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.
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