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Characters of Lie algebras #

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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.

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

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