The maximal solvable subgroups of the SU(p,q) groups and all subgroups of SU(2,1)

A general method is proposed for obtaining all conjugacy classes of maximal solvable subalgebras of an arbitrary semisimple Lie algebra over a zero characteristic field F. The method is applied to explicitly construct all q+1 maximal solvable subalgebras Sκ of the algebra of SU(p,q) for p≥q>0 (over the field of real numbers). The dimension of Sκ for 0≤κ≤q is (2κ+1)× (p+q−κ)−1 and it contains p+q−κ−1 compact elements. The low-dimensional pseudounitary groups with 0≤p+q≤4, 0≤q≤p are considered in detail and different realizations of the maximal solvable subalgebras are presented. Finally, all subalgebras of the physically interesting algebra SU(2,1) are found (not only the maximal solvable ones). The invariants of the subalgebras are found in all cases when they exist.