Lie algebra, maximal toral sub-algebra

Question

Is there a relation between number of roots of a finite dimensional semi-simple Lie algebra L and dimension of the maximal toral sub-algebra H(Cartan sub-algebra) of L?

Thanks!

Answer 1

With notations as in the question, 2 dim(H) <= no. of roots. This is true because roots span H*(Dual space of H) and roots always occur in pairs.(I mean x is a root iff -x is a root.)

I do not hope there is any other relation, in general. But, may be, if we consider some special classes of L, we may have some better results!

Answer 2

There is the relation $$ \dim (H)+\mid \Phi\mid =\dim (L), $$ where $\mid \Phi\mid$ denotes the cardinality of the root system of $L$. For example, with $L$ of type $A_n$ we have $\dim (H)=n$, $\mid \Phi\mid=n^2+n$ and $\dim(L)=n^2+2n$.