Weight (representation theory): Difference between revisions

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for all ''a'', ''b'' in ''A''. If an algebra ''A'' [[algebra representation|acts]] on a vector space ''V'' over '''F''' to any simultaneous eigenspace, this corresponds an [[algebra homomorphism]] from ''A'' to '''F''' assigning to each element of ''A'' its eigenvalue.

If ''A'' is a [[Lie algebra]],{{clarify|reason=Until now,(which is Agenerally wasnot an associative algebra.), Andthen theinstead definitionsof thatrequiring havemultiplicativity beenof seta upcharacter, soone farrequires don'tthat makeit sensemaps for aany Lie algebra.|date=Octoberbracket 2015}} thento the commutativitycorresponding of[[commutator]]; thebut fieldsince and'''F''' theis anticommutativitycommutative ofthis the Lie bracketsimply implymeans that this map must vanish on [[commutator]]sLie brackets: ''χ''([a,b])=0. A '''weight''' on a Lie algebra '''g''' over a field '''F''' is a linear map λ: '''g''' → '''F''' with λ([''x'', ''y''])=0 for all ''x'', ''y'' in '''g'''. Any weight on a Lie algebra '''g''' vanishes on the [[derived algebra]] ['''g''','''g'''] and hence descends to a weight on the [[abelian Lie algebra]] '''g'''/['''g''','''g''']. Thus weights are primarily of interest for abelian Lie algebras, where they reduce to the simple notion of a generalized eigenvalue for space of commuting linear transformations.

If ''G'' is a [[Lie group]] or an [[algebraic group]], then a multiplicative character θ: ''G'' → '''F'''^× induces a weight ''χ'' = dθ: '''g''' → '''F''' on its Lie algebra by differentiation. (For Lie groups, this is differentiation at the identity element of ''G'', and the algebraic group case is an abstraction using the notion of a derivation.)