Example 1.
We initialize the split real form of the exceptional Lie algebra and retrieve the root space decomposition and the list of all roots. We then calculate some root strings. The structure equations for are obtained using SimpleLieAlgebraData.
Initialize the Lie algebra with DGsetup.
The root space decomposition is retrieved, without calculation, using SimpleLieAlgebraProperties
Here is the root space decomposition and the list of all positive roots.
Define two roots and .
Calculate the -string through
With the optional keyword argument output = "stringlengths", we obtain the lengths of the -string through in the negative and positive directions.
Thus the -string through in is given explicitly by.
Example 2.
Here is another example of a root string for the exceptional Lie algebra .
The root -string through is now
and the string lengths are
Thus, the root string is explicitly given by
Example 3.
Root strings can also be calculated for abstract roots systems, that is, a set of vectors satisfying the standard axioms of a root system and not explicitly defined from the root space decomposition of a semi-simple Lie algebra. The positive roots of an abstract root systems can be calculated with the PositiveRoots command.
Here are are 2 roots and their root string.
