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Homotopy Batalin-Vilkovisky Algebras
Gálvez Carrillo, Maria Immaculada; Tonks, Andrew; Vallette, Bruno
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada III; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
This paper provides an explicit cofibrant resolution of the operad encoding Batalin-Vilkovisky algebras. Thus it defines the notion of homotopy Batalin-Vilkovisky algebras with the required homotopy properties. To define this resolution, we extend the theory of Koszul duality to operads and properads that are defined by quadratic and linear relations. The operad encoding Batalin-Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincaré-Birkhoff-Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal eld theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian-Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.
Peer Reviewed
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de categories; àlgebra homològica
Categories (Mathematics)
Algebra, Homological
Batalin-Vilkovisky algebra
Gerstenhaber algebra
homotopy algebras
Koszul duality theory
Maurer-Cartan equation
operad framed little disc
topological conformal field theory
vertex algebras
Categories (Matemàtica)
Àlgebra homològica
Classificació AMS::18 Category theory; homological algebra::18D Categories with structure
Classificació AMS::18 Category theory; homological algebra::18G Homological algebra
Classificació AMS::55 Algebraic topology::55P Homotopy theory
Classificació AMS::81 Quantum theory::81T Quantum field theory; related classical field theories
Classificació AMS::17 Nonassociative rings and algebras::17B Lie algebras and Lie superalgebras

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