Documentation

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

@[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.

Instances For
    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
    @[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
    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
    @[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
    @[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 := { toAddHom := ψ.toAddHom, map_smul' := (_ : ∀ (r : R) (x : L), AddHom.toFun ψ.toAddHom (r x) = ↑(RingHom.id R) r AddHom.toFun ψ.toAddHom x) }, map_lie' := (_ : ∀ {x y : L}, AddHom.toFun { toAddHom := ψ.toAddHom, map_smul' := (_ : ∀ (r : R) (x : L), AddHom.toFun ψ.toAddHom (r x) = ↑(RingHom.id R) r AddHom.toFun ψ.toAddHom x) }.toAddHom x, y = AddHom.toFun { toAddHom := ψ.toAddHom, map_smul' := (_ : ∀ (r : R) (x : L), AddHom.toFun ψ.toAddHom (r x) = ↑(RingHom.id R) r AddHom.toFun ψ.toAddHom x) }.toAddHom x, AddHom.toFun { toAddHom := ψ.toAddHom, map_smul' := (_ : ∀ (r : R) (x : L), AddHom.toFun ψ.toAddHom (r x) = ↑(RingHom.id R) r AddHom.toFun ψ.toAddHom x) }.toAddHom y) }

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

    Instances For