170 research outputs found
Pure Anderson Motives and Abelian \tau-Sheaves
Pure t-motives were introduced by G. Anderson as higher dimensional
generalizations of Drinfeld modules, and as the appropriate analogs of abelian
varieties in the arithmetic of function fields. In order to construct moduli
spaces for pure t-motives the second author has previously introduced the
concept of abelian \tau-sheaf. In this article we clarify the relation between
pure t-motives and abelian \tau-sheaves. We obtain an equivalence of the
respective quasi-isogeny categories. Furthermore, we develop the elementary
theory of both structures regarding morphisms, isogenies, Tate modules, and
local shtukas. The later are the analogs of p-divisible groups.Comment: final version as it appears in Mathematische Zeitschrif
Nilpotent action on the KdV variables and 2-dimensional Drinfeld-Sokolov reduction
We note that a version ``with spectral parameter'' of the Drinfeld-Sokolov
reduction gives a natural mapping from the KdV phase space to the group of
loops with values in ~: affine nilpotent and
principal commutative (or anisotropic Cartan) subgroup~; this mapping is
connected to the conserved densities of the hierarchy. We compute the
Feigin-Frenkel action of (defined in terms of screening
operators) on the conserved densities, in the case
Translation Invariance, Commutation Relations and Ultraviolet/Infrared Mixing
We show that the Ultraviolet/Infrared mixing of noncommutative field theories
with the Gronewold-Moyal product, whereby some (but not all) ultraviolet
divergences become infrared, is a generic feature of translationally invariant
associative products. We find, with an explicit calculation that the phase
appearing in the nonplanar diagrams is the one given by the commutator of the
coordinates, the semiclassical Poisson structure of the non commutative
spacetime. We do this with an explicit calculation for represented generic
products.Comment: 24 pages, 1 figur
Link Invariants and Combinatorial Quantization of Hamiltonian Chern-Simons Theory
We define and study the properties of observables associated to any link in
(where is a compact surface) using the
combinatorial quantization of hamiltonian Chern-Simons theory. These
observables are traces of holonomies in a non commutative Yang-Mills theory
where the gauge symmetry is ensured by a quantum group. We show that these
observables are link invariants taking values in a non commutative algebra, the
so called Moduli Algebra. When these link invariants are pure
numbers and are equal to Reshetikhin-Turaev link invariants.Comment: 39, latex, 7 figure
Quantum W-algebras and Elliptic Algebras
We define quantum W-algebras generalizing the results of Reshetikhin and the
second author, and Shiraishi-Kubo-Awata-Odake. The quantum W-algebra associated
to sl_N is an associative algebra depending on two parameters. For special
values of parameters it becomes the ordinary W-algebra of sl_N, or the
q-deformed classical W-algebra of sl_N. We construct free field realizations of
the quantum W-algebras and the screening currents. We also point out some
interesting elliptic structures arising in these algebras. In particular, we
show that the screening currents satisfy elliptic analogues of the Drinfeld
relations in U_q(n^).Comment: 26 pages, AMSLATE
Quantum principal commutative subalgebra in the nilpotent part of and lattice KdV variables
We propose a quantum lattice version of Feigin and E. Frenkel's
constructions, identifying the KdV differential polynomials with functions on a
homogeneous space under the nilpotent part of . We construct
an action of the nilpotent part of on
their lattice counterparts, and embed the lattice variables in a -module, coinduced from a quantum version of the principal commutative
subalgebra, which is defined using the identification of with
its coordinate algebra
Fusion and singular vectors in A1{(1)} highest weight cyclic modules
We show how the interplay between the fusion formalism of conformal field
theory and the Knizhnik--Zamolodchikov equation leads to explicit formulae for
the singular vectors in the highest weight representations of A1{(1)}.Comment: 42 page
Quantum function algebras as quantum enveloping algebras
Inspired by a result in [Ga], we locate two -integer forms of
, along with a presentation by generators and relations, and
prove that for they specialize to , where is the Lie bialgebra of the Poisson Lie group dual of ; moreover, we explain the relation with [loc. cit.]. In sight of
this, we prove two PBW-like theorems for , both related to the
classical PBW theorem for .Comment: 27 pages, AMS-TeX C, Version 3.0 - Author's file of the final
version, as it appears in the journal printed version, BUT for a formula in
Subsec. 3.5 and one in Subsec. 5.2 - six lines after (5.1) - that in this
very pre(post)print have been correcte
Braid Group Action and Quantum Affine Algebras
We lift the lattice of translations in the extended affine Weyl group to a
braid group action on the quantum affine algebra. This action fixes the
Heisenberg subalgebra pointwise. Loop like generators are found for the algebra
which satisfy the relations of Drinfeld's new realization. Coproduct
formulas are given and a PBW type basis is constructed.Comment: 15 page
Combinatorial expression for universal Vassiliev link invariant
The most general R-matrix type state sum model for link invariants is
constructed. It contains in itself all R-matrix invariants and is a generating
function for "universal" Vassiliev link invariants. This expression is more
simple than Kontsevich's expression for the same quantity, because it is
defined combinatorially and does not contain any integrals, except for an
expression for "the universal Drinfeld's associator".Comment: 20 page
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