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from <math>\mathfrak g</math> to the Lie algebra of [[endomorphism]]s on a [[vector space]] ''V'' (with the [[commutator]] as the Lie bracket), sending an element ''x'' of <math>\mathfrak g</math> to an element ''ρ''<sub>''x''</sub> of <math>\mathfrak{gl}(V)</math>.
Explicitly, this means that ''ρ'' is a linear map that satisfies
:<math>\rho_{[x,y]} = [\rho_x,\rho_y] = \rho_x\rho_y - \rho_y\rho_x\,</math>
for all ''x,y'' in <math>\mathfrak g</math>. The vector space ''V'', together with the representation ρ, is called a '''<math>\mathfrak g</math>-module'''. (Many authors abuse terminology and refer to ''V'' itself as the representation).
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