BGG CATEGORY FOR THE QUANTUM SCHRÖDINGER ALGEBRA

In 1996, a q-deformation of the universal enveloping algebra of the Schrödinger Lie algebra was introduced in Dobrev et al. [J. Phys. A 29 (1996) 5909–5918.]. This algebra is called the quantum Schrödinger algebra. In this paper, we study the Bernstein-Gelfand-Gelfand (BGG) category $\mathcal{O}$ for the quantum Schrödinger algebra $U_q(\mathfrak{s})$ , where q is a nonzero complex number which is not a root of unity. If the central charge $\dot z\neq 0$ , using the module $B_{\dot z}$ over the quantum Weyl algebra $H_q$ , we show that there is an equivalence between the full subcategory $\mathcal{O}[\dot Z]$ consisting of modules with the central charge $\dot z$ and the BGG category $\mathcal{O}^{(\mathfrak{sl}_2)}$ for the quantum group $U_q(\mathfrak{sl}_2)$ . In the case that $\dot z = 0$ , we study the subcategory $\mathcal{A}$ consisting of finite dimensional $U_q(\mathfrak{s})$ -modules of type 1 with zero action of Z. We directly construct an equivalence functor from $\mathcal{A}$ to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional $U_q(\mathfrak{s})$ -modules is wild.

Homology and Cohomology of the Super Schrödinger Algebra S(1/1) with Coefficients in the Trivial Module

In this paper we study the homology and cohomology groups of the super Schrödinger algebra [Formula: see text] in (1 + 1)-dimensional spacetime. We explicitly compute the homology groups of [Formula: see text] with coefficients in the trivial module. Then using duality, we finally obtain the dimensions of the cohomology groups of [Formula: see text] with coefficients in the trivial module.

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

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’.

Higher-Spin Symmetries and Deformed Schrödinger Algebra in Conformal Mechanics

The dynamical symmetries of 1+1-dimensional Matrix Partial Differential Equations with a Calogero potential (with/without the presence of an extra oscillatorial de Alfaro-Fubini-Furlan, DFF, term) are investigated. The first-order invariant differential operators induce several invariant algebras and superalgebras. Besides the sl(2)⊕u(1) invariance of the Calogero Conformal Mechanics, an osp2∣2 invariant superalgebra, realized by first-order and second-order differential operators, is obtained. The invariant algebras with an infinite tower of generators are given by the universal enveloping algebra of the deformed Heisenberg algebra, which is shown to be equivalent to a deformed version of the Schrödinger algebra. This vector space also gives rise to a higher-spin (gravity) superalgebra. We furthermore prove that the pure and DFF Matrix Calogero PDEs possess isomorphic dynamical symmetries, being related by a similarity transformation and a redefinition of the time variable.

Classification of Simple Weight Modules over the Schrödinger Algebra

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