View source for Gell-Mann formula

A prescription for constructing anti-Hermitian representations of a symmetric [[Lie algebra|Lie algebra]] (over the real numbers) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101101.png" /> from representations of an Inönü–Wigner contraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101102.png" />. One assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101103.png" /> is a direct sum of vector spaces and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101104.png" /></td> </tr></table>

Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101105.png" /> and there exists an isomorphism of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101106.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101109.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011011.png" />. In addition, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011012.png" />.

The best studied examples concern the (pseudo-) orthogonal algebras, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011013.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011015.png" /> [[#References|[a1]]], [[#References|[a2]]]. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011016.png" /> is an inhomogeneous Lie algebra with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011017.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011018.png" /> be the quadratic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011019.png" />-invariant element from the symmetric algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011021.png" /> is an anti-Hermitian representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011023.png" /> is a multiple of the unit operator, then the formula for the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011025.png" /> reads: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011026.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011027.png" />, and, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011028.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011029.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011030.png" /> is the second-degree Casimir element from the [[Universal enveloping algebra|universal enveloping algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011031.png" /> while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011033.png" /> are parameters. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011034.png" /> is real and arbitrary and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011035.png" /> is pure imaginary and depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011036.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Weimar,   "The range of validity of the Gell-Mann formula"  ''Nuovo Cim. Lett.'' , '''4'''  (1972)  pp. 43–50</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Hermann,   "Lie groups for physicists" , Benjamin  (1966)</TD></TR></table>