597 research outputs found
Global geometric deformations of current algebras as Krichever-Novikov type algebras
We construct algebraic-geometric families of genus one (i.e. elliptic)
current and affine Lie algebras of Krichever-Novikov type. These families
deform the classical current, respectively affine Kac-Moody Lie algebras. The
construction is induced by the geometric process of degenerating the elliptic
curve to singular cubics. If the finite-dimensional Lie algebra defining the
infinite dimensional current algebra is simple then, even if restricted to
local families, the constructed families are non-equivalent to the trivial
family. In particular, we show that the current algebra is geometrically not
rigid, despite its formal rigidity. This shows that in the infinite-dimensional
Lie algebra case the relations between geometric deformations, formal
deformations and Lie algebra two-cohomology are not that close as in the
finite-dimensional case. The constructed families are e.g. of relevance in the
global operator approach to the Wess-Zumino-Witten-Novikov models appearing in
the quantization of Conformal Field Theory.Comment: 35 pages, AMS-Late
Sugawara Construction for Higher Genus Riemann Surfaces
By the classical genus zero Sugawara construction one obtains from admissible
representations of affine Lie algebras (Kac-Moody algebras of affine type)
representations of the Virasoro algebra. In this lecture first the classical
construction is recalled. Then, after giving a review on the global multi-point
algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary
genus, the higher genus Sugawara construction is introduced. Finally, the
lecture reports on results obtained in joint work with O.K. Sheinman. We were
able to show that also in the higher genus, multi-point situation one obtains
from representations of the global algebras of affine type representations of a
centrally extended algebra of meromorphic vector fields on Riemann surfaces.
The latter algebra is the generalization of the Virasoro algebra to higher
genus.
Invited lecture at the XVI workshop on geometric methods in physics,
Bialowieza, Poland, June 30 -- July 6, 1997.Comment: 19 pages, latex
Global Geometric Deformations of the Virasoro algebra, current and affine algebras by Krichever-Novikov type algebra
In two earlier articles we constructed algebraic-geometric families of genus
one (i.e. elliptic) Lie algebras of Krichever-Novikov type. The considered
algebras are vector fields, current and affine Lie algebras. These families
deform the Witt algebra, the Virasoro algebra, the classical current, and the
affine Kac-Moody Lie algebras respectively. The constructed families are not
equivalent (not even locally) to the trivial families, despite the fact that
the classical algebras are formally rigid. This effect is due to the fact that
the algebras are infinite dimensional. In this article the results are reviewed
and developed further. The constructions are induced by the geometric process
of degenerating the elliptic curves to singular cubics. The algebras are of
relevance in the global operator approach to the Wess-Zumino-Witten-Novikov
models appearing in the quantization of Conformal Field Theory.Comment: 17 page
Elements of a Global Operator Approach to Wess-Zumino-Novikov-Witten Models
Elements of a global operator approach to the WZNW models for compact Riemann
surfaces of arbitrary genus g with N marked points were given by Schlichenmaier
and Sheinman. This contribution reports on the results. The approach is based
on the multi-point Krichever-Novikov algebras of global meromorphic functions
and vector fields, and the global algebras of affine type and their
representations. Using the global Sugawara construction and the identification
of a certain subspace of the vector field algebra with the tangent space to the
moduli space of the geometric data, Knizhnik-Zamalodchikov equations are
defined. Some steps of the approach of Tsuchia, Ueno and Yamada to WZNW models
are presented to compare it with our approach.Comment: 17 pages, Amslatex, Invited talk presented at the 3rd International
Workshop on "Lie Theory and Its Applications in Physics - Lie III", 11 - 14
July 1999, Clausthal, German
Some Concepts of Modern Algebraic Geometry: Point, Ideal and Homomorphism
Starting from classical algebraic geometry over the complex numbers (as it
can be found for example in Griffiths and Harris it was the goal of these
lectures to introduce some concepts of the modern point of view in algebraic
geometry. Of course, it was quite impossible even to give an introduction to
the whole subject in such a limited time. For this reason the lectures and now
the write-up concentrate on the substitution of the concept of classical points
by the notion of ideals and homomorphisms of algebras.Comment: 36 pages. This is a write-up of lectures given at the ``Kleine
Herbstschule 93'' of the Graduiertenkolleg ``Mathematik im Bereich Ihrer
Wechselwirkungen mit der Physik'' at the Ludwig-Maximilians-Universitaet
Muenche
Higher genus affine algebras of Krichever - Novikov type
For higher genus multi-point current algebras of Krichever-Novikov type
associated to a finite-dimensional Lie algebra, local Lie algebra two-cocycles
are studied. They yield as central extensions almost-graded higher genus affine
Lie algebras. In case that the Lie algebra is reductive a complete
classification is given. For a simple Lie algebra, like in the classical
situation, there is up to equivalence and rescaling only one non-trivial
almost-graded central extension. The classification is extended to the algebras
of meromorphic differential operators of order less or equal one on the
currents algebra.Comment: 35 page
Berezin-Toeplitz Quantization of compact Kaehler manifolds
Invited lecture at the XIV-th workshop on geometric methods in physics,
Bialowieza, Poland, July 9-15, 1995. In this lecture results are reviewed
obtained by the author together with Martin Bordemann and Eckhard Meinrenken on
the Berezin-Toeplitz quantization of compact Kaehler manifolds. Using global
Toeplitz operators, approximation results for the quantum operators are shown.
From them it follows that the quantum operators have the correct classical
limit. A star product deformation of the Poisson algebra is constructed.Comment: Amstex 2.1, 15 pages, minor changes, some annoying typos removed and
2 references adde
Differential Operator Algebras on compact Riemann Surfaces
Invited talk at the International Symposium on Generalized Symmetries in
Physics at the Arnold-Sommerfeld-Institute, Clausthal, Germany, July 26 -- July
29, 1993. This talk reviews results on the structure of algebras consisting of
meromorphic differential operators which are holomorphic outside a finite set
of points on compact Riemann surfaces. For each partition into two disjoint
subsets of the set of points where poles are allowed, a grading of the algebra
and of the modules of lambda - forms is introduced. With respect to this
grading the Lie structure of the algebra and of the modules are almost graded
ones. Central extensions and semi-infinite wedge representations are studied.
If one considers only differential operators of degree 1 then these algebras
are generalizations of the Virasoro algebra in genus zero, resp. of Krichever
Novikov algebras in higher genus.Comment: 11 pages, AmsTeX 2.1 and psbox macro
Local cocycles and central extensions for multi-point algebras of Krichever-Novikov type
Multi-point algebras of Krichever Novikov type for higher genus Riemann
surfaces are generalisations of the Virasoro algebra and its related algebras.
Complete existence and uniqueness results for local 2-cocycles defining
almost-graded central extensions of the functions algebra, the vector field
algebra, and the differential operator algebra (of degree \le 1) are shown.
This is applied to the higher genus, multi-point affine algebras to obtain
uniqueness for almost-graded central extensions of the current algebra of a
simple finite-dimensional Lie algebra. An earlier conjecture of the author
concerning the central extension of the differential operator algebra induced
by the semi-infinite wedge representations is proved.Comment: 38 pages, Amslatex, some minor changes in Section
- …