A001066 Dimensions (sorted, with duplicates removed) of real simple Lie algebras.

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

Offset: 1

Author

Richard E. Borcherds (reb(AT)math.berkeley.edu)

Keywords

• nonn
• nice
• easy

Comments

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.

Examples

			6 is the dimension of the real simple Lie algebra SL_2(C).
		

References

• Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652.
• N. Jacobson, Lie Algebras. Wiley, NY, 1962; see pp. 141-146.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003038.

Subsequences, apart from some initial terms: A000217, A000384, A002378, A005563, A014105.

Programs

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
-- Reinhard Zumkeller, Dec 16 2012

Mathematica

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 *)

Formula

Numbers n and 2n as n runs through A003038.

Extensions

Entry revised by N. J. A. Sloane, Mar 16 2007