  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.
