Quantized enveloping algebra

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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]. The notation was introduced by Drinfeld and independently by Jimbo.[2] 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].[3]

See also

References

  1. 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 
  2. Tjin 1992, § 5.
  3. 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, Bibcode1985LMaPh..10...63J 
  • Drinfel'd, V. G. (1987), "Quantum Groups", Proceedings of the International Congress of Mathematicians 986 (American Mathematical Society) 1: 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. doi:10.1142/S0217751X92002805. ISSN 0217-751X. Bibcode1992IJMPA...7.6175T. 

External links