×

Universal central extensions of elliptic affine Lie algebras. (English) Zbl 0839.17017

Author’s summary: Let \(\mathfrak g\) be a simple complex (finite dimensional) Lie algebra, and let \(R\) be the ring of regular functions on a compact complex algebraic curve with a finite number of points removed. Lie algebras of the form \({\mathfrak g} \otimes_\mathbb{C} R\) are considered; these generalize Kac-Moody loop algebras since for a curve of genus zero with two punctures \(R \simeq \mathbb{C}[t,t^{-1}]\). The universal central extension of \({\mathfrak g} \otimes R\) is analogous to an untwisted affine Kac-Moody algebra. By Kassel’s theorem the kernel of the universal central extension is linearly isomorphic to the Kähler differentials of \(R\) modulo exact differentials. The dimension of the kernel for any \(R\) is determined first. Restricting to hyperelliptic curves with 2, 3, or 4 special points removed, a basis for the kernel is determined. Restricting further to an elliptic curve with punctures at two points (of orders one and two in the group law) we explicitly determine the cocycles which give the commutation relations for the universal central extension. The results involve Pollaczek polynomials, which are a genus-one generalization of ultraspherical (Gegenbauer) polynomials.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
14H52 Elliptic curves
30F30 Differentials on Riemann surfaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1007/BF00147350 · Zbl 0704.17011 · doi:10.1007/BF00147350
[2] DOI: 10.3792/pjaa.61.179 · Zbl 0581.17010 · doi:10.3792/pjaa.61.179
[3] DOI: 10.1080/00927879208824523 · Zbl 0780.17026 · doi:10.1080/00927879208824523
[4] DOI: 10.2977/prims/1195173183 · Zbl 0693.17009 · doi:10.2977/prims/1195173183
[5] DOI: 10.4153/CMB-1994-004-8 · Zbl 0807.17019 · doi:10.4153/CMB-1994-004-8
[6] DOI: 10.1007/BF01078026 · Zbl 0634.17010 · doi:10.1007/BF01078026
[7] DOI: 10.1016/0022-4049(84)90040-9 · Zbl 0549.17009 · doi:10.1016/0022-4049(84)90040-9
[8] DOI: 10.1007/BF01077962 · Zbl 0715.17023 · doi:10.1007/BF01077962
[9] DOI: 10.1007/BF01075634 · Zbl 0820.17037 · doi:10.1007/BF01075634
[10] DOI: 10.1063/1.528652 · Zbl 0716.17026 · doi:10.1063/1.528652
[11] DOI: 10.1007/BF01045886 · Zbl 0691.30037 · doi:10.1007/BF01045886
[12] DOI: 10.1007/BF00429952 · Zbl 0703.30038 · doi:10.1007/BF00429952
[13] DOI: 10.1007/BF00417227 · Zbl 0703.30039 · doi:10.1007/BF00417227
[14] Schlichenmaier M., Mannheimer Manuskripte 164 (1993)
[15] DOI: 10.1007/BF02684779 · Zbl 0475.17004 · doi:10.1007/BF02684779
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.