Quantum enveloping algebra

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 ${\displaystyle {\mathfrak {sl}}_{2}}$. 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 ${\displaystyle {\mathfrak {sl}}_{2}}$.[2]

• Drinfel'd, V. G. (1987), "Quantum Groups", Proceedings of the International Congress of Mathematicians 986, American Mathematical Society, 1: 798–820

• Quantized enveloping algebra at the nLab
• Quantized enveloping algebras at ${\displaystyle q=1}$ at MathOverflow
• Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra ${\displaystyle U_{q}(g)}$? at MathOverflow