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

r""" 

Example of a finite dimensional algebra with basis 

""" 

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

# Copyright (C) 2008-2015 Franco Saliola <saliola@gmail.com> 

# 

# 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.categories.all import FiniteDimensionalAlgebrasWithBasis 

from sage.combinat.free_module import CombinatorialFreeModule 

class KroneckerQuiverPathAlgebra(CombinatorialFreeModule): 

r""" 

An example of a finite dimensional algebra with basis: the path 

algebra of the Kronecker quiver. 

This class illustrates a minimal implementation of a finite 

dimensional algebra with basis. See 

:class:`sage.quivers.algebra.PathAlgebra` for a full-featured 

implementation of path algebras. 

""" 

def __init__(self, base_ring): 

r""" 

EXAMPLES:: 

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

An example of a finite dimensional algebra with basis: 

the path algebra of the Kronecker quiver 

(containing the arrows a:x->y and b:x->y) over Rational Field 

sage: TestSuite(A).run() 

""" 

basis_keys = ['x', 'y', 'a', 'b'] 

self._nonzero_products = { 

'xx':'x', 'xa':'a', 'xb':'b', 

'yy':'y', 'ay':'a', 'by':'b' 

} 

CombinatorialFreeModule.__init__( 

self, base_ring, basis_keys, 

category=FiniteDimensionalAlgebrasWithBasis(base_ring)) 

def _repr_(self): 

r""" 

EXAMPLES:: 

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

An example of a finite dimensional algebra with basis: 

the path algebra of the Kronecker quiver 

(containing the arrows a:x->y and b:x->y) over Rational Field 

""" 

return "An example of a finite dimensional algebra with basis: " \ 

"the path algebra of the Kronecker quiver " \ 

"(containing the arrows a:x->y and b:x->y) over %s "%(self.base_ring()) 

def one(self): 

r""" 

Return the unit of this algebra. 

.. SEEALSO:: :meth:`AlgebrasWithBasis.ParentMethods.one_basis` 

EXAMPLES:: 

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

sage: A.one() 

x + y 

""" 

return self.sum_of_monomials(['x', 'y']) 

def product_on_basis(self, w1, w2): 

r""" 

Return the product of the two basis elements indexed by ``w1`` and ``w2``. 

.. SEEALSO:: :meth:`AlgebrasWithBasis.ParentMethods.product_on_basis`. 

EXAMPLES:: 

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

Here is the multiplication table for the algebra:: 

sage: matrix([[p*q for q in A.basis()] for p in A.basis()]) 

[x 0 a b] 

[0 y 0 0] 

[0 a 0 0] 

[0 b 0 0] 

Here we take some products of linear combinations of basis elements:: 

sage: x, y, a, b = A.basis() 

sage: a * (1-b)^2 * x 

0 

sage: x*a + b*y 

a + b 

sage: x*x 

x 

sage: x*y 

0 

sage: x*a*y 

a 

""" 

if w1+w2 in self._nonzero_products: 

return self.monomial(self._nonzero_products[w1+w2]) 

else: 

return self.zero() 

@cached_method 

def algebra_generators(self): 

r""" 

Return algebra generators for this algebra. 

.. SEEALSO:: :meth:`Algebras.ParentMethods.algebra_generators`. 

EXAMPLES:: 

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

An example of a finite dimensional algebra with basis: 

the path algebra of the Kronecker quiver 

(containing the arrows a:x->y and b:x->y) over Rational Field 

sage: A.algebra_generators() 

Finite family {'y': y, 'x': x, 'b': b, 'a': a} 

""" 

return self.basis() 

def _repr_term(self, p): 

r""" 

This method customizes the string representation of the basis 

element indexed by ``p``. 

In this example, we just return the string representation of 

``p`` itself. 

EXAMPLES:: 

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

sage: A.one() 

x + y 

""" 

return str(p) 

Example = KroneckerQuiverPathAlgebra