English
The adjoint action ad on an associative algebra A is equal to left multiplication minus right multiplication.
Русский
Адъектное отображение совпадает с левым минус правым умножением.
LaTeX
$$$\mathrm{ad}_{R,A} = \mathrm{mulLeft}_R - \mathrm{mulRight}_R$$$
Lean4
/-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A] (A' : Subalgebra R A) :
LieSubalgebra R A :=
{ Subalgebra.toSubmodule A' with
lie_mem' := fun {x y} hx hy => by
change ⁅x, y⁆ ∈ A'; change x ∈ A' at hx; change y ∈ A' at hy
rw [LieRing.of_associative_ring_bracket]
have hxy := A'.mul_mem hx hy
have hyx := A'.mul_mem hy hx
exact Submodule.sub_mem (Subalgebra.toSubmodule A') hxy hyx }
