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A263005 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A263005 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A263005" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA263005">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A263005 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A263005 </div> <div class=seqname> Dimensions of the simple Lie algebras over complex numbers (with repetitions), sorted nondecreasingly. </div> </div> <div class=scorerefs> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>3, 8, 10, 14, 15, 21, 21, 24, 28, 35, 36, 36, 45, 48, 55, 55, 57, 63, 66, 78, 78, 78, 80, 91, 99, 105, 105, 120, 120, 133, 136, 136, 143, 153, 168, 171, 171, 190, 195, 210, 210, 224, 231, 248, 253, 253, 255, 276, 288, 300, 300</div> <div class=seqdatalinks> (<a href="/A263005/list">list</a>; <a href="/A263005/graph">graph</a>; <a href="/search?q=A263005+-id:A263005">refs</a>; <a href="/A263005/listen">listen</a>; <a href="/history?seq=A263005">history</a>; <a href="/search?q=id:A263005&fmt=text">text</a>; <a href="/A263005/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>This sequence gives the dimensions of the (compact) simple Lie algebras A_l, l >= 1, B_l, l >= 2, C_l >= 3, D_l, l >= 4, E_6, E_7, E_8, F_4 and G_2 which are l*(l+2), l*(2*l + 1), l*(2*l + 1), l*(2*l - 1), 78, 133, 248, 52 and 14, respectively. These are also the dimensions of the adjoint representations of these Lie algebras. For the l-ranges see the Humphreys reference, p. 58, and for the dimensions, e.g., the Samelson link, Theorem A, p. 74.</div> <div class=sectline>The dimension duplications occur for the B_l and C_l series for l >= 3.</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>E. Cartan, Sur la structure des groupes de transformation finis et continus. Th猫se Paris 1894. Oeuvres Compl猫tes, I,1, pp. 137-287, Paris 1952.</div> <div class=sectline>J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1972.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A263005/b263005.txt">Table of n, a(n) for n=1..51.</a></div> <div class=sectline>W. Killing, Die Zusammensetzung der stetigen endlichen Transformationsgruppen, Mathematische Ann. I: 31 (1888) 252-290, II: 33 (1889) 1-48, III: 34 (1889) 57-122, IV: 36 (1890) 161-189: <a href="https://eudml.org/doc/157352">I</a>, <a href="https://eudml.org/doc/157397">II</a>, <a href="https://eudml.org/doc/157434">III</a>, <a href="https://eudml.org/doc/157490">IV</a>.</div> <div class=sectline>Hans Samelson, <a href="http://www.math.cornell.edu/~hatcher/Other/Samelson-LieAlg.pdf">Notes on Lie Algebras</a>.</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A104599" title="Dimensions of the irreducible representations of the simple Lie algebra of type G2 over the complex numbers, listed in incre...">A104599</a>, <a href="/A121214" title="Dimensions of the irreducible representations of the algebraic group SL4 (equivalently, simple Lie algebra of type A3) over ...">A121214</a>, <a href="/A121732" title="Dimensions of the irreducible representations of the simple Lie algebra of type E8 over the complex numbers, listed in incre...">A121732</a>, <a href="/A121736" title="Dimensions of the irreducible representations of the simple Lie algebra of type E7 over the complex numbers, listed in incre...">A121736</a>, <a href="/A121737" title="Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in incre...">A121737</a>, <a href="/A121738" title="Dimensions of the irreducible representations of the simple Lie algebra of type F4 over the complex numbers, listed in incre...">A121738</a>, <a href="/A121739" title="Dimensions of the irreducible representations of the simple Lie algebra of type D4 over the complex numbers, listed in incre...">A121739</a>, <a href="/A121741" title="Dimensions of the irreducible representations of the simple Lie algebra of type A2 (equivalently, the group SL3) over the co...">A121741</a>.</div> <div class=sectline>Sequence in context: <a href="/A287573" title="Positions of 1 in A287571.">A287573</a> <a href="/A122529" title="Complement of sequence A029903.">A122529</a> <a href="/A137920" title="Numbers k such that 24*k-1 and 24*k+1 are twin primes.">A137920</a> * <a href="/A126581" title="Freeman Dyson's list of values of k for which there is an identity of a certain type for the k-th power of the Dedekind eta ...">A126581</a> <a href="/A003038" title="Dimensions of split simple Lie algebras over any field of characteristic zero.">A003038</a> <a href="/A184870" title="Numbers m such that prime(m) is of the form floor[(k-1/2)*(2+2^(1/2))+1/2]; complement of A184867.">A184870</a></div> <div class=sectline>Adjacent sequences: <a href="/A263002" title="Expansion of (f(-x^5) / f(-x))^2 in powers of x where f() is a Ramanujan theta function.">A263002</a> <a href="/A263003" title="Partition array for the products of the hook lengths of Ferrers (Young) diagrams corresponding to the partitions of n, writt...">A263003</a> <a href="/A263004" title="Row sums of the partition array for the products of the hook lengths numbers of Ferrers (or Young) diagrams A263003.">A263004</a> * <a href="/A263006" title="First member R0(n) of the smallest positive pair (R0(n), S0(n)) for the n-th 1-happy number couple (B(n), C(n)).">A263006</a> <a href="/A263007" title="Second member S0(n) of the smallest positive pair (R0(n), S0(n)) for the n-th 1-happy number couple (B(n), C(n)).">A263007</a> <a href="/A263008" title="First member T0(n) of the smallest positive pair (T0(n), U0(n)) for the n-th 2-happy number couple (D(n), E(n)).">A263008</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="it is very easy to produce terms of sequence">easy</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Oct 23 2015</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified April 8 05:33 EDT 2025. Contains 382574 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>