The Infona portal uses cookies, i.e. strings of text saved by a browser on the user's device. The portal can access those files and use them to remember the user's data, such as their chosen settings (screen view, interface language, etc.), or their login data. By using the Infona portal the user accepts automatic saving and using this information for portal operation purposes. More information on the subject can be found in the Privacy Policy and Terms of Service. By closing this window the user confirms that they have read the information on cookie usage, and they accept the privacy policy and the way cookies are used by the portal. You can change the cookie settings in your browser.
We give a selective survey of topics in algebraic deformation theory ranging from its inception to current times. Throughout, the numerous contributions of Murray Gerstenhaber are emphasized, especially the common themes of cohomology, infinitesimal methods, and explicit global deformation formulas.
Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940s. Prompted by the failure of the Alexander–Whitney multiplication of cocycles...
The role of deformations in physics and mathematics, especially the theory of deformations of algebras developed by Gerstenhaber in the 1960s, led to the deformation philosophy promoted in mathematical physics by Flato since the 1970s, exemplified by deformation quantization and its manifold avatars, including quantum groups and the “dual” aspect of quantum spaces. Deforming Minkowski space-time and...
We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes due to W. Messing and the author [5], and give a more direct derivation of the associated cocycle equations. This leads us to a more restrictive definition than in [5] of the corresponding coboundary relations. We also show that the diagrammatic proofs of certain local curving and curvature...
In this article we relate the construction of Ricci-type symplectic connections by reduction to the construction of star products by reduction yielding rather explicit descriptions for the star product on the reduced space. AMS Classification (2010): 53D55, 53C07, 53D20
Dedicated to Murray Gerstenhaber and Jim Stasheff Kontsevich’s formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifoldMisL∞-quasi-isomorphic to its cohomology. The construction of theL∞-map is given in terms of integrals of differential forms on configuration...
To Murray Gerstenhaber on his 80th and to Jim Stasheff on his 70th birthday After an overview of noncommutative differential calculus, we construct parts of it explicitly and explain why this construction agrees with a fuller version obtained from the theory of operads. AMS 2010 Subject Codes: 19D55, 18G55
We study the notion of twisting elements da = a ⌣ 1a with respect to ⌣ 1 product when it is a part of homotopy Gerstenhaber algebra structure. This allows us to bring to one context the two classical concepts, the theory of deformation of algebras of M. Gerstenhaber, and A(∞)-algebras of J. Stasheff. AMS Classification (2010): 16E40, 18G55, 55S30
In this article, we discuss an application of homological perturbation theory (HPT) to homological mirror symmetry (HMS) based on Kontsevich and Soibelman’s proposal [Kontsevich, M., Soibelman, Y. (2001) Homological mirror symmetry and torus fibrations]. After a brief review of Morse theory, Morse homotopy and the corresponding Fukaya categories, we explain the idea of deriving a Fukaya category from...
This is a concise introduction to Fomin–Zelevinsky’s cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster variables and rigid indecomposable objects of the cluster category.
Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence...
We construct an A-infinity structure on the tensor product of two A-infinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AA-infinity based on the simplicial Stasheff polytope. The operad AA-infinity admits a coassociative diagonal and the operad A-infinity is a retract by deformation of it. We compare these constructions with...
This paper aims to give a short but self-contained introduction into the theory of (wheeled) props, properads, dioperads and operads, and illustrate some of its key ideas in terms of a prop(erad)ic interpretation of simplicial and permutahedra cell complexes with subsequent applications to the Hochschild–Kostant–Rosenberg type isomorphisms.
This article in French, with a large English introduction, is a survey about applications of bi-quantization theory in Lie theory. We focus on a conjecture of M. Duflo. Most of the applications are coming from our article with Alberto Cattaneo [Cattaneo, A.S., Torossian, C.: Quantification pour les paires symétriques et diagrammes de Kontsevich. Ann. Ecole Norm. Sup. 41, 787–852 (2008)] and some extensions...
At the 1996 conference honoring Jim Stasheff in the year of his 60th birthday, I initiated the search for A∞-bialgebras in a talk entitled “In Search of Higher Homotopy Hopf Algebras.” The idea in that talk was to think of a DG bialgebra as some (unknown) higher homotopy structure with trivial higher order structure and apply a graded version of Gerstenhaber and...
Set the date range to filter the displayed results. You can set a starting date, ending date or both. You can enter the dates manually or choose them from the calendar.