455 research outputs found
Hopf Algebras and the Penrose Polynomial
AbstractLet λ be a positive integer and let G be a plane graph. LetP (G, λ) be the Penrose polynomial of G. We will present an interpretation ofP (G, −λ) in terms of colourings of G. In order to prove our main theorem we construct a Hopf algebra A of graphs and a homomorphism of Hopf algebras Ψ fromA onto a Hopf algebra of polynomials in one indeterminate. If G is a plane graph, thenΨ (G) coincides with the Penrose polynomial of G
A Prehistory of n-Categorical Physics
This paper traces the growing role of categories and n-categories in physics,
starting with groups and their role in relativity, and leading up to more
sophisticated concepts which manifest themselves in Feynman diagrams, spin
networks, string theory, loop quantum gravity, and topological quantum field
theory. Our chronology ends around 2000, with just a taste of later
developments such as open-closed topological string theory, the
categorification of quantum groups, Khovanov homology, and Lurie's work on the
classification of topological quantum field theories.Comment: 129 pages, 8 eps figure
Hopf algebras and Tutte polynomials
By considering Tutte polynomials of Hopf algebras, we show how a Tutte
polynomial can be canonically associated with combinatorial objects that have
some notions of deletion and contraction. We show that several graph
polynomials from the literature arise from this framework. These polynomials
include the classical Tutte polynomial of graphs and matroids, Las Vergnas'
Tutte polynomial of the morphism of matroids and his Tutte polynomial for
embedded graphs, Bollobas and Riordan's ribbon graph polynomial, the Krushkal
polynomial, and the Penrose polynomial.
We show that our Tutte polynomials of Hopf algebras share common properties
with the classical Tutte polynomial, including deletion-contraction
definitions, universality properties, convolution formulas, and duality
relations. New results for graph polynomials from the literature are then
obtained as examples of the general results.
Our results offer a framework for the study of the Tutte polynomial and its
analogues in other settings, offering the means to determine the properties and
connections between a wide class of polynomial invariants.Comment: v2: change of title and some reorderin
Function Theory on a q-Analog of Complex Hyperbolic Space
This work deals with function theory on quantum complex hyperbolic spaces.
The principal notions are expounded. We obtain explicit formulas for invariant
integrals on `finite' functions on a quantum hyperbolic space and on the
associated quantum isotropic cone. Also we establish principal series of -modules related to this cone, and obtain the necessary
conditions for those modules to be equivalent.Comment: 21 page
Quantisation of twistor theory by cocycle twist
We present the main ingredients of twistor theory leading up to and including
the Penrose-Ward transform in a coordinate algebra form which we can then
`quantise' by means of a functorial cocycle twist. The quantum algebras for the
conformal group, twistor space CP^3, compactified Minkowski space CMh and the
twistor correspondence space are obtained along with their canonical quantum
differential calculi, both in a local form and in a global *-algebra
formulation which even in the classical commutative case provides a useful
alternative to the formulation in terms of projective varieties. We outline how
the Penrose-Ward transform then quantises. As an example, we show that the
pull-back of the tautological bundle on CMh pulls back to the basic instanton
on S^4\subset CMh and that this observation quantises to obtain the
Connes-Landi instanton on \theta-deformed S^4 as the pull-back of the
tautological bundle on our \theta-deformed CMh. We likewise quantise the
fibration CP^3--> S^4 and use it to construct the bundle on \theta-deformed
CP^3 that maps over under the transform to the \theta-deformed instanton.Comment: 68 pages 0 figures. Significant revision now has detailed formulae
for classical and quantum CP^
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