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 ana [[universal enveloping algebra]]. ItGiven an algebra <math>\mathfrak{g}</math>, the quantum enveloping algebra is typically denoted as <math>U_q(\mathfrak{g})</math>. Studying the <math>q \to 0</math> limit led to the discovery of [[crystal base]]s.

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

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

== See also ==

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

*[[crystal basis]]

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>.

== References ==

* {{Citation | last1last=KasselDrinfel'd | first1first=ChristianV. G. | title=Quantum groupsGroups | publisherjournal=[[Springer-Verlag]]Proceedings |of ___location=Berlin,the NewInternational YorkCongress |of series=GraduateMathematicians Texts in Mathematics986 | isbnvolume=978-0-387-94370-1 |mrpages=1321145798&ndash;820 | year=19951987 |publisher=[[American volumeMathematical Society]] |authorlink=155Vladimir Drinfeld}}

* {{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}}

* {{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}}

== External links ==

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

*http [https://mathoverflow.net/questions/93778126461/doesquantized-thereenveloping-existalgebras-anyat-quantum-lie-algebra-embeded-into-the-quantumq-1 Quantized enveloping-a algebras at <math>q = 1</math>] at [[MathOverflow]]

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

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