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 , the quantum enveloping algebra is typically denoted as . The notation was introduced by Drinfeld and independently by Jimbo.[2]
Among the applications, studying the limit led to the discovery of crystal bases.
The case of
Michio Jimbo considered the algebras with three generators related by the three commutators
When , these reduce to the commutators that define the special linear Lie algebra . In contrast, for nonzero , 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 .[3]
See also
- Quantum group
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 -difference analogue of 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 at MathOverflow
- Does there exist any "quantum Lie algebra" imbedded into the quantum enveloping algebra ? at MathOverflow
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