8 research outputs found
The diagrammatic Soergel category and sl(2) and sl(3) foams
We define two functors from Elias and Khovanov's diagrammatic Soergel
category, one targeting Clark-Morrison-Walker's category of disoriented sl(2)
cobordisms and the other the category of (universal) sl(3) foams.Comment: v4, minor changes, referee's comments implemented. v3, 20 pages, lots
of figures, remark about the general case rewritten, one redundant relation
removed from definition of SC1 and minor typo
Reidemeister/Roseman-type Moves to Embedded Foams in 4-dimensional Space
The dual to a tetrahedron consists of a single vertex at which four edges and
six faces are incident. Along each edge, three faces converge. A 2-foam is a
compact topological space such that each point has a neighborhood homeomorphic
to a neighborhood of that complex. Knotted foams in 4-dimensional space are to
knotted surfaces, as knotted trivalent graphs are to classical knots. The
diagram of a knotted foam consists of a generic projection into 4-space with
crossing information indicated via a broken surface. In this paper, a finite
set of moves to foams are presented that are analogous to the Reidemeister-type
moves for knotted graphs. These moves include the Roseman moves for knotted
surfaces. Given a pair of diagrams of isotopic knotted foams there is a finite
sequence of moves taken from this set that, when applied to one diagram
sequentially, produces the other diagram.Comment: 18 pages, 29 figures, Be aware: the figure on page 3 takes some time
to load. A higher resolution version is found at
http://www.southalabama.edu/mathstat/personal_pages/carter/Moves2Foams.pdf .
If you want to use to any drawings, please contact m
The Diagrammatic Soergel Category and s
For each N≥4, we define a monoidal functor from Elias and Khovanov's diagrammatic
version of Soergel's category of bimodules to the category of sl(N) foams defined by Mackaay, Stošić, and Vaz. We show that through these functors Soergel's category can be obtained
from the sl(N) foams
Extended graphical calculus for categorified quantum sl(2)
A categorification of the Beilinson-Lusztig-MacPherson form of the quantum
sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here
we enhance the graphical calculus introduced and developed in that paper to
include two-morphisms between divided powers one-morphisms and their
compositions. We obtain explicit diagrammatical formulas for the decomposition
of products of divided powers one-morphisms as direct sums of indecomposable
one-morphisms; the latter are in a bijection with the Lusztig canonical basis
elements. These formulas have integral coefficients and imply that one of the
main results of Lauda's paper---identification of the Grothendieck ring of his
2-category with the idempotented quantum sl(2)---also holds when the 2-category
is defined over the ring of integers rather than over a field.Comment: 72 pages, LaTeX2e with xypic and pstricks macro
A Diagrammatic Temperley-Lieb Categorification
The monoidal category of Soergel bimodules categorifies the Hecke algebra of
a finite Weyl group. In the case of the symmetric group, morphisms in this
category can be drawn as graphs in the plane. We define a quotient category,
also given in terms of planar graphs, which categorifies the Temperley-Lieb
algebra. Certain ideals appearing in this quotient are related both to the
1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We
demonstrate how further subquotients of this category will categorify the cell
modules of the Temperley-Lieb algebra.Comment: long awaited update to published versio
A diagrammatic categorification of the q-Schur algebra
In this paper we categorify the q-Schur algebra S(n,d) as a quotient of
Khovanov and Lauda's diagrammatic 2-category U(sln). We also show that our
2-category contains Soergel's monoidal category of bimodules of type A, which
categorifies the Hecke algebra H(d), as a full sub-2-category if d does not
exceed n. For the latter result we use Elias and Khovanov's diagrammatic
presentation of Soergel's monoidal category of type A.Comment: 60 pages, lots of figures. v3: Substantial changes. To appear in
Quantum Topolog