The category of weight modules for symplectic oscillator Lie algebras

Abstract

The rank $n$ symplectic oscillator Lie algebra ${\mathfrak{g}}_n$ is the semidirect product of the symplectic Lie algebra ${\mathfrak{sp}}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite dimensional weight spaces over ${\mathfrak{g}}_n$. When $\dot{z}\ne 0$, it is shown that there is an equivalence between the full subcategory ${\mathcal{O}}_{{\mathfrak{g}}_n}[\dot{z}]$ of the BGG category ${\mathcal{O}}_{{\mathfrak{g}}_n}$ for ${\mathfrak{g}}_n$ and the BGG category ${\mathcal{O}}_{{\mathfrak{sp}}_{2n}}$ for ${\mathfrak{sp}}_{2n}$. Then using the technique of localization and the structure of generalized highest weight modules, we also give the classification of simple weight modules over ${\mathfrak{g}}_n$ with finite-dimensional weight spaces.

References in corpus (3)

• Remarks on l-conformal extension of the Newton-Hooke algebra
• Classification of simple weight modules over affine Lie algebras
• Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball