instLieAlgebra — Mathlib

English

The space of derivations Derivation R A A carries a Lie algebra structure over R, induced by the bracket defined above and the natural action on A.

Русский

Область производных Derivation R A A имеет структуру билиевой алгебры над R, порождённую скобкой и естественным действием на A.

LaTeX

$$$\text{Derivation}(R,A,A) \text{ is a Lie algebra over } R \;\; (\text{bracket as above}).$$$

Lean4

instance instLieAlgebra : LieAlgebra R (Derivation R A A) :=
  { Derivation.instModule with
    lie_smul := fun r d e => by ext a; simp only [commutator_apply, map_smul, smul_sub, smul_apply] }

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