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 $U_{q}({\mathfrak {g}})$. 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 ${\mathfrak {sl}}_{2}$.

References

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

External links

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