uconf.models.coassociative

Coassociative cooperad: linear dual of the associative operad.

CoAssociative(n) is the arity-n component of Ass^*, spanned by the dual permutation basis with zero differential. The infinitesimal cocomposition Δ^{i;m,n} is the transpose of Associative.compose(·, i, ·).

Classes

CoAssociative(n, base_ring)

Coassociative cooperad component in fixed arity.

class uconf.models.coassociative.CoAssociative(n, base_ring)

Bases: Associative

Coassociative cooperad component in fixed arity.

This is the linear dual of uconf.associative.Associative. As a graded module it is identical to Associative; the additional cooperad structure is provided by counit(), reduced(), and infinitesimal_cocompose().

Parameters:

n (int)

static counit(x)

Cooperadic counit: extracts coefficient of the arity-1 generator.

The counit ε: CoAss(1) → k evaluates to 1 on the unique basis element (1,) and to 0 for all other arities.

Parameters:

x (Element)

static reduced(x)

Project to the reduced part (kills the arity-1 unit component).

Parameters:

x (Element)

Return type:

Element

static infinitesimal_cocompose(x, i, m, n)

Partial cocomposition dual to Associative.compose(·, i, ·).

For each basis element sigma ∈ S_{m+n-1} in x, returns Σ tau ⊗ rho summed over all (tau, rho) ∈ S_m × S_n with Ass.compose(tau, i, rho) == sigma.

Parameters:

• x (Element)

• i (int)

• m (int)

• n (int)

factory

alias of CoAssociative

CoAssociative.Element

alias of Element