Quantized enveloping algebra

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

Among the applications, studying the $$q \to 0$$ limit led to the discovery of crystal bases.

The case of $$\mathfrak{sl}_2$$
Michio Jimbo considered the algebras with three generators related by the three commutators
 * $$[h,e] = 2e,\ [h,f] = -2f,\ [e,f] = \sinh(\eta h)/\sinh \eta.$$

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