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 ${\mathfrak {g}}$ , the quantum enveloping algebra is typically denoted as $U_{q}({\mathfrak {g}})$ . The notation was introduced by Drinfeld and independently by Jimbo.^[2]

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

Notes

^ Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, vol. 155, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94370-1, MR 1321145
^ Tjin 1992, § 5.
^ Jimbo, Michio (1985), "A $q$ -difference analogue of $U({\mathfrak {g}})$ and the Yang–Baxter equation", Letters in Mathematical Physics, 10 (1): 63–69, Bibcode:1985LMaPh..10...63J, doi:10.1007/BF00704588, S2CID 123313856

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.