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Cartan–Gram determinants for the simple Lie groups

Wu, Alfred C. T.

Wu, Alfred C. T.

1982-11

Citation:Wu, Alfred C. T. (1982). "Cartan–Gram determinants for the simple Lie groups." Journal of Mathematical Physics 23(11): 2019-2021. <http://hdl.handle.net/2027.42/70011>

Abstract: 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.