  This paper generalizes the classification in a paper of Dimitrov and Penkov
of Borel subalgebras of gl_infty. Root-reductive Lie algebras are direct limits
of finite-dimensional reductive Lie algebras along inclusions preserving the
root spaces with respect to nested Cartan subalgebras. A Borel subalgebra of a
root-reductive Lie algebra is by definition a maximal locally solvable
subalgebra. The main general result of this paper is that a Borel subalgebra of
an infinite-dimensional indecomposable root-reductive Lie algebra is the
simultaneous stabilizer of a certain type of generalized flag in each of the
standard representations. For the three infinite-dimensional simple
root-reductive Lie algebras more precise results are obtained. The map sending
a maximal closed (isotropic) generalized flag in the standard representation to
its stabilizer hits Borel subalgebras, yielding a bijection in the cases of
sl_infty and sp_infty; in the case of so_infty the fibers are of size one and
two. A description is given of a nice class of toral subalgebras contained in
any Borel subalgebra. Finally, certain Borel subalgebras of a general
root-reductive Lie algebra are seen to correspond bijectively with Borel
subalgebras of the commutator subalgebra, which are understood in terms of the
special cases.

Representation Theory