Quantized enveloping algebra

Among the applications, studying the ${\displaystyle q\to 0}$ 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

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

When ${\displaystyle \eta \to 0}$, these reduce to the commutators that define the special linear Lie algebra ${\displaystyle {\mathfrak{s}\mathfrak{l}}_2}$. In contrast, for nonzero ${\displaystyle \eta }$, 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 ${\displaystyle {\mathfrak{s}\mathfrak{l}}_2}$.