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>

'''Quantum enveloping algebra''' is a ''q''-analog of an [[enveloping algebra]]{{dn|date=September 2017}}.

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

<math>U_q(\mathfrak{g})</math>

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

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

:<math>[h,e] = 2e,\ [h,f] = -2f,\ [e,f] = \sinh(\eta h)/\sinh \eta.</math>

When <math>\eta \to 0</math>, these reduce to the commutators that define the [[special linear Lie algebra]] <math>\mathfrak{sl}_2</math>. In contrast, for nonzero <math>\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>\mathfrak{sl}_2</math>.<ref name="jimbo">{{Citation | last=Jimbo |first=Michio |title=A <math>q</math>-difference analogue of <math>U(\mathfrak{g})</math> and the Yang&ndash;Baxter equation |journal=[[Letters in Mathematical Physics]] |volume=10 |year=1985 |number=1 |pages=63&ndash;69 |doi=10.1007/BF00704588 |authorlink=Michio Jimbo|bibcode=1985LMaPh..10...63J |s2cid=123313856 }}</ref>

== See also ==

*[[crystalQuantum basisgroup]]

==Notes==

{{reflist}}

== References ==

* {{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 ==

*http [https://mathoverflowncatlab.netorg/questionsnlab/126461show/quantized-+enveloping-algebras-+algebra Quantized enveloping algebra] at-q-1 the [[nLab]]

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

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

[[Category:Quantum groups]]

[[Category:Representation theory]]

[[Category:Mathematical quantization]]

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