The prime, completely prime, maximal, and primitive spectra are classified for the universal enveloping algebra of the Schrödinger algebra. The explicit generators are given for all of these ideals. A counterexample is constructed to the conjecture of Cheng and Zhang about nonexistence of simple singular Whittaker modules for the Schrödinger algebra (and all such modules are classified). It is proved that the conjecture holds ‘generically’.

A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer ${{C}_{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $S$ of the Schrödinger (Lie) algebra. The simple ${{C}_{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $S$ (over the centre). It is proved that some (prime) factor algebras of $S$ and ${{C}_{S}}(H)$ are tensor homological$/$Krull minimal.

Let G be an arbitrary non-zero additive subgroup of the complex number field ℂ, and let Vir[G] be the corresponding generalized Virasoro algebra over ℂ. In this paper we determine all irreducible weight modules with finite-dimensional weight spaces over Vir[G]. The classification strongly depends on the index group G. If G does not have a direct summand isomorphic to ℤ (the integers), then such irreducible modules over Vir[G] are only modules of intermediate series whose weight spaces are all one dimensional. Otherwise, there is one further class of modules that are constructed by using intermediate series modules over a generalized Virasoro subalgebra Vir[G0] of Vir[G] for a direct summand G0 of G with G = G0 ⊕ ℤb, where b ∈ G \ G0. This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.