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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