Locally finite Lie algebras¶with unitary highest weight representations

Abstract:

In this paper we essentially classify all locally finite Lie algebras with an involution and a compatible root decomposition which permit a faithful unitary highest weight representation. It turns out that these Lie algebras have many interesting relations to geometric structures such as infinite-dimensional bounded symmetric domains and coadjoint orbits of Banach–Lie groups which are strong Kähler manifolds. In the present paper we concentrate on the algebraic structure of these Lie algebras, such as the Levi decomposition, the structure of the almost reductive and locally nilpotent part, and the structure of the representation of the almost reductive algebra on the locally nilpotent ideal.