182 research outputs found
Nilpotency in type A cyclotomic quotients
We prove a conjecture made by Brundan and Kleshchev on the nilpotency degree
of cyclotomic quotients of rings that categorify one-half of quantum sl(k).Comment: 19 pages, 39 eps files. v3 simplifies antigravity moves and corrects
typo
Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras
In this paper, we prove Khovanov-Lauda's cyclotomic categorification
conjecture for all symmetrizable Kac-Moody algebras. Let be the
quantum group associated with a symmetrizable Cartan datum and let
be the irreducible highest weight -module with a dominant integral
highest weight . We prove that the cyclotomic Khovanov-Lauda-Rouquier
algebra gives a categorification of .Comment: Typoes correcte
Knot homology via derived categories of coherent sheaves II, sl(m) case
Using derived categories of equivariant coherent sheaves we construct a knot
homology theory which categorifies the quantum sl(m) knot polynomial. Our knot
homology naturally satisfies the categorified MOY relations and is
conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is
motivated by the geometric Satake correspondence and is related to Manolescu's
by homological mirror symmetry.Comment: 51 pages, 9 figure
Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial
An elementary introduction to Khovanov construction of superpolynomials.
Despite its technical complexity, this method remains the only source of a
definition of superpolynomials from the first principles and therefore is
important for development and testing of alternative approaches. In this first
part of the review series we concentrate on the most transparent and
unambiguous part of the story: the unreduced Jones superpolynomials in the
fundamental representation and consider the 2-strand braids as the main
example. Already for the 5_1 knot the unreduced superpolynomial contains more
items than the ordinary Jones.Comment: 33 page
Khovanov-Rozansky Homology and Topological Strings
We conjecture a relation between the sl(N) knot homology, recently introduced
by Khovanov and Rozansky, and the spectrum of BPS states captured by open
topological strings. This conjecture leads to new regularities among the sl(N)
knot homology groups and suggests that they can be interpreted directly in
topological string theory. We use this approach in various examples to predict
the sl(N) knot homology groups for all values of N. We verify that our
predictions pass some non-trivial checks.Comment: 25 pages, 2 figures, harvmac; minor corrections, references adde
Cyclic Foam Topological Field Theories
This paper proposes an axiomatic for Cyclic Foam Topological Field theories.
That is Topological Field theories, corresponding to String theories, where
particles are arbitrary graphs. World surfaces in this case are two-manifolds
with one-dimensional singularities. We proved that Cyclic Foam Topological
Field theories one-to-one correspond to graph-Cardy-Frobenius algebras, that
are families , where are families of
commutative associative Frobenius algebras, is an graduated by graphes, associative
algebras of Frobenius type and is a family of special representations. There are
constructed examples of Cyclic Foam Topological Field theories and its
graph-Cardy-Frobenius algebrasComment: 14 page
Enlargement of a low-dimensional stochastic web
We consider an archetypal example of a low-dimensional stochastic web, arising in a 1D oscillator driven by a plane wave of a frequency equal or close to a multiple of the oscillator’s natural frequency. We show that the web can be greatly enlarged by the introduction of a slow, very weak, modulation of the wave angle. Generalizations are discussed. An application to electron transport in a nanometre-scale semiconductor superlattice in electric and magnetic fields is suggested
A new approach to the treatment of Separatrix Chaos and its applications
We consider time-periodically perturbed 1D Hamiltonian systems possessing one or more separatrices. If the perturbation is weak, then the separatrix chaos is most developed when the perturbation frequency lies in the logarithmically small or moderate ranges: this corresponds to the involvement of resonance dynamics into the separatrix chaos. We develop a method matching the discrete chaotic dynamics of the separatrix map and the continuous regular dynamics of the resonance Hamiltonian. The method has allowed us to solve the long-standing problem of an accurate description of the maximum of the separatrix chaotic layer width as a function of the perturbation frequency. It has also allowed us to predict and describe
new phenomena including, in particular: (i) a drastic facilitation of the onset of global chaos between neighbouring separatrices, and (ii) a huge increase in the
size of the low-dimensional stochastic web
Trace as an alternative decategorification functor
Categorification is a process of lifting structures to a higher categorical
level. The original structure can then be recovered by means of the so-called
"decategorification" functor. Algebras are typically categorified to additive
categories with additional structure and decategorification is usually given by
the (split) Grothendieck group. In this expository article we study an
alternative decategorification functor given by the trace or the zeroth
Hochschild--Mitchell homology. We show that this form of decategorification
endows any 2-representation of the categorified quantum sl(n) with an action of
the current algebra U(sl(n)[t]) on its center.Comment: 47 pages with tikz figures. arXiv admin note: text overlap with
arXiv:1405.5920 by other author
The sl_3 web algebra
In this paper we use Kuperberg’s sl3-webs and Khovanov’s sl3-foams to define a new
algebra KS, which we call the sl3-web algebra. It is the sl3 analogue of Khovanov’s arc algebra.
We prove that KS is a graded symmetric Frobenius algebra. Furthermore, we categorify an
instance of q-skew Howe duality, which allows us to prove that KS
is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group K0
(WS )Q(q) , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein
variety, and to prove that KS is a graded cellular algebra.info:eu-repo/semantics/publishedVersio
- …