Category O over a deformation of the symplectic oscillator algebra

Abstract

We discuss the representation theory of $H_f$, which is a deformation of the symplectic oscillator algebra $sp(2n)⋉h_n$, where $h_n$ is the ((2n+1)-dimensional) Heisenberg algebra. We first look at a more general setup, involving an algebra with a triangular decomposition. Assuming the PBW theorem, and one other hypothesis, we show that the BGG category $\mathcal{O}$ is abelian, finite length, and self-dual. We decompose $\mathcal{O}$ as a direct sum of blocks $\calo(\la)$, and show that each block is a highest weight category. In the second part, we focus on the case $H_f$ for $n=1$, where we prove all these assumptions, as well as the PBW theorem.

42 pages, LaTeX, 11pt; Typos removed, references added, presentation improved, minor corrections and additions, Section 16 modified, and Standing Assumption added in Section 17; Final form, to appear in the Journal of Pure and Applied Algebra

References in corpus (2)

• On the category O for rational Cherednik algebras
• Quantized symplectic oscillator algebras of rank one

Cited by in corpus (15)

• Quantized symplectic oscillator algebras of rank one
• Faces of polytopes and Koszul algebras
• Drinfeld-Hecke algebras over cocommutative algebras
• Center and representations of infinitesimal Hecke algebras of sl_2
• Axiomatic framework for the BGG category O
• Generalized nil-Coxeter algebras, cocommutative algebras, and the PBW property
• Functoriality of the BGG Category O
• Completions of Infinitesimal Hecke Algebras of sl2
• Center of Infinitesimal Hecke Algebra of $s{l}_2$
• Reflection functors and representations for continuous wreath-product symplectic reflection algebras
• On infinitesimal Cherednik algebras of gl_2
• Category O over skew group rings
• Continuous Hecke algebras
• The category of weight modules for symplectic oscillator Lie algebras
• Infinitesimal Hecke algebra of sl_2 in positive characteristic