Down–Up Algebras

Abstract

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. ᮊ

PBW bases, non-degeneracy conditions and applications

“…We avoid the name PBW-algebras, since in general a factor-algebra of an algebra with a PBW basis does not have PBW basis itself. Example 3.6 (Examples of G-algebras) Quasi-commutative polynomial rings (for example, the quantum plane yx = q ·xy), universal enveloping algebras of finite dimensional Lie algebras, some iterated Ore extensions, some nonstandard quantum deformations ( [15], [18]), Weyl algebras and most of various flavors of quantizations of Weyl algebras, Witten's deformation of U (sl 2 ), Smith algebras, conformal sl 2 -algebras ( [5]), some of diffusion algebras ( [17]) and many more.…”

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“…Down-up algebras were defined by Benkart and Roby ( [2]) for combinatorial reasons. They are associative algebras A = A(α, β, γ) generated by u, d over C and satisfy the relations d 2 u = αdud + βud 2 + γd, (1.1)…”

“…where α, β, γ ∈ C. These have been studied by various authors, for example, in [2,3,5,6,7]. We consider the case when β = 0, so that the algebra A is Noetherian ( [6]).…”

“…We consider the case when β = 0, so that the algebra A is Noetherian ( [6]). These algebras have a rich representation theory; see [2,7]. We recall some basic structural properties.…”

“…Also, it is easy to see that C[ξ, η] embeds into A. We recall two other basic facts about this family of algebras: A has a PBW-basis {u i (du) j d k } ( [2,7]), and A is a domain if and only if β = 0. ( [6,7]).…”