Editing Weight (representation theory) (section)

==Motivation and general concept==

Given a set ''S'' of <math>n \times n</math> [[matrix (mathematics)|matrices]] over the same field, each of which is [[diagonalizable matrix|diagonalizable]], and any two of which [[commuting matrices|commute]], it is always possible to [[simultaneously diagonalize]] all of the elements of ''S''.<ref group="note">In fact, given a set of commuting matrices over an [[algebraically closed field]], they are [[simultaneously triangularizable]], without needing to assume that they are diagonalizable.</ref> Equivalently, for any set ''S'' of mutually commuting [[semisimple operator|semisimple]] [[linear transformation]]s of a [[dimension (vector space)|finite-dimensional]] [[vector space]] ''V'' there exists a [[basis (linear algebra)|basis]] of ''V'' consisting of ''{{anchor|simultaneous eigenvector}}simultaneous [[eigenvector]]s'' of all elements of ''S''. Each of these common eigenvectors ''v'' ∈ ''V'' defines a [[linear functional]] on the subalgebra ''U'' of End(''V''&thinsp;) generated by the set of [[endomorphism]]s ''S''; this functional is defined as the map which associates to each element of ''U'' its eigenvalue on the eigenvector ''v''. This map is also multiplicative, and sends the identity to 1; thus it is an algebra homomorphism from ''U'' to the base field. This "generalized eigenvalue" is a prototype for the notion of a weight.

The notion is closely related to the idea of a [[multiplicative character]] in [[group theory]], which is a [[group homomorphism|homomorphism]] ''χ'' from a group ''G'' to the [[multiplicative group]] of a field '''F'''. Thus ''χ'': ''G'' → '''F'''<sup>×</sup> satisfies ''χ''(''e'') = 1 (where ''e'' is the [[identity element]] of ''G'') and
:<math>\chi(gh) = \chi(g)\chi(h)</math> for all ''g'', ''h'' in ''G''.
Indeed, if ''G'' [[group representation|acts]] on a vector space ''V'' over '''F''', each simultaneous eigenspace for every element of ''G'', if such exists, determines a multiplicative character on ''G'': the eigenvalue on this common eigenspace of each element of the group.

The notion of multiplicative character can be extended to any algebra ''A'' over '''F''', by replacing ''χ'': ''G'' → '''F'''<sup>×</sup> by a [[linear map]] ''χ'': ''A'' → '''F''' with:
:<math>\chi(ab) = \chi(a)\chi(b)</math> 
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]] (which is generally not an [[associative algebra]]), then instead of requiring multiplicativity of a character, one requires that it maps any Lie bracket to the corresponding [[commutator]]; but since '''F''' is [[commutative]] this simply means that this map must vanish on Lie 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'''<sup>×</sup> 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.)