Quantum enveloping algebra
In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra.[1] Given a Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math], the quantum enveloping algebra is typically denoted as [math]\displaystyle{ U_q(\mathfrak{g}) }[/math]. Among the applications, studying the [math]\displaystyle{ q \to 0 }[/math] limit led to the discovery of crystal bases.
The case of [math]\displaystyle{ \mathfrak{sl}_2 }[/math]
Michio Jimbo considered the algebras with three generators related by the three commutators
- [math]\displaystyle{ [h,e] = 2e,\ [h,f] = -2f,\ [e,f] = \sinh(\eta h)/\sinh \eta. }[/math]
When [math]\displaystyle{ \eta \to 0 }[/math], these reduce to the commutators that define the special linear Lie algebra [math]\displaystyle{ \mathfrak{sl}_2 }[/math]. In contrast, for nonzero [math]\displaystyle{ \eta }[/math], 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 [math]\displaystyle{ \mathfrak{sl}_2 }[/math].[2]
References
- ↑ Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, 155, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94370-1, https://archive.org/details/quantumgroups0000kass
- ↑ Jimbo, Michio (1985), "A [math]\displaystyle{ q }[/math]-difference analogue of [math]\displaystyle{ U(\mathfrak{g}) }[/math] and the Yang–Baxter equation", Letters in Mathematical Physics 10 (1): 63–69, doi:10.1007/BF00704588, Bibcode: 1985LMaPh..10...63J
- Drinfel'd, V. G. (1987), "Quantum Groups", Proceedings of the International Congress of Mathematicians 986 (American Mathematical Society) 1: 798–820
External links
- Quantized enveloping algebra at the nLab
- Quantized enveloping algebras at [math]\displaystyle{ q = 1 }[/math] at MathOverflow
- Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra [math]\displaystyle{ U_q(g) }[/math]? at MathOverflow
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