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Down-up algebras. (English) Zbl 0922.17006
J. Algebra 209, No. 1, 305-344 (1998); addendum ibid. 213, No. 1, 378 (1999).
A down-up algebra $$A(\alpha, \beta, \gamma)$$ is the associative unital algebra (say over $$\mathbf C$$) presented by generators $$d$$ and $$u$$ and defining relations $$d^2u=\alpha dud+\beta ud^2+\gamma d$$, $$du^2=\alpha udu+\beta u^2d+\gamma u$$; where $$\alpha,\beta,\gamma$$ are arbitrary complex numbers. The authors motivation for this definition (and terminology) comes from the down and up operators on $$(q,r)$$-differential posets [see R. P. Stanley, J. Am. Math. Soc. 4, 919-961 (1988; Zbl 0658.05006), S. V. Fomin, Zap. Nauchn. Semin. LOMI Steklov 155, 156-175 (1986; Zbl 0698.05003)]. An extensive list of examples of (homomorphic images of) down-up algebras is given; including $$q$$-Weyl algebras, quantum planes, universal enveloping algebras of certain 3-dimensional Lie (super-)algebras, Witten’s deformations of $$U(sl_2)$$ [E. Witten, Commun. Math. Phys. 137, 29-66 (1991; Zbl 0717.53074)], conformal $$sl_2$$-algebras [L. Le Bruyn, Commun. Algebra 23, 1325-1362 (1995; Zbl 0824.17012)] and certain subalgebras of quantum enveloping algebras.
In the paper under review, the structure and representation theory of down-up algebras is studied. A PBW theorem is proved. “Highest weight” and “lowest weight” modules are explicitly constructed; the simple ones are determined and the exact expressions for all the weights are calculated. Two categories $$\mathcal O$$ and $$\mathcal O'$$, in the spirit of the Bernstein-Gelfand-Gelfand categories for representations of simple Lie algebras, are investigated.

##### MSC:
 17B05 Structure theory for Lie algebras and superalgebras 06A06 Partial orders, general 17B35 Universal enveloping (super)algebras
##### Keywords:
down-up algebras; PBW theorem
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##### References:
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