The Universal Enveloping Algebra of the Schrödinger Algebra and its Prime Spectrum

Abstract

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’.

The prime spectrum of the universal enveloping algebra of the 1-spatial ageing algebra and of U(gl2)

“…All the containments between primes are explicitly described, i.e., the Zariski-Jacobson topology on the spectra are described. In [6], the prime, completely prime, maximal and primitive ideals of the algebra U (s) are classified. Generating sets are given for all prime ideals of U (s) apart from an explicit set {I ′ n | n ∈ N + } (see [6,Theorem 3.3]).…”

“…In [6], the prime, completely prime, maximal and primitive ideals of the algebra U (s) are classified. Generating sets are given for all prime ideals of U (s) apart from an explicit set {I ′ n | n ∈ N + } (see [6,Theorem 3.3]). Primitive ideals of U (s) with nonzero-central charge were described by Dubsky, Lü, Mazorchuk and Zhao [13] in the following way: Each such ideal is the annihilator of a simple highest weight U (s)-module with nonzero central charge.…”

“…It was proved in [6] that the algebra A Z (the localization of A at the powers of the central element Z) is a tensor product of three algebras…”

Irreducible weight modules over the Schr{ö}dinger Lie algebra in $(n+1)$ dimensional space-time

“…In particular, all simple weight modules over s 1 with finite dimensional weight spaces were classified in [6], see also [10]. All simple weight modules with infinite dimensional weight spaces over s 1 were classified in [2,3]. The BGG category O of s 1 was studied in [7].…”

BGG Category for the Quantum Schrödinger Algebra

“…All simple weight modules with finite dimensional weight spaces were classified in [12], see also [17]. All simple weight modules of the Schrödinger algebra were classified in [5,6]. The BGG category O of s was studied in [13].…”