so(15), type \(B^{1}_7\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Page generated by the calculator project.
Lie algebra type: B^{1}_7.
Weyl group size: 645120.
A drawing of the root system in its corresponding Coxeter plane. Computations were carried out as explained by John Stembridge.
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The grey lines are the edges of the Weyl chamber.
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The root system has 98 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-1, -2, -2, -2, -2, -2, -2)-e_{1}-e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -2, -2, -2, -2, -2)-e_{1}-e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -2, -2, -2, -2, -2)-e_{2}-e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(-1, -1, -1, -2, -2, -2, -2)-e_{1}-e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}\)
(0, -1, -1, -2, -2, -2, -2)-e_{2}-e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}\)
(-1, -1, -1, -1, -2, -2, -2)-e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}\)
(0, 0, -1, -2, -2, -2, -2)-e_{3}-e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}s_{4}\)
(0, -1, -1, -1, -2, -2, -2)-e_{2}-e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}\)
(-1, -1, -1, -1, -1, -2, -2)-e_{1}-e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}\)
(0, 0, -1, -1, -2, -2, -2)-e_{3}-e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}\)
(0, -1, -1, -1, -1, -2, -2)-e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}\)
(-1, -1, -1, -1, -1, -1, -2)-e_{1}-e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}\)
(0, 0, 0, -1, -2, -2, -2)-e_{4}-e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}s_{6}s_{5}\)
(0, 0, -1, -1, -1, -2, -2)-e_{3}-e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}\)
(0, -1, -1, -1, -1, -1, -2)-e_{2}-e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}\)
(-1, -1, -1, -1, -1, -1, -1)-e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, -1, -1, -2, -2)-e_{4}-e_{6}\(s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}s_{6}\)
(0, 0, -1, -1, -1, -1, -2)-e_{3}-e_{7}\(s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}\)
(0, -1, -1, -1, -1, -1, -1)-e_{2}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, -1, -1, 0)-e_{1}+e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, -1, -2, -2)-e_{5}-e_{6}\(s_{6}s_{5}s_{7}s_{6}s_{5}s_{7}s_{6}\)
(0, 0, 0, -1, -1, -1, -2)-e_{4}-e_{7}\(s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}\)
(0, 0, -1, -1, -1, -1, -1)-e_{3}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, -1, -1, 0)-e_{2}+e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, -1, 0, 0)-e_{1}+e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, -1, -1, -2)-e_{5}-e_{7}\(s_{5}s_{7}s_{6}s_{5}s_{7}\)
(0, 0, 0, -1, -1, -1, -1)-e_{4}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, -1, 0)-e_{3}+e_{7}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, -1, 0, 0)-e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, 0, 0, 0)-e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, -1, -2)-e_{6}-e_{7}\(s_{7}s_{6}s_{7}\)
(0, 0, 0, 0, -1, -1, -1)-e_{5}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -1, 0)-e_{4}+e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, 0, 0)-e_{3}+e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, 0, 0, 0)-e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(-1, -1, -1, 0, 0, 0, 0)-e_{1}+e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, -1, -1)-e_{6}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, -1, -1, 0)-e_{5}+e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, -1, -1, 0, 0)-e_{4}+e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0, 0, 0)-e_{3}+e_{5}\(s_{3}s_{4}s_{3}\)
(0, -1, -1, 0, 0, 0, 0)-e_{2}+e_{4}\(s_{2}s_{3}s_{2}\)
(-1, -1, 0, 0, 0, 0, 0)-e_{1}+e_{3}\(s_{1}s_{2}s_{1}\)
(0, 0, 0, 0, 0, 0, -1)-e_{7}\(s_{7}\)
(0, 0, 0, 0, 0, -1, 0)-e_{6}+e_{7}\(s_{6}\)
(0, 0, 0, 0, -1, 0, 0)-e_{5}+e_{6}\(s_{5}\)
(0, 0, 0, -1, 0, 0, 0)-e_{4}+e_{5}\(s_{4}\)
(0, 0, -1, 0, 0, 0, 0)-e_{3}+e_{4}\(s_{3}\)
(0, -1, 0, 0, 0, 0, 0)-e_{2}+e_{3}\(s_{2}\)
(-1, 0, 0, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{1}\)
(1, 0, 0, 0, 0, 0, 0)e_{1}-e_{2}\(s_{1}\)
(0, 1, 0, 0, 0, 0, 0)e_{2}-e_{3}\(s_{2}\)
(0, 0, 1, 0, 0, 0, 0)e_{3}-e_{4}\(s_{3}\)
(0, 0, 0, 1, 0, 0, 0)e_{4}-e_{5}\(s_{4}\)
(0, 0, 0, 0, 1, 0, 0)e_{5}-e_{6}\(s_{5}\)
(0, 0, 0, 0, 0, 1, 0)e_{6}-e_{7}\(s_{6}\)
(0, 0, 0, 0, 0, 0, 1)e_{7}\(s_{7}\)
(1, 1, 0, 0, 0, 0, 0)e_{1}-e_{3}\(s_{1}s_{2}s_{1}\)
(0, 1, 1, 0, 0, 0, 0)e_{2}-e_{4}\(s_{2}s_{3}s_{2}\)
(0, 0, 1, 1, 0, 0, 0)e_{3}-e_{5}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 0, 0)e_{4}-e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 0)e_{5}-e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1)e_{6}\(s_{6}s_{7}s_{6}\)
(1, 1, 1, 0, 0, 0, 0)e_{1}-e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 0, 0, 0)e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 0, 0)e_{3}-e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 0)e_{4}-e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1)e_{5}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 2)e_{6}+e_{7}\(s_{7}s_{6}s_{7}\)
(1, 1, 1, 1, 0, 0, 0)e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 0, 0)e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 0)e_{3}-e_{7}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 1)e_{4}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 2)e_{5}+e_{7}\(s_{5}s_{7}s_{6}s_{5}s_{7}\)
(1, 1, 1, 1, 1, 0, 0)e_{1}-e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 0)e_{2}-e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 1)e_{3}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 2)e_{4}+e_{7}\(s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}\)
(0, 0, 0, 0, 1, 2, 2)e_{5}+e_{6}\(s_{6}s_{5}s_{7}s_{6}s_{5}s_{7}s_{6}\)
(1, 1, 1, 1, 1, 1, 0)e_{1}-e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 1)e_{2}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 2)e_{3}+e_{7}\(s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}\)
(0, 0, 0, 1, 1, 2, 2)e_{4}+e_{6}\(s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}s_{6}\)
(1, 1, 1, 1, 1, 1, 1)e_{1}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 2)e_{2}+e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}\)
(0, 0, 1, 1, 1, 2, 2)e_{3}+e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}\)
(0, 0, 0, 1, 2, 2, 2)e_{4}+e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 1, 2)e_{1}+e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}\)
(0, 1, 1, 1, 1, 2, 2)e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}\)
(0, 0, 1, 1, 2, 2, 2)e_{3}+e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 2, 2)e_{1}+e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}\)
(0, 1, 1, 1, 2, 2, 2)e_{2}+e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}\)
(0, 0, 1, 2, 2, 2, 2)e_{3}+e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 2, 2, 2)e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}\)
(0, 1, 1, 2, 2, 2, 2)e_{2}+e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}\)
(1, 1, 1, 2, 2, 2, 2)e_{1}+e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}\)
(0, 1, 2, 2, 2, 2, 2)e_{2}+e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 2, 2, 2, 2, 2)e_{1}+e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 2, 2, 2, 2, 2, 2)e_{1}+e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
Comma delimited list of roots: (-1, -2, -2, -2, -2, -2, -2), (-1, -1, -2, -2, -2, -2, -2), (0, -1, -2, -2, -2, -2, -2), (-1, -1, -1, -2, -2, -2, -2), (0, -1, -1, -2, -2, -2, -2), (-1, -1, -1, -1, -2, -2, -2), (0, 0, -1, -2, -2, -2, -2), (0, -1, -1, -1, -2, -2, -2), (-1, -1, -1, -1, -1, -2, -2), (0, 0, -1, -1, -2, -2, -2), (0, -1, -1, -1, -1, -2, -2), (-1, -1, -1, -1, -1, -1, -2), (0, 0, 0, -1, -2, -2, -2), (0, 0, -1, -1, -1, -2, -2), (0, -1, -1, -1, -1, -1, -2), (-1, -1, -1, -1, -1, -1, -1), (0, 0, 0, -1, -1, -2, -2), (0, 0, -1, -1, -1, -1, -2), (0, -1, -1, -1, -1, -1, -1), (-1, -1, -1, -1, -1, -1, 0), (0, 0, 0, 0, -1, -2, -2), (0, 0, 0, -1, -1, -1, -2), (0, 0, -1, -1, -1, -1, -1), (0, -1, -1, -1, -1, -1, 0), (-1, -1, -1, -1, -1, 0, 0), (0, 0, 0, 0, -1, -1, -2), (0, 0, 0, -1, -1, -1, -1), (0, 0, -1, -1, -1, -1, 0), (0, -1, -1, -1, -1, 0, 0), (-1, -1, -1, -1, 0, 0, 0), (0, 0, 0, 0, 0, -1, -2), (0, 0, 0, 0, -1, -1, -1), (0, 0, 0, -1, -1, -1, 0), (0, 0, -1, -1, -1, 0, 0), (0, -1, -1, -1, 0, 0, 0), (-1, -1, -1, 0, 0, 0, 0), (0, 0, 0, 0, 0, -1, -1), (0, 0, 0, 0, -1, -1, 0), (0, 0, 0, -1, -1, 0, 0), (0, 0, -1, -1, 0, 0, 0), (0, -1, -1, 0, 0, 0, 0), (-1, -1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, -1), (0, 0, 0, 0, 0, -1, 0), (0, 0, 0, 0, -1, 0, 0), (0, 0, 0, -1, 0, 0, 0), (0, 0, -1, 0, 0, 0, 0), (0, -1, 0, 0, 0, 0, 0), (-1, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1), (1, 1, 0, 0, 0, 0, 0), (0, 1, 1, 0, 0, 0, 0), (0, 0, 1, 1, 0, 0, 0), (0, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 0, 1, 1), (1, 1, 1, 0, 0, 0, 0), (0, 1, 1, 1, 0, 0, 0), (0, 0, 1, 1, 1, 0, 0), (0, 0, 0, 1, 1, 1, 0), (0, 0, 0, 0, 1, 1, 1), (0, 0, 0, 0, 0, 1, 2), (1, 1, 1, 1, 0, 0, 0), (0, 1, 1, 1, 1, 0, 0), (0, 0, 1, 1, 1, 1, 0), (0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 2), (1, 1, 1, 1, 1, 0, 0), (0, 1, 1, 1, 1, 1, 0), (0, 0, 1, 1, 1, 1, 1), (0, 0, 0, 1, 1, 1, 2), (0, 0, 0, 0, 1, 2, 2), (1, 1, 1, 1, 1, 1, 0), (0, 1, 1, 1, 1, 1, 1), (0, 0, 1, 1, 1, 1, 2), (0, 0, 0, 1, 1, 2, 2), (1, 1, 1, 1, 1, 1, 1), (0, 1, 1, 1, 1, 1, 2), (0, 0, 1, 1, 1, 2, 2), (0, 0, 0, 1, 2, 2, 2), (1, 1, 1, 1, 1, 1, 2), (0, 1, 1, 1, 1, 2, 2), (0, 0, 1, 1, 2, 2, 2), (1, 1, 1, 1, 1, 2, 2), (0, 1, 1, 1, 2, 2, 2), (0, 0, 1, 2, 2, 2, 2), (1, 1, 1, 1, 2, 2, 2), (0, 1, 1, 2, 2, 2, 2), (1, 1, 1, 2, 2, 2, 2), (0, 1, 2, 2, 2, 2, 2), (1, 1, 2, 2, 2, 2, 2), (1, 2, 2, 2, 2, 2, 2) The resulting Lie bracket pairing table follows.
Type B^{1}_7.The letter \(\displaystyle h\) stands for elements of the Cartan subalgebra,
the letter \(\displaystyle g\) stands for the Chevalley (root space) generators of non-zero weight.
The generator \(\displaystyle h_i\) is the element of the Cartan subalgebra dual to the
i^th simple root, that is, \(\displaystyle [h_i, g] =\langle \alpha_i , \gamma\rangle g\),
where g is a Chevalley generator, \(\displaystyle \gamma\) is its weight, and
\(\displaystyle \alpha_i\) is the i^th simple root.
The Lie bracket table is too large to be rendered in LaTeX, displaying in html format instead.
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(0, 0, 1, 1, 1, 1, 1)e_{3}g_{27}0-g_{-34}-g_{-31}000g_{-23}00g_{-18}000g_{-13}02g_{-8}0g_{-7}2g_{-2}0002h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}0002g_{3}-g_{7}0002g_{10}0-g_{13}002g_{16}00-g_{18}002g_{22}000-g_{23}000g_{27}-g_{27}00000-g_{31}0000-2g_{32}-g_{34}0000-2g_{36}0000-2g_{40}0000-2g_{43}00000002g_{47}002g_{48}000000000000000
(0, 0, 0, 1, 1, 1, 2)e_{4}+e_{7}g_{28}000-g_{-30}-g_{-26}0-g_{-22}0000g_{-14}g_{-12}0g_{-9}0g_{-6}g_{-3}0002h_{7}+h_{6}+h_{5}+h_{4}000g_{4}g_{7}000g_{11}0000000-g_{19}000-g_{23}00-g_{24}00000g_{28}-g_{28}0g_{28}-g_{28}00-g_{32}00-g_{33}00-g_{35}00-g_{37}0-g_{38}0000000g_{43}000g_{45}000g_{46}0000000000000000000
(0, 0, 0, 0, 1, 2, 2)e_{5}+e_{6}g_{29}00000-g_{-25}0-g_{-21}g_{-20}-g_{-16}g_{-15}0-g_{-11}g_{-10}00g_{-4}0002h_{7}+2h_{6}+h_{5}0000g_{6}0000-g_{12}g_{13}0000-g_{18}g_{19}00000-g_{24}00000000g_{29}0-g_{29}0000-g_{33}00000-g_{36}g_{37}000-g_{39}g_{40}000-g_{41}g_{42}000g_{44}000000000000000000000000
(1, 1, 1, 1, 1, 1, 0)e_{1}-e_{7}g_{30}g_{-35}g_{-32}0g_{-28}0g_{-24}00g_{-19}000000-g_{-7}000h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}000g_{1}-g_{6}00g_{8}0-g_{12}00g_{14}00-g_{17}0g_{20}000-g_{22}0g_{25}0000-g_{26}-g_{30}0000-g_{30}g_{30}000000g_{34}00000000000-g_{41}0000-g_{44}000-g_{46}000-g_{48}00-g_{49}00000000000000
(0, 1, 1, 1, 1, 1, 1)e_{2}g_{31}-g_{-34}0g_{-27}0g_{-23}00g_{-18}00g_{-13}000g_{-7}2g_{-1}002h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}0002g_{2}-g_{7}002g_{9}0-g_{13}002g_{15}00-g_{18}02g_{21}000-g_{23}02g_{26}0000-g_{27}0g_{31}-g_{31}00000-g_{34}00000-2g_{35}00000-2g_{39}0000-2g_{42}0000-2g_{45}000-2g_{47}0000002g_{49}000000000000000
(0, 0, 1, 1, 1, 1, 2)e_{3}+e_{7}g_{32}0-g_{-30}-g_{-26}000g_{-17}00g_{-12}0g_{-8}0g_{-6}g_{-2}002h_{7}+h_{6}+h_{5}+h_{4}+h_{3}000g_{3}g_{7}00g_{10}0000g_{16}00-g_{19}00000-g_{24}00-g_{27}000-g_{28}000g_{32}-g_{32}00g_{32}-g_{32}0-g_{35}000-g_{36}0-g_{38}000-g_{40}0000-g_{43}00000000g_{47}000g_{48}0000000000000000000
(0, 0, 0, 1, 1, 2, 2)e_{4}+e_{6}g_{33}000-g_{-25}-g_{-21}0-g_{-16}0g_{-14}0g_{-9}0g_{-5}g_{-3}002h_{7}+2h_{6}+h_{5}+h_{4}000g_{4}g_{6}0000g_{13}000-g_{17}0g_{19}000-g_{23}000000-g_{28}0-g_{29}00000g_{33}-g_{33}g_{33}-g_{33}000-g_{36}0-g_{37}000-g_{39}0000-g_{41}0g_{43}0000g_{45}000g_{46}000000000000000000000000
(1, 1, 1, 1, 1, 1, 1)e_{1}g_{34}g_{-31}g_{-27}0g_{-23}0g_{-18}00g_{-13}00g_{-7}0002h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}002g_{1}-g_{7}002g_{8}0-g_{13}02g_{14}00-g_{18}02g_{20}000-g_{23}2g_{25}0000-g_{27}2g_{30}00000-g_{31}-g_{34}000000000000-2g_{38}00000-2g_{41}0000-2g_{44}0000-2g_{46}000-2g_{48}000-2g_{49}000000000000000000
(0, 1, 1, 1, 1, 1, 2)e_{2}+e_{7}g_{35}-g_{-30}0g_{-22}0g_{-17}00g_{-12}00g_{-6}g_{-1}002h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}00g_{2}g_{7}00g_{9}000g_{15}00-g_{19}0g_{21}000-g_{24}00000-g_{28}0-g_{31}0000-g_{32}0g_{35}-g_{35}000g_{35}-g_{35}-g_{38}0000-g_{39}00000-g_{42}0000-g_{45}0000-g_{47}0000000g_{49}0000000000000000000
(0, 0, 1, 1, 1, 2, 2)e_{3}+e_{6}g_{36}0-g_{-25}-g_{-21}000g_{-11}0g_{-8}g_{-5}g_{-2}002h_{7}+2h_{6}+h_{5}+h_{4}+h_{3}00g_{3}g_{6}00g_{10}0g_{13}0000g_{19}00-g_{22}00000-g_{27}00-g_{29}000-g_{32}00-g_{33}000g_{36}-g_{36}0g_{36}-g_{36}00-g_{39}00-g_{40}00-g_{41}00-g_{43}000000000g_{47}000g_{48}000000000000000000000000
(0, 0, 0, 1, 2, 2, 2)e_{4}+e_{5}g_{37}000-g_{-20}-g_{-15}g_{-14}-g_{-10}g_{-9}0g_{-3}002h_{7}+2h_{6}+2h_{5}+h_{4}000g_{5}000-g_{11}g_{12}000-g_{17}g_{18}0000-g_{23}g_{24}0000-g_{28}g_{29}00000-g_{33}000000g_{37}0-g_{37}0000-g_{40}00000-g_{42}g_{43}000-g_{44}g_{45}0000g_{46}00000000000000000000000000000
(1, 1, 1, 1, 1, 1, 2)e_{1}+e_{7}g_{38}g_{-26}g_{-22}0g_{-17}0g_{-12}00g_{-6}002h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}00g_{1}g_{7}0g_{8}000g_{14}00-g_{19}g_{20}000-g_{24}g_{25}0000-g_{28}00000-g_{32}-g_{34}00000-g_{35}-g_{38}0000g_{38}-g_{38}00000-g_{41}00000-g_{44}0000-g_{46}0000-g_{48}000-g_{49}00000000000000000000000
(0, 1, 1, 1, 1, 2, 2)e_{2}+e_{6}g_{39}-g_{-25}0g_{-16}0g_{-11}00g_{-5}g_{-1}02h_{7}+2h_{6}+h_{5}+h_{4}+h_{3}+h_{2}00g_{2}g_{6}0g_{9}0g_{13}0g_{15}00g_{19}000000-g_{26}000-g_{29}0-g_{31}000-g_{33}00-g_{35}000-g_{36}0g_{39}-g_{39}00g_{39}-g_{39}0-g_{41}000-g_{42}00000-g_{45}0000-g_{47}00000000g_{49}000000000000000000000000
(0, 0, 1, 1, 2, 2, 2)e_{3}+e_{5}g_{40}0-g_{-20}-g_{-15}00g_{-8}g_{-4}g_{-2}02h_{7}+2h_{6}+2h_{5}+h_{4}+h_{3}00g_{3}g_{5}000g_{12}00-g_{16}0g_{18}00-g_{22}0g_{24}000-g_{27}0g_{29}000-g_{32}000000-g_{36}0-g_{37}000g_{40}-g_{40}g_{40}-g_{40}000-g_{42}0-g_{43}000-g_{44}000000g_{47}0000g_{48}00000000000000000000000000000
(1, 1, 1, 1, 1, 2, 2)e_{1}+e_{6}g_{41}g_{-21}g_{-16}0g_{-11}0g_{-5}002h_{7}+2h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}0g_{1}g_{6}0g_{8}0g_{13}g_{14}00g_{19}g_{20}00000000-g_{29}-g_{30}0000-g_{33}-g_{34}0000-g_{36}0-g_{38}0000-g_{39}-g_{41}000g_{41}-g_{41}00000-g_{44}00000-g_{46}0000-g_{48}0000-g_{49}0000000000000000000000000000
(0, 1, 1, 1, 2, 2, 2)e_{2}+e_{5}g_{42}-g_{-20}0g_{-10}0g_{-4}g_{-1}02h_{7}+2h_{6}+2h_{5}+h_{4}+h_{3}+h_{2}0g_{2}g_{5}0g_{9}0g_{12}000g_{18}0-g_{21}00g_{24}0-g_{26}00g_{29}00-g_{31}00000-g_{35}00-g_{37}000-g_{39}00-g_{40}0g_{42}-g_{42}0g_{42}-g_{42}00-g_{44}00-g_{45}00000-g_{47}000000000g_{49}00000000000000000000000000000
(0, 0, 1, 2, 2, 2, 2)e_{3}+e_{4}g_{43}0-g_{-14}-g_{-9}g_{-8}g_{-2}02h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}00g_{4}00-g_{10}g_{11}00-g_{16}g_{17}000-g_{22}g_{23}000-g_{27}g_{28}0000-g_{32}g_{33}0000-g_{36}g_{37}00000-g_{40}0000g_{43}0-g_{43}0000-g_{45}00000-g_{46}g_{47}0000g_{48}00000000000000000000000000000000000
(1, 1, 1, 1, 2, 2, 2)e_{1}+e_{5}g_{44}g_{-15}g_{-10}0g_{-4}02h_{7}+2h_{6}+2h_{5}+h_{4}+h_{3}+h_{2}+h_{1}0g_{1}g_{5}g_{8}0g_{12}g_{14}00g_{18}000g_{24}-g_{25}000g_{29}-g_{30}00000-g_{34}000-g_{37}0-g_{38}000-g_{40}00-g_{41}000-g_{42}-g_{44}00g_{44}-g_{44}00000-g_{46}00000-g_{48}0000-g_{49}0000000000000000000000000000000000
(0, 1, 1, 2, 2, 2, 2)e_{2}+e_{4}g_{45}-g_{-14}0g_{-3}g_{-1}2h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}0g_{2}g_{4}00g_{11}0-g_{15}0g_{17}0-g_{21}0g_{23}00-g_{26}0g_{28}00-g_{31}0g_{33}000-g_{35}0g_{37}000-g_{39}000000-g_{42}0-g_{43}0g_{45}-g_{45}g_{45}-g_{45}000-g_{46}0-g_{47}0000000000g_{49}00000000000000000000000000000000000
(1, 1, 1, 2, 2, 2, 2)e_{1}+e_{4}g_{46}g_{-9}g_{-3}02h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}+h_{1}g_{1}g_{4}g_{8}0g_{11}00g_{17}-g_{20}00g_{23}-g_{25}00g_{28}0-g_{30}00g_{33}0-g_{34}00g_{37}00-g_{38}00000-g_{41}00-g_{43}000-g_{44}00-g_{45}-g_{46}0g_{46}-g_{46}00000-g_{48}00000-g_{49}0000000000000000000000000000000000000000
(0, 1, 2, 2, 2, 2, 2)e_{2}+e_{3}g_{47}-g_{-8}g_{-1}2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}0g_{3}0-g_{9}g_{10}0-g_{15}g_{16}00-g_{21}g_{22}00-g_{26}g_{27}000-g_{31}g_{32}000-g_{35}g_{36}0000-g_{39}g_{40}0000-g_{42}g_{43}00000-g_{45}00g_{47}0-g_{47}0000-g_{48}000000g_{49}00000000000000000000000000000000000000000
(1, 1, 2, 2, 2, 2, 2)e_{1}+e_{3}g_{48}g_{-2}2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}+h_{1}g_{1}g_{3}0g_{10}-g_{14}0g_{16}-g_{20}0g_{22}0-g_{25}0g_{27}0-g_{30}0g_{32}00-g_{34}0g_{36}00-g_{38}0g_{40}000-g_{41}0g_{43}000-g_{44}000000-g_{46}0-g_{47}-g_{48}g_{48}-g_{48}00000-g_{49}00000000000000000000000000000000000000000000000
(1, 2, 2, 2, 2, 2, 2)e_{1}+e_{2}g_{49}2h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}g_{2}-g_{8}g_{9}-g_{14}g_{15}0-g_{20}g_{21}0-g_{25}g_{26}00-g_{30}g_{31}00-g_{34}g_{35}000-g_{38}g_{39}000-g_{41}g_{42}0000-g_{44}g_{45}0000-g_{46}g_{47}00000-g_{48}00-g_{49}000000000000000000000000000000000000000000000000000000
We define the symmetric Cartan matrix
by requesting that the entry in the i-th row and j-th column
be the scalar product of the i^th and j^th roots. The symmetric Cartan matrix is:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & 0 & -1 & 1\\ \end{pmatrix}\)
Let the (i, j)^{th} entry of the symmetric Cartan matrix be a_{ij}.
Then we define the co-symmetric Cartan matrix as the matrix whose (i, j)^{th} entry equals 4*a_{ij}/(a_{ii}*a_{jj}). In other words, the co-symmetric Cartan matrix is the symmetric Cartan matrix of the dual root system. The co-symmetric Cartan matrix equals:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -2\\ 0 & 0 & 0 & 0 & 0 & -2 & 4\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 1
Half sum of positive roots: (13/2, 12, 33/2, 20, 45/2, 24, 49/2)= \(\displaystyle 13/2\varepsilon_{1}+11/2\varepsilon_{2}+9/2\varepsilon_{3}+7/2\varepsilon_{4}+5/2\varepsilon_{5}+3/2\varepsilon_{6}+1/2\varepsilon_{7}\)
The fundamental weights (the j^th fundamental weight has scalar product 1
with the j^th simple root times 2 divided by the root length squared,
and 0 with the remaining simple roots):
(1, 1, 1, 1, 1, 1, 1) = \(\displaystyle \varepsilon_{1}\)
(1, 2, 2, 2, 2, 2, 2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}\)
(1, 2, 3, 3, 3, 3, 3) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\)
(1, 2, 3, 4, 4, 4, 4) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}\)
(1, 2, 3, 4, 5, 5, 5) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}\)
(1, 2, 3, 4, 5, 6, 6) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}\)
(1/2, 1, 3/2, 2, 5/2, 3, 7/2) = \(\displaystyle 1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}\)
Below is the simple basis realized in epsilon coordinates. Please note that the epsilon coordinate realizations do not have long roots of length of 2 in types G and C. This means that gramm matrix (w.r.t. the standard scalar product) of the epsilon coordinate realizations in types G and C does not equal the corresponding symmetric Cartan matrix.
(1, 0, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 1, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 1, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{5}-\varepsilon_{6}\)
(0, 0, 0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{6}-\varepsilon_{7}\)
(0, 0, 0, 0, 0, 0, 1) = \(\displaystyle \varepsilon_{7}\)