In this article, we show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic k-module and how the underlying simplicial homology gives rise to a Batalin–Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalized Lie derivative and a generalized (cyclic) cap product that obey a Cartan–Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Hochschild theory of associative algebras, for Poisson structures, but above all the calculus for general left Hopf algebroids with respect to general coefficients (in which the classical calculus of vector fields and differential forms is contained).

Gerstenhaber and Batalin-Vilkovisky Structures on Modules over Operads / Kowalzig, Niels. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1073-7928. - STAMPA. - 2015:22(2015), pp. 11694-11744. [10.1093/imrn/rnv034]

Gerstenhaber and Batalin-Vilkovisky Structures on Modules over Operads

KOWALZIG, NIELS
2015

Abstract

In this article, we show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic k-module and how the underlying simplicial homology gives rise to a Batalin–Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalized Lie derivative and a generalized (cyclic) cap product that obey a Cartan–Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Hochschild theory of associative algebras, for Poisson structures, but above all the calculus for general left Hopf algebroids with respect to general coefficients (in which the classical calculus of vector fields and differential forms is contained).
File allegati a questo prodotto
File Dimensione Formato  
Kowalzig_Gerstenhaber-and-Batalin_2015.pdf

accesso aperto

Note: https://doi.org/10.1093/imrn/rnv034
Tipologia: Documento in Post-print (versione successiva alla peer review e accettata per la pubblicazione)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 304.16 kB
Formato Adobe PDF
304.16 kB Adobe PDF Visualizza/Apri PDF

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/874475
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 6
  • ???jsp.display-item.citation.isi??? 6
social impact