  In the BRST quantization of gauge theories, the zero locus $Z_Q$ of the BRST
differential $Q$ carries an (anti)bracket whose parity is opposite to that of
the fundamental bracket. We show that the on-shell BFV/BV gauge symmetries are
in a 1:1 correspondence with Hamiltonian vector fields on $Z_Q$, and
observables of the BRST theory are in a 1:1 correspondence with characteristic
functions of the bracket on $Z_Q$. By reduction to the zero locus, we obtain
relations between bracket operations and differentials arising in different
complexes (the Gerstenhaber, Schouten, Berezin-Kirillov, and Sklyanin
brackets); the equation ensuring the existence of a nilpotent vector field on
the reduced manifold can be the classical Yang-Baxter equation. We also
generalize our constructions to the bi-QP-manifolds which from the BRST theory
viewpoint corresponds to the BRST-anti-BRST-symmetric quantization.
