Quantized enveloping algebra

<h2 id="the-case-of---------------------------------------------------------s--------------l--------------------------------------------2--------------------------------displaystyle-mathfrak-sl2">The case of 
  
    
      
        
          
            
              s
              l
            
          
          
            2
          
        
      
    
    {\displaystyle {\mathfrak {sl}}_{2}}
  
</h2>
<p><a href="/facts/Michio_Jimbo/Ij6q4FoL">Michio Jimbo</a> considered the algebras with three generators related by the three commutators
</p>

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

<p>When 
  
    
      
        η
        →
        0
      
    
    {\displaystyle \eta \to 0}
  
, these reduce to the commutators that define the <a href="/facts/Special_linear_Lie_algebra/IpHvcM2t">special linear Lie algebra</a> 
  
    
      
        
          
            
              s
              l
            
          
          
            2
          
        
      
    
    {\displaystyle {\mathfrak {sl}}_{2}}
  
. In contrast, for nonzero 
  
    
      
        η
      
    
    {\displaystyle \eta }
  
, the algebra defined by these relations is not a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> but instead an <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> that can be regarded as a deformation of the universal enveloping algebra of 
  
    
      
        
          
            
              s
              l
            
          
          
            2
          
        
      
    
    {\displaystyle {\mathfrak {sl}}_{2}}
  
.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Quantum_group/huNTuaXs">Quantum group</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Vladimir_Drinfeld/xGhJ8NAv">Drinfel'd, V. G.</a> (1987), "Quantum Groups", <i>Proceedings of the International Congress of Mathematicians 986</i>, 1, <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>: 798–820</li>
<li>Tjin, T. (10 October 1992). "An introduction to quantized Lie groups and algebras". <i>International Journal of Modern Physics A</i>. 07 (25): 6175–6213. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/hep-th/9111043">hep-th/9111043</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1992IJMPA...7.6175T">1992IJMPA...7.6175T</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2FS0217751X92002805">10.1142/S0217751X92002805</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0217-751X">0217-751X</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119087306">119087306</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://ncatlab.org/nlab/show/quantized+enveloping+algebra">Quantized enveloping algebra</a> at the <a href="/facts/NLab/lpPvRmgo">nLab</a></li>
<li><a href="https://mathoverflow.net/q/126461">Quantized enveloping algebras at 
  
    
      
        q
        =
        1
      
    
    {\displaystyle q=1}
  
</a> at <a href="/facts/MathOverflow/0XqLBHA5">MathOverflow</a></li>
<li><a href="https://mathoverflow.net/q/93778">Does there exist any "quantum Lie algebra" imbedded into the quantum enveloping algebra 
  
    
      
        
          U
          
            q
          
        
        (
        g
        )
      
    
    {\displaystyle U_{q}(g)}
  
?</a> at MathOverflow</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, vol. 155, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94370-1, MR 1321145 <a href="978-0-387-94370-1" target="_blank">978-0-387-94370-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Tjin 1992, § 5. - 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. <a href="https://arxiv.org/abs/hep-th/9111043" target="_blank">https://arxiv.org/abs/hep-th/9111043</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Jimbo, Michio (1985), "A 
  
    
      
        q
      
    
    {\displaystyle q}
  
-difference analogue of 
  
    
      
        U
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle U({\mathfrak {g}})}
  
 and the Yang–Baxter equation", Letters in Mathematical Physics, 10 (1): 63–69, Bibcode:1985LMaPh..10...63J, doi:10.1007/BF00704588, S2CID 123313856 <a href="/wiki/Michio_Jimbo" target="_blank">/wiki/Michio_Jimbo</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Quantized enveloping algebra open-in-new

Quantized enveloping algebra