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The PBW coordinate projection

Lax825748.Projection · concepts/Lax825748/Projection.lean · lax-825748

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    Natural Language Statement

    Theorem

    Lemma 7. For the PBW coordinates of a finite-codimension Lie subalgebra,

    θ(T)=∑I(T,PI)I!SI\theta(T)=\sum_I\frac{(T,P^I)}{I!}S^I

    is a formally continuous shuffle algebra homomorphism. It maps onto J⊥J^\perp and is the identity there.

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    In the paper

    • page 10 of this submission's paper

    Lean source view on GitHub

    1import Lax825748.Annihilator
    2
    3/-!
    4---
    5title: The PBW coordinate projection
    6type: theorem
    7---
    8Lemma 7. For the PBW coordinates of a finite-codimension Lie subalgebra,
    9θ(T)=∑I(T,PI)I!SI\theta(T)=\sum_I\frac{(T,P^I)}{I!}S^I
    10is a formally continuous shuffle algebra homomorphism. It maps onto
    11J⊥J^\perp and is the identity there.
    12-/
    13
    14noncomputable section
    15
    16namespace Lax825748.Projection
    17
    18open Series LieRank Monomials Shuffle FormalSeries Annihilator
    19
    20noncomputable def theta {X : Type} {d : ℕ} (P : Fin d → LiePolynomial X)
    21 (S : Fin d → NCSeries X) (T : NCSeries X) : NCSeries X :=
    22 substituteShuffle S (fun I =>
    23 pairing T (orderedPowers (fun j => (P j).val) I) / (multiFactorial I : ℝ))
    24
    25axiom projection {X : Type} {d : ℕ} (K : LieSubalgebraOfPolynomials X)
    26 (P : Fin d → LiePolynomial X) (hP : IsComplement K P)
    27 (S : Fin d → NCSeries X) (hS : IsCoordinateFamily K P S)
    28 (hPBW : IsPBWCoordinateFamily P S) :
    29 Series.FormallyContinuous (theta P S) ∧
    30 (∀ a : ℝ, theta P S (constantSeries a) = constantSeries a) ∧
    31 (∀ T U, theta P S (T + U) = theta P S T + theta P S U) ∧
    32 (∀ T U, theta P S (shuffle T U) = shuffle (theta P S T) (theta P S U)) ∧
    33 (∀ T, InAnnihilator K (theta P S T)) ∧
    34 (∀ T, InAnnihilator K T → theta P S T = T)
    35
    36end Lax825748.Projection
    37
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