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Mathlib.Algebra.Lie.Character

Characters of Lie algebras #

A character of a Lie algebra L over a commutative ring R is a morphism of Lie algebras L → R, where R is regarded as a Lie algebra over itself via the ring commutator. For an Abelian Lie algebra (e.g., a Cartan subalgebra of a semisimple Lie algebra) a character is just a linear form.

Main definitions #

Tags #

lie algebra, lie character

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@[inline, reducible]
abbrev LieAlgebra.LieCharacter (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] :
Type (max v u)

A character of a Lie algebra is a morphism to the scalars.

Equations
Instances For
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    theorem LieAlgebra.lieCharacter_apply_lie {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (χ : LieAlgebra.LieCharacter R L) (x : L) (y : L) :
    χ ⁅x, y⁆ = 0
    source
    @[simp]
    theorem LieAlgebra.lieCharacter_apply_lie' {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (χ : LieAlgebra.LieCharacter R L) (x : L) (y : L) :
    ⁅χ x, χ y⁆ = 0
    source
    theorem LieAlgebra.lieCharacter_apply_of_mem_derived {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (χ : LieAlgebra.LieCharacter R L) {x : L} (h : x ∈ LieAlgebra.derivedSeries R L 1) :
    χ x = 0
    source
    @[simp]
    theorem LieAlgebra.lieCharacterEquivLinearDual_apply {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] [IsLieAbelian L] (χ : LieAlgebra.LieCharacter R L) :
    LieAlgebra.lieCharacterEquivLinearDual χ = ↑χ
    source
    @[simp]
    theorem LieAlgebra.lieCharacterEquivLinearDual_symm_apply {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] [IsLieAbelian L] (ψ : Module.Dual R L) :
    LieAlgebra.lieCharacterEquivLinearDual.symm ψ = { toLinearMap := ψ, map_lie' := (_ : ∀ {x y : L}, ψ.toAddHom.toFun ⁅x, y⁆ = ⁅ψ.toAddHom.toFun x, ψ.toAddHom.toFun y⁆) }
    source
    def LieAlgebra.lieCharacterEquivLinearDual {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] [IsLieAbelian L] :
    LieAlgebra.LieCharacter R L ≃ Module.Dual R L

    For an Abelian Lie algebra, characters are just linear forms.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For