Path Algebras¶
-
sage.quivers.algebra.
PathAlgebra
¶ 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:
k
– field (or commutative ring), the base field of the path algebraP
– the path semigroup of a quiver \(Q\)order
– optional string, one of “negdegrevlex” (default), “degrevlex”, “negdeglex” or “deglex”, defining the monomial order to be used.
OUTPUT:
- the path algebra \(kP\) with the given monomial order
NOTE:
Monomial orders that are not degree orders are not supported.
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, field and monomial order:
sage: A is P.algebra(GF(7)) True sage: A is P.algebra(GF(7), order="degrevlex") False 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