Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras

Huebschmann, Johannes. "Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras." Annales de l'institut Fourier 48.2 (1998): 425-440. <http://eudml.org/doc/75288>.

@article{Huebschmann1998,
abstract = {For any Lie-Rinehart algebra $(A,L)$, B(atalin)-V(ilkovisky) algebra structures $\partial $ on the exterior $A$-algebra $\Lambda _A L$ correspond bijectively to right $(A,L)$-module structures on $A$; likewise, generators for the Gerstenhaber algebra $\Lambda _A L$ correspond bijectively to right $(A,L)$-connections on $A$. When $L$ is projective as an $A$-module, given a B-V algebra structure $\partial $ on $\Lambda _A L$, the homology of the B-V algebra $(\Lambda _A L,\partial )$ coincides with the homology of $L$ with coefficients in $A$ with reference to the right $(A,L)$-module structure determined by $\partial $. When $L$ is also of finite rank $n$, there are bijective correspondences between $(A,L)$-connections on $\Lambda _A^nL$ and right $(A,L)$-connections on $A$ and between left $(A,L)$-module structures on $\Lambda _A^nL$ and right $(A,L)$-module structures on $A$. Hence there are bijective correspondences between $(A,L)$-connections on $\Lambda _A^n L$ and generators for the Gerstenhaber bracket on $\Lambda _A L$ and between $(A,L)$-module structures on $\Lambda _A^n L$ and B-V algebra structures on $\Lambda _A L$. The homology of such a B-V algebra $(\Lambda _A L,\partial )$ coincides with the cohomology of $L$ with coefficients in $\Lambda _A^n L$, with reference to the left $(A,L)$-module structure determined by $\partial $. Some applications to Poisson structures and to differential geometry are discussed.},
author = {Huebschmann, Johannes},
journal = {Annales de l'institut Fourier},
keywords = {Lie-Rinehart algebras; Gertenhaber algebras; Batalin-Vilkovisky algebras; abstract left and right connections; cohomological duality for Lie-Rinehart algebras; Poisson structures},
language = {eng},
number = {2},
pages = {425-440},
publisher = {Association des Annales de l'Institut Fourier},
title = {Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras},
url = {http://eudml.org/doc/75288},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Huebschmann, Johannes
TI - Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 2
SP - 425
EP - 440
AB - For any Lie-Rinehart algebra $(A,L)$, B(atalin)-V(ilkovisky) algebra structures $\partial $ on the exterior $A$-algebra $\Lambda _A L$ correspond bijectively to right $(A,L)$-module structures on $A$; likewise, generators for the Gerstenhaber algebra $\Lambda _A L$ correspond bijectively to right $(A,L)$-connections on $A$. When $L$ is projective as an $A$-module, given a B-V algebra structure $\partial $ on $\Lambda _A L$, the homology of the B-V algebra $(\Lambda _A L,\partial )$ coincides with the homology of $L$ with coefficients in $A$ with reference to the right $(A,L)$-module structure determined by $\partial $. When $L$ is also of finite rank $n$, there are bijective correspondences between $(A,L)$-connections on $\Lambda _A^nL$ and right $(A,L)$-connections on $A$ and between left $(A,L)$-module structures on $\Lambda _A^nL$ and right $(A,L)$-module structures on $A$. Hence there are bijective correspondences between $(A,L)$-connections on $\Lambda _A^n L$ and generators for the Gerstenhaber bracket on $\Lambda _A L$ and between $(A,L)$-module structures on $\Lambda _A^n L$ and B-V algebra structures on $\Lambda _A L$. The homology of such a B-V algebra $(\Lambda _A L,\partial )$ coincides with the cohomology of $L$ with coefficients in $\Lambda _A^n L$, with reference to the left $(A,L)$-module structure determined by $\partial $. Some applications to Poisson structures and to differential geometry are discussed.
LA - eng
KW - Lie-Rinehart algebras; Gertenhaber algebras; Batalin-Vilkovisky algebras; abstract left and right connections; cohomological duality for Lie-Rinehart algebras; Poisson structures
UR - http://eudml.org/doc/75288
ER -