Dynkin index

In mathematics , the Dynkin index of an irreducible representation R is defined as ${\displaystyle T_{R}}$

${\displaystyle \mathrm {Spur} (T^{a}T^{b})=\delta ^{ab}T_{R}}$

where are the generators of the representation. The term takes its name in honor of the Russian mathematician Eugene Dynkin . ${\displaystyle T^{a}}$

For a representation of the Lie algebra with the highest weight , the Dynkin index is defined as ${\displaystyle |\lambda |}$ ${\displaystyle g}$ ${\displaystyle \lambda }$ ${\displaystyle \chi _{\lambda }}$

${\displaystyle \chi _{\lambda }={\frac {\dim(|\lambda |)}{2\dim(g)}}(\lambda ,\lambda +2\rho )}$

wherein the Weyl vector

${\displaystyle \rho ={\frac {1}{2}}\sum _{\alpha \in \Delta ^{+}}\alpha }$

is equal to half the sum of all positive roots of . If, as a special case, the greatest root, that is, is the adjoint representation , then the Dynkin index is equal to the dual Coxeter number . ${\displaystyle g}$ ${\displaystyle \lambda }$ ${\displaystyle |\lambda |}$ ${\displaystyle \chi _{\lambda }}$

literature

• Philippe Di Francesco, Pierre Mathieu, David Sénéchal: Conformal Field Theory. Springer-Verlag, New York 1997, ISBN 0-387-94785-X .