Quantized enveloping algebra: Difference between revisions

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In mathematics, a '''quantum''' or '''quantized enveloping algebra''' is a [[Q-analog|''q''-analog]] of a [[universal enveloping algebra]].<ref name="kassel">{{Citation | last1=Kassel | first1=Christian | title=Quantum groups | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-0-387-94370-1 | mr=1321145 | year=1995 | volume=155 | url-access=registration | url=https://archive.org/details/quantumgroups0000kass }}</ref> Given a [[Lie algebra]] <math>\mathfrak{g}</math>, the quantum enveloping algebra is typically denoted as <math>U_q(\mathfrak{g})</math>. The notation was introduced by Drinfeld and independently by Jimbo.<ref>{{harvnb|Tjin|1992|loc=§ 5.}}</ref>

Among the applications, studying the <math>q \to 0</math> limit led to the discovery of [[crystal base]]s.

== The case of <math>\mathfrak{sl}_2</math> ==

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== See also ==

*[[quantumQuantum group]]

==Notes==

{{reflist}}

== References ==

{{Reflist}}

* {{Citation | last=Drinfel'd |first=V. G. | title=Quantum Groups |journal=Proceedings of the International Congress of Mathematicians 986 |volume=1 |pages=798&ndash;820 |year=1987 |publisher=[[American Mathematical Society]] |authorlink=Vladimir Drinfeld}}

* {{cite journal |last1=Tjin |first1=T. |title=An introduction to quantized Lie groups and algebras |journal=International Journal of Modern Physics A |date=10 October 1992 |volume=07 |issue=25 |pages=6175–6213 |doi=10.1142/S0217751X92002805 |arxiv=hep-th/9111043 |bibcode=1992IJMPA...7.6175T |s2cid=119087306 |issn=0217-751X}}

== External links ==

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* [https://mathoverflow.net/q/126461 Quantized enveloping algebras at <math>q = 1</math>] at [[MathOverflow]]

* [https://mathoverflow.net/q/93778 Does there exist any "quantum Lie algebra" imbedded into the quantum enveloping algebra <math>U_q(g)</math>?] at MathOverflow

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[[Category:Quantum groups]]

[[Category:Representation theory]]

[[Category:Mathematical quantization]]

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