω has a group structure, denoted by , which is isomorphic to ℤ if ∃ d > 1 and {0} if . Let Bcd be a cd-homogeneous C*-algebra over of which no non-trivial matrix algebra can be factored out. The spherical noncommutative torus is defined by twisting in Bcd ⊗ C*(ℤm - 2) by a totally skew multiplier ρ on . It is shown that is isomorphic to if and only if the set of prime factors of cd is a subset of the set of prime factors of p.

On trivial and non-trivial n-homogeneousC * algebras

LetA be aC* — algebra for which all irreducible representations are of dimensional n. Then ([F], [TT], [V]) algebraA is isomorphic to algebra of all continuous sections of an appropriate algebraic bundle eA. The basisX of this bundle coincides with the compact of all maximal two-sided ideals ofA. We obtain some conditions which provide that eA is trivial and this yields thatA is isomorphic to the algebra of alln×n matrix functions continuous onX. In the case whenX=Sn is a sphere we describe the set of algebraic bundles overX and algebraic structures on this set. Some applications to algebras generated by idempotents are suggested.

The [formula omitted] conjecture and a vector bundle on the Hilbert scheme of [formula omitted] points in the plane

The focus of this paper is on algebraic vector bundles over P n and their applications to the Garsia–Haiman representation theoretic interpretation of the Macdonald symmetric polynomials. This interpretation involves a certain bigraded S n -module, H μ , indexed by partitions μ of n. Bergeron and Garsia (preprint) consider the relationships between H μ , for μ a partition of n+1, and the spaces H μ i , for μ i a partition of n contained in μ. They formulated conjectures regarding the sums and intersections of these spaces. This paper provides a geometric interpretation of these conjectures.

Trivialité des fibrés vectoriels équivariants pour les groupes abéliens finis

Let G be a finite abelian group, V a complex G-module, and A the algebra of all polynomial functions on V. We show that if M is a free A-module of finite rank, then there exists a G-module W such that M is isomorphic to A⊗W. As a consequence all algebraic G-vector bundles with base V are trivial, and the action of G on the total space of such a bundle is linearizable.

Formula omitted] geometry from Fedosov's deformation quantization

A geometric derivation of W ∞ gravity based on Fedosov's deformation quantization of symplectic manifolds is presented. To lowest order in Planck's constant it agrees with Hull's geometric formulation of classical non-chiral W ∞ gravity. The fundamental object is a W -valued connection one form belonging to the exterior algebra of the Weyl algebra bundle associated with the symplectic manifold. The W -valued analogs of the self-dual Yang-Mills equations, obtained from a zero curvature condition, naturally lead to the Moyal Plebanski equations, furnishing Moyal deformations of self-dual gravitational backgrounds associated with the complexified cotangent space of a two-dimensional Riemann surface. Deformation quantization of W ∞ gravity is retrieved upon the inclusion of all the ħ terms appearing in the Moyal bracket. Brief comments on non commutative geometry and M(atrix) theory are made.

Noncommutative Instantons and Twistor Transform

Recently N.Nekrasov and A.Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of the 4-dimensional real affine space. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one-to-one correspondences between three classes of objects: framed bundles on a noncommutative projective plane, certain complexes of sheaves on a noncommutative 3-dimensional projective space, and the modified ADHM data. The modified ADHM construction itself is interpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative projective plane has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative projective plane. The natural complex structures on the two moduli spaces do not coincide but are related by an SO(3) rotation. Finally, we propose a construction of instantons on a more general noncommutative R^4 than the one considered by Nekrasov and Schwarz (a q-deformed R^4).

Deformation Quantization of Symplectic Fibrations

A symplectic fibration is a fibre bundle in the symplectic category (a bundle of symplectic fibres over a symplectic base with a symplectic structure group). We find the relation between the deformation quantization of the base and the fibre, and that of the total space. We consider Fedosov's construction of deformation quantization. We generalize the Fedosov construction to the quantization with values in a bundle of algebras. We find that the characteristic class of deformation of a symplectic fibration is the weak coupling form of Guillemin, Lerman, and Sternberg. We also prove that the classical moment map could be quantized if there exists an equivariant connection.

A new construction of homogeneous quaternionic manifolds and related geometric structures

Introduction Extended Poincare algebras The homogeneous quaternionic manifold $(M,Q)$ associated to an extended Poincare algebra Bundles associated to the quaternionic manifold $(M,Q)$ Homogeneous quaternionic supermanifolds associated to superextended Poincare algebras Appendix. Supergeometry Bibliography.

Spin Spaces, Lipschitz Groups, and Spinor Bundles

It is shown that every bundle Σ → M of complex spinormodules over the Clifford bundle Cl(g) of a Riemannian space(M, g) with local model (V, h)is associated with an lpin(‘Lipschitz’) structure on M, this being a reduction of theO(h)-bundle of all orthonormal frames on M to the Lipschitzgroup Lpin(h) of all automorphisms of a suitably defined spinspace. An explicit construction is given of the total space of theLpin(h)-bundle defining such a structure. If the dimension mof M is even, then the Lipschitz group coincides with the complexClifford group and the lpin structure can be reduced to a pin c structure. If m = 2n − 1, then a spinor module Σ on M is of the Cartan type: its fibres are 2 n -dimensional anddecomposable at every point of M, but the homomorphism of bundlesof algebras Cl(g) → End Σ globally decomposes if, andonly if, M is orientable. Examples of such bundles are given. Thetopological condition for the existence of an lpin structure on anodd-dimensional Riemannian manifold is derived and illustrated by theexample of a manifold admitting such a structure, but no pin c structure.

Differential Graded Cohomology and Lie Algebras¶of Holomorphic Vector Fields

The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie algebras, known as the Fr\"olicher-Nijenhuis algebra $g$. We establish - following B. L. Feigin - an isomorphism between the differential graded cohomology of the space of global sections of $g$ and the hypercohomology of the sheaf of continuous cochain complexes of $Lie$. We calculate this cohomology up to the singular cohomology of some mapping space. We use and generalize results of N. Kawazumi on complex Gelfand-Fuks cohomology. Applications are - again following B. L. Feigin - in conformal field theory, and in the theory of deformations of complex structures. In an erratum to this paper, we admit that the sheaf of continuous cochains of a sheaf of vector fields with values in the ground fields does not make much sense. The most important cochains (like evaluations in a point or integrations over the manifold) do not come from sheaf homomorphisms. The main result of the above article (theorem 7) remains true.

Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces

LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let Xhol be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on Xhol (X normal algebraic surface) by algebraic vector bundles.

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.

Homology of real algebraic fiber bundles having circle as fiber or base.

For real algebraic setsX ⊆ R and Y ⊆ R , a map F : X→ Y is said to be entire rational if there exist fi, gi ∈R[x1, . . . , xr ], i = 1, . . . , s, such that each gi vanishes nowhere onX and F = (f1/g1, . . . , fs/gs). We sayX and Y are isomorphic to each other if there are entire rational maps F : X → Y and G : Y → X such that F B G = idY and G B F = idX. A complexification XC ⊆ CP of X will mean that X is a nonsingular algebraic subset of some RP and XC ⊆ CP is the complexification of the pair X ⊆ RP. We also require the complexification to be nonsingular (blow up XC along smooth centers away from X defined over reals if necessary). For basic definitions and facts about real algebraic geometry, we refer the reader to [2; 4]. Let KH∗(X,R) be the kernel of the induced map

Unconditional integrability for dual actions

The dual action of a locally compact abelian group, in the context of C*-algebraic bundles, is shown to satisfy an integrability property, similar to Rieffel's proper actions. The tools developed include a generalization of Bochner's integral as well as a Fourier inversion formula for operator valued maps.

Abstract Differential Geometry, Differential Algebras of Generalized Functions, and de Rham Cohomology

Abstract differential geometry is a recent extension of classical differential geometry on smooth manifolds which, however, does no longer use any notion of Calculus. Instead of smooth functions, one starts with a sheaf of algebras, i.e., the structure sheaf, considered on an arbitrary topological space, which is the base space of all the sheaves subsequently involved. Further, one deals with a sequence of sheaves of modules, interrelated with appropriate ‘differentials’, i.e., suitable ‘Leibniz’ sheaf morphisms, which will constitute the ‘differential complex’. This abstract approach captures much of the essence of classical differential geometry, since it places a powerful apparatus at our disposal which can reproduce and, therefore, extend fundamental classical results. The aim of this paper is to give an indication of the extent to which this apparatus can go beyond the classical framework by including the largest class of singularities dealt with so far. Thus, it is shown that, instead of the classical structure sheaf of algebras of smooth functions, one can start with a significantly larger, and nonsmooth, sheaf of so-called nowhere dense differential algebras of generalized functions. These latter algebras, which contain the Schwartz distributions, also provide global solutions for arbitrary analytic nonlinear PDEs. Moreover, unlike the distributions, and as a matter of physical interest, these algebras can deal with the vastly larger class of singularities which are concentrated on arbitrary closed, nowhere dense subsets and, hence, can have an arbitrary large positive Lebesgue measure. Within the abstract differential geometric context, it is shown that, starting with these nowhere dense differential algebras as a structure sheaf, one can recapture the exactness of the corresponding de Rham complex, and also obtain the short exponential sequence. These results are the two fundamental ingredients in developing differential geometry along classical, as well as abstract lines. Although the commutative framework is used here, one can easily deal with a class of singularities which is far larger than any other one dealt with so far, including in noncommutative theories.

On analytic torsion over C*-algebras

In this paper, we present an analytic definition for the relative torsion for flat C*-algebra bundles over a compact manifold. The advantage of such a relative torsion is that it is defined without any hypotheses on the flat C*-algebra bundle. In the case

Algebraic Kripke Sheaf Semantics for Non-Classical Predicate Logics

In so-called Kripke-type models, each sentence is assigned either to true or to false at each possible world. In this setting, every possible world has the two-valued Boolean algebra as the set of truth values. Instead, we take a collection of algebras each of which is attached to a world as the set of truth values at the world, and obtain an extended semantics based on the traditional Kripke-type semantics, which we call here the algebraic Kripke semantics. We introduce algebraic Kripke sheaf semantics for super-intuitionistic and modal predicate logics, and discuss some basic properties. We can state the Godel-McKinsey-Tarski translation theorem within this semantics. Further, we show new results on super-intuitionistic predicate logics. We prove that there exists a continuum of super-intuitionistic predicate logics each of which has both of the disjunction and existence properties and moreover the same propositional fragment as the intuitionistic logic.

Moduli spaces of vector bundles over ruled surfaces

AbstractWe study moduli spacesM(c1,c2,d, r) of isomorphism classes of algebraic 2-vector bundles with fixed numerical invariantsc1,c2,d,rover a ruled surface. These moduli spaces are independent of any ample line bundle on the surface. The main result gives necessary and sufficient conditions for the non-emptiness of the spaceM(c1,c2,d, r) and we apply this result to the moduli spacesML(c1,c2) of stable bundles, whereLis an ample line bundle on the ruled surface.