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

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

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

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

When ${\displaystyle \eta \to 0}${\displaystyle \eta \to 0}, these reduce to the commutators that define the special linear Lie algebra ${\displaystyle {\mathfrak {sl}}_{2}}${\displaystyle {\mathfrak {sl}}_{2}}. In contrast, for nonzero ${\displaystyle \eta }${\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 ${\displaystyle {\mathfrak {sl}}_{2}}${\displaystyle {\mathfrak {sl}}_{2}}.^[3]

• Quantum group

• 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.

• Quantized enveloping algebra at the nLab
• Quantized enveloping algebras at ${\displaystyle q=1}${\displaystyle q=1} at MathOverflow
• Does there exist any "quantum Lie algebra" imbedded into the quantum enveloping algebra ${\displaystyle U_{q}(g)}${\displaystyle U_{q}(g)}? at MathOverflow

• Quantum groups
• Representation theory
• Mathematical quantization
• Algebra stubs

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