A003038 Dimensions of split simple Lie algebras over any field of characteristic zero.
3, 8, 10, 14, 15, 21, 24, 28, 35, 36, 45, 48, 52, 55, 63, 66, 78, 80, 91, 99, 105, 120, 133, 136, 143, 153, 168, 171, 190, 195, 210, 224, 231, 248, 253, 255, 276, 288, 300, 323, 325, 351, 360, 378, 399, 406, 435, 440, 465, 483, 496, 528, 561, 575, 595, 624, 630
Offset: 1
Examples
The Lie algebras in question and their dimensions are the following: A_l: l(l+2), l >= 1, B_l: l(2l+1), l >= 2, C_l: l(2l+1), l >= 3, D_l: l(2l-1), l >= 4, G_2: 14, F_4: 52, E_6: 78, E_7: 133, E_8: 248.
References
- Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652.
- N. Jacobson, Lie Algebras. Wiley, NY, 1962; pp. 141-146.
- I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal., 13 (1982), 988-1007.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Haskell
import Data.Set (deleteFindMin, fromList, insert) a003038 n = a003038_list !! (n-1) a003038_list = f (fromList (3 : [14, 52, 78, 133, 248])) (drop 2 a005563_list) (drop 4 a000217_list) where f s (x:xs) (y:ys) = m : f (x `insert` (y `insert` s')) xs ys where (m, s') = deleteFindMin s -- Reinhard Zumkeller, Dec 16 2012 -
Maple
M:=4200; M2:=M^2; sa:=[seq(l*(l+2),l=1..M)]; sb:=[seq(l*(2*l+1),l=2..M)]; sd:=[seq(l*(2*l-1),l=4..M)]; se:=[14,52,78,133,248]; s:=convert(sa,set) union convert(sb,set) union convert(sd,set) union convert(se,set); t:=convert(s,list); for i from 1 to nops(t) do if t[i] <= M2 then lprint(i,t[i]); fi; od:
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Mathematica
max = 26; sa = Table[ k*(k+2), {k, 1, max}]; sb = Table[ k*(2k+1), {k, 2, max}]; sd:= Table[ k*(2k-1), {k, 4, max}]; se = {14, 52, 78, 133, 248}; Select[ Union[sa, sb, sd, se], # <= max^2 &](* Jean-François Alcover, Nov 18 2011, after Maple *)
Extensions
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004