Lie algebras

This file defines Lie rings and Lie algebras over a commutative ring together with their modules, morphisms and equivalences, as well as various lemmas to make these definitions usable.

Main definitions

• LieRing
• LieAlgebra
• LieRingModule
• LieModule
• LieHom
• LieEquiv
• LieModuleHom
• LieModuleEquiv

Notation

Working over a fixed commutative ring R, we introduce the notations:

• L →ₗ⁅R⁆ L' for a morphism of Lie algebras,
• L ≃ₗ⁅R⁆ L' for an equivalence of Lie algebras,
• M →ₗ⁅R,L⁆ N for a morphism of Lie algebra modules M, N over a Lie algebra L,
• M ≃ₗ⁅R,L⁆ N for an equivalence of Lie algebra modules M, N over a Lie algebra L.

Implementation notes

Lie algebras are defined as modules with a compatible Lie ring structure and thus, like modules, are partially unbundled.

References

• N. Bourbaki, Lie Groups and Lie Algebras, Chapters 1--3

Tags

lie bracket, jacobi identity, lie ring, lie algebra, lie module

class LieRing (L : Type v) extends AddCommGroup L, Bracket L L : Type v

A Lie ring is an additive group with compatible product, known as the bracket, satisfying the Jacobi identity.

• add : L → L → L
• add_assoc (a b c : L) : a + b + c = a + (b + c)
• zero : L
• zero_add (a : L) : 0 + a = a
• add_zero (a : L) : a + 0 = a
• nsmul : ℕ → L → L
• nsmul_zero (x : L) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : L) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : L → L
• sub : L → L → L
• sub_eq_add_neg (a b : L) : a - b = a + -b
• zsmul : ℤ → L → L
• zsmul_zero' (a : L) : SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' (n : ℕ) (a : L) : SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a
• zsmul_neg' (n : ℕ) (a : L) : SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a
• neg_add_cancel (a : L) : -a + a = 0
• add_comm (a b : L) : a + b = b + a
• bracket : L → L → L
• add_lie (x y z : L) : ⁅x + y, z⁆ = ⁅x, z⁆ + ⁅y, z⁆

  A Lie ring bracket is additive in its first component.

• lie_add (x y z : L) : ⁅x, y + z⁆ = ⁅x, y⁆ + ⁅x, z⁆

  A Lie ring bracket is additive in its second component.

• lie_self (x : L) : ⁅x, x⁆ = 0

  A Lie ring bracket vanishes on the diagonal in L × L.

• leibniz_lie (x y z : L) : ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆

  A Lie ring bracket satisfies a Leibniz / Jacobi identity.

class LieAlgebra (R : Type u) (L : Type v) [CommRing R] [LieRing L] extends Module R L : Type (max u v)

A Lie algebra is a module with compatible product, known as the bracket, satisfying the Jacobi identity. Forgetting the scalar multiplication, every Lie algebra is a Lie ring.

• smul : R → L → L
• one_smul (b : L) : 1 • b = b
• mul_smul (x y : R) (b : L) : (x * y) • b = x • y • b
• smul_zero (a : R) : a • 0 = 0
• smul_add (a : R) (x y : L) : a • (x + y) = a • x + a • y
• add_smul (r s : R) (x : L) : (r + s) • x = r • x + s • x
• zero_smul (x : L) : 0 • x = 0
• lie_smul (t : R) (x y : L) : ⁅x, t • y⁆ = t • ⁅x, y⁆

  A Lie algebra bracket is compatible with scalar multiplication in its second argument.

  The compatibility in the first argument is not a class property, but follows since every Lie algebra has a natural Lie module action on itself, see LieModule.

theorem LieAlgebra.ext_iff {R : Type u} {L : Type v} {inst✝ : CommRing R} {inst✝¹ : LieRing L} {x y : LieAlgebra R L} : x = y ↔ SMul.smul = SMul.smul

theorem LieAlgebra.ext {R : Type u} {L : Type v} {inst✝ : CommRing R} {inst✝¹ : LieRing L} {x y : LieAlgebra R L} (smul : SMul.smul = SMul.smul) : x = y

class LieRingModule (L : Type v) (M : Type w) [LieRing L] [AddCommGroup M] extends Bracket L M : Type (max v w)

A Lie ring module is an additive group, together with an additive action of a Lie ring on this group, such that the Lie bracket acts as the commutator of endomorphisms. (For representations of Lie algebras see LieModule.)

• bracket : L → M → M
• add_lie (x y : L) (m : M) : ⁅x + y, m⁆ = ⁅x, m⁆ + ⁅y, m⁆

  A Lie ring module bracket is additive in its first component.

• lie_add (x : L) (m n : M) : ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆

  A Lie ring module bracket is additive in its second component.

• leibniz_lie (x y : L) (m : M) : ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆

  A Lie ring module bracket satisfies a Leibniz / Jacobi identity.

class LieModule (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] : Prop

A Lie module is a module over a commutative ring, together with a linear action of a Lie algebra on this module, such that the Lie bracket acts as the commutator of endomorphisms.

• smul_lie (t : R) (x : L) (m : M) : ⁅t • x, m⁆ = t • ⁅x, m⁆

  A Lie module bracket is compatible with scalar multiplication in its first argument.

• lie_smul (t : R) (x : L) (m : M) : ⁅x, t • m⁆ = t • ⁅x, m⁆

  A Lie module bracket is compatible with scalar multiplication in its second argument.

class IsLieTower (L₁ : Type u_1) (L₂ : Type u_2) (M : Type u_3) [Bracket L₁ L₂] [Bracket L₁ M] [Bracket L₂ M] [Add M] : Prop

A tower of Lie bracket actions encapsulates the Leibniz rule for Lie bracket actions.

More precisely, it does so in a relative setting: Let L₁ and L₂ be two types with Lie bracket actions on a type M endowed with an addition, and additionally assume a Lie bracket action of L₁ on L₂. Then the Leibniz rule asserts for all x : L₁, y : L₂, and m : M that ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆ holds.

Common examples include the case where L₁ is a Lie subalgebra of L₂ and the case where L₂ is a Lie ideal of L₁.

• leibniz_lie (x : L₁) (y : L₂) (m : M) : ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆

theorem leibniz_lie {L₁ : Type u_1} {L₂ : Type u_2} {M : Type u_3} [Bracket L₁ L₂] [Bracket L₁ M] [Bracket L₂ M] [Add M] [IsLieTower L₁ L₂ M] (x : L₁) (y : L₂) (m : M) : ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆

theorem lie_swap_lie {L₁ : Type u_1} {L₂ : Type u_2} {M : Type u_3} [Bracket L₁ L₂] [Bracket L₁ M] [Bracket L₂ M] [Bracket L₂ L₁] [AddCommGroup M] [IsLieTower L₁ L₂ M] [IsLieTower L₂ L₁ M] (x : L₁) (y : L₂) (m : M) : ⁅⁅x, y⁆, m⁆ = -⁅⁅y, x⁆, m⁆

theorem LieAlgebra.toModule_injective (L : Type u_1) [LieRing L] : Function.Injective (@toModule ℚ L Rat.commRing inst✝)

instance instSubsingletonLieAlgebraRat (L : Type u_1) [LieRing L] : Subsingleton (LieAlgebra ℚ L)

@[simp]

theorem add_lie {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x y : L) (m : M) : ⁅x + y, m⁆ = ⁅x, m⁆ + ⁅y, m⁆

@[simp]

theorem lie_add {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m n : M) : ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆

@[simp]

theorem smul_lie {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (t : R) (x : L) (m : M) : ⁅t • x, m⁆ = t • ⁅x, m⁆

@[simp]

theorem lie_smul {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (t : R) (x : L) (m : M) : ⁅x, t • m⁆ = t • ⁅x, m⁆

instance instIsLieTower {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] : IsLieTower L L M

@[simp]

theorem lie_zero {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) : ⁅x, 0⁆ = 0

@[simp]

theorem zero_lie {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (m : M) : ⁅0, m⁆ = 0

@[simp]

theorem lie_self {L : Type v} [LieRing L] (x : L) : ⁅x, x⁆ = 0

instance lieRingSelfModule {L : Type v} [LieRing L] : LieRingModule L L

Equations

• lieRingSelfModule = { toBracket := inferInstance.toBracket, add_lie := ⋯, lie_add := ⋯, leibniz_lie := ⋯ }

@[simp]

theorem lie_skew {L : Type v} [LieRing L] (x y : L) : -⁅y, x⁆ = ⁅x, y⁆

instance lieAlgebraSelfModule {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : LieModule R L L

Every Lie algebra is a module over itself.

@[simp]

theorem neg_lie {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M) : ⁅-x, m⁆ = -⁅x, m⁆

@[simp]

theorem lie_neg {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M) : ⁅x, -m⁆ = -⁅x, m⁆

@[simp]

theorem sub_lie {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x y : L) (m : M) : ⁅x - y, m⁆ = ⁅x, m⁆ - ⁅y, m⁆

@[simp]

theorem lie_sub {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m n : M) : ⁅x, m - n⁆ = ⁅x, m⁆ - ⁅x, n⁆

@[simp]

theorem nsmul_lie {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M) (n : ℕ) : ⁅n • x, m⁆ = n • ⁅x, m⁆

@[simp]

theorem lie_nsmul {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M) (n : ℕ) : ⁅x, n • m⁆ = n • ⁅x, m⁆

theorem zsmul_lie {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M) (a : ℤ) : ⁅a • x, m⁆ = a • ⁅x, m⁆

theorem lie_zsmul {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M) (a : ℤ) : ⁅x, a • m⁆ = a • ⁅x, m⁆

@[simp]

theorem lie_lie {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x y : L) (m : M) : ⁅⁅x, y⁆, m⁆ = ⁅x, ⁅y, m⁆⁆ - ⁅y, ⁅x, m⁆⁆

theorem lie_jacobi {L : Type v} [LieRing L] (x y z : L) : ⁅x, ⁅y, z⁆⁆ + ⁅y, ⁅z, x⁆⁆ + ⁅z, ⁅x, y⁆⁆ = 0

instance LieRing.instLieAlgebra {L : Type v} [LieRing L] : LieAlgebra ℤ L

Equations

• LieRing.instLieAlgebra = { toModule := AddCommGroup.toIntModule L, lie_smul := ⋯ }

instance instLieModuleInt {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] : LieModule ℤ L M

instance LinearMap.instLieRingModule {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [AddCommGroup N] [Module R N] [LieRingModule L N] [LieModule R L N] : LieRingModule L (M →ₗ[R] N)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem LieHom.lie_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [AddCommGroup N] [Module R N] [LieRingModule L N] [LieModule R L N] (f : M →ₗ[R] N) (x : L) (m : M) : ⁅x, f⁆ m = ⁅x, f m⁆ - f ⁅x, m⁆

instance LinearMap.instLieModule {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [AddCommGroup N] [Module R N] [LieRingModule L N] [LieModule R L N] : LieModule R L (M →ₗ[R] N)

instance Module.Dual.instLieRingModule {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] : LieRingModule L (M →ₗ[R] R)

We could avoid defining this by instead defining a LieRingModule L R instance with a zero bracket and relying on LinearMap.instLieRingModule. We do not do this because in the case that L = R we would have a non-defeq diamond via Ring.instBracket.

Equations

• Module.Dual.instLieRingModule = { bracket := fun (x : L) (f : M →ₗ[R] R) => { toFun := fun (m : M) => -f ⁅x, m⁆, map_add' := ⋯, map_smul' := ⋯ }, add_lie := ⋯, lie_add := ⋯, leibniz_lie := ⋯ }

@[simp]

theorem Module.Dual.lie_apply {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L) (m : M) (f : M →ₗ[R] R) : ⁅x, f⁆ m = -f ⁅x, m⁆

instance Module.Dual.instLieModule {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] : LieModule R L (M →ₗ[R] R)

def LieRing.toNonUnitalNonAssocRing (L : Type v) [LieRing L] : NonUnitalNonAssocRing L

It is sometimes useful to regard a LieRing as a NonUnitalNonAssocRing.

Equations

• LieRing.toNonUnitalNonAssocRing L = { toAddCommGroup := inst✝.toAddCommGroup, mul := Bracket.bracket, left_distrib := ⋯, right_distrib := ⋯, zero_mul := ⋯, mul_zero := ⋯ }

theorem sum_lie {L : Type v} [LieRing L] {ι : Type u_1} (s : Finset ι) (f : ι → L) (a : L) : ⁅∑ i ∈ s, f i, a⁆ = ∑ i ∈ s, ⁅f i, a⁆

theorem lie_sum {L : Type v} [LieRing L] {ι : Type u_1} (s : Finset ι) (f : ι → L) (a : L) : ⁅a, ∑ i ∈ s, f i⁆ = ∑ i ∈ s, ⁅a, f i⁆

theorem sum_lie_sum {L : Type v} [LieRing L] {ι : Type u_1} {κ : Type u_3} (s : Finset ι) (t : Finset κ) (f : ι → L) (g : κ → L) : ⁅∑ i ∈ s, f i, ∑ j ∈ t, g j⁆ = ∑ i ∈ s, ∑ j ∈ t, ⁅f i, g j⁆

structure LieHom (R : Type u_1) (L : Type u_2) (L' : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] extends L →ₗ[R] L' : Type (max u_2 u_3)

A morphism of Lie algebras (denoted as L₁ →ₗ⁅R⁆ L₂) is a linear map respecting the bracket operations.

• toFun : L → L'
• 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
• map_lie' {x y : L} : (↑self).toFun ⁅x, y⁆ = ⁅(↑self).toFun x, (↑self).toFun y⁆

  A morphism of Lie algebras is compatible with brackets.

def «term_→ₗ⁅_⁆_» : Lean.TrailingParserDescr

A morphism of Lie algebras (denoted as L₁ →ₗ⁅R⁆ L₂) is a linear map respecting the bracket operations.

Equations

• One or more equations did not get rendered due to their size.

instance LieHom.instCoeLinearMapId {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] : Coe (L₁ →ₗ⁅R⁆ L₂) (L₁ →ₗ[R] L₂)

Equations

• LieHom.instCoeLinearMapId = { coe := LieHom.toLinearMap }

instance LieHom.instFunLike {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] : FunLike (L₁ →ₗ⁅R⁆ L₂) L₁ L₂

Equations

• LieHom.instFunLike = { coe := fun (f : L₁ →ₗ⁅R⁆ L₂) => (↑f).toFun, coe_injective' := ⋯ }

@[simp]

theorem LieHom.coe_toLinearMap {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) : ⇑↑f = ⇑f

@[simp]

theorem LieHom.toFun_eq_coe {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) : (↑f).toFun = ⇑f

@[simp]

theorem LieHom.map_smul {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (c : R) (x : L₁) : f (c • x) = c • f x

@[simp]

theorem LieHom.map_add {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁) : f (x + y) = f x + f y

@[simp]

theorem LieHom.map_sub {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁) : f (x - y) = f x - f y

@[simp]

theorem LieHom.map_neg {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (x : L₁) : f (-x) = -f x

@[simp]

theorem LieHom.map_lie {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁) : f ⁅x, y⁆ = ⁅f x, f y⁆

@[simp]

theorem LieHom.map_zero {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) : f 0 = 0

def LieHom.id {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] : L₁ →ₗ⁅R⁆ L₁

The identity map is a morphism of Lie algebras.

Equations

• LieHom.id = { toLinearMap := LinearMap.id, map_lie' := ⋯ }

@[simp]

theorem LieHom.coe_id {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] : ⇑id = _root_.id

theorem LieHom.id_apply {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] (x : L₁) : id x = x

instance LieHom.instZero {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] : Zero (L₁ →ₗ⁅R⁆ L₂)

The constant 0 map is a Lie algebra morphism.

Equations

• LieHom.instZero = { zero := let __src := 0; { toLinearMap := __src, map_lie' := ⋯ } }

@[simp]

theorem LieHom.coe_zero {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] : ⇑0 = 0

theorem LieHom.zero_apply {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (x : L₁) : 0 x = 0

instance LieHom.instOne {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] : One (L₁ →ₗ⁅R⁆ L₁)

The identity map is a Lie algebra morphism.

Equations

• LieHom.instOne = { one := LieHom.id }

@[simp]

theorem LieHom.coe_one {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] : ⇑1 = _root_.id

theorem LieHom.one_apply {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] (x : L₁) : 1 x = x

instance LieHom.instInhabited {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] : Inhabited (L₁ →ₗ⁅R⁆ L₂)

Equations

• LieHom.instInhabited = { default := 0 }

theorem LieHom.coe_injective {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] : Function.Injective DFunLike.coe

theorem LieHom.ext {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] {f g : L₁ →ₗ⁅R⁆ L₂} (h : ∀ (x : L₁), f x = g x) : f = g

theorem LieHom.ext_iff {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] {f g : L₁ →ₗ⁅R⁆ L₂} : f = g ↔ ∀ (x : L₁), f x = g x

theorem LieHom.congr_fun {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] {f g : L₁ →ₗ⁅R⁆ L₂} (h : f = g) (x : L₁) : f x = g x

@[simp]

theorem LieHom.mk_coe {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (h₁ : ∀ (x y : L₁), f (x + y) = f x + f y) (h₂ : ∀ (m : R) (x : L₁), { toFun := ⇑f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) m • { toFun := ⇑f, map_add' := h₁ }.toFun x) (h₃ : ∀ {x y : L₁}, { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }.toFun ⁅x, y⁆ = ⁅{ toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }.toFun x, { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }.toFun y⁆) : { toFun := ⇑f, map_add' := h₁, map_smul' := h₂, map_lie' := h₃ } = f

@[simp]

theorem LieHom.coe_mk {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ → L₂) (h₁ : ∀ (x y : L₁), f (x + y) = f x + f y) (h₂ : ∀ (m : R) (x : L₁), { toFun := f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) m • { toFun := f, map_add' := h₁ }.toFun x) (h₃ : ∀ {x y : L₁}, { toFun := f, map_add' := h₁, map_smul' := h₂ }.toFun ⁅x, y⁆ = ⁅{ toFun := f, map_add' := h₁, map_smul' := h₂ }.toFun x, { toFun := f, map_add' := h₁, map_smul' := h₂ }.toFun y⁆) : ⇑{ toFun := f, map_add' := h₁, map_smul' := h₂, map_lie' := h₃ } = f

def LieHom.comp {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [LieRing L₃] [LieAlgebra R L₃] (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) : L₁ →ₗ⁅R⁆ L₃

The composition of morphisms is a morphism.

Equations

• f.comp g = { toLinearMap := ↑f ∘ₗ ↑g, map_lie' := ⋯ }

theorem LieHom.comp_apply {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [LieRing L₃] [LieAlgebra R L₃] (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) (x : L₁) : (f.comp g) x = f (g x)

@[simp]

theorem LieHom.coe_comp {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [LieRing L₃] [LieAlgebra R L₃] (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) : ⇑(f.comp g) = ⇑f ∘ ⇑g

@[simp]

theorem LieHom.toLinearMap_comp {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [LieRing L₃] [LieAlgebra R L₃] (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) : ↑(f.comp g) = ↑f ∘ₗ ↑g

@[deprecated LieHom.toLinearMap_comp (since := "2024-12-30")]

theorem LieHom.coe_linearMap_comp {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [LieRing L₃] [LieAlgebra R L₃] (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) : ↑(f.comp g) = ↑f ∘ₗ ↑g

Alias of LieHom.toLinearMap_comp.

@[simp]

theorem LieHom.comp_id {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) : f.comp id = f

@[simp]

theorem LieHom.id_comp {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) : id.comp f = f

def LieHom.inverse {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (g : L₂ → L₁) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) : L₂ →ₗ⁅R⁆ L₁

The inverse of a bijective morphism is a morphism.

Equations

• f.inverse g h₁ h₂ = { toLinearMap := (↑f).inverse g h₁ h₂, map_lie' := ⋯ }

def LieRingModule.compLieHom {R : Type u} {L₁ : Type v} {L₂ : Type w} (M : Type w₁) [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [AddCommGroup M] [LieRingModule L₂ M] (f : L₁ →ₗ⁅R⁆ L₂) : LieRingModule L₁ M

A Lie ring module may be pulled back along a morphism of Lie algebras.

See note [reducible non-instances].

Equations

• LieRingModule.compLieHom M f = { bracket := fun (x : L₁) (m : M) => ⁅f x, m⁆, add_lie := ⋯, lie_add := ⋯, leibniz_lie := ⋯ }

theorem LieRingModule.compLieHom_apply {R : Type u} {L₁ : Type v} {L₂ : Type w} (M : Type w₁) [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [AddCommGroup M] [LieRingModule L₂ M] (f : L₁ →ₗ⁅R⁆ L₂) (x : L₁) (m : M) : ⁅x, m⁆ = ⁅f x, m⁆

theorem LieModule.compLieHom {R : Type u} {L₁ : Type v} {L₂ : Type w} (M : Type w₁) [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [AddCommGroup M] [LieRingModule L₂ M] (f : L₁ →ₗ⁅R⁆ L₂) [Module R M] [LieModule R L₂ M] : LieModule R L₁ M

A Lie module may be pulled back along a morphism of Lie algebras.

structure LieEquiv (R : Type u) (L : Type v) (L' : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] extends L →ₗ⁅R⁆ L' : Type (max v w)

An equivalence of Lie algebras (denoted as L₁ ≃ₗ⁅R⁆ L₂) is a morphism which is also a linear equivalence. We could instead define an equivalence to be a morphism which is also a (plain) equivalence. However, it is more convenient to define via linear equivalence to get .toLinearEquiv for free.

• toFun : L → L'
• map_add' (x y : L) : (↑self.toLieHom).toFun (x + y) = (↑self.toLieHom).toFun x + (↑self.toLieHom).toFun y
• map_smul' (m : R) (x : L) : (↑self.toLieHom).toFun (m • x) = (RingHom.id R) m • (↑self.toLieHom).toFun x
• map_lie' {x y : L} : (↑self.toLieHom).toFun ⁅x, y⁆ = ⁅(↑self.toLieHom).toFun x, (↑self.toLieHom).toFun y⁆
• invFun : L' → L

  The inverse function of an equivalence of Lie algebras

• left_inv : Function.LeftInverse self.invFun (↑self.toLieHom).toFun

  The inverse function of an equivalence of Lie algebras is a left inverse of the underlying function.

• right_inv : Function.RightInverse self.invFun (↑self.toLieHom).toFun

  The inverse function of an equivalence of Lie algebras is a right inverse of the underlying function.

def «term_≃ₗ⁅_⁆_» : Lean.TrailingParserDescr

An equivalence of Lie algebras (denoted as L₁ ≃ₗ⁅R⁆ L₂) is a morphism which is also a linear equivalence. We could instead define an equivalence to be a morphism which is also a (plain) equivalence. However, it is more convenient to define via linear equivalence to get .toLinearEquiv for free.

Equations

• One or more equations did not get rendered due to their size.

def LieEquiv.toLinearEquiv {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (f : L₁ ≃ₗ⁅R⁆ L₂) : L₁ ≃ₗ[R] L₂

Consider an equivalence of Lie algebras as a linear equivalence.

Equations

• f.toLinearEquiv = { toLinearMap := ↑f.toLieHom, invFun := f.invFun, left_inv := ⋯, right_inv := ⋯ }

instance LieEquiv.hasCoeToLieHom {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] : Coe (L₁ ≃ₗ⁅R⁆ L₂) (L₁ →ₗ⁅R⁆ L₂)

Equations

• LieEquiv.hasCoeToLieHom = { coe := LieEquiv.toLieHom }

instance LieEquiv.hasCoeToLinearEquiv {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] : Coe (L₁ ≃ₗ⁅R⁆ L₂) (L₁ ≃ₗ[R] L₂)

Equations

• LieEquiv.hasCoeToLinearEquiv = { coe := LieEquiv.toLinearEquiv }

instance LieEquiv.instEquivLike {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] : EquivLike (L₁ ≃ₗ⁅R⁆ L₂) L₁ L₂

Equations

• LieEquiv.instEquivLike = { coe := fun (f : L₁ ≃ₗ⁅R⁆ L₂) => (↑f.toLieHom).toFun, inv := fun (f : L₁ ≃ₗ⁅R⁆ L₂) => f.invFun, left_inv := ⋯, right_inv := ⋯, coe_injective' := ⋯ }

theorem LieEquiv.coe_toLieHom {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : ⇑e.toLieHom = ⇑e

@[deprecated LieEquiv.coe_toLieHom (since := "2024-12-30")]

theorem LieEquiv.coe_to_lieHom {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : ⇑e.toLieHom = ⇑e

Alias of LieEquiv.coe_toLieHom.

@[simp]

theorem LieEquiv.coe_toLinearEquiv {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : ⇑e.toLinearEquiv = ⇑e

@[simp]

theorem LieEquiv.coe_coe {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : ⇑e.toLieHom = ⇑e

@[deprecated LieEquiv.coe_toLinearEquiv (since := "2024-12-30")]

theorem LieEquiv.coe_to_linearEquiv {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : ⇑e.toLinearEquiv = ⇑e

Alias of LieEquiv.coe_toLinearEquiv.

@[simp]

theorem LieEquiv.toLinearEquiv_mk {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (g : L₂ → L₁) (h₁ : Function.LeftInverse g (↑f).toFun) (h₂ : Function.RightInverse g (↑f).toFun) : { toLieHom := f, invFun := g, left_inv := h₁, right_inv := h₂ }.toLinearEquiv = { toLinearMap := ↑f, invFun := g, left_inv := h₁, right_inv := h₂ }

@[deprecated LieEquiv.toLinearEquiv_mk (since := "2024-12-30")]

theorem LieEquiv.to_linearEquiv_mk {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (g : L₂ → L₁) (h₁ : Function.LeftInverse g (↑f).toFun) (h₂ : Function.RightInverse g (↑f).toFun) : { toLieHom := f, invFun := g, left_inv := h₁, right_inv := h₂ }.toLinearEquiv = { toLinearMap := ↑f, invFun := g, left_inv := h₁, right_inv := h₂ }

Alias of LieEquiv.toLinearEquiv_mk.

theorem LieEquiv.toLinearEquiv_injective {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] : Function.Injective toLinearEquiv

@[deprecated LieEquiv.toLinearEquiv_injective (since := "2024-12-30")]

theorem LieEquiv.coe_linearEquiv_injective {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] : Function.Injective toLinearEquiv

Alias of LieEquiv.toLinearEquiv_injective.

theorem LieEquiv.coe_injective {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] : Function.Injective DFunLike.coe

theorem LieEquiv.ext {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] {f g : L₁ ≃ₗ⁅R⁆ L₂} (h : ∀ (x : L₁), f x = g x) : f = g

theorem LieEquiv.ext_iff {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] {f g : L₁ ≃ₗ⁅R⁆ L₂} : f = g ↔ ∀ (x : L₁), f x = g x

instance LieEquiv.instOne {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] : One (L₁ ≃ₗ⁅R⁆ L₁)

Equations

• LieEquiv.instOne = { one := let __src := 1; { toLinearMap := ↑__src, map_lie' := ⋯, invFun := __src.invFun, left_inv := ⋯, right_inv := ⋯ } }

@[simp]

theorem LieEquiv.one_apply {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] (x : L₁) : 1 x = x

instance LieEquiv.instInhabited {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] : Inhabited (L₁ ≃ₗ⁅R⁆ L₁)

Equations

• LieEquiv.instInhabited = { default := 1 }

theorem LieEquiv.map_lie {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) (x y : L₁) : e ⁅x, y⁆ = ⁅e x, e y⁆

def LieEquiv.refl {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] : L₁ ≃ₗ⁅R⁆ L₁

Lie algebra equivalences are reflexive.

Equations

• LieEquiv.refl = 1

@[simp]

theorem LieEquiv.refl_apply {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] (x : L₁) : refl x = x

def LieEquiv.symm {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : L₂ ≃ₗ⁅R⁆ L₁

Lie algebra equivalences are symmetric.

Equations

• e.symm = { toLieHom := e.inverse e.invFun ⋯ ⋯, invFun := e.toLinearEquiv.symm.invFun, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem LieEquiv.symm_symm {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : e.symm.symm = e

theorem LieEquiv.symm_bijective {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] : Function.Bijective symm

@[simp]

theorem LieEquiv.apply_symm_apply {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) (x : L₂) : e (e.symm x) = x

@[simp]

theorem LieEquiv.symm_apply_apply {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) (x : L₁) : e.symm (e x) = x

@[simp]

theorem LieEquiv.refl_symm {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] : refl.symm = refl

def LieEquiv.trans {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [CommRing R] [LieRing L₁] [LieRing L₂] [LieRing L₃] [LieAlgebra R L₁] [LieAlgebra R L₂] [LieAlgebra R L₃] (e₁ : L₁ ≃ₗ⁅R⁆ L₂) (e₂ : L₂ ≃ₗ⁅R⁆ L₃) : L₁ ≃ₗ⁅R⁆ L₃

Lie algebra equivalences are transitive.

Equations

• e₁.trans e₂ = { toLieHom := e₂.comp e₁.toLieHom, invFun := (e₁.toLinearEquiv.trans e₂.toLinearEquiv).invFun, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem LieEquiv.self_trans_symm {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : e.trans e.symm = refl

@[simp]

theorem LieEquiv.symm_trans_self {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : e.symm.trans e = refl

@[simp]

theorem LieEquiv.trans_apply {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [CommRing R] [LieRing L₁] [LieRing L₂] [LieRing L₃] [LieAlgebra R L₁] [LieAlgebra R L₂] [LieAlgebra R L₃] (e₁ : L₁ ≃ₗ⁅R⁆ L₂) (e₂ : L₂ ≃ₗ⁅R⁆ L₃) (x : L₁) : (e₁.trans e₂) x = e₂ (e₁ x)

@[simp]

theorem LieEquiv.symm_trans {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [CommRing R] [LieRing L₁] [LieRing L₂] [LieRing L₃] [LieAlgebra R L₁] [LieAlgebra R L₂] [LieAlgebra R L₃] (e₁ : L₁ ≃ₗ⁅R⁆ L₂) (e₂ : L₂ ≃ₗ⁅R⁆ L₃) : (e₁.trans e₂).symm = e₂.symm.trans e₁.symm

theorem LieEquiv.bijective {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : Function.Bijective ⇑e.toLieHom

theorem LieEquiv.injective {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : Function.Injective ⇑e.toLieHom

theorem LieEquiv.surjective {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) : Function.Surjective ⇑e.toLieHom

noncomputable def LieEquiv.ofBijective {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (h : Function.Bijective ⇑f) : L₁ ≃ₗ⁅R⁆ L₂

A bijective morphism of Lie algebras yields an equivalence of Lie algebras.

Equations

• LieEquiv.ofBijective f h = { toFun := ⇑f, map_add' := ⋯, map_smul' := ⋯, map_lie' := ⋯, invFun := (LinearEquiv.ofBijective (↑f) h).invFun, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem LieEquiv.ofBijective_toFun {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (h : Function.Bijective ⇑f) (a : L₁) : (ofBijective f h) a = f a

@[simp]

theorem LieEquiv.ofBijective_invFun {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (h : Function.Bijective ⇑f) (a✝ : L₂) : (ofBijective f h).invFun a✝ = (LinearEquiv.ofBijective (↑f) h).symm a✝

structure LieModuleHom (R : Type u) (L : Type v) (M : Type w) (N : Type w₁) [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] extends M →ₗ[R] N : Type (max w w₁)

A morphism of Lie algebra modules (denoted as M →ₗ⁅R,L⁆ N) is a linear map which commutes with the action of the Lie algebra.

• toFun : M → N
• map_add' (x y : M) : (↑self).toFun (x + y) = (↑self).toFun x + (↑self).toFun y
• map_smul' (m : R) (x : M) : (↑self).toFun (m • x) = (RingHom.id R) m • (↑self).toFun x
• map_lie' {x : L} {m : M} : (↑self).toFun ⁅x, m⁆ = ⁅x, (↑self).toFun m⁆

  A module of Lie algebra modules is compatible with the action of the Lie algebra on the modules.

def «term_→ₗ⁅_,_⁆_» : Lean.TrailingParserDescr

A morphism of Lie algebra modules (denoted as M →ₗ⁅R,L⁆ N) is a linear map which commutes with the action of the Lie algebra.

Equations

• One or more equations did not get rendered due to their size.

instance LieModuleHom.instCoeOutLinearMapId {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : CoeOut (M →ₗ⁅R,L⁆ N) (M →ₗ[R] N)

Equations

• LieModuleHom.instCoeOutLinearMapId = { coe := LieModuleHom.toLinearMap }

instance LieModuleHom.instFunLike {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : FunLike (M →ₗ⁅R,L⁆ N) M N

Equations

• LieModuleHom.instFunLike = { coe := fun (f : M →ₗ⁅R,L⁆ N) => (↑f).toFun, coe_injective' := ⋯ }

@[simp]

theorem LieModuleHom.coe_toLinearMap {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) : ⇑↑f = ⇑f

@[simp]

theorem LieModuleHom.map_smul {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) (c : R) (x : M) : f (c • x) = c • f x

@[simp]

theorem LieModuleHom.map_add {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) (x y : M) : f (x + y) = f x + f y

@[simp]

theorem LieModuleHom.map_sub {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) (x y : M) : f (x - y) = f x - f y

@[simp]

theorem LieModuleHom.map_neg {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) (x : M) : f (-x) = -f x

@[simp]

theorem LieModuleHom.map_lie {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) (x : L) (m : M) : f ⁅x, m⁆ = ⁅x, f m⁆

theorem LieModuleHom.map_lie₂ {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} {P : Type w₂} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [LieRingModule L M] [LieRingModule L N] [LieRingModule L P] [LieAlgebra R L] [LieModule R L N] [LieModule R L P] (f : M →ₗ⁅R,L⁆ N →ₗ[R] P) (x : L) (m : M) (n : N) : ⁅x, (f m) n⁆ = (f ⁅x, m⁆) n + (f m) ⁅x, n⁆

@[simp]

theorem LieModuleHom.map_zero {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) : f 0 = 0

def LieModuleHom.id {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] : M →ₗ⁅R,L⁆ M

The identity map is a morphism of Lie modules.

Equations

• LieModuleHom.id = { toLinearMap := LinearMap.id, map_lie' := ⋯ }

@[simp]

theorem LieModuleHom.coe_id {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] : ⇑id = _root_.id

theorem LieModuleHom.id_apply {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (x : M) : id x = x

instance LieModuleHom.instZero {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : Zero (M →ₗ⁅R,L⁆ N)

The constant 0 map is a Lie module morphism.

Equations

• LieModuleHom.instZero = { zero := let __src := 0; { toLinearMap := __src, map_lie' := ⋯ } }

@[simp]

theorem LieModuleHom.coe_zero {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : ⇑0 = 0

theorem LieModuleHom.zero_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (m : M) : 0 m = 0

instance LieModuleHom.instOne {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] : One (M →ₗ⁅R,L⁆ M)

The identity map is a Lie module morphism.

Equations

• LieModuleHom.instOne = { one := LieModuleHom.id }

instance LieModuleHom.instInhabited {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : Inhabited (M →ₗ⁅R,L⁆ N)

Equations

• LieModuleHom.instInhabited = { default := 0 }

theorem LieModuleHom.coe_injective {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : Function.Injective DFunLike.coe

theorem LieModuleHom.ext {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] {f g : M →ₗ⁅R,L⁆ N} (h : ∀ (m : M), f m = g m) : f = g

theorem LieModuleHom.ext_iff {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] {f g : M →ₗ⁅R,L⁆ N} : f = g ↔ ∀ (m : M), f m = g m

theorem LieModuleHom.congr_fun {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] {f g : M →ₗ⁅R,L⁆ N} (h : f = g) (x : M) : f x = g x

@[simp]

theorem LieModuleHom.mk_coe {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) (h : ∀ {x : L} {m : M}, (↑f).toFun ⁅x, m⁆ = ⁅x, (↑f).toFun m⁆) : { toLinearMap := ↑f, map_lie' := h } = f

@[simp]

theorem LieModuleHom.coe_mk {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ[R] N) (h : ∀ {x : L} {m : M}, f.toFun ⁅x, m⁆ = ⁅x, f.toFun m⁆) : ⇑{ toLinearMap := f, map_lie' := h } = ⇑f

theorem LieModuleHom.coe_linear_mk {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ[R] N) (h : ∀ {x : L} {m : M}, f.toFun ⁅x, m⁆ = ⁅x, f.toFun m⁆) : ↑{ toLinearMap := f, map_lie' := h } = f

def LieModuleHom.comp {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} {P : Type w₂} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [LieRingModule L M] [LieRingModule L N] [LieRingModule L P] (f : N →ₗ⁅R,L⁆ P) (g : M →ₗ⁅R,L⁆ N) : M →ₗ⁅R,L⁆ P

The composition of Lie module morphisms is a morphism.

Equations

• f.comp g = { toLinearMap := ↑f ∘ₗ ↑g, map_lie' := ⋯ }

theorem LieModuleHom.comp_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} {P : Type w₂} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [LieRingModule L M] [LieRingModule L N] [LieRingModule L P] (f : N →ₗ⁅R,L⁆ P) (g : M →ₗ⁅R,L⁆ N) (m : M) : (f.comp g) m = f (g m)

@[simp]

theorem LieModuleHom.coe_comp {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} {P : Type w₂} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [LieRingModule L M] [LieRingModule L N] [LieRingModule L P] (f : N →ₗ⁅R,L⁆ P) (g : M →ₗ⁅R,L⁆ N) : ⇑(f.comp g) = ⇑f ∘ ⇑g

@[simp]

theorem LieModuleHom.toLinearMap_comp {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} {P : Type w₂} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [LieRingModule L M] [LieRingModule L N] [LieRingModule L P] (f : N →ₗ⁅R,L⁆ P) (g : M →ₗ⁅R,L⁆ N) : ↑(f.comp g) = ↑f ∘ₗ ↑g

@[deprecated LieModuleHom.toLinearMap_comp (since := "2024-12-30")]

theorem LieModuleHom.coe_linearMap_comp {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} {P : Type w₂} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [LieRingModule L M] [LieRingModule L N] [LieRingModule L P] (f : N →ₗ⁅R,L⁆ P) (g : M →ₗ⁅R,L⁆ N) : ↑(f.comp g) = ↑f ∘ₗ ↑g

Alias of LieModuleHom.toLinearMap_comp.

def LieModuleHom.inverse {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) (g : N → M) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) : N →ₗ⁅R,L⁆ M

The inverse of a bijective morphism of Lie modules is a morphism of Lie modules.

Equations

• f.inverse g h₁ h₂ = { toLinearMap := (↑f).inverse g h₁ h₂, map_lie' := ⋯ }

instance LieModuleHom.instAdd {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : Add (M →ₗ⁅R,L⁆ N)

Equations

• LieModuleHom.instAdd = { add := fun (f g : M →ₗ⁅R,L⁆ N) => let __src := ↑f + ↑g; { toLinearMap := __src, map_lie' := ⋯ } }

instance LieModuleHom.instSub {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : Sub (M →ₗ⁅R,L⁆ N)

Equations

• LieModuleHom.instSub = { sub := fun (f g : M →ₗ⁅R,L⁆ N) => let __src := ↑f - ↑g; { toLinearMap := __src, map_lie' := ⋯ } }

instance LieModuleHom.instNeg {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : Neg (M →ₗ⁅R,L⁆ N)

Equations

• LieModuleHom.instNeg = { neg := fun (f : M →ₗ⁅R,L⁆ N) => let __src := -↑f; { toLinearMap := __src, map_lie' := ⋯ } }

@[simp]

theorem LieModuleHom.coe_add {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f g : M →ₗ⁅R,L⁆ N) : ⇑(f + g) = ⇑f + ⇑g

theorem LieModuleHom.add_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f g : M →ₗ⁅R,L⁆ N) (m : M) : (f + g) m = f m + g m

@[simp]

theorem LieModuleHom.coe_sub {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f g : M →ₗ⁅R,L⁆ N) : ⇑(f - g) = ⇑f - ⇑g

theorem LieModuleHom.sub_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f g : M →ₗ⁅R,L⁆ N) (m : M) : (f - g) m = f m - g m

@[simp]

theorem LieModuleHom.coe_neg {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) : ⇑(-f) = -⇑f

theorem LieModuleHom.neg_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) (m : M) : (-f) m = -f m

instance LieModuleHom.hasNSMul {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : SMul ℕ (M →ₗ⁅R,L⁆ N)

Equations

• LieModuleHom.hasNSMul = { smul := fun (n : ℕ) (f : M →ₗ⁅R,L⁆ N) => let __src := n • ↑f; { toLinearMap := __src, map_lie' := ⋯ } }

@[simp]

theorem LieModuleHom.coe_nsmul {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (n : ℕ) (f : M →ₗ⁅R,L⁆ N) : ⇑(n • f) = n • ⇑f

theorem LieModuleHom.nsmul_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (n : ℕ) (f : M →ₗ⁅R,L⁆ N) (m : M) : (n • f) m = n • f m

instance LieModuleHom.hasZSMul {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : SMul ℤ (M →ₗ⁅R,L⁆ N)

Equations

• LieModuleHom.hasZSMul = { smul := fun (z : ℤ) (f : M →ₗ⁅R,L⁆ N) => let __src := z • ↑f; { toLinearMap := __src, map_lie' := ⋯ } }

@[simp]

theorem LieModuleHom.coe_zsmul {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (z : ℤ) (f : M →ₗ⁅R,L⁆ N) : ⇑(z • f) = z • ⇑f

theorem LieModuleHom.zsmul_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (z : ℤ) (f : M →ₗ⁅R,L⁆ N) (m : M) : (z • f) m = z • f m

instance LieModuleHom.instAddCommGroup {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : AddCommGroup (M →ₗ⁅R,L⁆ N)

Equations

• LieModuleHom.instAddCommGroup = Function.Injective.addCommGroup DFunLike.coe ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance LieModuleHom.instSMul {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] [LieAlgebra R L] [LieModule R L N] : SMul R (M →ₗ⁅R,L⁆ N)

Equations

• LieModuleHom.instSMul = { smul := fun (t : R) (f : M →ₗ⁅R,L⁆ N) => let __src := t • ↑f; { toLinearMap := __src, map_lie' := ⋯ } }

@[simp]

theorem LieModuleHom.coe_smul {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] [LieAlgebra R L] [LieModule R L N] (t : R) (f : M →ₗ⁅R,L⁆ N) : ⇑(t • f) = t • ⇑f

theorem LieModuleHom.smul_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] [LieAlgebra R L] [LieModule R L N] (t : R) (f : M →ₗ⁅R,L⁆ N) (m : M) : (t • f) m = t • f m

instance LieModuleHom.instModule {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] [LieAlgebra R L] [LieModule R L N] : Module R (M →ₗ⁅R,L⁆ N)

Equations

• LieModuleHom.instModule = Function.Injective.module R { toFun := fun (f : M →ₗ⁅R,L⁆ N) => (↑f).toFun, map_zero' := ⋯, map_add' := ⋯ } ⋯ ⋯

structure LieModuleEquiv (R : Type u) (L : Type v) (M : Type w) (N : Type w₁) [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] extends M →ₗ⁅R,L⁆ N : Type (max w w₁)

An equivalence of Lie algebra modules (denoted as M ≃ₗ⁅R,L⁆ N) is a linear equivalence which is also a morphism of Lie algebra modules.

• toFun : M → N
• map_add' (x y : M) : (↑self.toLieModuleHom).toFun (x + y) = (↑self.toLieModuleHom).toFun x + (↑self.toLieModuleHom).toFun y
• map_smul' (m : R) (x : M) : (↑self.toLieModuleHom).toFun (m • x) = (RingHom.id R) m • (↑self.toLieModuleHom).toFun x
• map_lie' {x : L} {m : M} : (↑self.toLieModuleHom).toFun ⁅x, m⁆ = ⁅x, (↑self.toLieModuleHom).toFun m⁆
• invFun : N → M

  The inverse function of an equivalence of Lie modules

• left_inv : Function.LeftInverse self.invFun (↑self.toLieModuleHom).toFun

  The inverse function of an equivalence of Lie modules is a left inverse of the underlying function.

• right_inv : Function.RightInverse self.invFun (↑self.toLieModuleHom).toFun

  The inverse function of an equivalence of Lie modules is a right inverse of the underlying function.

def «term_≃ₗ⁅_,_⁆_» : Lean.TrailingParserDescr

An equivalence of Lie algebra modules (denoted as M ≃ₗ⁅R,L⁆ N) is a linear equivalence which is also a morphism of Lie algebra modules.

Equations

• One or more equations did not get rendered due to their size.

def LieModuleEquiv.toLinearEquiv {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : M ≃ₗ[R] N

View an equivalence of Lie modules as a linear equivalence.

Equations

• e.toLinearEquiv = { toLinearMap := ↑e.toLieModuleHom, invFun := e.invFun, left_inv := ⋯, right_inv := ⋯ }

def LieModuleEquiv.toEquiv {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : M ≃ N

View an equivalence of Lie modules as a type level equivalence.

Equations

• e.toEquiv = { toFun := (↑e.toLieModuleHom).toFun, invFun := e.invFun, left_inv := ⋯, right_inv := ⋯ }

instance LieModuleEquiv.hasCoeToEquiv {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : CoeOut (M ≃ₗ⁅R,L⁆ N) (M ≃ N)

Equations

• LieModuleEquiv.hasCoeToEquiv = { coe := LieModuleEquiv.toEquiv }

instance LieModuleEquiv.hasCoeToLieModuleHom {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : Coe (M ≃ₗ⁅R,L⁆ N) (M →ₗ⁅R,L⁆ N)

Equations

• LieModuleEquiv.hasCoeToLieModuleHom = { coe := LieModuleEquiv.toLieModuleHom }

instance LieModuleEquiv.hasCoeToLinearEquiv {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : CoeOut (M ≃ₗ⁅R,L⁆ N) (M ≃ₗ[R] N)

Equations

• LieModuleEquiv.hasCoeToLinearEquiv = { coe := LieModuleEquiv.toLinearEquiv }

instance LieModuleEquiv.instEquivLike {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : EquivLike (M ≃ₗ⁅R,L⁆ N) M N

Equations

• LieModuleEquiv.instEquivLike = { coe := fun (f : M ≃ₗ⁅R,L⁆ N) => (↑f.toLieModuleHom).toFun, inv := fun (f : M ≃ₗ⁅R,L⁆ N) => f.invFun, left_inv := ⋯, right_inv := ⋯, coe_injective' := ⋯ }

@[simp]

theorem LieModuleEquiv.coe_coe {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : ⇑e.toLieModuleHom = ⇑e

theorem LieModuleEquiv.injective {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : Function.Injective ⇑e

theorem LieModuleEquiv.surjective {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : Function.Surjective ⇑e

@[simp]

theorem LieModuleEquiv.toEquiv_mk {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) (g : N → M) (h₁ : Function.LeftInverse g (↑f).toFun) (h₂ : Function.RightInverse g (↑f).toFun) : { toLieModuleHom := f, invFun := g, left_inv := h₁, right_inv := h₂ }.toEquiv = { toFun := ⇑f, invFun := g, left_inv := h₁, right_inv := h₂ }

@[simp]

theorem LieModuleEquiv.coe_mk {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (f : M →ₗ⁅R,L⁆ N) (invFun : N → M) (h₁ : Function.LeftInverse invFun (↑f).toFun) (h₂ : Function.RightInverse invFun (↑f).toFun) : ⇑{ toLieModuleHom := f, invFun := invFun, left_inv := h₁, right_inv := h₂ } = ⇑f

theorem LieModuleEquiv.coe_toLieModuleHom {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : ⇑e.toLieModuleHom = ⇑e

@[deprecated LieModuleEquiv.coe_toLieModuleHom (since := "2024-12-30")]

theorem LieModuleEquiv.coe_to_lieModuleHom {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : ⇑e.toLieModuleHom = ⇑e

Alias of LieModuleEquiv.coe_toLieModuleHom.

@[simp]

theorem LieModuleEquiv.coe_toLinearEquiv {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : ⇑e.toLinearEquiv = ⇑e

@[deprecated LieModuleEquiv.coe_toLinearEquiv (since := "2024-12-30")]

theorem LieModuleEquiv.coe_to_linearEquiv {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : ⇑e.toLinearEquiv = ⇑e

Alias of LieModuleEquiv.coe_toLinearEquiv.

theorem LieModuleEquiv.toEquiv_injective {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : Function.Injective toEquiv

theorem LieModuleEquiv.ext {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e₁ e₂ : M ≃ₗ⁅R,L⁆ N) (h : ∀ (m : M), e₁ m = e₂ m) : e₁ = e₂

theorem LieModuleEquiv.ext_iff {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] {e₁ e₂ : M ≃ₗ⁅R,L⁆ N} : e₁ = e₂ ↔ ∀ (m : M), e₁ m = e₂ m

instance LieModuleEquiv.instOne {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] : One (M ≃ₗ⁅R,L⁆ M)

Equations

• LieModuleEquiv.instOne = { one := let __src := 1; { toLinearMap := ↑__src, map_lie' := ⋯, invFun := __src.invFun, left_inv := ⋯, right_inv := ⋯ } }

@[simp]

theorem LieModuleEquiv.one_apply {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (m : M) : 1 m = m

instance LieModuleEquiv.instInhabited {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] : Inhabited (M ≃ₗ⁅R,L⁆ M)

Equations

• LieModuleEquiv.instInhabited = { default := 1 }

def LieModuleEquiv.refl {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] : M ≃ₗ⁅R,L⁆ M

Lie module equivalences are reflexive.

Equations

• LieModuleEquiv.refl = 1

@[simp]

theorem LieModuleEquiv.refl_apply {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (m : M) : refl m = m

def LieModuleEquiv.symm {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : N ≃ₗ⁅R,L⁆ M

Lie module equivalences are symmetric.

Equations

• e.symm = { toLieModuleHom := e.inverse e.invFun ⋯ ⋯, invFun := e.toLinearEquiv.symm.invFun, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem LieModuleEquiv.apply_symm_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) (x : N) : e (e.symm x) = x

@[simp]

theorem LieModuleEquiv.symm_apply_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) (x : M) : e.symm (e x) = x

theorem LieModuleEquiv.apply_eq_iff_eq_symm_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] {m : M} {n : N} (e : M ≃ₗ⁅R,L⁆ N) : e m = n ↔ m = e.symm n

@[simp]

theorem LieModuleEquiv.symm_symm {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : e.symm.symm = e

theorem LieModuleEquiv.symm_bijective {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] : Function.Bijective symm

def LieModuleEquiv.trans {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} {P : Type w₂} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [LieRingModule L M] [LieRingModule L N] [LieRingModule L P] (e₁ : M ≃ₗ⁅R,L⁆ N) (e₂ : N ≃ₗ⁅R,L⁆ P) : M ≃ₗ⁅R,L⁆ P

Lie module equivalences are transitive.

Equations

• e₁.trans e₂ = { toLieModuleHom := e₂.comp e₁.toLieModuleHom, invFun := (e₁.toLinearEquiv.trans e₂.toLinearEquiv).invFun, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem LieModuleEquiv.trans_apply {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} {P : Type w₂} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [LieRingModule L M] [LieRingModule L N] [LieRingModule L P] (e₁ : M ≃ₗ⁅R,L⁆ N) (e₂ : N ≃ₗ⁅R,L⁆ P) (m : M) : (e₁.trans e₂) m = e₂ (e₁ m)

@[simp]

theorem LieModuleEquiv.symm_trans {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} {P : Type w₂} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [LieRingModule L M] [LieRingModule L N] [LieRingModule L P] (e₁ : M ≃ₗ⁅R,L⁆ N) (e₂ : N ≃ₗ⁅R,L⁆ P) : (e₁.trans e₂).symm = e₂.symm.trans e₁.symm

@[simp]

theorem LieModuleEquiv.self_trans_symm {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : e.trans e.symm = refl

@[simp]

theorem LieModuleEquiv.symm_trans_self {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) : e.symm.trans e = refl