Weak Projections onto a Braided Hopf Algebra

We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) sub-bialgebra with antipode; see Theorem 1.12. In the second part of the paper we prove that bialgebras with weak projections are cross product bialgebras; see Theorem 2.12. In the particular case when the bialgebra $A$ is cocommutative and a certain cocycle associated to the weak projection is trivial we prove that $A$ is a double cross product, or biproduct in Madjid's terminology. The last result is based on a universal property of double cross products which, by Theorem 2.15, works in braided monoidal categories. We also investigate the situation when the right action of the associated matched pair is trivial

Mini-Workshop: Infinite Dimensional Hopf Algebras

This is a report of the above mini-workshop. It contains brief accounts of all 17 talks given at the meeting, with commentary on their interconnections. A selection of the numerous open questions discussed at and generated by the meeting is provided in a separate section. The cumulative references listed for each of the talks together provide an up-to-date guide to the fast-growing literature on the topics covered

Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology

In this monograph, we extend S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore we establish an explicit description of an isomorphism by A. Neeman and V. Retakh, which links $\mathrm{Ext}$-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid. As a main result, we show that our construction behaves well with respect to structure preserving functors between exact monoidal categories. We use our main result to conclude, that both the Lie bracket and the squaring map in Hochschild cohomology are invariants under Morita equivalence. For quasi-triangular bialgebras, we further determine a significant part of the Lie bracket's kernel, and thereby prove a conjecture by L. Menichi. Along the way, we introduce $n$-extension closed and entirely extension closed subcategories of abelian categories, and study some of their properties.Comment: Modified version of author's PhD thesis (Bielefeld University, December 2013). 159 pages. --- Final version, to appear in "Memoirs of the American Mathematical Society". Corrected a mistake in Section 6 (the main results are not affected) and made minor changes according to the suggestions of the referee

Graduate School: Course Decriptions, 1972-73

Official publication of Cornell University V.64 1972/7