| import OAI.Probability.InvariantIsing.Haar.HaarPlaneCasimir |
|
|
| namespace OAI |
|
|
| /-! The exact Frobenius bracket contraction for the special orthogonal algebra. -/ |
| noncomputable section |
| open Matrix |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| /-- The real, unnormalized Frobenius bilinear form. -/ |
| def matrixFrobeniusPair {N : ℕ} (A B : Matrix (Fin N) (Fin N) ℝ) : ℝ := |
| (A.transpose * B).trace |
|
|
| lemma matrixFrobeniusPair_self {N : ℕ} (A : Matrix (Fin N) (Fin N) ℝ) : |
| matrixFrobeniusPair A A = ∑ i, ∑ j, (A i j)^2 := by |
| simp only [matrixFrobeniusPair,Matrix.trace,Matrix.diag_apply,Matrix.mul_apply, |
| Matrix.transpose_apply,pow_two] |
| exact Finset.sum_comm |
|
|
| lemma matrixFrobeniusPair_self_nonneg {N : ℕ} (A : Matrix (Fin N) (Fin N) ℝ) : |
| 0 ≤ matrixFrobeniusPair A A := by |
| rw [matrixFrobeniusPair_self] |
| exact Finset.sum_nonneg fun i _ => Finset.sum_nonneg fun j _ => sq_nonneg _ |
|
|
| lemma planeGenerator_transpose {N : ℕ} (i j : Fin N) : |
| (planeGenerator i j).transpose = -planeGenerator i j := by |
| simp [planeGenerator,Matrix.transpose_sub,Matrix.transpose_single,neg_sub] |
|
|
| lemma matrixFrobeniusPair_bracket {N : ℕ} |
| (E A B : Matrix (Fin N) (Fin N) ℝ) (hE : E.transpose = -E) : |
| matrixFrobeniusPair (E*A-A*E) B = -matrixFrobeniusPair A (E*B-B*E) := by |
| simp only [matrixFrobeniusPair,Matrix.transpose_sub,Matrix.transpose_mul,hE, |
| Matrix.mul_neg,Matrix.neg_mul,Matrix.sub_mul,Matrix.mul_sub,Matrix.trace_sub, |
| Matrix.trace_neg] |
| rw [← mul_assoc A.transpose E B,Matrix.trace_mul_cycle E A.transpose B] |
| rw [Matrix.trace_mul_cycle B E A.transpose] |
| simp only [mul_assoc] |
| ring |
|
|
| /-- Ordered coordinate planes count each orthogonal tangent direction twice. -/ |
| theorem sum_planeGenerator_bracket_norm_sq {N : ℕ} |
| (A : Matrix (Fin N) (Fin N) ℝ) (hA : A.transpose = -A) : |
| (∑ i : Fin N, ∑ j : Fin N, |
| matrixFrobeniusPair (planeGenerator i j*A-A*planeGenerator i j) |
| (planeGenerator i j*A-A*planeGenerator i j)) = |
| (4*((N : ℝ)-2))*matrixFrobeniusPair A A := by |
| simp_rw [matrixFrobeniusPair_bracket _ _ _ (planeGenerator_transpose _ _)] |
| simp only [Finset.sum_neg_distrib] |
| have hs : (∑ i : Fin N, ∑ j : Fin N, |
| matrixFrobeniusPair A |
| (planeGenerator i j*(planeGenerator i j*A-A*planeGenerator i j)- |
| (planeGenerator i j*A-A*planeGenerator i j)*planeGenerator i j)) = |
| matrixFrobeniusPair A (∑ i : Fin N, ∑ j : Fin N, |
| (planeGenerator i j*(planeGenerator i j*A-A*planeGenerator i j)- |
| (planeGenerator i j*A-A*planeGenerator i j)*planeGenerator i j)) := by |
| simp only [matrixFrobeniusPair,Matrix.mul_sum,Matrix.trace_sum] |
| rw [hs,sum_planeGenerator_double_commutator A hA] |
| simp only [matrixFrobeniusPair,Matrix.mul_smul,Matrix.trace_smul] |
| ring |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
