Bases: sage.combinat.free_module.CombinatorialFreeModule
Create the path algebra of a quiver over a given field.
Given a quiver \(Q\) and a field \(k\), the path algebra \(kQ\) is defined as follows. As a vector space it has basis the set of all paths in \(Q\). Multiplication is defined on this basis and extended bilinearly. If \(p\) is a path with terminal vertex \(t\) and \(q\) is a path with initial vertex \(i\) then the product \(p*q\) is defined to be the composition of the paths \(p\) and \(q\) if \(t = i\) and \(0\) otherwise.
INPUT:
OUTPUT:
EXAMPLES:
sage: P = DiGraph({1:{2:['a']}, 2:{3:['b']}}).path_semigroup()
sage: A = P.algebra(GF(7))
sage: A
Path algebra of Multi-digraph on 3 vertices over Finite Field of size 7
sage: A.variable_names()
('e_1', 'e_2', 'e_3', 'a', 'b')
Note that path algebras are uniquely defined by their quiver and field:
sage: A is P.algebra(GF(7))
True
sage: A is P.algebra(RR)
False
sage: A is DiGraph({1:{2:['a']}}).path_semigroup().algebra(GF(7))
False
The path algebra of an acyclic quiver has a finite basis:
sage: A.dimension()
6
sage: list(A.basis())
[e_1, e_2, e_3, a, b, a*b]
The path algebra can create elements from paths or from elements of the base ring:
sage: A(5)
5*e_1 + 5*e_2 + 5*e_3
sage: S = A.semigroup()
sage: S
Partial semigroup formed by the directed paths of Multi-digraph on 3 vertices
sage: p = S([(1, 2, 'a')])
sage: r = S([(2, 3, 'b')])
sage: e2 = S([(2, 2)])
sage: x = A(p) + A(e2)
sage: x
a + e_2
sage: y = A(p) + A(r)
sage: y
a + b
Path algebras are graded algebras. The grading is given by assigning to each basis element the length of the path corresponding to that basis element:
sage: x.is_homogeneous()
False
sage: x.degree()
Traceback (most recent call last):
...
ValueError: Element is not homogeneous.
sage: y.is_homogeneous()
True
sage: y.degree()
1
sage: A[1]
Free module spanned by [a, b] over Finite Field of size 7
sage: A[2]
Free module spanned by [a*b] over Finite Field of size 7
TESTS:
sage: TestSuite(A).run()
Bases: sage.combinat.free_module.CombinatorialFreeModuleElement
Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.
TESTS:
sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
sage: f == loads(dumps(f))
True
The degree of self, if self is homogeneous.
EXAMPLES:
sage: A = DiGraph({1:{2:['a', 'b']}, 2:{3:['c']}}).path_semigroup().algebra(QQ)
sage: A(1).degree()
0
sage: (A('a') + A('b')).degree()
1
An error is raised if the element is not homogeneous:
sage: (A(1) + A('a')).degree()
Traceback (most recent call last):
...
ValueError: Element is not homogeneous.
Return True if and only if this element is homogeneous.
EXAMPLES:
sage: A = DiGraph({1:{2:['a', 'b']}, 2:{3:['c']}}).path_semigroup().algebra(QQ)
sage: (A('a') + A('b')).is_homogeneous()
True
sage: (A(1) + A('a')).is_homogeneous()
False
Return the arrows of this algebra (corresponding to edges of the underlying quiver).
EXAMPLES:
sage: P = DiGraph({1:{2:['a']}, 2:{3:['b', 'c']}, 4:{}}).path_semigroup()
sage: A = P.algebra(GF(5))
sage: A.arrows()
(a, b, c)
Return the degree of the monomial specified by the path p.
EXAMPLES:
sage: A = DiGraph({1:{2:['a']}, 2:{3:['b']}}).path_semigroup().algebra(QQ)
sage: A.degree_on_basis([(1, 1)])
0
sage: A.degree_on_basis('a')
1
sage: A.degree_on_basis([(1, 2, 'a'), (2, 3, 'b')])
2
Return the \(i\)-th generator of this algebra.
This is an idempotent (corresponding to a trivial path at a vertex) if \(i < n\) (where \(n\) is the number of vertices of the quiver), and a single-edge path otherwise.
EXAMPLES:
sage: P = DiGraph({1:{2:['a']}, 2:{3:['b', 'c']}, 4:{}}).path_semigroup()
sage: A = P.algebra(GF(5))
sage: A.gens()
(e_1, e_2, e_3, e_4, a, b, c)
sage: A.gen(2)
e_3
sage: A.gen(5)
b
Return the generators of this algebra (idempotents and arrows).
EXAMPLES:
sage: P = DiGraph({1:{2:['a']}, 2:{3:['b', 'c']}, 4:{}}).path_semigroup()
sage: A = P.algebra(GF(5))
sage: A.variable_names()
('e_1', 'e_2', 'e_3', 'e_4', 'a', 'b', 'c')
sage: A.gens()
(e_1, e_2, e_3, e_4, a, b, c)
Return the \(n\)-th homogeneous piece of the path algebra.
INPUT:
OUTPUT:
EXAMPLES:
sage: P = DiGraph({1:{2:['a'], 3:['b']}, 2:{4:['c']}, 3:{4:['d']}}).path_semigroup()
sage: A = P.algebra(GF(7))
sage: A.homogeneous_component(2)
Free module spanned by [a*c, b*d] over Finite Field of size 7
sage: D = DiGraph({1: {2: 'a'}, 2: {3: 'b'}, 3: {1: 'c'}})
sage: P = D.path_semigroup()
sage: A = P.algebra(ZZ)
sage: A.homogeneous_component(3)
Free module spanned by [a*b*c, b*c*a, c*a*b] over Integer Ring
Return the non-zero homogeneous components of self.
EXAMPLES:
sage: Q = DiGraph([[1,2,'a'],[2,3,'b'],[3,4,'c']])
sage: PQ = Q.path_semigroup()
sage: A = PQ.algebra(GF(7))
sage: A.homogeneous_components()
[Free module spanned by [e_1, e_2, e_3, e_4] over Finite Field of size 7,
Free module spanned by [a, b, c] over Finite Field of size 7,
Free module spanned by [a*b, b*c] over Finite Field of size 7,
Free module spanned by [a*b*c] over Finite Field of size 7]
Warning
Backward incompatible change: since trac ticket #12630 and until trac ticket #8678, this feature was implemented under the syntax list(A) by means of A.__iter__. This was incorrect since A.__iter__, when defined for a parent, should iterate through the elements of \(A\).
Return the idempotents of this algebra (corresponding to vertices of the underlying quiver).
EXAMPLES:
sage: P = DiGraph({1:{2:['a']}, 2:{3:['b', 'c']}, 4:{}}).path_semigroup()
sage: A = P.algebra(GF(5))
sage: A.idempotents()
(e_1, e_2, e_3, e_4)
Number of generators of this algebra.
EXAMPLES:
sage: P = DiGraph({1:{2:['a']}, 2:{3:['b', 'c']}, 4:{}}).path_semigroup()
sage: A = P.algebra(GF(5))
sage: A.ngens()
7
Return the multiplicative identity element.
The multiplicative identity of a path algebra is the sum of the basis elements corresponding to the trivial paths at each vertex.
EXAMPLES:
sage: A = DiGraph({1:{2:['a']}, 2:{3:['b']}}).path_semigroup().algebra(QQ)
sage: A.one()
e_1 + e_2 + e_3
Return the product p1*p2 in the path algebra.
INPUT:
OUTPUT:
EXAMPLES:
sage: Q = DiGraph({1:{2:['a']}, 2:{3:['b']}, 3:{4:['c']}}).path_semigroup()
sage: p1 = Q('a')
sage: p2 = Q([(2, 3, 'b'), (3, 4, 'c')])
sage: A = Q.algebra(QQ)
sage: A.product_on_basis(p1, p2)
a*b*c
sage: A.product_on_basis(p2, p1)
0
Return the quiver from which the algebra self was formed.
OUTPUT:
EXAMPLES:
sage: P = DiGraph({1:{2:[‘a’, ‘b’]}}).path_semigroup() sage: A = P.algebra(GF(3)) sage: A.quiver() is P.quiver() True
Return the (partial) semigroup from which the algebra self was constructed.
Note
The partial semigroup is formed by the paths of a quiver, multiplied by concatenation. If the quiver has more than a single vertex, then multiplication in the path semigroup is not always defined.
OUTPUT:
EXAMPLES:
sage: P = DiGraph({1:{2:[‘a’, ‘b’]}}).path_semigroup() sage: A = P.algebra(GF(3)) sage: A.semigroup() is P True