Higher-Dimensional generalizations of Affine Kac-Moody and Virasoro Lie Algebras

We discuss the higher dimensional generalizations of the Virasoro and Affine Kac-Moody Lie algebras. We present an explicit construction for a central extensions of the Lie Algebra $Map (X, \g)$ where $\g$ is a finite-dimensional Lie algebra and $X$ is a complex manifold that can be described as a "right" higher-dimensional generalization of $C^*$ from the point of view of a corresponding group action. The constructed algebras have most of the good properties of finite dimensional semi-simple Lie algebras and are a new class of generalized Kac-Moody algebras. These algebras have description in terms of higher dimensional local fields.