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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 , 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 [edit]

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[edit]

Notes[edit]

  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 -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

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