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

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

Notes

1. Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, vol. 155, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94370-1, MR 1321145
2. Tjin 1992, § 5.
3. Jimbo, Michio (1985), "A ${\displaystyle q}$-difference analogue of ${\displaystyle U({\mathfrak {g}})}$ 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.

External links

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