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

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 ${\mathfrak {sl}}_{2}$.

See also

• quantum group

References

• Drinfel'd, V. G. (1987), "Quantum Groups", Proceedings of the International Congress of Mathematicians 986, 1, American Mathematical Society: 798–820
• Tjin, T. (10 October 1992). "An introduction to quantized Lie groups and algebras". International Journal of Modern Physics A. 07 (25): 6175–6213. arXiv:hep-th/9111043. Bibcode:1992IJMPA...7.6175T. doi:10.1142/S0217751X92002805. ISSN 0217-751X. S2CID 119087306.

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