Description
Let I be an ideal in a commutative (associative) algebra O. Starting
from a resolution of O/I as an O-module, we construct a Koszul-Tate
resolution for this quotient, i.e.\ a graded symmetric algebra over O
with a differential which provides simultaneously a resolution as an
O-module. This algebra resolution has the beautiful structure of a
forest of decorated trees and is related to an A-infinity algebra on the
original module resolution.
Considering O to be a Poisson algebra
and I a finitely generated Poisson subalgebra, we use the above
construction to obtain the corresponding BFV formulation. Its cohomology
at degree zero is proven to coincide with the reduced Poisson algebra
N(I)/I, where N(I) is the normaliser of I inside O, thus
generalising ordinary coisotropic reduction to the singular setting. As
an illustration we use the example where O consists of functions on
T^*(\R^3) and I is the ideal generated by angular momenta.
This is joint work with Aliaksandr Hancharuk and, in part, with Camille
Laurent-Gengoux.
