Journal of Lie Theory 18 (2008), No. 1, 215--241
Copyright Heldermann Verlag 2008
Borel Subalgebras of Root-Reductive Lie Algebras
Elizabeth Dan-Cohen
Dept. of Mathematics, University of California, Berkeley, CA 94720, U.S.A.
edc@math.berkeley.edu
[Abstract-pdf]
\def\g{{\frak g}} \def\l{{\frak l}} \def\o{{\frak o}} \def\s{{\frak s}} This paper generalizes the classification of I. Dimitrov and I. Penkov [{\it Borel subalgebras of $\g\l(\infty)$}, Resenhas 6 (2004) 153--163] of Borel subalgebras of $\g\l_\infty$. 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. \par 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 $\s\l_\infty$ and $\s\p_\infty$; in the case of $\s\o_\infty$ 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.
Keywords: Locally finite Lie algebra, root-reductive Lie algebra, Borel subalgebra, maximal locally solvable subalgebra.
MSC: 17B65, 17B30, 17B05
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