3,773 research outputs found
Cluster structures for 2-Calabi-Yau categories and unipotent groups
We investigate cluster tilting objects (and subcategories) in triangulated
2-Calabi-Yau categories and related categories. In particular we construct a
new class of such categories related to preprojective algebras of non Dynkin
quivers associated with elements in the Coxeter group. This class of
2-Calabi-Yau categories contains the cluster categories and the stable
categories of preprojective algebras of Dynkin graphs as special cases. For
these 2-Calabi-Yau categories we construct cluster tilting objects associated
with each reduced expression. The associated quiver is described in terms of
the reduced expression. Motivated by the theory of cluster algebras, we
formulate the notions of (weak) cluster structure and substructure, and give
several illustrations of these concepts. We give applications to cluster
algebras and subcluster algebras related to unipotent groups, both in the
Dynkin and non Dynkin case.Comment: 49 pages. For the third version the presentation is revised,
especially Chapter III replaces the old Chapter III and I
On algebraic structures of the Hochschild complex
We first review various known algebraic structures on the Hochschild
(co)homology of a differential graded algebras under weak Poincar{\'e} duality
hypothesis, such as Calabi-Yau algebras, derived Poincar{\'e} duality algebras
and closed Frobenius algebras. This includes a BV-algebra structure on
or , which in the latter case is an extension of
the natural Gerstenhaber structure on . As an example, after proving
that the chain complex of the Moore loop space of a manifold is a
CY-algebra and using Burghelea-Fiedorowicz-Goodwillie theorem we obtain a
BV-structure on the homology of the free space. In Sections 6 we prove that
these BV/coBVstructures can be indeed defined for the Hochschild homology of a
symmetric open Frobenius DG-algebras. In particular we prove that the
Hochschild homology and cohomology of a symmetric open Frobenius algebra is a
BV and coBV-algebra. In Section 7 we exhibit a BV structure on the shifted
relative Hochschild homology of a symmetric commutative Frobenius algebra. The
existence of a BV-structure on the relative Hochschild homology was expected in
the light of Chas-Sullivan and Goresky-Hingston results for free loop spaces.
In Section 8 we present an action of Sullivan diagrams on the Hochschild
(co)chain complex of a closed Frobenius DG-algebra. This recovers
Tradler-Zeinalian \cite{TZ} result for closed Froebenius algebras using the
isomorphism .Comment: This is the final version. Many improvements and corrections have
been made.To appear in Free Loop Spaces in Geometry and Topology, IRMA
Lectures in Mathematicsand Theoretical Physics, to be published by EMS-P
Dual Feynman transform for modular operads
We introduce and study the notion of a dual Feynman transform of a modular
operad. This generalizes and gives a conceptual explanation of Kontsevich's
dual construction producing graph cohomology classes from a contractible
differential graded Frobenius algebra. The dual Feynman transform of a modular
operad is indeed linear dual to the Feynman transform introduced by Getzler and
Kapranov when evaluated on vacuum graphs. In marked contrast to the Feynman
transform, the dual notion admits an extremely simple presentation via
generators and relations; this leads to an explicit and easy description of its
algebras. We discuss a further generalization of the dual Feynman transform
whose algebras are not necessarily contractible. This naturally gives rise to a
two-colored graph complex analogous to the Boardman-Vogt topological tree
complex.Comment: 27 pages. A few conceptual changes in the last section; in particular
we prove that the two-colored graph complex is a resolution of the
corresponding modular operad. It is now called 'BV-resolution' as suggested
by Sasha Vorono
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