Downup 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 infinitedimensional associative algebras called downup algebras. We show that downup algebras exhibit many of the important features of the universal enveloping algebra $U(\fsl)$ of the Lie algebra $\fsl$ including a PoincaréBirkhoffWitt type basis and a wellbehaved representation theory. We investigate the structure and representations of downup 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:

arXiv Mathematics eprints
 Pub Date:
 March 1998
 DOI:
 10.48550/arXiv.math/9803159
 arXiv:
 arXiv:math/9803159
 Bibcode:
 1998math......3159B
 Keywords:

 Mathematics  Representation Theory