The CROCs, non-commutative deformations, and (co)associative bialgebras

Details The CROCs, non-commutative deformations, and (co)associative bialgebras The CROCs, non-commutative deformations, and (co)associative bialgebras

2003-06-09

arxiv.org/abs/math/0306143v1

Mathematics

Quantum Algebra

18 pages

Scientific paper

We compactify the spaces $K(m,n)$ introduced by Maxim Kontsevich. The initial idea was to construct an $L_\infty$ algebra governing the deformations of a (co)associative bialgebra. However, this compactification leads not to a resolution of the PROP of (co)associative bialgebras, but to a new algebraic structure we call here a CROC. It turns out that these constructions are related to the non-commutative deformations of (co)associative bialgebras. We construct an associative dg algebra conjecturally governing the non-commutative deformations of a bialgebra. Then, using the Quillen duality, we construct a dg Lie algebra conjecturally governing the commutative (usual) deformations of a (co)associative bialgebra. Philosophically, the main point is that for the associative bialgebras the non-commutative deformations is maybe a more fundamental object than the usual commutative ones.

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