uconf.constructions.bar_algebra

Bar construction for a P-algebra: B_α(A).

Given a twisting morphism α: C → P and a P-algebra (A, γ), the bar construction is the cofree conilpotent C-coalgebra

B_α(A) = (T^c_C(A), d_{T^c} + d_α)

where:

• T^c_C(A) = ⊕_{n≥1} C(n) ⊗ A^{⊗n} is the cofree conilpotent C-coalgebra on A (see uconf.algebraic.cofree_coalgebra).

• d_{T^c} = d_C + d_A is the Leibniz differential on the cofree coalgebra (Koszul sign rule).

• d_α is the extra twisting differential.

The twisting differential d_α acts only on corollas (basis keys (c, (a_1, …, a_n)) where n ≥ 2):

d_α(c ⊗ a_1 ⊗ … ⊗ a_n) = γ(α(c); a_1, …, a_n)

where γ is the P-algebra structure map and α(c) ∈ P(n) is the image of c under the twisting morphism. The result is a single-leaf element (id_C, (γ(α(c); a_1,…,a_n),)).

The sign convention is: d_α carries a Koszul sign (-1)^{|c|} since α has degree -1 and passes past c of degree |c| in the bar ordering.

Maurer-Cartan equation ∂α + α ⋆ α = 0 ensures d² = 0.

B_α(A) is a dg-C-coalgebra.

Reference: Loday-Vallette “Algebraic Operads”, Section 11.2.

Classes

BarAlgebra(alpha, algebra)	Bar construction B_α(A) as a C-coalgebra.
BarAlgebraModule(alpha, algebra)	Underlying dg-module of the bar construction B_α(A).

class uconf.constructions.bar_algebra.BarAlgebraModule(alpha, algebra)

Bases: CofreeCoalgebraModule

Underlying dg-module of the bar construction B_α(A).

Extends CofreeCoalgebraModule with the twisting differential d_α. Basis keys are (c_key, m_tuple) pairs inherited from the cofree coalgebra, where c_key ∈ C(n) and m_tuple has length n.

The total differential is d = d_{T^c} + d_α where d_{T^c} is the cofree-coalgebra Leibniz differential and d_α contracts corollas via the algebra action.

Parameters:

• alpha (TwistingMorphism)

• algebra (OperadAlgebra)

d_cofree(elem)

Apply only the cofree coalgebra differential d_{T^c}.

d_alpha(elem)

Apply only the twisting differential d_α.

BarAlgebraModule.Element

alias of Element

class uconf.constructions.bar_algebra.BarAlgebra(alpha, algebra)

Bases: CooperadCoalgebra

Bar construction B_α(A) as a C-coalgebra.

Wraps BarAlgebraModule with the canonical C-coalgebra structure inherited from the cofree coalgebra.

Parameters:

• alpha (TwistingMorphism) – A twisting morphism α: C → P.

• algebra (OperadAlgebra) – A P-algebra (A, γ).

project(x)

Coprojection π: B_α(A) → A, onto weight-1 generators.

Returns the image in the algebra module A. Non-zero only for (id_key, (a,)) terms.