A class of algebras similar to the enveloping algebra of $\textrm {sl}(2)$
Author:
S. P. Smith
Journal:
Trans. Amer. Math. Soc. 322 (1990), 285314
MSC:
Primary 17B35; Secondary 16S30
DOI:
https://doi.org/10.1090/S00029947199009727065
MathSciNet review:
972706
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Abstract  References  Similar Articles  Additional Information
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 finitedimensional $R$module is semisimple.

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© Copyright 1990
American Mathematical Society