Documentation

Mathlib.Algebra.Lie.Derivation.Basic

Lie derivations #

This file defines Lie derivations and establishes some basic properties.

Main definitions #

Main statements #

Implementation notes #

structure LieDerivation (R : Type u_1) (L : Type u_2) (M : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] extends LinearMap :
Type (max u_2 u_3)

A Lie derivation D from the Lie R-algebra L to the L-module M is an R-linear map that satisfies the Leibniz rule D [a, b] = [a, D b] - [b, D a].

  • toFun : LM
  • map_add' : ∀ (x y : L), (↑self).toFun (x + y) = (↑self).toFun x + (↑self).toFun y
  • map_smul' : ∀ (m : R) (x : L), (↑self).toFun (m x) = (RingHom.id R) m (↑self).toFun x
  • leibniz' : ∀ (a b : L), self a, b = a, self b - b, self a
Instances For
theorem LieDerivation.leibniz' {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (self : LieDerivation R L M) (a : L) (b : L) :
self a, b = a, self b - b, self a
instance LieDerivation.instFunLike {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Equations
  • LieDerivation.instFunLike = { coe := fun (D : LieDerivation R L M) => (↑D).toFun, coe_injective' := }
instance LieDerivation.instLinearMapClass {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Equations
  • =
theorem LieDerivation.toFun_eq_coe {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) :
(↑D).toFun = D
def LieDerivation.Simps.apply {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) :
LM

See Note [custom simps projection]

Equations
instance LieDerivation.instCoeToLinearMap {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Coe (LieDerivation R L M) (L →ₗ[R] M)
Equations
  • LieDerivation.instCoeToLinearMap = { coe := fun (D : LieDerivation R L M) => D }
@[simp]
theorem LieDerivation.mk_coe {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (f : L →ₗ[R] M) (h₁ : ∀ (a b : L), f a, b = a, f b - b, f a) :
{ toLinearMap := f, leibniz' := h₁ } = f
@[simp]
theorem LieDerivation.coeFn_coe {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (f : LieDerivation R L M) :
f = f
theorem LieDerivation.coe_injective {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Function.Injective DFunLike.coe
theorem LieDerivation.ext_iff {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 : LieDerivation R L M} {D2 : LieDerivation R L M} :
D1 = D2 ∀ (a : L), D1 a = D2 a
theorem LieDerivation.ext {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 : LieDerivation R L M} {D2 : LieDerivation R L M} (H : ∀ (a : L), D1 a = D2 a) :
D1 = D2
theorem LieDerivation.congr_fun {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 : LieDerivation R L M} {D2 : LieDerivation R L M} (h : D1 = D2) (a : L) :
D1 a = D2 a
@[simp]
theorem LieDerivation.apply_lie_eq_sub {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) (a : L) (b : L) :
D a, b = a, D b - b, D a
theorem LieDerivation.apply_lie_eq_add {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (D : LieDerivation R L L) (a : L) (b : L) :
D a, b = a, D b + D a, b

For a Lie derivation from a Lie algebra to itself, the usual Leibniz rule holds.

theorem LieDerivation.eqOn_lieSpan {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 : LieDerivation R L M} {D2 : LieDerivation R L M} {s : Set L} (h : Set.EqOn (⇑D1) (⇑D2) s) :
Set.EqOn D1 D2 (LieSubalgebra.lieSpan R L s)

Two Lie derivations equal on a set are equal on its Lie span.

theorem LieDerivation.ext_of_lieSpan_eq_top {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 : LieDerivation R L M} {D2 : LieDerivation R L M} (s : Set L) (hs : LieSubalgebra.lieSpan R L s = ) (h : Set.EqOn (⇑D1) (⇑D2) s) :
D1 = D2

If the Lie span of a set is the whole Lie algebra, then two Lie derivations equal on this set are equal on the whole Lie algebra.

theorem LieDerivation.iterate_apply_lie {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (D : LieDerivation R L L) (n : ) (a : L) (b : L) :
(⇑D)^[n] a, b = ijFinset.antidiagonal n, n.choose ij.1 (⇑D)^[ij.1] a, (⇑D)^[ij.2] b

The general Leibniz rule for Lie derivatives.

theorem LieDerivation.iterate_apply_lie' {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (D : LieDerivation R L L) (n : ) (a : L) (b : L) :
(⇑D)^[n] a, b = iFinset.range (n + 1), n.choose i (⇑D)^[i] a, (⇑D)^[n - i] b

Alternate version of the general Leibniz rule for Lie derivatives.

instance LieDerivation.instZero {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Equations
  • LieDerivation.instZero = { zero := { toLinearMap := 0, leibniz' := } }
@[simp]
theorem LieDerivation.coe_zero {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
0 = 0
@[simp]
theorem LieDerivation.coe_zero_linearMap {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
0 = 0
theorem LieDerivation.zero_apply {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (a : L) :
0 a = 0
instance LieDerivation.instInhabited {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Equations
  • LieDerivation.instInhabited = { default := 0 }
instance LieDerivation.instAdd {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Equations
  • LieDerivation.instAdd = { add := fun (D1 D2 : LieDerivation R L M) => { toLinearMap := D1 + D2, leibniz' := } }
@[simp]
theorem LieDerivation.coe_add {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D1 : LieDerivation R L M) (D2 : LieDerivation R L M) :
(D1 + D2) = D1 + D2
@[simp]
theorem LieDerivation.coe_add_linearMap {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D1 : LieDerivation R L M) (D2 : LieDerivation R L M) :
(D1 + D2) = D1 + D2
theorem LieDerivation.add_apply {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 : LieDerivation R L M} {D2 : LieDerivation R L M} (a : L) :
(D1 + D2) a = D1 a + D2 a
theorem LieDerivation.map_neg {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) (a : L) :
D (-a) = -D a
theorem LieDerivation.map_sub {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) (a : L) (b : L) :
D (a - b) = D a - D b
instance LieDerivation.instNeg {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Equations
  • LieDerivation.instNeg = { neg := fun (D : LieDerivation R L M) => { toLinearMap := -D, leibniz' := } }
@[simp]
theorem LieDerivation.coe_neg {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) :
(-D) = -D
@[simp]
theorem LieDerivation.coe_neg_linearMap {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) :
(-D) = -D
theorem LieDerivation.neg_apply {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D : LieDerivation R L M) (a : L) :
(-D) a = -D a
instance LieDerivation.instSub {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Equations
  • LieDerivation.instSub = { sub := fun (D1 D2 : LieDerivation R L M) => { toLinearMap := D1 - D2, leibniz' := } }
@[simp]
theorem LieDerivation.coe_sub {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D1 : LieDerivation R L M) (D2 : LieDerivation R L M) :
(D1 - D2) = D1 - D2
@[simp]
theorem LieDerivation.coe_sub_linearMap {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (D1 : LieDerivation R L M) (D2 : LieDerivation R L M) :
(D1 - D2) = D1 - D2
theorem LieDerivation.sub_apply {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (a : L) {D1 : LieDerivation R L M} {D2 : LieDerivation R L M} :
(D1 - D2) a = D1 a - D2 a
class LieDerivation.SMulBracketCommClass (S : Type u_4) (L : Type u_5) (α : Type u_6) [SMul S α] [LieRing L] [AddCommGroup α] [LieRingModule L α] :

A typeclass mixin saying that scalar multiplication and Lie bracket are left commutative.

  • smul_bracket_comm : ∀ (s : S) (l : L) (a : α), s l, a = l, s a

    and ⁅⬝, ⬝⁆ are left commutative

Instances
theorem LieDerivation.SMulBracketCommClass.smul_bracket_comm {S : Type u_4} {L : Type u_5} {α : Type u_6} [SMul S α] [LieRing L] [AddCommGroup α] [LieRingModule L α] [self : LieDerivation.SMulBracketCommClass S L α] (s : S) (l : L) (a : α) :
s l, a = l, s a

and ⁅⬝, ⬝⁆ are left commutative

instance LieDerivation.instSMul {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {S : Type u_4} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [LieDerivation.SMulBracketCommClass S L M] :
Equations
  • LieDerivation.instSMul = { smul := fun (r : S) (D : LieDerivation R L M) => { toLinearMap := r D, leibniz' := } }
@[simp]
theorem LieDerivation.coe_smul {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {S : Type u_4} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [LieDerivation.SMulBracketCommClass S L M] (r : S) (D : LieDerivation R L M) :
(r D) = r D
@[simp]
theorem LieDerivation.coe_smul_linearMap {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {S : Type u_4} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [LieDerivation.SMulBracketCommClass S L M] (r : S) (D : LieDerivation R L M) :
(r D) = r D
theorem LieDerivation.smul_apply {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (a : L) {S : Type u_4} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [LieDerivation.SMulBracketCommClass S L M] (r : S) (D : LieDerivation R L M) :
(r D) a = r D a
instance LieDerivation.instSMulBase {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Equations
  • =
instance LieDerivation.instAddCommGroup {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Equations
def LieDerivation.coeFnAddMonoidHom {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
LieDerivation R L M →+ LM

coe_fn as an AddMonoidHom.

Equations
  • LieDerivation.coeFnAddMonoidHom = { toFun := DFunLike.coe, map_zero' := , map_add' := }
Equations
instance LieDerivation.instIsScalarTower {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {S : Type u_4} {T : Type u_5} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [LieDerivation.SMulBracketCommClass S L M] [Monoid T] [DistribMulAction T M] [SMulCommClass R T M] [LieDerivation.SMulBracketCommClass T L M] [SMul S T] [IsScalarTower S T M] :
Equations
  • =
Equations
  • =
instance LieDerivation.instModule {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {S : Type u_4} [Semiring S] [Module S M] [SMulCommClass R S M] [LieDerivation.SMulBracketCommClass S L M] :
Equations
instance LieDerivation.instBracket {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] :

The commutator of two Lie derivations on a Lie algebra is a Lie derivation.

Equations
  • LieDerivation.instBracket = { bracket := fun (D1 D2 : LieDerivation R L L) => { toLinearMap := D1, D2, leibniz' := } }
@[simp]
theorem LieDerivation.commutator_coe_linear_map {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {D1 : LieDerivation R L L} {D2 : LieDerivation R L L} :
D1, D2 = D1, D2
theorem LieDerivation.commutator_apply {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {D1 : LieDerivation R L L} {D2 : LieDerivation R L L} (a : L) :
D1, D2 a = D1 (D2 a) - D2 (D1 a)
instance LieDerivation.instLieRing {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] :
Equations
instance LieDerivation.instLieAlgebra {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] :

The set of Lie derivations from a Lie algebra L to itself is a Lie algebra.

Equations
@[simp]
theorem LieDerivation.lie_apply {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (D₁ : LieDerivation R L L) (D₂ : LieDerivation R L L) (x : L) :
D₁, D₂ x = D₁ (D₂ x) - D₂ (D₁ x)

The Lie algebra morphism from Lie derivations into linear endormophisms.

Equations

The map from Lie derivations to linear endormophisms is injective.

instance LieDerivation.instNoetherian (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] [IsNoetherian R L] :

Lie derivations over a Noetherian Lie algebra form a Noetherian module.

Equations
  • =
@[simp]
theorem LieDerivation.inner_apply_apply (R : Type u_1) (L : Type u_2) (M : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (m : M) (m : L) :
((LieDerivation.inner R L M) m✝) m = m, m✝
def LieDerivation.inner (R : Type u_1) (L : Type u_2) (M : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :

The natural map from a Lie module to the derivations taking values in it.

Equations
  • One or more equations did not get rendered due to their size.
instance LieDerivation.instLieRingModule (R : Type u_1) (L : Type u_2) (M : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Equations
@[simp]
theorem LieDerivation.lie_lieDerivation_apply (R : Type u_1) (L : Type u_2) (M : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L) (y : L) (D : LieDerivation R L M) :
x, D y = y, D x
@[simp]
theorem LieDerivation.lie_coe_lieDerivation_apply (R : Type u_1) (L : Type u_2) (M : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L) (D : LieDerivation R L M) :
x, D = x, D
instance LieDerivation.instLieModule (R : Type u_1) (L : Type u_2) (M : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] :
Equations
  • =
theorem LieDerivation.leibniz_lie (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (x : L) (D₁ : LieDerivation R L L) (D₂ : LieDerivation R L L) :
x, D₁, D₂ = x, D₁, D₂ + D₁, x, D₂