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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}}</ref> Given 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.▼
▲In mathematics, a '''quantum''' or '''quantized enveloping algebra''' is a [[Q-analog|''q''-analog]] of a [[universal enveloping algebra]]. Given 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> ==
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[[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–Baxter equation |journal=[[Letters in Mathematical Physics]] |volume=10 |year=1985 |number=1 |pages=63–69 |doi=10.1007/BF00704588 |authorlink=Michio Jimbo}}</ref>
== References ==
{{Reflist}}
* {{Citation | last=Drinfel'd |first=V. G. | title=Quantum Groups |journal=Proceedings of the International Congress of Mathematicians 986 |volume=1 |pages=798–820 |year=1987 |publisher=[[American Mathematical Society]] |authorlink=Vladimir Drinfeld}}
== External links ==
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