Related papers: Bialgebra cohomology and exact sequences
We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We…
Quasishuffle Hopf algebras, usually defined on a commutative monoid, can be more generally defined on any associative algebra V. If V is a commutative and cocommutative bialgebra, the associated quasishuffle bialgebra QSh(V) inherits a…
We prove a universal characterization of Hopf algebras among cocommutative bialgebras over a field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why…
Let A be any finite dimensional Hopf algebra over a field k. We specify the Tate and Tate-Hochschild cohomology for A and introduce cup products that make them become graded rings. We establish the relationship between these rings. In…
This article introduces a method, which starting from simple and quite general mathematical data, allows to construct linear algebras of operators which are, each of them, endowed with a bialgebra structure (coproduct and counity). Moreover…
We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study…
We recapture Kuperberg's numerical invariant of 3-manifolds associated to a semisimple and cosemisimple Hopf algebra through a `planar algebra construction'. A result of possibly independent interest, used during the proof, which relates…
A condition is identified which guarantees that the coinvariants of a coaction of a Hopf algebra on an algebra form a subalgebra, even though the coaction may fail to be an algebra homomorphism. A Hilbert Theorem (finite generation of the…
We give the explicit construction of the product of an arbitrary family of coalgebras, bialgebras and Hopf algebras: it turns out that the product of an arbitrary family of coalgebras (resp. bialgebras, Hopf algebras) is the sum of a family…
A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of…
Primitive cohomology of a Hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra. Among many of its applications, two classifications are presented. Firstly we classify all non locally PI,…
We study twisted bialgebras and double twisted bialgebras, that is to say bialgebras in the category of linear species, or in the category of species in the category of coalgebras. We define the notion of cofree twisted coalgebra and…
We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed…
It is known that there is a Hopf algebra structure on the vector space with basis all heap-ordered trees. We give a new bialgebra structure on the space with basis all permutations and show that there is a direct bialgebra isomorphism…
We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by…
We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum…
We use deformation sequences of (Hopf) algebras, extending the results of Negron and Pevtsova, to show that bosonizations of some suitable braided Hopf algebras by some suitable finite-dimensional Hopf algebras have finitely generated…
We present a rich source of Hopf algebras starting from a cofinite central extension of a Noetherian Hopf algebra and a subgroup of the algebraic group of characters of the central Hopf subalgebra. The construction is transparent from a…
We show that the cohomology for some 2-generated selfinjective algebras is finitely generated. This applies in particular to the algebras $A5(\beta)$ for $\beta\neq 0$ in the classification of connected Hopf algebras of dimension $p^3$ over…
We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that…