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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

def lie_algebra.lie_character (R : Type u) (L : Type v) [comm_ring R] [lie_ring L] [lie_algebra R L] :
Type (max v u)

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

@[simp]
theorem lie_algebra.lie_character_apply_lie {R : Type u} {L : Type v} [comm_ring R] [lie_ring L] [lie_algebra R L] (χ : lie_algebra.lie_character R L) (x y : L) :
χ x,y = 0
theorem lie_algebra.lie_character_apply_of_mem_derived {R : Type u} {L : Type v} [comm_ring R] [lie_ring L] [lie_algebra R L] (χ : lie_algebra.lie_character R L) {x : L} (h : x lie_algebra.derived_series R L 1) :
χ x = 0

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

Equations