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

Among the applications, studying the q → 0 {\displaystyle q\to 0} limit led to the discovery of crystal bases.

The case of s l 2 {\displaystyle {\mathfrak {sl}}_{2}}

Michio Jimbo considered the algebras with three generators related by the three commutators

[ h , e ] = 2 e ,   [ h , f ] = − 2 f ,   [ e , f ] = sinh ⁡ ( η h ) / sinh ⁡ η . {\displaystyle [h,e]=2e,\ [h,f]=-2f,\ [e,f]=\sinh(\eta h)/\sinh \eta .}

When η → 0 {\displaystyle \eta \to 0} , these reduce to the commutators that define the special linear Lie algebra s l 2 {\displaystyle {\mathfrak {sl}}_{2}} . In contrast, for nonzero η {\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 s l 2 {\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 q {\displaystyle q} -difference analogue of U ( g ) {\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.


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