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.
We give a notation of Krichever-Novikov vertex algebras on compact Riemann surfaces which is a bit weaker, but quite similar to vertex algebras. As example, we construct Krichever-Novikov vertex algebras of generalized Heisenberg algebras on arbitrary compact Riemann surfaces, which are reduced to be Heisenberg vertex algebra when restricted on Riemann spheres.
It is shown that a certain representation of the Heisenberg type Krichever-Novikov algebra gives rise to a state field correspondence that is quite similar to the vertex algebra structure of the usual Heisenberg algebra. Finally a definition of Krichever-Novikov type vertex algebras is proposed and its relation to vertex algebras is discussed.
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 curr…
By considering non-trivial global deformations of the Witt (and the Virasoro) algebra given by geometric constructions it is shown that, despite their infinitesimal and formal rigidity, they are globally not rigid. This shows the need of a clear indication of the type of deformations considered. The families appearing are constructed as families of algebras of Krichever-Novikov type. They show up in a natural way in the global operator approach to the quantization of two-dimensional conformal field theory. In addition, a proof of the infinitesimal and formal rigidity of the Witt algebra is pre…
This paper continues the same-named article, Part I (math.QA/9812083). We give a global operator approach to the WZWN theory for compact Riemann surfaces of an arbitrary genus g 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 elements of this global approach are described in Part I. In the present paper we give the construction of conformal blocks and the projective flat connection on the bundle constituted by them.
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…
We show how the recently again discussed $N$-point Witt, Virasoro, and affine Lie algebras are genus zero examples of the multi-point versions of Krichever--Novikov type algebras as introduced and studied by Schlichenmaier. Using this more general point of view, useful structural insights and an easier access to calculations can be obtained. The concept of almost-grading will yield information about triangular decompositions which are of importance in the theory of representations. As examples the algebra of functions, vector fields, differential operators, current algebras, affine Lie algebra…
Classically important examples of Lie superalgebras have been constructed starting from the Witt and Virasoro algebra. In this article we consider Lie superalgebras of Krichever-Novikov type. These algebras are multi-point and higher genus equivalents. The grading in the classical case is replaced by an almost-grading. The almost-grading is determined by a splitting of the set of points were poles are allowed into two disjoint subsets. With respect to a fixed splitting, or equivalently with respect to an almost-grading, it is shown that there is up to rescaling and equivalence a unique non-tri…
The recent focus on deformations of algebras called quantum algebras can be attributed to the fact that they appear to be the basic algebraic structures underlying an amazingly diverse set of physical situations. To date many interesting features of these algebras have been found and they are now known to belong to a class of algebras called Hopf algebras [1]. The remarkable aspect of these structures is that they can be regarded as deformations of the usual Lie algebras. Of late, there has been a considerable interest in the deformation of the Virasoro algebra and the underlying Heisenberg al…
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…
Elements of a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus $g$ are given. Sheaves of representations of affine Krichever-Novikov algebras over a dense open subset of the moduli space of Riemann surfaces (respectively of smooth, projective complex curves) with $N$ marked points are introduced. It is shown that the tangent space of the moduli space at an arbitrary moduli point is isomorphic to a certain subspace of the Krichever-Novikov vector field algebra given by the data of the moduli point. This subspace is complementary to the direct sum of th…
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 two-point case. Schlichenmaier studied the most general multi-poin…
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 re…
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 …
Lax operator algebras constitute a new class of infinite dimensional Lie algebras of geometric origin. More precisely, they are algebras of matrices whose entries are meromorphic functions on a compact Riemann surface. They generalize classical current algebras and current algebras of Krichever-Novikov type. Lax operators for $\gl(n)$, with the spectral parameter on a Riemann surface, were introduced by Krichever. In joint works of Krichever and Sheinman their algebraic structure was revealed and extended to more general groups. These algebras are almost-graded. In this article their definitio…
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 situa…
In this contributed presentation, we discuss and compare the mutually opposite procedures of deformations and contractions of Lie algebras. We suggest that with appropriate combinations of both procedures one may construct new Lie algebras. We first discuss low-dimensional Lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation, the converse is not true in general. Also we note that some Lie algebras belonging to parameterized families are singled out by the irreversibility of deformations and contractions. After reminding that global deformati…
Recently, Lax operator algebras appeared as a new class of higher genus current type algebras. Based on I.Krichever's theory of Lax operators on algebraic curves they were introduced by I. Krichever and O. Sheinman. These algebras are almost-graded Lie algebras of currents on Riemann surfaces with marked points (in-points, out-points, and Tyurin points). In a previous joint article of the author with Sheinman the local cocycles and associated almost-graded central extensions are classified in the case of one in-point and one out-point. It was shown that the almost-graded extension is essential…
