An explicit deformation theory of (co)associative bialgebras

We find here another (not of the CROC-type) compactification of the Kontsevich spaces K(m,n). This compactification is a Stasheff-type compactification, in particular, it is exactly the Stasheff compactification when n = 1 or m = 1. The boundary strata of this compactification are products of the spaces K(m, n) and of the Stasheff polyhedra. Then we construct a dg Lie algebra naturally associated with this compactification. This dg Lie algebra describes some deformation theory in which the deformation theory of (co)associative bialgebras can be naturally imbedded. The compactification constructed here allows us to construct a minimal model of some 1 2 CROC (a concept very close to the concept of dioperad) closely related to the theory of (co)associative bialgebras. We define the 1 2 CROC End(A) for a bialgebra A and finally apply the Markl’s construction from [M1] to prove that our construction gives indeed a dg Lie algerbra. 1 A Stasheff-type compactification of the Kontsevich spaces K(m, n) First of all, recall the definition of the spaces K(m,n) due to Maxim Kontsevich (see also [Sh]). We show in the sequel that these spaces and its compactification introduced below play a crucial role in the deformation theory of (co)associative bialgebras. First define the space Conf(m,n). By definition, m,n ≥ 1, m+ n ≥ 3, and Conf(m,n) = {p1, . . . , pm ∈ R , pi < pj for i < j; q1, . . . , qn ∈ R , qi < qj for i < j} (1) Here we denote by R(1) and by R(2) two different copies of a real line R.