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A formula for topology/deformations and its significance

Ruth Lawrence, Dennis Sullivan (2014)

Fundamenta Mathematicae

The formula is e = ( a d e ) b + i = 0 ( B i ) / i ! ( a d e ) i ( b - a ) , with ∂a + 1/2 [a,a] = 0 and ∂b + 1/2 [b,b] = 0, where a, b and e in degrees -1, -1 and 0 are the free generators of a completed free graded Lie algebra L[a,b,e]. The coefficients are defined by x / ( e x - 1 ) = n = 0 B / n ! x . The theorem is that ∙ this formula for ∂ on generators extends to a derivation of square zero on L[a,b,e]; ∙ the formula for ∂e is unique satisfying the first property, once given the formulae for ∂a and ∂b, along with the condition that the “flow” generated by e moves a to b in unit...

A Riemann-Roch theorem for dg algebras

François Petit (2013)

Bulletin de la Société Mathématique de France

Given a smooth proper dg algebra A , a perfect dg A -module M and an endomorphism f of M , we define the Hochschild class of the pair ( M , f ) with values in the Hochschild homology of the algebra A . Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.

Batalin-Vilkovisky algebra structures on Hochschild cohomology

Luc Menichi (2009)

Bulletin de la Société Mathématique de France

Let M be any compact simply-connected oriented d -dimensional smooth manifold and let 𝔽 be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of M , H H * ( S * ( M ) , S * ( M ) ) , extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjectured to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on M , H * + d ( L M ) introduced by Chas and Sullivan. We also show that the negative cyclic cohomology H C - * ( S * ( M ) ) ...

Cluster categories for algebras of global dimension 2 and quivers with potential

Claire Amiot (2009)

Annales de l’institut Fourier

Let k be a field and A a finite-dimensional k -algebra of global dimension 2 . We construct a triangulated category 𝒞 A associated to A which, if  A is hereditary, is triangle equivalent to the cluster category of A . When 𝒞 A is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also...

Coalgebraic Approach to the Loday Infinity Category, Stem Differential for 2 n -ary Graded and Homotopy Algebras

Mourad Ammar, Norbert Poncin (2010)

Annales de l’institut Fourier

We define a graded twisted-coassociative coproduct on the tensor algebra the desuspension space of a graded vector space V . The coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences of multilinear maps on V (resp. graded Loday structures on V , sequences that we call Loday infinity structures on V ). We prove a minimal model theorem for Loday infinity algebras and observe that the Lod category contains the L category as...

Compact corigid objects in triangulated categories and co-t-structures

David Pauksztello (2008)

Open Mathematics

In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, C , of a triangulated category, 𝒯 , which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on 𝒯 whose heart is equivalent to Mod(End( C )op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave...

Gerstenhaber and Batalin-Vilkovisky algebras; algebraic, geometric, and physical aspects

Claude Roger (2009)

Archivum Mathematicum

We shall give a survey of classical examples, together with algebraic methods to deal with those structures: graded algebra, cohomologies, cohomology operations. The corresponding geometric structures will be described(e.g., Lie algebroids), with particular emphasis on supergeometry, odd supersymplectic structures and their classification. Finally, we shall explain how BV-structures appear in Quantum Field Theory, as a version of functional integral quantization.

Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs

Dennis Sullivan (2009)

Banach Center Publications

Using the algebraic theory of homotopies between maps of dga's we obtain a homotopy theory for algebraic structures defined by collections of multiplications and comultiplications. This is done by expressing these structures and resolved versions of them in terms of dga maps. This same homotopy theory of dga maps applies to extract invariants beyond homological periods from systems of moduli spaces that determine systems of chains that satisfy master equations like dX + X*X = 0. Minimal models of...

Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications

Jian Qiu, Maxim Zabzine (2011)

Archivum Mathematicum

These notes are intended to provide a self-contained introduction to the basic ideas of finite dimensional Batalin-Vilkovisky (BV) formalism and its applications. A brief exposition of super- and graded geometries is also given. The BV–formalism is introduced through an odd Fourier transform and the algebraic aspects of integration theory are stressed. As a main application we consider the perturbation theory for certain finite dimensional integrals within BV-formalism. As an illustration we present...

Les ( a , b ) -algèbres à homotopie près

Walid Aloulou (2010)

Annales mathématiques Blaise Pascal

On étudie dans cet article les notions d’algèbre à homotopie près pour une structure définie par deux opérations . et [ , ] . Ayant déterminé la structure des G algèbres et des P algèbres, on généralise cette construction et on définit la stucture des ( a , b ) -algèbres à homotopie près. Etant donnée une structure d’algèbre commutative et de Lie différentielle graduée pour deux décalages des degrés donnés par a et b , on donnera une construction explicite de l’algèbre à homotopie près associée et on précisera...

On restrictions of indecomposables of tame algebras

R. Bautista, E. Pérez, L. Salmerón (2011)

Colloquium Mathematicae

We continue the study of ditalgebras, an acronym for "differential tensor algebras", and of their categories of modules. We examine extension/restriction interactions between module categories over a ditalgebra and a proper subditalgebra. As an application, we prove a result on representations of finite-dimensional tame algebras Λ over an algebraically closed field, which gives information on the extension/restriction interaction between module categories of some special algebras Λ₀, called convex...

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