A001066 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

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

FORMULA

Numbers n and 2n as n runs through A003038.

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Cf. A003038.

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