A class of algebras similar to the enveloping algebra of 𝑠𝑙(2)

Abstract

Fix f ∈ C [ X ] f \in {\mathbf {C}}[X] . Define R = C [ A , B , H ] R = {\mathbf {C}}[A,B,H] subject to the relations \[ H A − A H = A , H B − B H = − B , A B − B A = f ( H ) HA - AH = A,\quad HB - BH = - B,\quad AB - BA = f(H) \] . We study these algebras (for different f f ) and in particular show how they are similar to (and different from) U ( sl ( 2 ) ) U({\text {sl}}(2)) , the enveloping algebra of sl ( 2 , C ) {\text {sl}}(2,{\mathbf {C}}) . There is a notion of highest weight modules and a category O \mathcal {O} for such R R . For each n > 0 n > 0 , if f ( x ) = ( x + 1 ) n + 1 − x n + 1 f(x) = {(x + 1)^{n + 1}} - {x^{n + 1}} , then R R has precisely n n simple modules in each finite dimension, and every finite-dimensional R R -module is semisimple.