Quantized enveloping algebra

In mathematics, a quantum or quantized enveloping algebra is a "q"-analog of a universal enveloping algebra. Given a Lie algebra formula_1, the quantum enveloping algebra is typically denoted as formula_2. Among the applications, studying the formula_3 limit led to the discovery of crystal bases.
The case of formula_4.
Michio Jimbo considered the algebras with three generators related by the three commutators
When formula_6, these reduce to the commutators that define the special linear Lie algebra formula_4. In contrast, for nonzero formula_8, 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 formula_4.