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A001066
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Dimensions (sorted, with duplicates removed) of real simple Lie algebras.
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4
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3, 6, 8, 10, 14, 15, 16, 20, 21, 24, 28, 30, 35, 36, 42, 45, 48, 52, 55, 56, 63, 66, 70, 72, 78, 80, 90, 91, 96, 99, 104, 105, 110, 120, 126, 132, 133, 136, 143, 153, 156, 160, 168, 171, 182, 190, 195, 198, 210, 224, 231, 240, 248, 253, 255, 266, 272, 276, 286, 288, 300, 306
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OFFSET
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1,1
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COMMENTS
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The possible dimensions of real simple Lie algebras are the numbers n and 2n where n runs through the dimensions of the complex simple Lie algebras.
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REFERENCES
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Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652.
N. Jacobson, Lie Algebras. Wiley, NY, 1962; see pp. 141-146.
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LINKS
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FORMULA
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Numbers n and 2n as n runs through A003038.
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EXAMPLE
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6 is the dimension of the real simple Lie algebra SL_2(C).
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MATHEMATICA
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max = 18; 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, 2*sa, sb, 2*sb, sd, 2*sd, se, 2*se], # <= max^2 &] (* Jean-François Alcover, Apr 02 2012, after A003038 *)
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PROG
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(Haskell)
import Data.Set (deleteFindMin, fromList, insert)
a001066 n = a001066_list !! (n-1)
a001066_list = f (fromList [h, 2 * h]) $ tail a003038_list where
h = head a003038_list
f s (x:xs) = m : f (x `insert` (( 2 * x) `insert` s')) xs where
(m, s') = deleteFindMin s
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Richard E. Borcherds (reb(AT)math.berkeley.edu)
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EXTENSIONS
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STATUS
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approved
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