38,852 research outputs found
Normal forms in Poisson geometry
No full textThe structure of Poisson manifolds is highly nontrivial even locally. The first important result in this direction is Conn's linearization theorem around fixed points. One of the main results of this thesis (Theorem 2) is a normal form theorem in Poisson geometry, which is the Poisson-geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry). The result generalizes Conn's theorem from one-point leaves to arbitrary symplectic leaves. We present two proofs of this result: a geometric one relying heavily on the theory of Lie algebroids and Lie groupoids (similar to the new proof of Conn's theorem by Crainic and Fernandes), and an analytic one using the Nash-Moser fast convergence method (more in the spirit of Conn's original proof). The analytic approach gives much more, we prove a local rigidity result (Theorem 4) around compact Poisson submanifolds, which is the first of this kind in Poisson geometry. Yet more surprising is an application of this result to smooth deformation of Poisson structures: in Theorem 5 we compute the Poisson-moduli space around the Lie-Poisson sphere (i.e. the invariant unit sphere inside the linear Poisson manifold corresponding to a compact semisimple Lie algebra). This can be described as the space of invariant functions on the sphere modulo the finite group of outer automorphism of the Lie algebra, thus it is infinite dimensional. This is the first such computation of a Poisson moduli space in dimension greater or equal to three around a degenerate (i.e. non-symplectic) Poisson structure. Other results presented in the thesis are: a new proof to the existence of symplectic realizations (Theorem 0), a normal form theorem for symplectic foliations (Theorem 1), a formal normal form/rigidity result around Poisson submanifolds (Theorem 3), and a general construction of tame homotopy operators for Lie algebroid cohomology (the Tame Vanishing Lemma)
Homological Algebra for Superalgebras of Differentiable Functions
No full textThis is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra to define, in particular, the notion of a differential graded algebra in the world of C-infinity rings. The opposite of the category of differential graded C-infinity algebras contains the category of differential graded manifolds as a full subcategory. More generally, this notion of differential graded algebra makes sense for algebras over any (super) Fermat theory, and hence one also arrives at the definition of a differential graded algebra appropriate for the study of derived real and complex analytic manifolds and other variants. We go on to show that, for any super Fermat theory S which admits integration, a concept we define and show is satisfied by all important examples, the category of differential graded S-algebras supports a Quillen model structure naturally extending the classical one on differential graded algebras, both in the bounded and unbounded case (as well as differential algebras with no grading). Finally, we show that, under the same assumptions, any of these categories of differential graded S-algebras have a simplicial enrichment, compatible in a suitable sense with the model structure
Bookreview Geometry of algebraic curves. Volume II (Grundlehren der mathematischen Wissenschaften, 268)
No full textGeometry of algebraic curves. Volume II (Grundlehren der mathematischen Wissenschaften, 268
Zeta function rigidity : a view from noncommutative geometry
No full textIn this thesis we study the zeta function formalism of finitely summable spectral triples in noncommutative geometry introduced by Alain Connes in 1995. In particular we study how these zeta functions, that naturally come in different classes, can be used to classify objects. The thesis starts with the study of finite, connected, unoriented graphs with Betti number at least 2 and valencies at least 3. We start by constructing a finitely summable spectral triple for these and prove that the first class of zeta functions determines the graph. Next closed smooth Riemannian manifolds are studied. The ideas of finitely summable spectral triples are applied to the Laplacian. We prove that the first class together with the diagonal of the second class of zeta functions determines the Riemannian manifold. This is in contrast with the usual zeta function where the phenomenon of isospectrality occurs: different objects can have the same spectrum (which is equivalent of having the same usual zeta function). Finally, the last result is formalized by introducing the concept of length categories and distances. We construct the length of a smooth map between smooth Riemannian manifolds and this in turn induces a distance between them. We conclude by proving that this distance induces the topology of uniform convergence of smooth Riemannian manifolds
Classification and equivalences of noncommutative tori and quantum lens spaces
No full textIn noncommutative geometry, one studies abstract spaces through their, possibly noncommutative, algebras of continuous functions. Through these function algebras, and certain operators interacting with them, one can derive much geometrical information of the underlying space, even though this space does not exist as a classical topological space if the algebra is noncommutative. In this thesis the classification of certain geometrical structures, spin structures, is extended to several kinds of noncommutative spaces: noncommutative tori and quantum lens spaces. It is found that this classification yields very similar results to the classical geometrical classification, even though the methods are completely different. Also, equivalences between these spaces and spinstructures are considered. It is proven that a certain proposed equivalence, Morita equivalence of spectral triples, is really an equivalence in the case of noncommutative tori. In general, it is not known when a Morita equivalence of spectral triples is really an equivalence relation
Irreducible highest weight representations of the simple n-Lie algebra
No full textA. Dzhumadil’daev classified all irreducible finite dimensional representations of the simple n-Lie algebra. Using a slightly different approach, we obtain in this paper a complete classification of all irreducible, highest weight modules, including the infinite-dimensional ones. As a corollary we find all primitive ideals of the universal enveloping algebra of this simple n-Lie algebr
On the derived category of an algebra over an operad.
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129001.pdf (author's version ) (Open Access
The (n,1)-Reduced DKP Hierarchy, the String Equation and W Constraints
Full text linkThe total descendent potential of a simple singularity satisfies the Kac-Wakimoto principal hierarchy. Bakalov and Milanov showed recently that it is also a highest weight vector for the corresponding W-algebra. This was used by Liu, Yang and Zhang to prove its uniqueness. We construct this principal hierarchy of type D in a different way, viz. as a reduction of some DKP hierarchy. This gives a Lax type and a Grassmannian formulation of this hierarchy. We show in particular that the string equation induces a large part of the W constraints of Bakalov and Milanov. These constraints are not only given on the tau function, but also in terms of the Lax and Orlov-Schulman operators
The Weil algebra and the Van Est isomorphism
Full text linkThis paper belongs to a series of papers devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman’s BRST model, here we introduce the Weil algebra W(A) associated to any Lie algebroid A. We then show that this Weil algebra is related to the Bott-Shulman complex (computing the cohomology of the classifying space) via a Van Est map and we prove a Van Est isomorphism theorem. As application, we generalize and find a simpler more conceptual proof of the main result of [6] on the reconstructions of multiplicative forms and of a result of [21, 9] on the reconstruction of connection 1-forms. This reveals the relevance of the Weil algebra and Van Est maps to the integration and the pre-quantization of Poisson (and Dirac) manifolds
On Theories of Superalgebras of Differentiable Functions
No full textThis is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras for which infinitely differentiable functions can be evaluated on elements. Such a theory is called a super Fermat theory. Any category of superspaces and smooth functions has an associated such theory. This includes both real and complex supermanifolds, as well as algebraic superschemes. In particular, there is a super Fermat theory of C-infinity superalgebras. C-infinity superalgebras are the appropriate notion of supercommutative algebras in the world of C-infinity rings, the latter being of central importance both to synthetic differential geometry and to all existing models of derived smooth manifolds. A super Fermat theory is a natural generalization of the concept of a Fermat theory introduced by E. Dubuc and A. Kock. We show that any Fermat theory admits a canonical superization, however not every super Fermat theory arises in this way. For a fixed super Fermat theory, we go on to study a special subcategory of algebras called near-point determined algebras, and derive many of their algebraic properties
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