Realization of compact Lie algebras in K\"ahler manifolds

TL;DR

This paper explores how compact Lie algebras can be explicitly realized within Kähler manifolds, linking geometric structures with quantum and classical symmetries through Berezin quantization and differential operators.

Contribution

It provides explicit constructions of Kähler potentials, cocycle functions, and Killing vector fields for manifolds associated with all compact semi-simple Lie groups, advancing geometric quantization methods.

Findings

Explicit Kähler potentials for all compact semi-simple Lie groups.

Construction of Lie algebra representations via differential operators.

Derivation of cocycle functions and Killing vector fields.

Abstract

The Berezin quantization on a simply connected homogeneous K\"{a}hler manifold, which is considered as a phase space for a dynamical system, enables a description of the quantal system in a (finite-dimensional) Hilbert space of holomorphic functions corresponding to generalized coherent states. The Lie algebra associated with the manifold symmetry group is given in terms of first-order differential operators. In the classical theory, the Lie algebra is represented by the momentum maps which are functions on the manifold, and the Lie product is the Poisson bracket given by the K\"{a}hler structure. The K\"{a}hler potentials are constructed for the manifolds related to all compact semi-simple Lie groups. The complex coordinates are introduced by means of the Borel method. The K\"{a}hler structure is obtained explicitly for any unitary group representation. The cocycle functions for the Lie algebra and the Killing vector fields on the manifold are also obtained.