Quantized enveloping algebra

In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra.^[1] Given a Lie algebra ${\displaystyle {\mathfrak {g}}}$, the quantum enveloping algebra is typically denoted as ${\displaystyle U_{q}({\mathfrak {g}})}$. The notation was introduced by Drinfeld and independently by Jimbo.^[2]

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

The case of s l 2 {\displaystyle {\mathfrak {sl}}_{2}}

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 ${\displaystyle {\mathfrak {sl}}_{2}}$.^[3]

See also

• Quantum group