uconf.wrappers.shifted_operad

Shifted operad construction with standard Koszul/sign-representation twists.

For an operad P and integer shift d, this module implements a wrapper ShiftedOperad(P, d) whose arity-n component is shifted by d (n - 1). The symmetric-group action and composition follow the usual suspension-style sign rules from standard operad references.

Classes

ShiftedOperad(operad_cls, shift_degree)

Factory for shifted operad components.
class uconf.wrappers.shifted_operad.ShiftedOperad(operad_cls, shift_degree)[source]

Bases: UniqueRepresentation

Factory for shifted operad components.

Parameters:

  • operad_cls (OperadLike) – Base operad class (e.g. Lie or Surjection).

  • shift_degree (int) – Integer d in the arity-dependent shift d (n-1).

The construction uses the following conventions for homogeneous terms:

  • Symmetric action twist by sgn(sigma)^d.

  • Composition twist (-1)^(d * ((i-1)(n-1) + (m-1)|y|)) with m = arity(x), n = arity(y), and |y| in the base operad.

EXAMPLES:

sage: from sage.all import QQ
sage: from uconf import Lie, ShiftedOperad
sage: sLie = ShiftedOperad(Lie, 1)
sage: x = sLie(2, QQ)((1,))
sage: x.arity()
2
property connectivity: int

Connectivity of the shifted operad.

ShiftedOperad(P, d)(n) lives in degrees >= (k + d)*(n-1) where k is the connectivity of P, because the degree-shift by d*(n-1) raises the lower bound by d*(n-1).

unit(base_ring)[source]

Return the shifted operadic unit over base_ring.

Return type:

Element

unit_key()[source]

Return the basis key of the unit element in arity 1.

Return type:

object

compose(x, i, y)[source]

Compose two shifted-operad elements in this wrapper.

Parameters:
Return type:

Element

ShiftedOperad.Component[source]

alias of Component

ShiftedOperad.Element[source]

alias of Element