Representation

class discopy.hopf.Representation(dim=None, action=None)

Bases: discopy.abc.Result, discopy.frobenius.Dim

A finite-dimensional (left) module over the class parameter algebra: given a Algebra H, the class Representation[H] is the type of objects of \(\mathrm{Rep}(H)\), the category Intertwiner[H], and algebra is accessible on both the class and its instances. A representation is a tensor.Dim — inside is the underlying vector space \(V\) — carrying its action diagram \(H \otimes V \to V\) (with V = action.cod), which the ribbon classmethods of Intertwiner read off.

Parameters:

• dim – The underlying vector space \(V\) as a single tensor.Dim, Dim(1) by default.

• action – The action diagram \(H \otimes V \to V\); when not given, the trivial action algebra.counit @ Id(dim).

The product of representations (tensor()) acts through the comultiplication and the adjoints l, r are the dual module with the action twisted by \(S^{-1}\) and \(S\). Equality is that of the underlying tensor.Dim: two modules on the same space are distinguished by their action, not by ==.

A representation satisfies the two module axioms as diagram equations (is_module()): the action is associative over mult and unital over unit. Here is the associativity axiom (mult @ V) >> action == (H @ action) >> action for a direct sum of two anyon modules of \(D(\mathbb{Z}/2)\), with the left-hand side drawn as the tensor.CMap it contracts to:

>>> D = Double(Algebra.cyclic(2))
>>> e = Representation[D].anyon(0, -1)
>>> m = Representation[D].anyon(1, 1)
>>> V = Representation[D].direct_sum([e, m])
>>> assert V.algebra == D
>>> assert V.is_module() and V == Dim(2)
>>> ty = V.action.cod
>>> (D.mult @ ty >> V.action).to_map().draw(  
...     doctest='docs/_static/hopf/module.dot')

tensor(*others)

The product of representations: the underlying spaces concatenate as for a tensor.Dim and the product of modules acts through the comultiplication, \(\rho_{V \otimes W} = (\rho_V \otimes \rho_W) (\Delta \otimes 1_{V \otimes W})\). A plain tensor.Dim factor is taken as the trivial representation.

>>> D = Double(Algebra.cyclic(2))
>>> e = Representation[D].anyon(0, -1)
>>> m = Representation[D].anyon(1, 1)
>>> assert (e @ m).is_module() and (e @ Dim(2)).is_module()

dual(antipode)

The dual module \(V^*\) with \(\rho^*(h) = \rho(S h)^T\) for the given antipode diagram \(S\): the transpose() of the twisted action, with the algebra wire bent back to the domain. The legs of \(V^*\) come out in reversed order.

property r

The right dual, dual() for the antipode.

>>> H = Algebra.sweedler()
>>> assert Representation[H].regular().r.is_module()

property l

The left dual, dual() for the inverse antipode — it differs from r unless \(S^2 = 1\).

>>> H = Algebra.sweedler()
>>> assert Representation[H].regular().l.is_module()

qdim()

The quantum dimension: the value of a loop coloured by V, i.e. the (co)evaluation \(\cup \circ \cap\) in Intertwiner.

is_module()

Whether action is a representation: the two module axioms hold as diagram equations — the action is associative over mult and unital over unit. See the axiom drawn in the class docstring.

classmethod regular()

The regular representation, acting on H by mult.

classmethod anyon(flux, charge)

A one-dimensional anyon module of the quantum double of a cyclic group algebra: the group element e_a acts by charge ** a in the flux sector flux.

classmethod direct_sum(reps)

The direct sum of modules over one algebra, acting block-diagonally.

>>> D = Double(Algebra.cyclic(2))
>>> e = Representation[D].anyon(0, -1)
>>> m = Representation[D].anyon(1, 1)
>>> V = Representation[D].direct_sum([e, m])
>>> assert V.is_module() and V == Dim(2)