Category O over a deformation of the symplectic oscillator algebra

Details Category O over a deformation of the symplectic oscillator algebra Category O over a deformation of the symplectic oscillator algebra

2003-09-15

arxiv.org/abs/math/0309251v3

Journal of Pure and Applied Algebra 195 (2005), no.2, 131-166

Mathematics

Representation Theory

42 pages, LaTeX, 11pt; Typos removed, references added, presentation improved, minor corrections and additions, Section 16 mod

Scientific paper

We discuss the representation theory of $H_f$, which is a deformation of the symplectic oscillator algebra $sp(2n) \ltimes h_n$, 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.

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