28 research outputs found
Bicommutant categories from fusion categories
Bicommutant categories are higher categorical analogs of von Neumann algebras
that were recently introduced by the first author. In this article, we prove
that every unitary fusion category gives an example of a bicommutant category.
This theorem categorifies the well known result according to which a finite
dimensional *-algebra that can be faithfully represented on a Hilbert space is
in fact a von Neumann algebra.Comment: Updated to the published version + fixed some small typo
Knot Theory: from Fox 3-colorings of links to Yang-Baxter homology and Khovanov homology
This paper is an extended account of my "Introductory Plenary talk at Knots
in Hellas 2016" conference We start from the short introduction to Knot Theory
from the historical perspective, starting from Heraclas text (the first century
AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of
Geometria Situs (1679), and J.B.Listing (student of Gauss) work of 1847. We
spend some space on Ralph H. Fox (1913-1973) elementary introduction to diagram
colorings (1956). In the second section we describe how Fox work was
generalized to distributive colorings (racks and quandles) and eventually in
the work of Jones and Turaev to link invariants via Yang-Baxter operators, here
the importance of statistical mechanics to topology will be mentioned. Finally
we describe recent developments which started with Mikhail Khovanov work on
categorification of the Jones polynomial. By analogy to Khovanov homology we
build homology of distributive structures (including homology of Fox colorings)
and generalize it to homology of Yang-Baxter operators. We speculate, with
supporting evidence, on co-cycle invariants of knots coming from Yang-Baxter
homology. Here the work of Fenn-Rourke-Sanderson (geometric realization of
pre-cubic sets of link diagrams) and Carter-Kamada-Saito (co-cycle invariants
of links) will be discussed and expanded.
Dedicated to Lou Kauffman for his 70th birthday.Comment: 35 pages, 31 figures, for Knots in Hellas II Proceedings, Springer,
part of the series Proceedings in Mathematics & Statistics (PROMS
Current algebras and categorified quantum groups
We identify the trace, or 0th Hochschild homology, of type ADE categorified
quantum groups with the corresponding current algebra of the same type. To
prove this, we show that 2-representations defined using categories of modules
over cyclotomic (or deformed cyclotomic) quotients of KLR-algebras correspond
to local (or global) Weyl modules. We also investigate the implications for
centers of categories in 2-representations of categorified quantum groups.Comment: 30 pages, with tikz and xypic diagrams. v2: changed discussion of
cyclicity to include all simply-laced Kac-Moody algebra
On Doctrines and Cartesian Bicategories
We study the relationship between cartesian bicategories and a specialisation
of Lawvere's hyperdoctrines, namely elementary existential doctrines. Both
provide different ways of abstracting the structural properties of logical
systems: the former in algebraic terms based on a string diagrammatic calculus,
the latter in universal terms using the fundamental notion of adjoint functor.
We prove that these two approaches are related by an adjunction, which can be
strengthened to an equivalence by imposing further constraints on doctrines