In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra.[1] Given a Lie algebra, the quantum enveloping algebra is typically denoted as . The notation was introduced by Drinfeld and independently by Jimbo.[2]
Among the applications, studying the limit led to the discovery of crystal bases.
Michio Jimbo considered the algebras with three generators related by the three commutators
When , these reduce to the commutators that define the special linear Lie algebra. In contrast, for nonzero , 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 .[3]