Global geometric deformations of current algebras as Krichever-Novikov type algebras

Abstract

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.

35 pages, AMS-Latex

References in corpus (3)

• Deformation quantization of algebraic varieties
• The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras, I
• Degenerations of Generalized Krichever Novikov algebras on Tori

Cited by in corpus (7)

• On Rigidity of 3d Asymptotic Symmetry Algebras
• Global Geometric Deformations of the Virasoro algebra, current and affine algebras by Krichever-Novikov type algebra
• Realizations of the three point algebra $\mathfrak{sl}(2,\mathcal{R})\oplus \left({\Omega }_{\mathcal{R}}/d\mathcal{R}\right)$
• Extensions of superalgebras of Krichever-Novikov type
• The 3-point Virasoro algebra and its action on a Fock space
• The Low-Dimensional Algebraic Cohomology of the Virasoro Algebra
• A classification of automorphic Lie algebras on complex tori