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Dimension formulas for Lie algebras

  1. May 8, 2012 #1

    1. Can anyone definitively tell me what the dimension formula for the classical Lie algebras?

    For example, I know for SO(2n) or D_n, the dimension formula is

    SO(N)--> (N*(N-1))/2

    E.g. SO(8) is 8*7/2 = 28.

    Ok, so what about SU(N+1) i.e. A_n, SO(2n+1) i.e. B_N and Sp(n) i.e. C_n ?

    2. I have (using # of simple roots = the rank )

    that # total roots = dim (above) - rank

    and therefore the # of positive roots is half of this quantity (i.e. (dim-rank)/2 )

    Any help would be great. Thanks!
  2. jcsd
  3. May 8, 2012 #2

    I'm not sure, but I'd say most decent books in Lie Algebras mention this. Anyway, Humphreys's "Int. to Lie Algebras and Repres. Theory" does.

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