Left invariant complex structures on U(2) and SU(2)xSU(2) revisited

TL;DR

This paper classifies and analyzes left-invariant complex structures on certain Lie groups, providing new insights into their integrability, automorphism classes, and manifold structures, with extensions and representations explored.

Contribution

It offers a new determination of integrable complex structures on u(2) and su(2)xsu(2), and shows these form differentiable manifolds, including extensions and representation considerations.

Findings

Set of complex structures forms a differentiable manifold

Extensions of complex structures are explicitly constructed

Representation analysis of complex structures is performed

Abstract

We compute the torsion-free linear maps from the Lie algebra su(2) into itself, deduce a new determination of the integrable complex structures and their equivalence classes under the action of the automorphism group for u(2) and su(2)xsu(2), and prove that in both cases the set of complex structures is a differentiable manifold. u(2)x u(2), su(2)^N and u(2)^N are also considered. Extensions of complex structures from u(2) to su(2)xsu(2) are studied, local holomorphic charts given, and attention is paid to what representations of u(2) we can get from a substitute to the regular representation on a space of holomorphic functions for the complex structure.