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

• LieAlgebra.LieCharacter
• LieAlgebra.lieCharacterEquivLinearDual

Tags

lie algebra, lie character

@[reducible, inline]

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

• LieAlgebra.LieCharacter R L = (L →ₗ⁅R⁆ R)

theorem LieAlgebra.lieCharacter_apply_lie {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (χ : LieCharacter R L) (x 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 : LieCharacter R L) (x 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 : LieCharacter R L) {x : L} (h : x ∈ derivedSeries R L 1) : χ x = 0

def LieAlgebra.lieCharacterEquivLinearDual {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] [IsLieAbelian L] : 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.

@[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) : lieCharacterEquivLinearDual.symm ψ = { toLinearMap := ψ, map_lie' := ⋯ }

@[simp]

theorem LieAlgebra.lieCharacterEquivLinearDual_apply {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] [IsLieAbelian L] (χ : LieCharacter R L) : lieCharacterEquivLinearDual χ = ↑χ