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

Mourad Ammar[1]; Norbert Poncin[1]

  • [1] University of Luxembourg Campus Limpertsberg 162A, avenue de la Faïencerie 1511 Luxembourg City (Grand-Duchy of Luxembourg)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 1, page 355-387
  • ISSN: 0373-0956

Abstract

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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 a subcategory. Moreover, the graded Lie bracket of coderivations gives rise to a graded Lie “stem” bracket on the cochain spaces of graded Loday, Loday infinity, and 2 n -ary graded Loday algebras. This stem bracket restricts to the graded Nijenhuis-Richardson and Grabowski-Marmo brackets, and it encodes, beyond the already mentioned cohomologies, those of graded Lie, graded Poisson, graded Jacobi, Lie infinity, as well as that of 2 n -ary graded Lie algebras.

How to cite

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Ammar, Mourad, and Poncin, Norbert. "Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras." Annales de l’institut Fourier 60.1 (2010): 355-387. <http://eudml.org/doc/116272>.

@article{Ammar2010,
abstract = {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 $\text\{Lod\}_\{\!\infty \}$ category contains the $\text\{L\}_\{\!\infty \}$ category as a subcategory. Moreover, the graded Lie bracket of coderivations gives rise to a graded Lie “stem” bracket on the cochain spaces of graded Loday, Loday infinity, and $2n$-ary graded Loday algebras. This stem bracket restricts to the graded Nijenhuis-Richardson and Grabowski-Marmo brackets, and it encodes, beyond the already mentioned cohomologies, those of graded Lie, graded Poisson, graded Jacobi, Lie infinity, as well as that of $2n$-ary graded Lie algebras.},
affiliation = {University of Luxembourg Campus Limpertsberg 162A, avenue de la Faïencerie 1511 Luxembourg City (Grand-Duchy of Luxembourg); University of Luxembourg Campus Limpertsberg 162A, avenue de la Faïencerie 1511 Luxembourg City (Grand-Duchy of Luxembourg)},
author = {Ammar, Mourad, Poncin, Norbert},
journal = {Annales de l’institut Fourier},
keywords = {Zinbiel coalgebra; graded Loday; Lie; Poisson; Jacobi structure; strongly homotopy algebra; square-zero element method; graded cohomology; Schouten-Nijenhuis; Nijenhuis-Richardson; Grabowski-Marmo bracket; deformation theory; coalgebras; graded Loday structures; cohomologies; stem},
language = {eng},
number = {1},
pages = {355-387},
publisher = {Association des Annales de l’institut Fourier},
title = {Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras},
url = {http://eudml.org/doc/116272},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Ammar, Mourad
AU - Poncin, Norbert
TI - Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 1
SP - 355
EP - 387
AB - 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 $\text{Lod}_{\!\infty }$ category contains the $\text{L}_{\!\infty }$ category as a subcategory. Moreover, the graded Lie bracket of coderivations gives rise to a graded Lie “stem” bracket on the cochain spaces of graded Loday, Loday infinity, and $2n$-ary graded Loday algebras. This stem bracket restricts to the graded Nijenhuis-Richardson and Grabowski-Marmo brackets, and it encodes, beyond the already mentioned cohomologies, those of graded Lie, graded Poisson, graded Jacobi, Lie infinity, as well as that of $2n$-ary graded Lie algebras.
LA - eng
KW - Zinbiel coalgebra; graded Loday; Lie; Poisson; Jacobi structure; strongly homotopy algebra; square-zero element method; graded cohomology; Schouten-Nijenhuis; Nijenhuis-Richardson; Grabowski-Marmo bracket; deformation theory; coalgebras; graded Loday structures; cohomologies; stem
UR - http://eudml.org/doc/116272
ER -

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