Krichever–Novikov algebras on Riemann surfaces of genus zero and one with N punctures (original) (raw)
Related papers
Krichever-Novikov algebras for more than two points
Letters in Mathematical Physics, 1990
Krlchever Novikov algebras of meromorphlc vector fields with more than two poles on higher genus Riemann surfaces are introduced. The structure of these algebras and their induced modules of forms of weight 2 is studied.
HIGHER GENUS AFFINE LIE ALGEBRAS OF KRICHEVER – NOVIKOV TYPE
Difference Equations, Special Functions and Orthogonal Polynomials - Proceedings of the International Conference, 2007
Classical affine Lie algebras appear e.g. as symmetries of infinite dimensional integrable systems and are related to certain differential equations. They are central extensions of current algebras associated to finite-dimensional Lie algebras g. In geometric terms these current algebras might be described as Lie algebra valued meromorphic functions on the Riemann sphere with two possible poles. They carry a natural grading. In this talk the generalization to higher genus compact Riemann surfaces and more poles is reviewed. In case that the Lie algebra g is reductive (e.g. g is simple, semi-simple, abelian, ...) a complete classification of (almost-) graded central extensions is given. In particular, for g simple there exists a unique non-trivial (almost-)graded extension class. The considered algebras are related to difference equations, special functions and play a role in Conformal Field Theory.
From the Virasoro Algebra to Krichever–Novikov Type Algebras and Beyond
Harmonic and Complex Analysis and its Applications, 2013
Starting from the Virasoro algebra and its relatives the generalization to higher genus compact Riemann surfaces was initiated by Krichever and Novikov. The elements of these algebras are meromorphic objects which are holomorphic outside a finite set of points. A crucial and non-trivial point is to establish an almost-grading replacing the honest grading in the Virasoro case. Such an almost-grading is given by splitting the set of points of possible poles into two non-empty disjoint subsets. Krichever and Novikov considered the twopoint case. Schlichenmaier studied the most general multi-point situation with arbitrary splittings. Here we will review the path of developments from the Virasoro algebra to its higher genus and multi-point analogs. The starting point will be a Poisson algebra structure on the space of meromorphic forms of all weights. As sub-structures the vector field algebras, function algebras, Lie superalgebras and the related current algebras show up. All these algebras will be almost-graded. In detail almost-graded central extensions are classified. In particular, for the vector field algebra it is essentially unique. The defining cocycle are given in geometric terms. Some applications, including the semi-infinite wedge form representations are recalled. Finally, some remarks on the by Krichever and Sheinman recently introduced Lax operator algebras are made.
A Krichever-Novikov formulation of classical W algebras on Riemann surfaces
Physics Letters B, 1994
It is shown how the theory of classical W -algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. An intriguing relation between the theory of A 1 embeddings into simple Lie algebras and the holomorphic geometry of Riemann surfaces is exihibited.
Krichever{Novikov Type Algebras. An Introduction
2016
Krichever-Novikov type algebras are generalizations of the Witt, Virasoro, affine Lie algebras, and their relatives to Riemann surfaces of ar- bitrary genus. We give the most important results about their structure, almost-grading and central extensions. This contribution is based on a se- quence of introductory lectures delivered by the author at the Southeast Lie Theory Workshop 2012 in Charleston, U.S.A.
Krichever-Novikov Type Algebras — Personal Recollections of Julius Wess
International Journal of Modern Physics: Conference Series, 2012
The author's research directions on Krichever-Novikov algebras was inspired by Julius Wess. After some recollections of Julius Wess, mainly from his Karlsruhe time, an overview over these (multi-point) algebras is given. They generalize the Witt, Virasoro, and related algebras to higher genus Riemann surfaces. Differential operator algebras, current algebras and their central extensions are discussed. For the two point situation some of these algebras were introduced by Krichever and Novikov in 1986. The multi-point generalization and the general setting is due to the author. References for further directions and applications are given.
DIFFERENTIAL OPERATOR ALGEBRAS ON COMPACT RIEMANN SURFACES1
1994
This talk reviews results on the structure of algebras consisting of meromor- phic 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 offorms 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.
Differential Operator Algebras on compact Riemann Surfaces
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.
Knizhnik-Zamolodchikov equations for positive genus and Krichever-Novikov algebras
Russian Mathematical Surveys, 2004
We give a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus with marked points. Globality means here that we use Krichever-Novikov algebras of gauge and conformal symmetries (i.e. algebras of global symmetries) instead of loop and Virasoro algebras (which are local in this context). The basic elements of this global approach are described in a previous article of the authors (Russ. Math. Surv., (54)(1)). In the present article we construct the conformal blocks and the projectively flat connection on the bundle constituted by them.
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 λ−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. 1 invited talk at the International Symposium on Generalized Symmetries in Physics at the Arnold-Sommerfeld-Institute, Clausthal, Germany, July 26 – July 29, 1993