J un 2 00 3 The CROCs , non - commutative deformations , and ( co ) associative bialgebras
نویسنده
چکیده
We compactify the spaces K(m,n) introduced by Maxim Kontsevich. The initial idea was to construct an L∞ 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.
منابع مشابه
(Pure) transcendence bases in $\phi$-deformed shuffle bialgebras
Computations with integro-differential operators are often carried out in an associative algebra with unit and they are essentially non-commutative computations. By adjoining a cocommutative co-product, one can have those operators perform on a bialgebra isomorphic to an enveloping algebra. That gives an adequate framework for a computer-algebra implementation via monoidal factorization, (pure)...
متن کاملDeformation Theory of Representations of Prop(erad)s I
In this paper and its follow-up [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a non-linear framework. The associated chain complex is endowed with an L∞-algebra structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these r...
متن کاملA simple construction of bialgebra deformations∗
Let A denote a bialgebra over a field k and let At = A[[t]] denote the ring of formal power series with coefficients in A. Assume that A is also isomorphic to a free, associative algebra over k. We give a simple construction which makes At a bialgebra deformation of A. In typical applications, At is neither commutative nor cocommutative. This construction yields bialgebra deformations associate...
متن کامل23PROP and deformation theory of (co)associative bialgebras
We introduce a concept of 2 3 PROP generalizing the Kontsevich concept of 1 2 PROP. We prove that some Stasheff-type compactification of the Kontsevich spaces K(m,n) defines a topological 2 3 PROP structure. The corresponding chain complex is a minimal model for its cohomology (both are considered as 2 3 PROPs). We construct a 2 3 PROP End(V ) for a vector space V . Finally, we construct a dg L...
متن کامل2 8 N ov 2 00 8 On the deformation theory of structure constants for associative algebras
Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a deformation driving algebras (DDAs) is proposed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative commutative three-dimensional algebras with the DDA being a three-dimensional Lie algebra and their connection with integrable system...
متن کامل