View source for Gell-Mann-Dashen algebra

An infinite-dimensional [[Lie algebra|Lie algebra]] occurring in [[Quantum field theory|quantum field theory]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g1101001.png" /> be a finite-dimensional Lie algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g1101002.png" /> the space of Schwartz test functions (cf. [[Generalized functions, space of|Generalized functions, space of]]). The Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g1101003.png" /> is defined by

<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/g110100/g1101004.png" /></td> </tr></table>

and can be interpreted as the Lie algebra of the group of gauge transformations (cf. [[Gauge transformation|Gauge transformation]]) [[#References|[a1]]]. Representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g1101005.png" /> are called current algebras in [[Quantum field theory|quantum field theory]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g1101006.png" /> be a homomorphism of Lie algebras and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g1101007.png" /> be a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g1101008.png" /> with structure constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g1101009.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g11010010.png" /> defines an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g11010011.png" />-valued distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g11010012.png" /> and it is true that

<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/g110100/g11010013.png" /></td> </tr></table>

Passing to the Fourier image one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g11010014.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g11010015.png" />; then

<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/g110100/g11010016.png" /></td> </tr></table>

R. Dashen and M. Gell-Mann (1966) studied and applied the latter commutation relations in the particular case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110100/g11010017.png" />, [[#References|[a2]]].

General references for current algebras are [[#References|[a3]]], [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Hermann,   "Lie algebras and quantum mechanics" , Benjamin  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Dashen,   M. Gell-Mann,   "Representation of local current algebra at infinite momentum"  ''Phys. Rev. Lett.'' , '''17'''  (1966)  pp. 340–343</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.L. Adler,   R. Dashen,   "Current algebras" , Benjamin  (1968)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Renner,   "Current algebras and their applications" , Pergamon  (1968)</TD></TR></table>