3,221 research outputs found
Reduction Groups and Automorphic Lie Algebras
We study a new class of infinite dimensional Lie algebras, which has
important applications to the theory of integrable equations. The construction
of these algebras is very similar to the one for automorphic functions and this
motivates the name automorphic Lie algebras. For automorphic Lie algebras we
present bases in which they are quasigraded and all structure constants can be
written out explicitly. These algebras have a useful factorisations on two
subalgebras similar to the factorisation of the current algebra on the positive
and negative parts.Comment: 21 pages, standard LaTeX2e, corrected typos, accepted for publication
in CMP - Communications in Mathematical Physic
Noncommutative Geometry and Arithmetic
This is an overview of recent results aimed at developing a geometry of
noncommutative tori with real multiplication, with the purpose of providing a
parallel, for real quadratic fields, of the classical theory of elliptic curves
with complex multiplication for imaginary quadratic fields. This talk
concentrates on two main aspects: the relation of Stark numbers to the geometry
of noncommutative tori with real multiplication, and the shadows of modular
forms on the noncommutative boundary of modular curves, that is, the moduli
space of noncommutative tori. To appear in Proc. ICM 2010.Comment: 16 pages, LaTe
Examples of noncommutative manifolds: complex tori and spherical manifolds
We survey some aspects of the theory of noncommutative manifolds focusing on
the noncommutative analogs of two-dimensional tori and low-dimensional spheres.
We are particularly interested in those aspects of the theory that link the
differential geometry and the algebraic geometry of these spaces.Comment: Survey article. Final version. To appear in the proceedings volume of
the "International Workshop on Noncommutative Geometry", IPM, Tehran 200
Algebraic conformal quantum field theory in perspective
Conformal quantum field theory is reviewed in the perspective of Axiomatic,
notably Algebraic QFT. This theory is particularly developped in two spacetime
dimensions, where many rigorous constructions are possible, as well as some
complete classifications. The structural insights, analytical methods and
constructive tools are expected to be useful also for four-dimensional QFT.Comment: Review paper, 40 pages. v2: minor changes and references added, so as
to match published versio
New elliptic solutions of the Yang-Baxter equation
We consider finite-dimensional reductions of an integral operator with the
elliptic hypergeometric kernel describing the most general known solution of
the Yang-Baxter equation with a rank 1 symmetry algebra. The reduced
R-operators reproduce at their bottom the standard Baxter's R-matrix for the
8-vertex model and Sklyanin's L-operator. The general formula has a remarkably
compact form and yields new elliptic solutions of the Yang-Baxter equation
based on the finite-dimensional representations of the elliptic modular double.
The same result is also derived using the fusion formalism.Comment: 34 pages, to appear in Commun. Math. Phy
On Hamiltonian structure of the spin Ruijsenaars-Schneider model
The Hamiltonian structure of spin generalization of the rational
Ruijsenaars-Schneider model is found by using the Hamiltonian reduction
technique. It is shown that the model possesses the current algebra symmetry.
The possibility of generalizing the found Poisson structure to the
trigonometric case is discussed and degeneration to the Euler-Calogero-Moser
system is examined.Comment: latex, 16 pages, references are adde
Lie Groups, Cluster Variables and Integrable Systems
We discuss the Poisson structures on Lie groups and propose an explicit
construction of the integrable models on their appropriate Poisson
submanifolds. The integrals of motion for the SL(N)-series are computed in
cluster variables via the Lax map. This construction, when generalised to the
co-extended loop groups, gives rise not only to several alternative
descriptions of relativistic Toda systems, but allows to formulate in general
terms some new class of integrable models.Comment: Based on talks given at Versatility of integrability, Columbia
University, May 2011; Simons Summer Workshop on Geometry and Physics, Stony
Brook, July-August 2011; Classical and Quantum Integrable Systems, Dubna,
January 2012; Progress in Quantum Field Theory and String Theory, Osaka,
April 2012; Workshop on Combinatorics of Moduli Spaces and Cluster Algebras,
Moscow, May-June 201
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