Maximal abelian subalgebras of real and complex symplectic Lie algebras

Abstract

We provide guidelines for classifying maximal abelian subalgebras (MASA’s) of the symplectic Lie algebras sp(2n, R) and sp(2n, C) into conjugacy classes under the Lie groups Sp(2n, R) and Sp(2n, C), respectively. The task of classifying all MASA’s is reduced to the classification of orthogonally indecomposable (OID) MASA’s. Two types of orthogonally indecomposable MASA’s of sp(2n, C) exist: 1. Indecomposable maximal abelian nilpotent subalgebras (MANS’s). 2. Decomposable MASA’s [their classification reduces to a classification of MANS’s of sl(n, C)]. Four types of orthogonally indecomposable MASA’s of sp(2n, R) exist: 1. Absolutely indecomposable MASA’s (MANS’s). 2. Relatively indecomposable MASA’s [their classification reduces to a classification of MANS’s of su( p, q) for p+q=n]. 3. Decomposable absolutely OID MASA’s [involving MANS’s of sl(n,R)]. 4. Decomposable relatively OID MASA’s [involving MANS’s of sl(n/2, C), for n even]. Low-dimensional cases of sp(2n, F) (n=1, 2, 3, F=R or C) are treated exhaustively. The algebras sp(2, R), sp(4, R), and sp(6, R) have 3, 10, and 30 classes of MASA’s, respectively; sp(2, C), sp(4, C), and sp(6, C) have 2, 5, and 14 classes of MASA’s, respectively. For n≥4 infinitely many classes of MASA’s exist.