8,291 research outputs found
Graphical Calculus for the Double Affine Q-Dependent Braid Group
We define a double affine -dependent braid group. This group is
constructed by appending to the braid group a set of operators , before
extending it to an affine -dependent braid group. We show specifically that
the elliptic braid group and the double affine Hecke algebra (DAHA) can be
obtained as quotient groups. Complementing this we present a pictorial
representation of the double affine -dependent braid group based on ribbons
living in a toroid. We show that in this pictorial representation we can fully
describe any DAHA. Specifically, we graphically describe the parameter upon
which this algebra is dependent and show that in this particular representation
corresponds to a twist in the ribbon
A_k Generalization of the O(1) Loop Model on a Cylinder: Affine Hecke Algebra, q-KZ Equation and the Sum Rule
We study the A_k generalized model of the O(1) loop model on a cylinder. The
affine Hecke algebra associated with the model is characterized by a vanishing
condition, the cylindric relation. We present two representations of the
algebra: the first one is the spin representation, and the other is in the
vector space of states of the A_k generalized model. A state of the model is a
natural generalization of a link pattern. We propose a new graphical way of
dealing with the Yang-Baxter equation and -symmetrizers by the use of the
rhombus tiling. The relation between two representations and the meaning of the
cylindric relations are clarified. The sum rule for this model is obtained by
solving the q-KZ equation at the Razumov-Stroganov point.Comment: 43 pages, 22 figures, LaTeX, (ver 2) Introduction rewritten and
Section 4.3 adde
The center of the affine nilTemperley-Lieb algebra
We give a description of the center of the affine nilTemperley-Lieb algebra
based on a certain grading of the algebra and on a faithful representation of
it on fermionic particle configurations. We present a normal form for
monomials, hence construct a basis of the algebra, and use this basis to show
that the affine nilTemperley-Lieb algebra is finitely generated over its
center. As an application, we obtain a natural embedding of the affine
nilTemperley-Lieb algebra on N generators into the affine nilTemperley-Lieb
algebra on N + 1 generators.Comment: 27 pages, 5 figures, comments welcom
A survey of Heisenberg categorification via graphical calculus
In this expository paper we present an overview of various graphical
categorifications of the Heisenberg algebra and its Fock space representation.
We begin with a discussion of "weak" categorifications via modules for Hecke
algebras and "geometrizations" in terms of the cohomology of the Hilbert
scheme. We then turn our attention to more recent "strong" categorifications
involving planar diagrammatics and derived categories of coherent sheaves on
Hilbert schemes.Comment: 23 pages; v2: Some typos corrected and other minor improvements made;
v3: Some small errors corrected; v4: Code corrected to fix problem with
missing arrows on some diagram
Discrete holomorphicity and quantized affine algebras
We consider non-local currents in the context of quantized affine algebras,
following the construction introduced by Bernard and Felder. In the case of
and , these currents can be identified with
configurations in the six-vertex and Izergin--Korepin nineteen-vertex models.
Mapping these to their corresponding Temperley--Lieb loop models, we directly
identify non-local currents with discretely holomorphic loop observables. In
particular, we show that the bulk discrete holomorphicity relation and its
recently derived boundary analogue are equivalent to conservation laws for
non-local currents
The Quantum Double in Integrable Quantum Field Theory
Various aspects of recent works on affine quantum group symmetry of
integrable 2d quantum field theory are reviewed and further clarified. A
geometrical meaning is given to the quantum double, and other properties of
quantum groups. Multiplicative presentations of the Yangian double are
analyzed.Comment: 43 page
Heisenberg categorification and Hilbert schemes
Given a finite subgroup G of SL(2,C) we define an additive 2-category H^G
whose Grothendieck group is isomorphic to an integral form of the Heisenberg
algebra. We construct an action of H^G on derived categories of coherent
sheaves on Hilbert schemes of points on the minimal resolutions of C^2/G.Comment: 53 page
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