Gell-Mann formula
A prescription for constructing anti-Hermitian representations of a symmetric Lie algebra (over the real numbers) from representations of an Inönü–Wigner contraction
. One assumes that
is a direct sum of vector spaces and
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Then and there exists an isomorphism of vector spaces
such that
,
and
for all
,
. In addition, one has
.
The best studied examples concern the (pseudo-) orthogonal algebras, when or
and
[a1], [a2]. Then
is an inhomogeneous Lie algebra with
. Let
be the quadratic
-invariant element from the symmetric algebra of
. If
is an anti-Hermitian representation of
such that
is a multiple of the unit operator, then the formula for the representation
of
reads:
for all
, and, for all
,
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where is the second-degree Casimir element from the universal enveloping algebra of
while
and
are parameters. Here,
is real and arbitrary and
is pure imaginary and depends on
.
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) |
Gell-Mann formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gell-Mann_formula&oldid=47062