Down-up Algebras
Abstract
The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down-up algebras. We show that down-up algebras exhibit many of the important features of the universal enveloping algebra $U(\fsl)$ of the Lie algebra $\fsl$ including a Poincaré-Birkhoff-Witt type basis and a well-behaved representation theory. We investigate the structure and representations of down-up algebras and focus especially on Verma modules, highest weight representations, and category $\mathcal O$ modules for them. We calculate the exact expressions for all the weights, since that information has proven to be particularly useful in determining structural results about posets.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- March 1998
- DOI:
- 10.48550/arXiv.math/9803159
- arXiv:
- arXiv:math/9803159
- Bibcode:
- 1998math......3159B
- Keywords:
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- Mathematics - Representation Theory