Quantized enveloping algebra

In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra. Given a Lie algebra $𝔤$, the quantum enveloping algebra is typically denoted as $U_q(𝔤)$. The notation was introduced by Drinfeld and independently by Jimbo.

Among the applications, studying the $q\to 0$ limit led to the discovery of crystal bases.

The case of ${𝔰𝔩}_2$

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 ${𝔰𝔩}_2$. 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 ${𝔰𝔩}_2$.

See also

• Quantum group

References

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External links

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

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