In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra. Given a Lie algebra g {\displaystyle {\mathfrak {g}}}, the quantum enveloping algebra is typically denoted as U q ( g ) {\displaystyle U_{q}({\mathfrak {g}})}. The notation was introduced by Drinfeld and independently by Jimbo.

Among the applications, studying the q → 0 {\displaystyle q\to 0} limit led to the discovery of crystal bases.

The case of s l 2 {\displaystyle {\mathfrak {sl}}_{2}}

Michio Jimbo considered the algebras with three generators related by the three commutators

[ h , e ] = 2 e , [ h , f ] = − 2 f , [ e , f ] = sinh ⁡ ( η h ) / sinh ⁡ η . {\displaystyle [h,e]=2e,\ [h,f]=-2f,\ [e,f]=\sinh(\eta h)/\sinh \eta .}

When η → 0 {\displaystyle \eta \to 0}, these reduce to the commutators that define the special linear Lie algebra s l 2 {\displaystyle {\mathfrak {sl}}_{2}}. In contrast, for nonzero η {\displaystyle \eta }, the algebra defined by these relations is not a Lie algebra but instead an associative algebra that can be regarded as a deformation of the universal enveloping algebra of s l 2 {\displaystyle {\mathfrak {sl}}_{2}}.

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