statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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Eb_val (h : j ≠ i) : (Eb R i j h).val = matrix.std_basis_matrix i j 1 | rfl | lemma | lie_algebra.special_linear.Eb_val | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"matrix.std_basis_matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sl_non_abelian [fintype n] [nontrivial R] (h : 1 < fintype.card n) :
¬is_lie_abelian ↥(sl n R) | begin
rcases fintype.exists_pair_of_one_lt_card h with ⟨j, i, hij⟩,
let A := Eb R i j hij,
let B := Eb R j i hij.symm,
intros c,
have c' : A.val ⬝ B.val = B.val ⬝ A.val, by { rw [← sub_eq_zero, ← sl_bracket, c.trivial], refl },
simpa [std_basis_matrix, matrix.mul_apply, hij] using congr_fun (congr_fun c' ... | lemma | lie_algebra.special_linear.sl_non_abelian | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"fintype.card",
"fintype.exists_pair_of_one_lt_card",
"is_lie_abelian",
"matrix.mul_apply",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sp [fintype l] : lie_subalgebra R (matrix (l ⊕ l) (l ⊕ l) R) | skew_adjoint_matrices_lie_subalgebra (matrix.J l R) | def | lie_algebra.symplectic.sp | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"lie_subalgebra",
"matrix",
"matrix.J",
"skew_adjoint_matrices_lie_subalgebra"
] | The symplectic Lie algebra: skew-adjoint matrices with respect to the canonical skew-symmetric
bilinear form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
so [fintype n] : lie_subalgebra R (matrix n n R) | skew_adjoint_matrices_lie_subalgebra (1 : matrix n n R) | def | lie_algebra.orthogonal.so | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"lie_subalgebra",
"matrix",
"skew_adjoint_matrices_lie_subalgebra"
] | The definite orthogonal Lie subalgebra: skew-adjoint matrices with respect to the symmetric
bilinear form defined by the identity matrix. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_so [fintype n] (A : matrix n n R) : A ∈ so n R ↔ Aᵀ = -A | begin
erw mem_skew_adjoint_matrices_submodule,
simp only [matrix.is_skew_adjoint, matrix.is_adjoint_pair, matrix.mul_one, matrix.one_mul],
end | lemma | lie_algebra.orthogonal.mem_so | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"matrix",
"matrix.is_adjoint_pair",
"matrix.is_skew_adjoint",
"matrix.mul_one",
"matrix.one_mul",
"mem_skew_adjoint_matrices_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indefinite_diagonal : matrix (p ⊕ q) (p ⊕ q) R | matrix.diagonal $ sum.elim (λ _, 1) (λ _, -1) | def | lie_algebra.orthogonal.indefinite_diagonal | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"matrix",
"matrix.diagonal",
"sum.elim"
] | The indefinite diagonal matrix with `p` 1s and `q` -1s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
so' [fintype p] [fintype q] : lie_subalgebra R (matrix (p ⊕ q) (p ⊕ q) R) | skew_adjoint_matrices_lie_subalgebra $ indefinite_diagonal p q R | def | lie_algebra.orthogonal.so' | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"lie_subalgebra",
"matrix",
"skew_adjoint_matrices_lie_subalgebra"
] | The indefinite orthogonal Lie subalgebra: skew-adjoint matrices with respect to the symmetric
bilinear form defined by the indefinite diagonal matrix. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pso (i : R) : matrix (p ⊕ q) (p ⊕ q) R | matrix.diagonal $ sum.elim (λ _, 1) (λ _, i) | def | lie_algebra.orthogonal.Pso | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"matrix",
"matrix.diagonal",
"sum.elim"
] | A matrix for transforming the indefinite diagonal bilinear form into the definite one, provided
the parameter `i` is a square root of -1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pso_inv {i : R} (hi : i*i = -1) : (Pso p q R i) * (Pso p q R (-i)) = 1 | begin
ext x y, rcases x; rcases y,
{ -- x y : p
by_cases h : x = y; simp [Pso, indefinite_diagonal, h], },
{ -- x : p, y : q
simp [Pso, indefinite_diagonal], },
{ -- x : q, y : p
simp [Pso, indefinite_diagonal], },
{ -- x y : q
by_cases h : x = y; simp [Pso, indefinite_diagonal, h, hi], },
end | lemma | lie_algebra.orthogonal.Pso_inv | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible_Pso {i : R} (hi : i*i = -1) : invertible (Pso p q R i) | invertible_of_right_inverse _ _ (Pso_inv p q R hi) | def | lie_algebra.orthogonal.invertible_Pso | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"invertible"
] | There is a constructive inverse of `Pso p q R i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indefinite_diagonal_transform {i : R} (hi : i*i = -1) :
(Pso p q R i)ᵀ ⬝ (indefinite_diagonal p q R) ⬝ (Pso p q R i) = 1 | begin
ext x y, rcases x; rcases y,
{ -- x y : p
by_cases h : x = y; simp [Pso, indefinite_diagonal, h], },
{ -- x : p, y : q
simp [Pso, indefinite_diagonal], },
{ -- x : q, y : p
simp [Pso, indefinite_diagonal], },
{ -- x y : q
by_cases h : x = y; simp [Pso, indefinite_diagonal, h, hi], },
end | lemma | lie_algebra.orthogonal.indefinite_diagonal_transform | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
so_indefinite_equiv {i : R} (hi : i*i = -1) : so' p q R ≃ₗ⁅R⁆ so (p ⊕ q) R | begin
apply (skew_adjoint_matrices_lie_subalgebra_equiv
(indefinite_diagonal p q R) (Pso p q R i) (invertible_Pso p q R hi)).trans,
apply lie_equiv.of_eq,
ext A, rw indefinite_diagonal_transform p q R hi, refl,
end | def | lie_algebra.orthogonal.so_indefinite_equiv | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"lie_equiv.of_eq",
"skew_adjoint_matrices_lie_subalgebra_equiv"
] | An equivalence between the indefinite and definite orthogonal Lie algebras, over a ring
containing a square root of -1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
so_indefinite_equiv_apply {i : R} (hi : i*i = -1) (A : so' p q R) :
(so_indefinite_equiv p q R hi A : matrix (p ⊕ q) (p ⊕ q) R) =
(Pso p q R i)⁻¹ ⬝ (A : matrix (p ⊕ q) (p ⊕ q) R) ⬝ (Pso p q R i) | by erw [lie_equiv.trans_apply, lie_equiv.of_eq_apply,
skew_adjoint_matrices_lie_subalgebra_equiv_apply] | lemma | lie_algebra.orthogonal.so_indefinite_equiv_apply | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"lie_equiv.of_eq_apply",
"lie_equiv.trans_apply",
"matrix",
"skew_adjoint_matrices_lie_subalgebra_equiv_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
JD : matrix (l ⊕ l) (l ⊕ l) R | matrix.from_blocks 0 1 1 0 | def | lie_algebra.orthogonal.JD | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"matrix",
"matrix.from_blocks"
] | A matrix defining a canonical even-rank symmetric bilinear form.
It looks like this as a `2l x 2l` matrix of `l x l` blocks:
[ 0 1 ]
[ 1 0 ] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
type_D [fintype l] | skew_adjoint_matrices_lie_subalgebra (JD l R) | def | lie_algebra.orthogonal.type_D | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"skew_adjoint_matrices_lie_subalgebra"
] | The classical Lie algebra of type D as a Lie subalgebra of matrices associated to the matrix
`JD`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
PD : matrix (l ⊕ l) (l ⊕ l) R | matrix.from_blocks 1 (-1) 1 1 | def | lie_algebra.orthogonal.PD | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"matrix",
"matrix.from_blocks"
] | A matrix transforming the bilinear form defined by the matrix `JD` into a split-signature
diagonal matrix.
It looks like this as a `2l x 2l` matrix of `l x l` blocks:
[ 1 -1 ]
[ 1 1 ] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
S | indefinite_diagonal l l R | def | lie_algebra.orthogonal.S | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [] | The split-signature diagonal matrix. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
S_as_blocks : S l R = matrix.from_blocks 1 0 0 (-1) | begin
rw [← matrix.diagonal_one, matrix.diagonal_neg, matrix.from_blocks_diagonal],
refl,
end | lemma | lie_algebra.orthogonal.S_as_blocks | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"matrix.diagonal_neg",
"matrix.diagonal_one",
"matrix.from_blocks",
"matrix.from_blocks_diagonal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
JD_transform [fintype l] : (PD l R)ᵀ ⬝ (JD l R) ⬝ (PD l R) = (2 : R) • (S l R) | begin
have h : (PD l R)ᵀ ⬝ (JD l R) = matrix.from_blocks 1 1 1 (-1) := by
{ simp [PD, JD, matrix.from_blocks_transpose, matrix.from_blocks_multiply], },
erw [h, S_as_blocks, matrix.from_blocks_multiply, matrix.from_blocks_smul],
congr; simp [two_smul],
end | lemma | lie_algebra.orthogonal.JD_transform | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"matrix.from_blocks",
"matrix.from_blocks_multiply",
"matrix.from_blocks_smul",
"matrix.from_blocks_transpose",
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
PD_inv [fintype l] [invertible (2 : R)] : (PD l R) * (⅟(2 : R) • (PD l R)ᵀ) = 1 | begin
have h : ⅟(2 : R) • (1 : matrix l l R) + ⅟(2 : R) • 1 = 1 := by
rw [← smul_add, ← (two_smul R _), smul_smul, inv_of_mul_self, one_smul],
erw [matrix.from_blocks_transpose, matrix.from_blocks_smul, matrix.mul_eq_mul,
matrix.from_blocks_multiply],
simp [h],
end | lemma | lie_algebra.orthogonal.PD_inv | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"inv_of_mul_self",
"invertible",
"matrix",
"matrix.from_blocks_multiply",
"matrix.from_blocks_smul",
"matrix.from_blocks_transpose",
"matrix.mul_eq_mul",
"one_smul",
"smul_add",
"smul_smul",
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible_PD [fintype l] [invertible (2 : R)] : invertible (PD l R) | invertible_of_right_inverse _ _ (PD_inv l R) | instance | lie_algebra.orthogonal.invertible_PD | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_D_equiv_so' [fintype l] [invertible (2 : R)] :
type_D l R ≃ₗ⁅R⁆ so' l l R | begin
apply (skew_adjoint_matrices_lie_subalgebra_equiv (JD l R) (PD l R) (by apply_instance)).trans,
apply lie_equiv.of_eq,
ext A,
rw [JD_transform, ← coe_unit_of_invertible (2 : R), ←units.smul_def, lie_subalgebra.mem_coe,
mem_skew_adjoint_matrices_lie_subalgebra_unit_smul],
refl,
end | def | lie_algebra.orthogonal.type_D_equiv_so' | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"invertible",
"lie_equiv.of_eq",
"lie_subalgebra.mem_coe",
"mem_skew_adjoint_matrices_lie_subalgebra_unit_smul",
"skew_adjoint_matrices_lie_subalgebra_equiv"
] | An equivalence between two possible definitions of the classical Lie algebra of type D. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
JB | matrix.from_blocks ((2 : R) • 1 : matrix unit unit R) 0 0 (JD l R) | def | lie_algebra.orthogonal.JB | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"matrix",
"matrix.from_blocks"
] | A matrix defining a canonical odd-rank symmetric bilinear form.
It looks like this as a `(2l+1) x (2l+1)` matrix of blocks:
[ 2 0 0 ]
[ 0 0 1 ]
[ 0 1 0 ]
where sizes of the blocks are:
[`1 x 1` `1 x l` `1 x l`]
[`l x 1` `l x l` `l x l`]
[`l x 1` `l x l` `l x l`] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
type_B [fintype l] | skew_adjoint_matrices_lie_subalgebra(JB l R) | def | lie_algebra.orthogonal.type_B | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"skew_adjoint_matrices_lie_subalgebra"
] | The classical Lie algebra of type B as a Lie subalgebra of matrices associated to the matrix
`JB`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
PB | matrix.from_blocks (1 : matrix unit unit R) 0 0 (PD l R) | def | lie_algebra.orthogonal.PB | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"matrix",
"matrix.from_blocks"
] | A matrix transforming the bilinear form defined by the matrix `JB` into an
almost-split-signature diagonal matrix.
It looks like this as a `(2l+1) x (2l+1)` matrix of blocks:
[ 1 0 0 ]
[ 0 1 -1 ]
[ 0 1 1 ]
where sizes of the blocks are:
[`1 x 1` `1 x l` `1 x l`]
[`l x 1` `l x l` `l x l`]
[`l x 1... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
PB_inv [invertible (2 : R)] : PB l R * matrix.from_blocks 1 0 0 (⅟(PD l R)) = 1 | begin
rw [PB, matrix.mul_eq_mul, matrix.from_blocks_multiply, matrix.mul_inv_of_self],
simp only [matrix.mul_zero, matrix.mul_one, matrix.zero_mul, zero_add, add_zero,
matrix.from_blocks_one]
end | lemma | lie_algebra.orthogonal.PB_inv | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"invertible",
"matrix.from_blocks",
"matrix.from_blocks_multiply",
"matrix.from_blocks_one",
"matrix.mul_eq_mul",
"matrix.mul_inv_of_self",
"matrix.mul_one",
"matrix.mul_zero",
"matrix.zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible_PB [invertible (2 : R)] : invertible (PB l R) | invertible_of_right_inverse _ _ (PB_inv l R) | instance | lie_algebra.orthogonal.invertible_PB | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
JB_transform : (PB l R)ᵀ ⬝ (JB l R) ⬝ (PB l R) = (2 : R) • matrix.from_blocks 1 0 0 (S l R) | by simp [PB, JB, JD_transform, matrix.from_blocks_transpose, matrix.from_blocks_multiply,
matrix.from_blocks_smul] | lemma | lie_algebra.orthogonal.JB_transform | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"matrix.from_blocks",
"matrix.from_blocks_multiply",
"matrix.from_blocks_smul",
"matrix.from_blocks_transpose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indefinite_diagonal_assoc :
indefinite_diagonal (unit ⊕ l) l R =
matrix.reindex_lie_equiv (equiv.sum_assoc unit l l).symm
(matrix.from_blocks 1 0 0 (indefinite_diagonal l l R)) | begin
ext i j,
rcases i with ⟨⟨i₁ | i₂⟩ | i₃⟩;
rcases j with ⟨⟨j₁ | j₂⟩ | j₃⟩;
simp only [indefinite_diagonal, matrix.diagonal_apply, equiv.sum_assoc_apply_inl_inl,
matrix.reindex_lie_equiv_apply, matrix.submatrix_apply, equiv.symm_symm, matrix.reindex_apply,
sum.elim_inl, if_true, eq_self_iff_true, mat... | lemma | lie_algebra.orthogonal.indefinite_diagonal_assoc | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"dmatrix.zero_apply",
"equiv.sum_assoc",
"equiv.sum_assoc_apply_inl_inl",
"equiv.sum_assoc_apply_inl_inr",
"equiv.sum_assoc_apply_inr",
"equiv.symm_symm",
"matrix.diagonal_apply",
"matrix.from_blocks",
"matrix.from_blocks_apply₁₁",
"matrix.from_blocks_apply₁₂",
"matrix.from_blocks_apply₂₁",
"m... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_B_equiv_so' [invertible (2 : R)] :
type_B l R ≃ₗ⁅R⁆ so' (unit ⊕ l) l R | begin
apply (skew_adjoint_matrices_lie_subalgebra_equiv (JB l R) (PB l R) (by apply_instance)).trans,
symmetry,
apply (skew_adjoint_matrices_lie_subalgebra_equiv_transpose
(indefinite_diagonal (unit ⊕ l) l R)
(matrix.reindex_alg_equiv _ (equiv.sum_assoc punit l l)) (matrix.transpose_reindex _ _)).trans,
... | def | lie_algebra.orthogonal.type_B_equiv_so' | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"equiv.sum_assoc",
"invertible",
"lie_equiv.of_eq",
"lie_subalgebra.mem_coe",
"matrix.reindex_alg_equiv",
"matrix.transpose_reindex",
"mem_skew_adjoint_matrices_lie_subalgebra_unit_smul",
"skew_adjoint_matrices_lie_subalgebra_equiv",
"skew_adjoint_matrices_lie_subalgebra_equiv_transpose"
] | An equivalence between two possible definitions of the classical Lie algebra of type B. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_module_bracket_apply (x : L) (m : ⨁ i, M i) (i : ι) :
⁅x, m⁆ i = ⁅x, m i⁆ | map_range_apply _ _ m i | lemma | direct_sum.lie_module_bracket_apply | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_module_of [decidable_eq ι] (j : ι) : M j →ₗ⁅R,L⁆ ⨁ i, M i | { map_lie' := λ x m,
begin
ext i, by_cases h : j = i,
{ rw ← h, simp, },
{ simp [lof, single_eq_of_ne h], },
end,
..lof R ι M j } | def | direct_sum.lie_module_of | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [] | The inclusion of each component into a direct sum as a morphism of Lie modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_module_component (j : ι) : (⨁ i, M i) →ₗ⁅R,L⁆ M j | { map_lie' := λ x m,
by simp only [component, lapply_apply, lie_module_bracket_apply, linear_map.to_fun_eq_coe],
..component R ι M j } | def | direct_sum.lie_module_component | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [
"linear_map.to_fun_eq_coe"
] | The projection map onto one component, as a morphism of Lie modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ring : lie_ring (⨁ i, L i) | { bracket := zip_with (λ i, λ x y, ⁅x, y⁆) (λ i, lie_zero 0),
add_lie := λ x y z, by { ext, simp only [zip_with_apply, add_apply, add_lie], },
lie_add := λ x y z, by { ext, simp only [zip_with_apply, add_apply, lie_add], },
lie_self := λ x, by { ext, simp only [zip_with_apply, add_apply, lie_self, ... | instance | direct_sum.lie_ring | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [
"add_comm_group",
"add_lie",
"leibniz_lie",
"lie_add",
"lie_ring",
"lie_self",
"lie_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bracket_apply (x y : ⨁ i, L i) (i : ι) :
⁅x, y⁆ i = ⁅x i, y i⁆ | zip_with_apply _ _ x y i | lemma | direct_sum.bracket_apply | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra : lie_algebra R (⨁ i, L i) | { lie_smul := λ c x y, by { ext, simp only [
zip_with_apply, smul_apply, bracket_apply, lie_smul] },
..(infer_instance : module R _) } | instance | direct_sum.lie_algebra | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [
"lie_algebra",
"lie_smul",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra_of [decidable_eq ι] (j : ι) : L j →ₗ⁅R⁆ ⨁ i, L i | { to_fun := of L j,
map_lie' := λ x y, by
{ ext i, by_cases h : j = i,
{ rw ← h, simp [of], },
{ simp [of, single_eq_of_ne h], }, },
..lof R ι L j, } | def | direct_sum.lie_algebra_of | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [] | The inclusion of each component into the direct sum as morphism of Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_algebra_component (j : ι) : (⨁ i, L i) →ₗ⁅R⁆ L j | { to_fun := component R ι L j,
map_lie' := λ x y,
by simp only [component, bracket_apply, lapply_apply, linear_map.to_fun_eq_coe],
..component R ι L j } | def | direct_sum.lie_algebra_component | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [
"linear_map.to_fun_eq_coe"
] | The projection map onto one component, as a morphism of Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_algebra_ext {x y : ⨁ i, L i}
(h : ∀ i, lie_algebra_component R ι L i x = lie_algebra_component R ι L i y) : x = y | dfinsupp.ext h | lemma | direct_sum.lie_algebra_ext | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [
"dfinsupp.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_of_of_ne [decidable_eq ι] {i j : ι} (hij : j ≠ i) (x : L i) (y : L j) :
⁅of L i x, of L j y⁆ = 0 | begin
apply lie_algebra_ext R ι L, intros k,
rw lie_hom.map_lie,
simp only [component, of, lapply_apply, single_add_hom_apply, lie_algebra_component_apply,
single_apply, zero_apply],
by_cases hik : i = k,
{ simp only [dif_neg, not_false_iff, lie_zero, hik.symm, hij], },
{ simp only [dif_neg, not_false_i... | lemma | direct_sum.lie_of_of_ne | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [
"lie_hom.map_lie",
"lie_zero",
"zero_lie"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_of_of_eq [decidable_eq ι] {i j : ι} (hij : j = i) (x : L i) (y : L j) :
⁅of L i x, of L j y⁆ = of L i ⁅x, hij.rec_on y⁆ | begin
have : of L j y = of L i (hij.rec_on y), { exact eq.drec (eq.refl _) hij, },
rw [this, ← lie_algebra_of_apply R ι L i ⁅x, hij.rec_on y⁆, lie_hom.map_lie,
lie_algebra_of_apply, lie_algebra_of_apply],
end | lemma | direct_sum.lie_of_of_eq | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [
"lie_hom.map_lie"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_of [decidable_eq ι] {i j : ι} (x : L i) (y : L j) :
⁅of L i x, of L j y⁆ =
if hij : j = i then lie_algebra_of R ι L i ⁅x, hij.rec_on y⁆ else 0 | begin
by_cases hij : j = i,
{ simp only [lie_of_of_eq R ι L hij x y, hij, dif_pos, not_false_iff, lie_algebra_of_apply], },
{ simp only [lie_of_of_ne R ι L hij x y, hij, dif_neg, not_false_iff], },
end | lemma | direct_sum.lie_of | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lie_algebra [decidable_eq ι] (L' : Type w₁) [lie_ring L'] [lie_algebra R L']
(f : Π i, L i →ₗ⁅R⁆ L') (hf : ∀ (i j : ι), i ≠ j → ∀ (x : L i) (y : L j), ⁅f i x, f j y⁆ = 0) :
(⨁ i, L i) →ₗ⁅R⁆ L' | { to_fun := to_module R ι L' (λ i, (f i : L i →ₗ[R] L')),
map_lie' := λ x y,
begin
let f' := λ i, (f i : L i →ₗ[R] L'),
/- The goal is linear in `y`. We can use this to reduce to the case that `y` has only one
non-zero component. -/
suffices : ∀ (i : ι) (y : L i), to_module R ι L' f' ⁅... | def | direct_sum.to_lie_algebra | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [
"eq_or_ne",
"lie_algebra",
"lie_algebra.ad_apply",
"lie_hom.coe_to_linear_map",
"lie_hom.map_lie",
"lie_ring",
"lie_skew",
"linear_map.comp_apply",
"linear_map.map_neg",
"linear_map.to_add_monoid_hom_coe"
] | Given a family of Lie algebras `L i`, together with a family of morphisms of Lie algebras
`f i : L i →ₗ⁅R⁆ L'` into a fixed Lie algebra `L'`, we have a natural linear map:
`(⨁ i, L i) →ₗ[R] L'`. If in addition `⁅f i x, f j y⁆ = 0` for any `x ∈ L i` and `y ∈ L j` (`i ≠ j`)
then this map is a morphism of Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ring_of_ideals : lie_ring (⨁ i, I i) | direct_sum.lie_ring (λ i, ↥(I i)) | instance | direct_sum.lie_ring_of_ideals | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [
"direct_sum.lie_ring",
"lie_ring"
] | The fact that this instance is necessary seems to be a bug in typeclass inference. See
[this Zulip thread](https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/
Typeclass.20resolution.20under.20binders/near/245151099). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_algebra_of_ideals : lie_algebra R (⨁ i, I i) | direct_sum.lie_algebra (λ i, ↥(I i)) | instance | direct_sum.lie_algebra_of_ideals | algebra.lie | src/algebra/lie/direct_sum.lean | [
"algebra.direct_sum.module",
"algebra.lie.of_associative",
"algebra.lie.submodule",
"algebra.lie.basic"
] | [
"direct_sum.lie_algebra",
"lie_algebra"
] | See `direct_sum.lie_ring_of_ideals` comment. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_smul_add_of_span_sup_eq_top (y : L) : ∃ (t : R) (z ∈ I), y = t • x + z | begin
have hy : y ∈ (⊤ : submodule R L) := submodule.mem_top,
simp only [← hxI, submodule.mem_sup, submodule.mem_span_singleton] at hy,
obtain ⟨-, ⟨t, rfl⟩, z, hz, rfl⟩ := hy,
exact ⟨t, z, hz, rfl⟩,
end | lemma | lie_submodule.exists_smul_add_of_span_sup_eq_top | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"submodule",
"submodule.mem_span_singleton",
"submodule.mem_sup",
"submodule.mem_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_top_eq_of_span_sup_eq_top (N : lie_submodule R L M) :
(↑⁅(⊤ : lie_ideal R L), N⁆ : submodule R M) =
(N : submodule R M).map (to_endomorphism R L M x) ⊔ (↑⁅I, N⁆ : submodule R M) | begin
simp only [lie_ideal_oper_eq_linear_span', submodule.sup_span, mem_top, exists_prop,
exists_true_left, submodule.map_coe, to_endomorphism_apply_apply],
refine le_antisymm (submodule.span_le.mpr _) (submodule.span_mono (λ z hz, _)),
{ rintros z ⟨y, n, hn : n ∈ N, rfl⟩,
obtain ⟨t, z, hz, rfl⟩ := exist... | lemma | lie_submodule.lie_top_eq_of_span_sup_eq_top | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"exists_prop",
"exists_true_left",
"lie_ideal",
"lie_smul",
"lie_submodule",
"set_like.mem_coe",
"submodule",
"submodule.map_coe",
"submodule.mem_sup",
"submodule.span_mono",
"submodule.span_union",
"submodule.subset_span",
"submodule.sup_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcs_le_lcs_of_is_nilpotent_span_sup_eq_top {n i j : ℕ} (hxn : (to_endomorphism R L M x)^n = 0)
(hIM : lower_central_series R L M i ≤ I.lcs M j) :
lower_central_series R L M (i + n) ≤ I.lcs M (j + 1) | begin
suffices : ∀ l, ((⊤ : lie_ideal R L).lcs M (i + l) : submodule R M) ≤
(I.lcs M j : submodule R M).map
((to_endomorphism R L M x)^l) ⊔ (I.lcs M (j + 1) : submodule R M),
{ simpa only [bot_sup_eq, lie_ideal.incl_coe, submodule.map_zero, hxn] using this n, },
intros l,
induction l with l ih,
{ simp... | lemma | lie_submodule.lcs_le_lcs_of_is_nilpotent_span_sup_eq_top | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"bot_sup_eq",
"ih",
"le_sup_of_le_left",
"le_sup_of_le_right",
"lie_ideal",
"lie_ideal.incl_coe",
"lie_ideal.lcs_succ",
"linear_map.mul_eq_comp",
"linear_map.one_eq_id",
"lower_central_series",
"pow_succ",
"pow_zero",
"submodule",
"submodule.map_comp",
"submodule.map_id",
"submodule.ma... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_of_is_nilpotent_span_sup_eq_top
(hnp : is_nilpotent $ to_endomorphism R L M x) (hIM : is_nilpotent R I M) :
is_nilpotent R L M | begin
obtain ⟨n, hn⟩ := hnp,
unfreezingI { obtain ⟨k, hk⟩ := hIM, },
have hk' : I.lcs M k = ⊥,
{ simp only [← coe_to_submodule_eq_iff, I.coe_lcs_eq, hk, bot_coe_submodule], },
suffices : ∀ l, lower_central_series R L M (l * n) ≤ I.lcs M l,
{ use k * n,
simpa [hk'] using this k, },
intros l,
inductio... | lemma | lie_submodule.is_nilpotent_of_is_nilpotent_span_sup_eq_top | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"ih",
"is_nilpotent",
"lower_central_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.is_engelian : Prop | ∀ (M : Type u₄) [add_comm_group M], by exactI ∀ [module R M] [lie_ring_module L M], by exactI ∀
[lie_module R L M], by exactI ∀ (h : ∀ (x : L), is_nilpotent (to_endomorphism R L M x)),
lie_module.is_nilpotent R L M | def | lie_algebra.is_engelian | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"add_comm_group",
"is_nilpotent",
"lie_module",
"lie_module.is_nilpotent",
"lie_ring_module",
"module"
] | A Lie algebra `L` is said to be Engelian if a sufficient condition for any `L`-Lie module `M` to
be nilpotent is that the image of the map `L → End(M)` consists of nilpotent elements.
Engel's theorem `lie_algebra.is_engelian_of_is_noetherian` states that any Noetherian Lie algebra is
Engelian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_algebra.is_engelian_of_subsingleton [subsingleton L] : lie_algebra.is_engelian R L | begin
intros M _i1 _i2 _i3 _i4 h,
use 1,
suffices : (⊤ : lie_ideal R L) = ⊥, { simp [this], },
haveI := (lie_submodule.subsingleton_iff R L L).mpr infer_instance,
apply subsingleton.elim,
end | lemma | lie_algebra.is_engelian_of_subsingleton | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"lie_algebra.is_engelian",
"lie_ideal",
"lie_submodule.subsingleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.surjective.is_engelian
{f : L →ₗ⁅R⁆ L₂} (hf : function.surjective f) (h : lie_algebra.is_engelian.{u₁ u₂ u₄} R L) :
lie_algebra.is_engelian.{u₁ u₃ u₄} R L₂ | begin
introsI M _i1 _i2 _i3 _i4 h',
letI : lie_ring_module L M := lie_ring_module.comp_lie_hom M f,
letI : lie_module R L M := comp_lie_hom M f,
have hnp : ∀ x, is_nilpotent (to_endomorphism R L M x) := λ x, h' (f x),
have surj_id : function.surjective (linear_map.id : M →ₗ[R] M) := function.surjective_id,
... | lemma | function.surjective.is_engelian | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"is_nilpotent",
"lie_module",
"lie_module.is_nilpotent",
"lie_ring_module",
"lie_ring_module.comp_lie_hom",
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_equiv.is_engelian_iff (e : L ≃ₗ⁅R⁆ L₂) :
lie_algebra.is_engelian.{u₁ u₂ u₄} R L ↔ lie_algebra.is_engelian.{u₁ u₃ u₄} R L₂ | ⟨e.surjective.is_engelian, e.symm.surjective.is_engelian⟩ | lemma | lie_equiv.is_engelian_iff | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.exists_engelian_lie_subalgebra_of_lt_normalizer
{K : lie_subalgebra R L} (hK₁ : lie_algebra.is_engelian.{u₁ u₂ u₄} R K) (hK₂ : K < K.normalizer) :
∃ (K' : lie_subalgebra R L) (hK' : lie_algebra.is_engelian.{u₁ u₂ u₄} R K'), K < K' | begin
obtain ⟨x, hx₁, hx₂⟩ := set_like.exists_of_lt hK₂,
let K' : lie_subalgebra R L :=
{ lie_mem' := λ y z, lie_subalgebra.lie_mem_sup_of_mem_normalizer hx₁,
.. (R ∙ x) ⊔ (K : submodule R L) },
have hxK' : x ∈ K' := submodule.mem_sup_left (submodule.subset_span (set.mem_singleton _)),
have hKK' : K ≤ K' ... | lemma | lie_algebra.exists_engelian_lie_subalgebra_of_lt_normalizer | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"le_sup_right",
"lie_algebra.is_engelian",
"lie_equiv.of_eq",
"lie_ideal.coe_to_lie_subalgebra_to_submodule",
"lie_subalgebra",
"lie_subalgebra.coe_set_eq",
"lie_subalgebra.coe_submodule_le_coe_submodule",
"lie_subalgebra.equiv_of_le",
"lie_subalgebra.exists_nested_lie_ideal_of_le_normalizer",
"li... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.is_engelian_of_is_noetherian : lie_algebra.is_engelian R L | begin
introsI M _i1 _i2 _i3 _i4 h,
rw ← is_nilpotent_range_to_endomorphism_iff,
let L' := (to_endomorphism R L M).range,
replace h : ∀ (y : L'), is_nilpotent (y : module.End R M),
{ rintros ⟨-, ⟨y, rfl⟩⟩,
simp [h], },
change lie_module.is_nilpotent R L' M,
let s := { K : lie_subalgebra R L' | lie_alge... | lemma | lie_algebra.is_engelian_of_is_noetherian | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"by_contra",
"is_nilpotent",
"is_noetherian",
"is_noetherian_of_surjective",
"lie_algebra.exists_engelian_lie_subalgebra_of_lt_normalizer",
"lie_algebra.is_engelian",
"lie_algebra.is_nilpotent_ad_of_is_nilpotent",
"lie_module.is_nilpotent",
"lie_subalgebra",
"lie_subalgebra.to_endomorphism_eq",
... | *Engel's theorem*.
Note that this implies all traditional forms of Engel's theorem via
`lie_module.nontrivial_max_triv_of_is_nilpotent`, `lie_module.is_nilpotent_iff_forall`,
`lie_algebra.is_nilpotent_iff_forall`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_module.is_nilpotent_iff_forall :
lie_module.is_nilpotent R L M ↔ ∀ x, is_nilpotent $ to_endomorphism R L M x | ⟨begin
introsI h,
obtain ⟨k, hk⟩ := nilpotent_endo_of_nilpotent_module R L M,
exact λ x, ⟨k, hk x⟩,
end,
λ h, lie_algebra.is_engelian_of_is_noetherian M h⟩ | lemma | lie_module.is_nilpotent_iff_forall | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"is_nilpotent",
"lie_algebra.is_engelian_of_is_noetherian",
"lie_module.is_nilpotent"
] | Engel's theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_algebra.is_nilpotent_iff_forall :
lie_algebra.is_nilpotent R L ↔ ∀ x, is_nilpotent $ lie_algebra.ad R L x | lie_module.is_nilpotent_iff_forall | lemma | lie_algebra.is_nilpotent_iff_forall | algebra.lie | src/algebra/lie/engel.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"is_nilpotent",
"lie_algebra.ad",
"lie_algebra.is_nilpotent",
"lie_module.is_nilpotent_iff_forall"
] | Engel's theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel : lib R X → lib R X → Prop
| lie_self (a : lib R X) : rel (a * a) 0
| leibniz_lie (a b c : lib R X) : rel (a * (b * c)) (((a * b) * c) + (b * (a * c)))
| smul (t : R) {a b : lib R X} : rel a b → rel (t • a) (t • b)
| add_right {a b : lib R X} (c : lib R X) : rel a b → rel (a + c) (b + c)
| mul_left (a : lib R X) {b... | inductive | free_lie_algebra.rel | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"leibniz_lie",
"lie_self",
"rel"
] | The quotient of `lib R X` by the equivalence relation generated by this relation will give us
the free Lie algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel.add_left (a : lib R X) {b c : lib R X} (h : rel R X b c) : rel R X (a + b) (a + c) | by { rw [add_comm _ b, add_comm _ c], exact h.add_right _, } | lemma | free_lie_algebra.rel.add_left | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel.neg {a b : lib R X} (h : rel R X a b) : rel R X (-a) (-b) | by simpa only [neg_one_smul] using h.smul (-1) | lemma | free_lie_algebra.rel.neg | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"neg_one_smul",
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel.sub_left (a : lib R X) {b c : lib R X} (h : rel R X b c) : rel R X (a - b) (a - c) | by simpa only [sub_eq_add_neg] using h.neg.add_left a | lemma | free_lie_algebra.rel.sub_left | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel.sub_right {a b : lib R X} (c : lib R X) (h : rel R X a b) : rel R X (a - c) (b - c) | by simpa only [sub_eq_add_neg] using h.add_right (-c) | lemma | free_lie_algebra.rel.sub_right | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel.smul_of_tower {S : Type*} [monoid S] [distrib_mul_action S R] [is_scalar_tower S R R]
(t : S) (a b : lib R X)
(h : rel R X a b) : rel R X (t • a) (t • b) | begin
rw [←smul_one_smul R t a, ←smul_one_smul R t b],
exact h.smul _,
end | lemma | free_lie_algebra.rel.smul_of_tower | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"distrib_mul_action",
"is_scalar_tower",
"monoid",
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_lie_algebra | quot (free_lie_algebra.rel R X) | def | free_lie_algebra | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"free_lie_algebra.rel"
] | The free Lie algebra on the type `X` with coefficients in the commutative ring `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of : X → free_lie_algebra R X | λ x, quot.mk _ (lib.of R x) | def | free_lie_algebra.of | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"free_lie_algebra"
] | The embedding of `X` into the free Lie algebra of `X` with coefficients in the commutative ring
`R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_aux (f : X → commutator_ring L) | lib.lift R f | def | free_lie_algebra.lift_aux | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"commutator_ring"
] | An auxiliary definition used to construct the equivalence `lift` below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_aux_map_smul (f : X → L) (t : R) (a : lib R X) :
lift_aux R f (t • a) = t • lift_aux R f a | non_unital_alg_hom.map_smul _ t a | lemma | free_lie_algebra.lift_aux_map_smul | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"non_unital_alg_hom.map_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_aux_map_add (f : X → L) (a b : lib R X) :
lift_aux R f (a + b) = (lift_aux R f a) + (lift_aux R f b) | non_unital_alg_hom.map_add _ a b | lemma | free_lie_algebra.lift_aux_map_add | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"non_unital_alg_hom.map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_aux_map_mul (f : X → L) (a b : lib R X) :
lift_aux R f (a * b) = ⁅lift_aux R f a, lift_aux R f b⁆ | non_unital_alg_hom.map_mul _ a b | lemma | free_lie_algebra.lift_aux_map_mul | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"non_unital_alg_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_aux_spec (f : X → L) (a b : lib R X) (h : free_lie_algebra.rel R X a b) :
lift_aux R f a = lift_aux R f b | begin
induction h,
case rel.lie_self : a'
{ simp only [lift_aux_map_mul, non_unital_alg_hom.map_zero, lie_self], },
case rel.leibniz_lie : a' b' c'
{ simp only [lift_aux_map_mul, lift_aux_map_add, sub_add_cancel, lie_lie], },
case rel.smul : t a' b' h₁ h₂
{ simp only [lift_aux_map_smul, h₂], },
case rel... | lemma | free_lie_algebra.lift_aux_spec | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"free_lie_algebra.rel",
"lie_lie",
"lie_self",
"non_unital_alg_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk : lib R X →ₙₐ[R] commutator_ring (free_lie_algebra R X) | { to_fun := quot.mk (rel R X),
map_smul' := λ t a, rfl,
map_zero' := rfl,
map_add' := λ a b, rfl,
map_mul' := λ a b, rfl, } | def | free_lie_algebra.mk | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"commutator_ring",
"free_lie_algebra",
"rel"
] | The quotient map as a `non_unital_alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : (X → L) ≃ (free_lie_algebra R X →ₗ⁅R⁆ L) | { to_fun := λ f,
{ to_fun := λ c, quot.lift_on c (lift_aux R f) (lift_aux_spec R f),
map_add' := by { rintros ⟨a⟩ ⟨b⟩, rw ← lift_aux_map_add, refl, },
map_smul' := by { rintros t ⟨a⟩, rw ← lift_aux_map_smul, refl, },
map_lie' := by { rintros ⟨a⟩ ⟨b⟩, rw ← lift_aux_map_mul, refl, }, },
in... | def | free_lie_algebra.lift | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"free_lie_algebra",
"inv_fun",
"lie_hom.coe_mk",
"lift",
"non_unital_alg_hom.congr_fun",
"quot.lift_on_mk"
] | The functor `X ↦ free_lie_algebra R X` from the category of types to the category of Lie
algebras over `R` is adjoint to the forgetful functor in the other direction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_symm_apply (F : free_lie_algebra R X →ₗ⁅R⁆ L) : (lift R).symm F = F ∘ (of R) | rfl | lemma | free_lie_algebra.lift_symm_apply | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"free_lie_algebra",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp_lift (f : X → L) : (lift R f) ∘ (of R) = f | (lift R).left_inv f | lemma | free_lie_algebra.of_comp_lift | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique (f : X → L) (g : free_lie_algebra R X →ₗ⁅R⁆ L) :
g ∘ (of R) = f ↔ g = lift R f | (lift R).symm_apply_eq | lemma | free_lie_algebra.lift_unique | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"free_lie_algebra",
"lift",
"lift_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_of_apply (f : X → L) (x) : lift R f (of R x) = f x | by rw [← function.comp_app (lift R f) (of R) x, of_comp_lift] | lemma | free_lie_algebra.lift_of_apply | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_of (F : free_lie_algebra R X →ₗ⁅R⁆ L) : lift R (F ∘ (of R)) = F | by { rw ← lift_symm_apply, exact (lift R).apply_symm_apply F, } | lemma | free_lie_algebra.lift_comp_of | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"free_lie_algebra",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext {F₁ F₂ : free_lie_algebra R X →ₗ⁅R⁆ L} (h : ∀ x, F₁ (of R x) = F₂ (of R x)) :
F₁ = F₂ | have h' : (lift R).symm F₁ = (lift R).symm F₂, { ext, simp [h], },
(lift R).symm.injective h' | lemma | free_lie_algebra.hom_ext | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"free_lie_algebra",
"hom_ext",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universal_enveloping_equiv_free_algebra :
universal_enveloping_algebra R (free_lie_algebra R X) ≃ₐ[R] free_algebra R X | alg_equiv.of_alg_hom
(universal_enveloping_algebra.lift R $ free_lie_algebra.lift R $ free_algebra.ι R)
(free_algebra.lift R $ (universal_enveloping_algebra.ι R) ∘ (free_lie_algebra.of R))
(by { ext, simp, })
(by { ext, simp, }) | def | free_lie_algebra.universal_enveloping_equiv_free_algebra | algebra.lie | src/algebra/lie/free.lean | [
"algebra.lie.of_associative",
"algebra.lie.non_unital_non_assoc_algebra",
"algebra.lie.universal_enveloping",
"algebra.free_non_unital_non_assoc_algebra"
] | [
"alg_equiv.of_alg_hom",
"free_algebra",
"free_algebra.lift",
"free_algebra.ι",
"free_lie_algebra",
"free_lie_algebra.lift",
"free_lie_algebra.of",
"universal_enveloping_algebra",
"universal_enveloping_algebra.lift",
"universal_enveloping_algebra.ι"
] | The universal enveloping algebra of the free Lie algebra is just the free unital associative
algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_bracket : has_bracket (lie_ideal R L) (lie_submodule R L M) | ⟨λ I N, lie_span R L { m | ∃ (x : I) (n : N), ⁅(x : L), (n : M)⁆ = m }⟩ | instance | lie_submodule.has_bracket | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"has_bracket",
"lie_ideal",
"lie_submodule"
] | Given a Lie module `M` over a Lie algebra `L`, the set of Lie ideals of `L` acts on the set
of submodules of `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ideal_oper_eq_span :
⁅I, N⁆ = lie_span R L { m | ∃ (x : I) (n : N), ⁅(x : L), (n : M)⁆ = m } | rfl | lemma | lie_submodule.lie_ideal_oper_eq_span | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ideal_oper_eq_linear_span :
(↑⁅I, N⁆ : submodule R M) = submodule.span R { m | ∃ (x : I) (n : N), ⁅(x : L), (n : M)⁆ = m } | begin
apply le_antisymm,
{ let s := {m : M | ∃ (x : ↥I) (n : ↥N), ⁅(x : L), (n : M)⁆ = m},
have aux : ∀ (y : L) (m' ∈ submodule.span R s), ⁅y, m'⁆ ∈ submodule.span R s,
{ intros y m' hm', apply submodule.span_induction hm',
{ rintros m'' ⟨x, n, hm''⟩, rw [← hm'', leibniz_lie],
refine submodule... | lemma | lie_submodule.lie_ideal_oper_eq_linear_span | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"aux",
"leibniz_lie",
"lie_add",
"lie_smul",
"lie_submodule",
"lie_zero",
"submodule",
"submodule.add_mem",
"submodule.smul_mem",
"submodule.span",
"submodule.span_induction",
"submodule.subset_span",
"submodule.zero_mem"
] | See also `lie_submodule.lie_ideal_oper_eq_linear_span'` and
`lie_submodule.lie_ideal_oper_eq_tensor_map_range`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ideal_oper_eq_linear_span' :
(↑⁅I, N⁆ : submodule R M) = submodule.span R { m | ∃ (x ∈ I) (n ∈ N), ⁅x, n⁆ = m } | begin
rw lie_ideal_oper_eq_linear_span,
congr,
ext m,
split,
{ rintros ⟨⟨x, hx⟩, ⟨n, hn⟩, rfl⟩,
exact ⟨x, hx, n, hn, rfl⟩, },
{ rintros ⟨x, hx, n, hn, rfl⟩,
exact ⟨⟨x, hx⟩, ⟨n, hn⟩, rfl⟩, },
end | lemma | lie_submodule.lie_ideal_oper_eq_linear_span' | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"submodule",
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_le_iff : ⁅I, N⁆ ≤ N' ↔ ∀ (x ∈ I) (m ∈ N), ⁅x, m⁆ ∈ N' | begin
rw [lie_ideal_oper_eq_span, lie_submodule.lie_span_le],
refine ⟨λ h x hx m hm, h ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩, _⟩,
rintros h _ ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩,
exact h x hx m hm,
end | lemma | lie_submodule.lie_le_iff | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"lie_submodule.lie_span_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_coe_mem_lie (x : I) (m : N) : ⁅(x : L), (m : M)⁆ ∈ ⁅I, N⁆ | by { rw lie_ideal_oper_eq_span, apply subset_lie_span, use [x, m], } | lemma | lie_submodule.lie_coe_mem_lie | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_mem_lie {x : L} {m : M} (hx : x ∈ I) (hm : m ∈ N) : ⁅x, m⁆ ∈ ⁅I, N⁆ | N.lie_coe_mem_lie I ⟨x, hx⟩ ⟨m, hm⟩ | lemma | lie_submodule.lie_mem_lie | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_comm : ⁅I, J⁆ = ⁅J, I⁆ | begin
suffices : ∀ (I J : lie_ideal R L), ⁅I, J⁆ ≤ ⁅J, I⁆, { exact le_antisymm (this I J) (this J I), },
clear I J, intros I J,
rw [lie_ideal_oper_eq_span, lie_span_le], rintros x ⟨y, z, h⟩, rw ← h,
rw [← lie_skew, ← lie_neg, ← lie_submodule.coe_neg],
apply lie_coe_mem_lie,
end | lemma | lie_submodule.lie_comm | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"lie_ideal",
"lie_neg",
"lie_skew",
"lie_submodule.coe_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_le_right : ⁅I, N⁆ ≤ N | begin
rw [lie_ideal_oper_eq_span, lie_span_le], rintros m ⟨x, n, hn⟩, rw ← hn,
exact N.lie_mem n.property,
end | lemma | lie_submodule.lie_le_right | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_le_left : ⁅I, J⁆ ≤ I | by { rw lie_comm, exact lie_le_right I J, } | lemma | lie_submodule.lie_le_left | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_le_inf : ⁅I, J⁆ ≤ I ⊓ J | by { rw le_inf_iff, exact ⟨lie_le_left I J, lie_le_right J I⟩, } | lemma | lie_submodule.lie_le_inf | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"le_inf_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_bot : ⁅I, (⊥ : lie_submodule R L M)⁆ = ⊥ | by { rw eq_bot_iff, apply lie_le_right, } | lemma | lie_submodule.lie_bot | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"eq_bot_iff",
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_lie : ⁅(⊥ : lie_ideal R L), N⁆ = ⊥ | begin
suffices : ⁅(⊥ : lie_ideal R L), N⁆ ≤ ⊥, { exact le_bot_iff.mp this, },
rw [lie_ideal_oper_eq_span, lie_span_le], rintros m ⟨⟨x, hx⟩, n, hn⟩, rw ← hn,
change x ∈ (⊥ : lie_ideal R L) at hx, rw mem_bot at hx, simp [hx],
end | lemma | lie_submodule.bot_lie | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_eq_bot_iff : ⁅I, N⁆ = ⊥ ↔ ∀ (x ∈ I) (m ∈ N), ⁅(x : L), m⁆ = 0 | begin
rw [lie_ideal_oper_eq_span, lie_submodule.lie_span_eq_bot_iff],
refine ⟨λ h x hx m hm, h ⁅x, m⁆ ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩, _⟩,
rintros h - ⟨⟨x, hx⟩, ⟨⟨n, hn⟩, rfl⟩⟩,
exact h x hx n hn,
end | lemma | lie_submodule.lie_eq_bot_iff | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"lie_submodule.lie_span_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_lie (h₁ : I ≤ J) (h₂ : N ≤ N') : ⁅I, N⁆ ≤ ⁅J, N'⁆ | begin
intros m h,
rw [lie_ideal_oper_eq_span, mem_lie_span] at h, rw [lie_ideal_oper_eq_span, mem_lie_span],
intros N hN, apply h, rintros m' ⟨⟨x, hx⟩, ⟨n, hn⟩, hm⟩, rw ← hm, apply hN,
use [⟨x, h₁ hx⟩, ⟨n, h₂ hn⟩], refl,
end | lemma | lie_submodule.mono_lie | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_lie_left (h : I ≤ J) : ⁅I, N⁆ ≤ ⁅J, N⁆ | mono_lie _ _ _ _ h (le_refl N) | lemma | lie_submodule.mono_lie_left | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_lie_right (h : N ≤ N') : ⁅I, N⁆ ≤ ⁅I, N'⁆ | mono_lie _ _ _ _ (le_refl I) h | lemma | lie_submodule.mono_lie_right | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_sup : ⁅I, N ⊔ N'⁆ = ⁅I, N⁆ ⊔ ⁅I, N'⁆ | begin
have h : ⁅I, N⁆ ⊔ ⁅I, N'⁆ ≤ ⁅I, N ⊔ N'⁆,
{ rw sup_le_iff, split; apply mono_lie_right; [exact le_sup_left, exact le_sup_right], },
suffices : ⁅I, N ⊔ N'⁆ ≤ ⁅I, N⁆ ⊔ ⁅I, N'⁆, { exact le_antisymm this h, }, clear h,
rw [lie_ideal_oper_eq_span, lie_span_le], rintros m ⟨x, ⟨n, hn⟩, h⟩, erw lie_submodule.mem_s... | lemma | lie_submodule.lie_sup | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"le_sup_left",
"le_sup_right",
"lie_submodule.mem_sup",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_lie : ⁅I ⊔ J, N⁆ = ⁅I, N⁆ ⊔ ⁅J, N⁆ | begin
have h : ⁅I, N⁆ ⊔ ⁅J, N⁆ ≤ ⁅I ⊔ J, N⁆,
{ rw sup_le_iff, split; apply mono_lie_left; [exact le_sup_left, exact le_sup_right], },
suffices : ⁅I ⊔ J, N⁆ ≤ ⁅I, N⁆ ⊔ ⁅J, N⁆, { exact le_antisymm this h, }, clear h,
rw [lie_ideal_oper_eq_span, lie_span_le], rintros m ⟨⟨x, hx⟩, n, h⟩, erw lie_submodule.mem_sup,
... | lemma | lie_submodule.sup_lie | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"le_sup_left",
"le_sup_right",
"lie_submodule.mem_sup",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_inf : ⁅I, N ⊓ N'⁆ ≤ ⁅I, N⁆ ⊓ ⁅I, N'⁆ | by { rw le_inf_iff, split; apply mono_lie_right; [exact inf_le_left, exact inf_le_right], } | lemma | lie_submodule.lie_inf | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"inf_le_left",
"inf_le_right",
"le_inf_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_lie : ⁅I ⊓ J, N⁆ ≤ ⁅I, N⁆ ⊓ ⁅J, N⁆ | by { rw le_inf_iff, split; apply mono_lie_left; [exact inf_le_left, exact inf_le_right], } | lemma | lie_submodule.inf_lie | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"inf_le_left",
"inf_le_right",
"le_inf_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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