Abstract
In this note we give a homological explanation of "pure spinors" in YM theories
with minimal amount of supersymmetries. We construct A_{\infty} algebras A for every
dimension D=3,4,6,10, which for D=10 coincides with homogeneous coordinate ring of pure
spinors with coordinate lambda^{alpha}. These algebras are Bar-dual to Lie algebras
generated by supersymmetries, written in components. The algebras have a finite number of
higher multiplications. The main result of the present note is that in dimension D=3,6,10
the algebra A\otimes \Lambda[\theta^{\alpha}]\otimes Mat_n with a differential D is
equivalent to Batalin-Vilkovisky algebra of minimally supersymmetric YM theory in dimension
D reduced to a point. This statement can be extended to nonreduced theories.
