Cartan–Gram determinants for the simple Lie groups
Copyright policy )Cartan–Gram determinants for the simple Lie groups
The Cartan–Gram determinants for the simple root systems are evaluated for the simple Lie groups An, Bn, Cn, Dn, and Ek (k=6,7,8). The determinants satisfy a linear recursion relation which turns out to be the same for all these groups. For the En family, the Cartan–Gram determinant contains an explicit factor of (9−n) which vanishes for n=9 and is negative for n>9. This gives a simple explanation why the En family terminates at E8. The Cartan–Gram determinant affords a systematic explanation for the nonexistence of the forbidden Dynkin diagrams.
- University of Michigan United States
- University of Michigan–Flint United States
Semisimple Lie groups and their representations, Physics, Science, simple Lie groups, Lie algebras of Lie groups, gram determinant, Cartan matrices, Dynkin diagrams
Semisimple Lie groups and their representations, Physics, Science, simple Lie groups, Lie algebras of Lie groups, gram determinant, Cartan matrices, Dynkin diagrams
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