Dimension formulas for Lie algebras

[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!

 

[DonAntonio]

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!

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