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Transactions of the American Mathematical Society

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A class of algebras similar to the enveloping algebra of $ {\rm sl}(2)$

Author: S. P. Smith
Journal: Trans. Amer. Math. Soc. 322 (1990), 285-314
MSC: Primary 17B35; Secondary 16S30
MathSciNet review: 972706
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Abstract: Fix $ f \in {\mathbf{C}}[X]$. Define $ R = {\mathbf{C}}[A,B,H]$ subject to the relations

$\displaystyle HA - AH = A,\quad HB - BH = - B,\quad AB - BA = f(H)$

. We study these algebras (for different $ f$) and in particular show how they are similar to (and different from) $ U({\text{sl}}(2))$, the enveloping algebra of $ {\text{sl}}(2,{\mathbf{C}})$. There is a notion of highest weight modules and a category $ \mathcal{O}$ for such $ R$. For each $ n > 0$, if $ f(x) = {(x + 1)^{n + 1}} - {x^{n + 1}}$, then $ R$ has precisely $ n$ simple modules in each finite dimension, and every finite-dimensional $ R$-module is semisimple.

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