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

r""" 

Examples of algebras with basis 

""" 

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

# Copyright (C) 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net> 

# 

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

# http://www.gnu.org/licenses/ 

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

from sage.misc.cachefunc import cached_method 

from sage.sets.family import Family 

from sage.categories.all import HopfAlgebrasWithBasis 

from sage.combinat.free_module import CombinatorialFreeModule 

from sage.categories.all import tensor 

class MyGroupAlgebra(CombinatorialFreeModule): 

r""" 

An of a Hopf algebra with basis: the group algebra of a group 

This class illustrates a minimal implementation of a Hopf algebra with basis. 

""" 

def __init__(self, R, G): 

""" 

EXAMPLES:: 

sage: from sage.categories.examples.hopf_algebras_with_basis import MyGroupAlgebra 

sage: A = MyGroupAlgebra(QQ, DihedralGroup(6)) 

sage: A.category() 

Category of finite dimensional hopf algebras with basis over Rational Field 

sage: TestSuite(A).run() 

""" 

self._group = G 

CombinatorialFreeModule.__init__(self, R, G, category = HopfAlgebrasWithBasis(R)) 

def _repr_(self): 

""" 

EXAMPLES:: 

sage: HopfAlgebrasWithBasis(QQ).example() # indirect doctest 

An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field 

""" 

return "An example of Hopf algebra with basis: the group algebra of the %s over %s"%(self._group, self.base_ring()) 

@cached_method 

def one_basis(self): 

""" 

Returns the one of the group, which index the one of this algebra, 

as per :meth:`AlgebrasWithBasis.ParentMethods.one_basis`. 

EXAMPLES:: 

sage: A = HopfAlgebrasWithBasis(QQ).example() 

sage: A.one_basis() 

() 

sage: A.one() 

B[()] 

""" 

return self._group.one() 

def product_on_basis(self, g1, g2): 

r""" 

Product, on basis elements, as per 

:meth:`AlgebrasWithBasis.ParentMethods.product_on_basis`. 

The product of two basis elements is induced by the product of 

the corresponding elements of the group. 

EXAMPLES:: 

sage: A = HopfAlgebrasWithBasis(QQ).example() 

sage: (a, b) = A._group.gens() 

sage: a*b 

(1,2) 

sage: A.product_on_basis(a, b) 

B[(1,2)] 

""" 

return self.basis()[g1 * g2] 

@cached_method 

def algebra_generators(self): 

r""" 

Return the generators of this algebra, as per :meth:`~.magmatic_algebras.MagmaticAlgebras.ParentMethods.algebra_generators`. 

They correspond to the generators of the group. 

EXAMPLES:: 

sage: A = HopfAlgebrasWithBasis(QQ).example(); A 

An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field 

sage: A.algebra_generators() 

Finite family {(1,3): B[(1,3)], (1,2,3): B[(1,2,3)]} 

""" 

return Family(self._group.gens(), self.monomial) 

def coproduct_on_basis(self, g): 

r""" 

Coproduct, on basis elements, as per :meth:`HopfAlgebrasWithBasis.ParentMethods.coproduct_on_basis`. 

The basis elements are group like: `\Delta(g) = g \otimes g`. 

EXAMPLES:: 

sage: A = HopfAlgebrasWithBasis(QQ).example() 

sage: (a, b) = A._group.gens() 

sage: A.coproduct_on_basis(a) 

B[(1,2,3)] # B[(1,2,3)] 

""" 

g = self.monomial(g) 

return tensor([g, g]) 

def counit_on_basis(self, g): 

r""" 

Counit, on basis elements, as per :meth:`HopfAlgebrasWithBasis.ParentMethods.counit_on_basis`. 

The counit on the basis elements is 1. 

EXAMPLES:: 

sage: A = HopfAlgebrasWithBasis(QQ).example() 

sage: (a, b) = A._group.gens() 

sage: A.counit_on_basis(a) 

1 

""" 

return self.base_ring().one() 

def antipode_on_basis(self, g): 

r""" 

Antipode, on basis elements, as per :meth:`HopfAlgebrasWithBasis.ParentMethods.antipode_on_basis`. 

It is given, on basis elements, by `\nu(g) = g^{-1}` 

EXAMPLES:: 

sage: A = HopfAlgebrasWithBasis(QQ).example() 

sage: (a, b) = A._group.gens() 

sage: A.antipode_on_basis(a) 

B[(1,3,2)] 

""" 

return self.monomial(~g)