A Cacti theoretical interpretation of the axioms of bialgebras and H-module algebras (original) (raw)

The Cohomology Structure of an Algebra Entwined with a Coalgebra

Journal of Algebra, 2001

Two cochain complexes are constructed for an algebra A and a coalgebra C entwined with each other via the map ψ : C ⊗ A → A ⊗ C. One complex is associated to an A-bimodule, the other to a C-bicomodule. In the former case the resulting complex can be considered as a ψ-twisted Hochschild complex of A, while for the latter one obtains a ψ-twist of the Cartier complex of C. The notion of a weak comp algebra is introduced by weakening the axioms of the Gerstenhaber comp algebra. It is shown that such a weak comp algebra is a cochain complex with two cup products that descend to the cohomology. It is also shown that the complexes associated to an entwining structure and A or C are examples of a weak comp algebra. Finally both complexes are combined in a double complex whose role in the deformation theory of entwining structures is outlined.

A Categorical Approach to Algebras and Coalgebras

International Electronic Journal of Algebra, 2018

Algebraic and coalgebraic structures are often handled independently. In this survey we want to show that they both show up naturally when introducing them from a categorical point of you. Azumaya, Frobenius, separable, and Hopf algebras are obtained when both notions are combined. The starting point and guiding lines for this approach are given by adjoint pairs of functors and their elementary properties.

Fe b 20 03 Plus-construction of algebras over an operad , cyclic and Hochschild homologies up to homotopy ∗

2003

In this paper we generalize the plus-construction given by M. Livernet for algebras over rational differential graded Koszul operads to the framework of admissible operads (the category of algebras over such operads admits a closed model category structure). We follow the modern approach of J. Berrick and C. Casacuberta defining topological plus-construction as a nullification with respect to a universal acyclic space. Similarly, we construct a universal H ∗ -acyclic algebra U and we define A −→ A+ as the U -nullification of the algebra A. This map induces an isomorphism on Quillen homology and quotients out the maximal perfect ideal of π0(A). As an application, we consider for any associative algebra R the plusconstructions of gl(R) in the categories of Lie and Leibniz algebras up to homotopy. This gives rise to two new homology theories for associative algebras, namely cyclic and Hochschild homologies up to homotopy. In particular, these theories coincide with the classical cyclic...

Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bialgebras

Eprint Arxiv Math 0212326, 2002

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. In order to do that we define the Hochschild cohomology of an algebra in an abelian monoidal category. Then we characterize those algebras which have dimension less than or equal to 1 with respect to Hochschild cohomology. Now let us consider a Hopf algebra A such that its Jacobson radical J is a nilpotent Hopf ideal and H := A/J is a semisimple algebra. By using our homological results, we prove that the canonical projection of A on H has a section which is an H-colinear algebra map. Furthermore, if H is cosemisimple too, then we can choose this section to be an (H, H)-bicolinear algebra morphism. This fact allows us to describe A as a 'generalized bosonization' of a certain algebra R in the category of Yetter-Drinfeld modules over H. As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. In this case, many results that we obtain hold true for a large enough class of H-module coalgebras, where H is a cosemisimple Hopf algebra.

A ] 19 N ov 2 00 3 A concept of 23 PROP and deformation theory of ( co ) associative bialgebras

2003

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 Lie algebra controlling the deformations of a (co)associative bialgebra. Philosophically, this construction is a version of the Markl’s operadic construction from [M1] applied to minimal models of 2 3 PROPs.

The Fukaya Type Categories For Associative Algebras

this paper, however, we will show that, if one takes for End A the differential graded algebra C (A; A) of Hochschild cochains, the maps (1), in a sense, still exist. More precisely, they exist if one passes from the category of algebras to the category of complexes by means of some well known homological functors. Let A be an associative unital algebra over a commutative unital ground ring k. Consider A as a bimodule over itself; by E A we denote the differential graded algebra (C (A; A); ffi; ) which is the standard complex for computing Ext A\Omega A ffi (A; A) = H (A; A) (the Hochschild cohomology) equipped with the Yoneda product (Sect. 2). We construct the map of complexes ffl : C (A)\Omega C (E A ) ! C (A) (2) ffl : C (E A )\Omega C (E A ) ! C (E A ) (3) Here C stands for the Hochschild complex computing HH (A) or for the periodic cyclic complex. For the case of bar complex an analogous operation was discovered by Getzler--Jones and by Gerstenhaber--Voronov [11, 9]. Therefore...