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Down–Up Algebras

✍ Scribed by Georgia Benkart; Tom Roby

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
403 KB
Volume
209
Category
Article
ISSN
0021-8693

No coin nor oath required. For personal study only.

✦ Synopsis

The algebra generated by the down and up operators on a differential or Ž . uniform 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 Ž . of the important features of the universal enveloping algebra U ᒐ l of the Lie 2 algebra ᒐ ᒉ including a Poincare᎐BirkhoffᎏWitt type basis and a well-behaved 2 representation theory. We investigate the structure and representations of down᎐up algebras and focus especially on Verma modules, highest weight representations, and category O 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.

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