Gell-Mann formula

A prescription for constructing anti-Hermitian representations of a symmetric Lie algebra (over the real numbers) $ \mathfrak g $ from representations of an Inönü–Wigner contraction $ {\overline{\mathfrak g}\; } $. One assumes that $ \mathfrak g = \mathfrak k + \mathfrak p $ is a direct sum of vector spaces and

$$ [ \mathfrak k, \mathfrak k ] \subset \mathfrak k, [ \mathfrak k, \mathfrak p ] \subset \mathfrak p, [ \mathfrak p, \mathfrak p ] \subset \mathfrak k. $$

Then $ {\overline{\mathfrak g}\; } = {\overline{\mathfrak k}\; } + {\overline{\mathfrak p}\; } $ and there exists an isomorphism of vector spaces $ \pi : \mathfrak g \rightarrow { {\overline{\mathfrak g}\; } } $ such that $ \pi ( \mathfrak k ) = {\overline{\mathfrak k}\; } $, $ \pi ( \mathfrak p ) = {\overline{\mathfrak p}\; } $ and $ [ \pi ( X ) , \pi ( Y ) ] = \pi ( [ X,Y ] ) $ for all $ X \in \mathfrak k $, $ Y \in \mathfrak g $. In addition, one has $ [ {\overline{\mathfrak p}\; } , {\overline{\mathfrak p}\; } ] = 0 $.

The best studied examples concern the (pseudo-) orthogonal algebras, when $ \mathfrak g = \mathfrak s \mathfrak o ( m + 1,n ) $ or $ \mathfrak g = \mathfrak s \mathfrak o ( m,n + 1 ) $ and $ \mathfrak k = \mathfrak s \mathfrak o ( m,n ) $[a1], [a2]. Then $ {\overline{\mathfrak g}\; } = \mathfrak i \mathfrak s \mathfrak o ( m,n ) $ is an inhomogeneous Lie algebra with $ {\overline{\mathfrak p}\; } = \mathbf R ^ {m + n } $. Let $ M ^ {2} $ be the quadratic $ {\overline{\mathfrak k}\; } $- invariant element from the symmetric algebra of $ {\overline{\mathfrak p}\; } $. If $ {\overline \rho \; } $ is an anti-Hermitian representation of $ {\overline{\mathfrak g}\; } $ such that $ {\overline \rho \; } ( M ^ {2} ) $ is a multiple of the unit operator, then the formula for the representation $ \rho $ of $ \mathfrak g $ reads: $ \rho ( X ) = {\overline \rho \; } ( \pi ( X ) ) $ for all $ X \in \mathfrak k $, and, for all $ Y \in \mathfrak p $,

$$ \rho ( Y ) = \lambda {\overline \rho \; } ( \pi ( Y ) ) + a [ {\overline \rho \; } ( \Delta ) , {\overline \rho \; } ( \pi ( Y ) ) ] , $$

where $ \Delta $ is the second-degree Casimir element from the universal enveloping algebra of $ {\overline{\mathfrak k}\; } $ while $ \lambda $ and $ a $ are parameters. Here, $ \lambda $ is real and arbitrary and $ a $ is pure imaginary and depends on $ {\overline \rho \; } ( M ^ {2} ) $.

References

[a1]	E. Weimar, "The range of validity of the Gell-Mann formula" Nuovo Cim. Lett. , 4 (1972) pp. 43–50
[a2]	R. Hermann, "Lie groups for physicists" , Benjamin (1966)