Quantized enveloping algebra

In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra. Given an algebra ${\mathfrak {g}}$ , the quantum enveloping algebra is typically denoted as $U_{q}({\mathfrak {g}})$ . 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 ${\mathfrak {sl}}_{2}$ .

References

Drinfel'd, V. G. (1987), "Quantum Groups", Proceedings of the International Congress of Mathematicians 986, 1, American Mathematical Society: 798–820
Jimbo, Michio (1985), "A $q$ -difference analogue of $U({\mathfrak {g}})$ and the Yang–Baxter equation", Letters in Mathematical Physics, 10 (1): 63–69, doi:10.1007/BF00704588
Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, vol. 155, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94370-1, MR 1321145