Operads: Hopf algebras and coloured Koszul duality

P.P.I. van der Laan

Research output: ThesisDoctoral thesis 1 (Research UU / Graduation UU)

Abstract

Operads are tools designed to study not mathematical objects themselves,
but operations
on these. A simplified example: instead of integers, one studies
multiplication.
Multiplication is a map that takes two integers as input and gives one new
integer as output (2*3 = 6 says that the inputs 2 and 3 give 6 as
output). The origin
of operads is to look at operations with an arbitrary number of inputs
and one output
(2*3*4*5=120 describes an operation with 4 inputs that with inputs 2, 3,
4, and 5
gives output 120). One should think of an operad as a system of such
maps together
with composition rules (in our example (2*3)*(4*5) = 2*3*4*5 is a
composition rule).
This point of view seems rather straightforward, but has led to
important developments
in several branches of mathematics. Operads enable handling complicated
combinatorics where many operations appear.

Imagine a surface together with a rule called multiplication, that
assigns to every
two points on the surface a third point. One might ask what are the
possible ways
to change the multiplication a little bit. The study of such changes is
a branch of
Deformation Theory. Deformation theory can be extended to a purely algebraic
context, where the surface in the example above is replaced by an
algebraic object.
In fact, for systems of operations described by an operad one can often
develop a
deformation theory. The interplay between operads and Deformation Theory has
contributed to a large extent to the understanding of one of the major
results in
mathematics in the previous decennium: Kontsevich’ theorem on the
Quantisation
Deformation of Poisson Manifolds. This result can be interpreted as a
theorem
about the correspondence between Classical and Quantum Mechanics. More
precisely: it is an instance of Bohr’s Correspondence Principle, which
says that to
every system in Classical Mechanics there exists a system in Quantum
Mechanics
that looks the same if one observes it at sufficiently large scale.
My research consists in part in generalising existing theory on
deformations on the
basis of operads, and finding new applications and relations to
classical results.

Furthermore, the author is currently studying not deformations of
structures described
by operads, but deformations of operads themselves. From a mathematical
point of view this is a natural thing to do. Moreover, this approach
reveals a subtle
interplay of deformation results at different levels.
Apart from the above, I am studying applications of operads in
Combinatorics.
These applications are related to so called Hopf Algebras. Hopf algebras
have applications
in Perturbative Quantum Field Theory. The essential problem is that
computations of physical quantities from Feynmann integrals often give
infinity as
a result. Hopf Algebras suggest a solution to this problem. My work
shows that
the Hopf Algebras used in renormalisation often stem from operads. This
approach
simplifies the combinatorics involved, and shows relations to other Hopf
Algebras.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Utrecht University
Supervisors/Advisors
  • Moerdijk, Ieke, Primary supervisor
Award date12 Jan 2004
Publisher
Publication statusPublished - 12 Jan 2004

Keywords

  • Wiskunde en Informatica (WIIN)
  • Other mathematical specialities
  • Wiskunde en computerwetenschappen
  • Wiskunde: algemeen
  • Operads
  • Homological Algebra
  • Hopf Algebras
  • Koszul Duality
  • trees
  • L-Infinity Algebras
  • Homotopy Algebras

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