Coverage for local/lib/python2.7/site-packages/sage/categories/examples/graded_connected_hopf_algebras_with_basis.py : 100%

r""" 

Examples of graded connected Hopf algebras with basis 

""" 

#***************************************************************************** 

# Copyright (C) 2015 Jean-Baptiste Priez <jbp@kerios.fr> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.categories.graded_hopf_algebras_with_basis import GradedHopfAlgebrasWithBasis 

from sage.combinat.free_module import CombinatorialFreeModule 

from sage.functions.other import binomial 

from sage.misc.cachefunc import cached_method 

from sage.sets.non_negative_integers import NonNegativeIntegers 

class GradedConnectedCombinatorialHopfAlgebraWithPrimitiveGenerator(CombinatorialFreeModule): 

r""" 

This class illustrates an implementation of a graded Hopf algebra 

with basis that has one primitive generator of degree 1 and basis 

elements indexed by non-negative integers. 

This Hopf algebra example differs from what topologists refer to as 

a graded Hopf algebra because the twist operation in the tensor rule 

satisfies 

.. MATH:: 

(\mu \otimes \mu) \circ (id \otimes \tau \otimes id) \circ 

(\Delta \otimes \Delta) = \Delta \circ \mu 

where `\tau(x\otimes y) = y\otimes x`. 

""" 

def __init__(self, base_ring): 

""" 

EXAMPLES:: 

sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() 

sage: TestSuite(H).run() 

""" 

CombinatorialFreeModule.__init__(self, base_ring, NonNegativeIntegers(), 

category=GradedHopfAlgebrasWithBasis(base_ring).Connected()) 

@cached_method 

def one_basis(self): 

""" 

Returns 0, which index the unit of the Hopf algebra. 

OUTPUT: 

- the non-negative integer 0 

EXAMPLES:: 

sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() 

sage: H.one_basis() 

0 

sage: H.one() 

P0 

""" 

return self.basis().keys()(0) 

def degree_on_basis(self, i): 

""" 

The degree of a non-negative integer is itself 

INPUT: 

- ``i`` -- a non-negative integer 

OUTPUT: 

- a non-negative integer 

TESTS:: 

sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() 

sage: H.degree_on_basis(45) 

45 

""" 

return i 

def _repr_(self): 

""" 

Representation of the graded connected Hopf algebra 

EXAMPLES:: 

sage: GradedHopfAlgebrasWithBasis(QQ).Connected().example() 

An example of a graded connected Hopf algebra with basis over Rational Field 

""" 

return "An example of a graded connected Hopf algebra with basis over %s" % self.base_ring() 

def _repr_term(self, i): 

""" 

Representation for the basis element indexed by the integer ``i``. 

EXAMPLES:: 

sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() 

sage: H._repr_term(45) 

'P45' 

""" 

return 'P' + repr(i) 

def product_on_basis(self, i, j): 

""" 

The product of two basis elements. 

The product of elements of degree ``i`` and ``j`` is an element 

of degree ``i+j``. 

INPUT: 

- ``i``, ``j`` -- non-negative integers 

OUTPUT: 

- a basis element indexed by ``i+j`` 

TESTS:: 

sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() 

sage: H.monomial(4) * H.monomial(5) 

P9 

""" 

return self.monomial(i+j) 

def coproduct_on_basis(self, i): 

""" 

The coproduct of a basis element. 

.. MATH:: 

\Delta(P_i) = \sum_{j=0}^i P_{i-j} \otimes P_j 

INPUT: 

- ``i`` -- a non-negative integer 

OUTPUT: 

- an element of the tensor square of ``self`` 

TESTS:: 

sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() 

sage: H.monomial(3).coproduct() 

P0 # P3 + 3*P1 # P2 + 3*P2 # P1 + P3 # P0 

""" 

return self.sum_of_terms( 

((i-j, j), binomial(i, j)) 

for j in range(i+1) 

) 

Example = GradedConnectedCombinatorialHopfAlgebraWithPrimitiveGenerator