Integrals for bialgebras

Abstract: An extremely familiar notion in Hopf algebra theory is that of an \emph{integral}. If $G$ is a compact topological group, then a Haar integral on $G$ is a linear functional $\mathcal{C}(G) \to \mathbb{R}$ which is translation invariant. % A well-known result by Larson and Sweedler shows that integrals on a Hopf algebra can be obtained by applying the Structure Theorem of Hopf modules to the rational part of its linear dual, i.e., the space of integrals comes from a right adjoint functor from a category of modules to the category of vector spaces. % In this seminar we will discuss how this construction can be carried out even in the absence of an antipode, offering a novel perspective on integrals which differs profoundly from their classical description as ``colinear forms''. Our approach leads to a new notion of integrals for bialgebras which does not require, and does not imply, the existence of an antipode.

Time permitting, we will seize this opportunity to say a few words about Hopf envelopes of bialgebras, since in many cases of interest integrals for a bialgebra are in bijection with integrals for its Hopf envelope. % The Hopf envelope of a bialgebra B is a certain universal Hopf algebra that we can associate with B and that plays for it the same role that the universal enveloping group plays for a monoid. %In categorical terms, the Hopf envelope is the left adjoint to the forgetful functor from Hopf algebras to bialgebras, hence it may be legitimately called the free Hopf algebra generated by B. Its existence is a well-known fact in Hopf algebra theory, but its construction is very technical. Nevertheless, there are a number of cases where we can realize the Hopf envelope of a bialgebra B as a suitable quotient of B itself and we can take advantage of it to study integrals for the corresponding bialgebra.

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\centering This talk is based on an ongoing project with A.\ Ardizzoni and C.\ Menini.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar