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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: S. P. Smith
Journal: Trans. Amer. Math. Soc. 322 (1990), 285-314
MSC: Primary 17B35; Secondary 16S30
DOI: https://doi.org/10.1090/S0002-9947-1990-0972706-5
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 \[ HA - AH = A,\quad HB - BH = - B,\quad AB - BA = f(H)\]. 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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Article copyright: © Copyright 1990 American Mathematical Society