### 4.2 Roots

The adjoint action of the Cartan subalgebra on and is diagonal. Explicitly,
for any element , where is the linear form on (i.e., the element of the dual ) defined by
. The ’s are called the simple roots. Similarly,
and, if is non-zero, one says that is a positive root. On the
negative side, , one has
and is called a negative root when is non-zero. This
occurs if and only if is non-zero: is a negative root if and only if is a
positive root.
We see from the construction that the roots (linear forms such that has nonzero
solutions ) are either positive (linear combinations of the simple roots with integer
non-negative coefficients) or negative (linear combinations of the simple roots with integer non-positive
coefficients). The set of positive roots is denoted by ; that of negative roots by . The
set of all roots is , so we have . The simple roots are positive and form
a basis of . One sometimes denotes the by (and thus, etc).
Similarly, one also uses the notation for the standard pairing between and its dual
, i.e., . In this notation the entries of the Cartan matrix can be written as

Finally, the root lattice is the set of linear combinations with integer coefficients of the simple roots,

All roots belong to the root lattice, of course, but the converse is not true: There are elements of
that are not roots.