# Dimension formulas for Lie algebras

1. May 8, 2012

### eherrtelle59

Hi.

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. May 8, 2012

### DonAntonio

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.

DonAntonio