Quiver Representations

AUTHORS:

  • Jim Stark (2012-03-04): Initial implementation of acyclic quivers without relations.
  • Simon King (2013-05, 2014-02): Split code up. Allow cyclic quivers where possible.

A quiver is a directed graph used for representation theory. In our representation theoretic code, it is assumed that

  • the vertices of the quiver are labelled by integers, and
  • each edge of the quiver is labelled with a nonempty string. The label cannot begin with 'e_' or contain '*' and distinct edges must have distinct labels.

As far as the DiGraph class is concerned, a path is a finite list of pairwise distinct vertices \(v_1, ..., v_n\) such that there exists an edge from \(v_i\) to \(v_{i + 1}\). If there are multiple edges between the same two vertices this does not contribute additional paths as listed by the DiGraph class; for example only two paths are listed from 1 to 3 in Q:

sage: Q = DiGraph({1:{2:['a','b'], 3:['c']}, 2:{3:['d']}})
sage: Q.edges()
[(1, 2, 'a'), (1, 2, 'b'), (1, 3, 'c'), (2, 3, 'd')]
sage: Q.all_paths(1, 3)
[[1, 2, 3], [1, 3]]

The notion of a path in a quiver (in representation theory) is fundamentally different in several aspects. First, paths are no longer required to have distinct vertices, or even distinct edges; thus, “path” in quiver theory is closer to the notion of “walk” in graph theory. Furthermore, paths in quiver theory “know” their edges, so parallel edges between the same two vertices of a Quiver make different paths. But paths in quiver theory also “know” their vertices, so that a length-\(0\) path from \(a\) to \(a\) is not the same as a length-\(0\) path from \(b\) to \(b\) for \(a \neq b\). Formally, we say that a path is given by two vertices, start and end, and a finite (possibly empty) list of edges \(e_1, e_2, \ldots, e_n\) such that the initial vertex of \(e_1\) is start, the final vertex of \(e_i\) is the initial vertex of \(e_{i + 1}\), and the final vertex of \(e_n\) is end. In the case where no edges are specified, we must have start = end and the path is called the trivial path at the given vertex.

Quiver paths in the sense stated above correspond to the elements of a partial semigroup, with multiplication of paths given by concatenation. Hence, rather than overloading the method name inherited from DiGraph or inventing a new method name, we move this functionality to this so-called path semigroup. Note that with this definition there are three paths from 1 to 3 in our example:

sage: Q.path_semigroup().all_paths(1, 3)
[a*d, b*d, c]

The returned paths are of type QuiverPath, which are elements in the path semigroup that is associated with the quiver (a partial semigroup, which does not generally have a neutral element). You can specify a QuiverPath by giving an edge or a list of edges, passed as arguments to the path semigroup containing this path. Here an edge is a tuple of the form (i, j, l), where i and j are vertices and l is the label of an edge from i to j:

sage: p = Q.path_semigroup()([(1, 2, 'a'), (2, 3, 'd')])
sage: p
a*d

Trivial paths are indicated by passing the tuple (vertex, vertex):

sage: Q.path_semigroup()((6, 6))
e_6

Trivial “edges” can occur in the input. They are simply deleted if their vertex matches the start and end vertex of adjacent edges. Here is an alternative way to define a path:

sage: PQ = Q.path_semigroup()
sage: q = PQ([(1, 1), (1, 2, 'a'), (2, 2), (2, 3, 'd'), (3, 3)])
sage: p == q
True

If the vertex of a trivial path does not match with adjacent edges, or if two adjacent edges do not match, an error is raised.

sage: inv1 = PQ([(1, 2, 'a'), (1, 1)])
Traceback (most recent call last):
...
ValueError: Cannot interpret [(1, 2, 'a'), (1, 1)] as element of
 Partial semigroup formed by the directed paths of Multi-digraph on 3 vertices
sage: inv2 = PQ([(1, 2, 'a'), (1, 2, 'a')])
Traceback (most recent call last):
...
ValueError: Cannot interpret [(1, 2, 'a'), (1, 2, 'a')] as element of
 Partial semigroup formed by the directed paths of Multi-digraph on 3 vertices
sage: inv3 = PQ([(1, 2, 'x')])
Traceback (most recent call last):
...
ValueError: Cannot interpret [(1, 2, 'x')] as element of
 Partial semigroup formed by the directed paths of Multi-digraph on 3 vertices

The * operator is concatenation of paths. If the two paths do not compose, then the result is None (whence the “partial” in “partial semigroup”).

sage: print p*q
None

Let us now construct a larger quiver:

sage: Qbig = DiGraph({1:{2:['a','b'], 3:['c']}, 2:{3:['d']}, 3:{4:['e']}, 4:{5:['f']}, 5:{1:['g']} })
sage: Pbig = Qbig.path_semigroup()

Since Q is a sub-digraph of Qbig, we have a coercion of the associated path semigroups:

sage: Pbig.has_coerce_map_from(PQ)
True

In particular, p is considered to be an element of Pbig, and can be composed with paths that were defined for the larger quiver:

sage: p in Pbig
True
sage: p*Pbig([(3, 4, 'e')])
a*d*e
sage: Pbig([(4, 5, 'f'), (5, 1, 'g')])*p
f*g*a*d

The length of a path is the number of edges in that path:

sage: len(p)
2
sage: triv = PQ((1, 1))
sage: len(triv)
0

List index and slice notation can be used to access the edges in a path. QuiverPaths can also be iterated over. Trivial paths have no elements:

sage: for x in p: print x
(1, 2, 'a')
(2, 3, 'd')
sage: triv[:]
[]

There are methods giving the initial and terminal vertex of a path:

sage: p.initial_vertex()
1
sage: p.terminal_vertex()
3

QuiverPath form the basis of the quiver algebra of a quiver. Given a field \(k\) and a quiver \(Q\), the quiver algebra \(kQ\) is, as a vector space, the free \(k\)-vector space whose basis is the set of all paths in \(Q\). Multiplication is defined on this basis and extended bilinearly. The product of two basis elements is given by path composition when it makes sense and is set to be zero otherwise. Specifically, if the terminal vertex of the left path equals the initial vertex of the right path, then their product is the concatenation of the two paths, and otherwise their product is zero. In sage, quiver algebras are handled by the QuiverAlgebra class:

sage: A = PQ.algebra(GF(7))
sage: A
Path algebra of Multi-digraph on 3 vertices over Finite Field of size 7

Quivers have a method that creates their algebra over a given field (or, more generally, commutative ring). Note that QuiverAlgebras are uniquely defined by their quiver and field, and play nicely with coercions of the underlying path semigroups:

sage: A is PQ.algebra(GF(7))
True
sage: A is PQ.algebra(RR)
False
sage: Q1 = Q.copy()
sage: Q1.add_vertex(4)
sage: PQ1 = Q1.path_semigroup()
sage: A is PQ1.algebra(GF(7))
False
sage: Pbig.algebra(GF(7)).has_coerce_map_from(A)
True

The QuiverAlgebra can create elements from QuiverPaths or from elements of the base ring:

sage: A(5)
5*e_1 + 5*e_2 + 5*e_3
sage: r = PQ([(1, 2, 'b'), (2, 3, 'd')])
sage: e2 = PQ((2, 2))
sage: x = A(p) + A(e2)
sage: x
e_2 + a*d
sage: y = A(p) + A(r)
sage: y
a*d + b*d

QuiverAlgebras are \(\NN\)-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()
2
sage: A[1]
Free module spanned by [a, b, c, d] over Finite Field of size 7
sage: A[2]
Free module spanned by [a*d, b*d] over Finite Field of size 7

The category of right modules over a given quiver algebra is equivalent to the category of representations of that quiver. A quiver representation is a diagram in the category of vector spaces whose underlying graph is the quiver. So to each vertex of the quiver we assign a vector space and to each edge of the quiver a linear map between the vector spaces assigned to the start and end vertices of that edge. To create the zero representation we just specify the base ring and the path semigroup:

sage: Z = Q1.path_semigroup().representation(GF(5))
sage: Z.is_zero()
True

To each vertex of a quiver there is associated a simple module, an indecomposable projective, and an indecomposable injective, and these can be created from the qQuiver:

sage: S = PQ.S(GF(3), 1)
sage: I = PQ.I(QQ, 2)
sage: P = PQ.P(GF(3), 1)

Radicals, socles, tops, and quotients can all be computed and we can test if modules are simple or semisimple, get their dimension, and test for equality. Like quivers, QuiverRep objects are unique and therefore equal if and only if they are identical:

sage: P.is_simple()
False
sage: P.dimension()
6
sage: R = P.radical()
sage: P.socle()
Representation with dimension vector (0, 0, 3)
sage: (P/R).is_simple()
True
sage: P == R
False
sage: P.top() is P/R
True

There are special methods to deal with modules that are given as right ideals in the quiver algebra. To create such a module pass the keyword option='paths' along with a path or list of paths that generate the desired ideal:

sage: M = PQ.representation(QQ, [[(1, 1)], [(1, 2, 'a')]], option='paths')
sage: M.dimension_vector()
(1, 2, 3)

There are also special methods to deal with modules that are given as the linear dual of a right ideal in the quiver algebra. To create such a module, pass the keyword option='dual paths' to the constructor along with a path or list of paths. The module returned is the dual of the ideal created in the opposite quiver by the reverses of the given paths:

sage: D = PQ.representation(QQ, [[(1, 1)], [(1, 2, 'a')]], option='dual paths')
sage: D.dimension_vector()
(2, 0, 0)

For modules that are not a standard module or an ideal of the quiver algebra QuiverRep can take as input two dictionaries. The first associates to each vertex a vector space or an integer (the desired dimension of the vector space), the second associates to each edge a map or a matrix or something from which sage can construct a map:

sage: PQ2 = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: M2 = PQ2.representation(QQ, {1: QQ^2, 2: QQ^1}, {(1, 2, 'a'): [1, 0], (1, 2, 'b'): [0, 1]})
sage: M.get_space(2)
Vector space of dimension 2 over Rational Field
sage: M2.get_space(2)
Vector space of dimension 1 over Rational Field
sage: M.get_map((1, 2, 'a'))
Vector space morphism represented by the matrix:
[1 0]
Domain: Vector space of dimension 1 over Rational Field
Codomain: Vector space of dimension 2 over Rational Field

A homomorphism between two quiver representations is given by homomorphisms between the spaces assigned to the vertices of those representations such that those homomorphisms commute with the edge maps of the representations. The homomorphisms are created in the usual Sage syntax, the defining data given by a dictionary associating maps to vertices:

sage: P2 = PQ2.P(QQ, 1)
sage: f = P2.hom({1:[1, 1], 2:[[1], [1]]}, M2)

When the domain is given as a right ideal in the quiver algebra we can also create a homomorphism by just giving a single element in the codomain. The map is then induced by acting on that element:

sage: x = P2.gens('x')[0]
sage: x
x_0
sage: f == P2.hom(f(x), M2)
True

As you can see, the above homomorphisms can be applied to elements. Just like elements, addition is defined via the + operator. On elements scalar multiplication is defined via the * operator but on homomorphisms * defines composition, so scalar multiplication is done using a method:

sage: g = f + f
sage: g == f.scalar_mult(2)
True
sage: g == 2*f       # This multiplies the map with the scalar 2
True
sage: g(x) == 2*f(x) # This applies the map, then multiplies by the scalar
True

The direct_sum method for modules returns only the resulting module by default. But can also return the projection and inclusion homomorphisms into the various factors:

sage: N2, inclusions, projections = M2.direct_sum([P2], return_maps=True)
sage: inclusions[0].domain() is M2
True
sage: projections[0].codomain() is M2
True
sage: (projections[0]*inclusions[0]).is_isomorphism()
True

As you see above we can determine if a given map is an isomorphism. Testing for injectivity and surjectivity works as well:

sage: f.is_injective()
False
sage: f.is_surjective()
False

We can create all the standard modules associated to maps:

sage: f.kernel()
Representation with dimension vector (0, 1)
sage: f.cokernel()
Representation with dimension vector (1, 0)
sage: im = f.image()
sage: im
Representation with dimension vector (1, 1)

These methods, as well as the submodule and quotient methods that are defined for representations, return only the resulting representation. To get the inclusion map of a submodule or the factor homomorphism of a quotient use coerce_map_from:

sage: incl = M2.coerce_map_from(im)
sage: incl.domain() is im
True
sage: incl.codomain() is M2
True
sage: incl.is_injective()
True

Both QuiverRep objects and QuiverRepHom objects have linear_dual and algebraic_dual methods. The linear_dual method applies the functor \(Hom_k(..., k)\) where \(k\) is the base ring of the representation, and the algebraic_dual method applies the functor \(Hom_Q(..., kQ)\) where \(kQ\) is the quiver algebra. Both these functors yield left modules. A left module is equivalent to a right module over the opposite algebra, and the opposite of a quiver algebra is the algebra of the opposite quiver, so both these methods yield modules and representations of the opposite quiver:

sage: f.linear_dual()
Homomorphism of representations of Reverse of (): Multi-digraph on 2 vertices
sage: D = M2.algebraic_dual()
sage: D.quiver() is PQ2.reverse().quiver()
True

Todo

Change the wording Reverse of () into something more meaningful.

There is a method returning the projective cover of any module. Note that this method returns the homomorphism; to get the module take the domain of the homomorphism:

sage: cov = M2.projective_cover()
sage: cov
Homomorphism of representations of Multi-digraph on 2 vertices
sage: cov.domain()
Representation with dimension vector (2, 4)

As projective covers are computable, so are the transpose and Auslander-Reiten translates of modules:

sage: M2.transpose()
Representation with dimension vector (4, 3)
sage: PQ2.I(QQ, 1).AR_translate()
Representation with dimension vector (3, 2)

We have already used the gens method above to get an element of a quiver representation. An element of a quiver representation is simply a choice of element from each of the spaces assigned to the vertices of the quiver. Addition, subtraction, and scalar multiplication are performed pointwise and implemented by the usual operators:

sage: M2.dimension_vector()
(2, 1)
sage: x, y, z = M2.gens('xyz')
sage: 2*x + y != x + 2*y
True

To create a specific element of a given representation we just specify the representation and a dictionary associating to each vertex an element of the space associated to that vertex in the representation:

sage: w = M2({1:(1, -1), 2:(3,)})
sage: w.get_element(1)
(1, -1)

The right action of a quiver algebra on an element is implemented via the * operator:

sage: A2 = x.quiver().path_semigroup().algebra(QQ)
sage: a = A2((1, 2, 'a'))
sage: x*a == z
True
class sage.quivers.representation.QuiverRepElement(parent, elements=None, name=None)

Bases: sage.structure.element.ModuleElement

An element of a quiver representation is a choice of element from each of the spaces assigned to the vertices of the quiver. Addition, subtraction, and scalar multiplication of these elements is done pointwise within these spaces.

INPUT:

  • moduleQuiverRep (default: None), the module to which the element belongs
  • elements - dict (default: empty), a dictionary associating to each vertex a vector or an object from which sage can create a vector. Not all vertices must be specified, unspecified vertices will be assigned the zero vector of the space associated to that vertex in the given module. Keys that do not correspond to a vertex are ignored.
  • name – string (default: None), the name of the element

OUTPUT:

Note

The constructor needs to know the quiver in order to create an element of a representation over that quiver. The default is to read this information from module as well as to fill in unspecified vectors with the zeros of the spaces in module. If module is None then quiver MUST be a quiver and each vertex MUST be specified or an error will result. If both module and quiver are given then quiver is ignored.

EXAMPLES:

sage: Q = DiGraph({1:{2:['a'], 3:['b']}, 2:{3:['c']}}).path_semigroup()
sage: spaces = dict((v, GF(3)^2) for v in Q.quiver())
sage: M = Q.representation(GF(3), spaces)
sage: elems = {1: (1, 0), 2: (0, 1), 3: (2, 1)}
sage: M(elems)
Element of quiver representation
sage: v = M(elems, 'v')
sage: v
v
sage: (v + v + v).is_zero()
True
copy()

Return a copy of self.

EXAMPLES:

sage: Q = DiGraph({1:{2:['a'], 3:['b']}, 2:{3:['c']}}).path_semigroup()
sage: spaces = dict((v, GF(3)^2) for v in Q.quiver())
sage: M = Q.representation(GF(3), spaces)
sage: elems = {1: (1, 0), 2: (0, 1), 3: (2, 1)}
sage: v = M(elems)
sage: w = v.copy()
sage: w._set_element((0, 0), 1)
sage: w.get_element(1)
(0, 0)
sage: v.get_element(1)
(1, 0)
get_element(vertex)

Return the element at the given vertex.

INPUT:

  • vertex – integer, a vertex of the quiver

OUTPUT:

  • vector, the vector assigned to the given vertex

EXAMPLES:

sage: Q = DiGraph({1:{2:['a'], 3:['b']}, 2:{3:['c']}}).path_semigroup()
sage: spaces = dict((v, GF(3)^2) for v in Q.quiver())
sage: M = Q.representation(GF(3), spaces)
sage: elems = {1: (1, 0), 2: (0, 1), 3: (2, 1)}
sage: v = M(elems)
sage: v.get_element(1)
(1, 0)
sage: v.get_element(3)
(2, 1)
is_zero()

Test whether self is zero.

OUTPUT:

  • bool, True if the element is the zero element, False otherwise

EXAMPLES:

sage: Q = DiGraph({1:{2:['a'], 3:['b']}, 2:{3:['c']}}).path_semigroup()
sage: spaces = dict((v, GF(3)^2) for v in Q.quiver())
sage: M = Q.representation(GF(3), spaces)
sage: elems = {1: (1, 0), 2: (0, 1), 3: (2, 1)}
sage: v = M(elems)
sage: v.is_zero()
False
sage: w = M()
sage: w.is_zero()
True

TESTS:

sage: M = Q.P(QQ, 1)
sage: M.zero().is_zero()
True
quiver()

Return the quiver of the representation.

OUTPUT:

  • DiGraph, the quiver of the representation

EXAMPLES:

sage: Q = DiGraph({1:{2:['a'], 3:['b']}, 2:{3:['c']}}).path_semigroup()
sage: P = Q.P(QQ, 1)
sage: v = P.an_element()
sage: v.quiver() is Q.quiver()
True
support()

Return the support of self as a list.

The support is the set of vertices to which a nonzero vector is associated.

OUTPUT

  • list, the support

EXAMPLES:

sage: Q = DiGraph({1:{2:['a'], 3:['b']}, 2:{3:['c']}}).path_semigroup()
sage: spaces = dict((v, GF(3)^2) for v in Q.quiver())
sage: M = Q.representation(GF(3), spaces)
sage: elems = {1: (1, 0), 2: (0, 0), 3: (2, 1)}
sage: v = M(elems)
sage: v.support()
[1, 3]
class sage.quivers.representation.QuiverRepFactory

Bases: sage.structure.factory.UniqueFactory

A quiver representation is a diagram in the category of vector spaces whose underlying graph is the quiver. Giving a finite dimensional representation is equivalent to giving a finite dimensional right module for the path algebra of the quiver.

INPUT:

The first two arguments specify the base ring and the quiver, and they are always required:

  • k – ring, the base ring of the representation
  • P – the partial semigroup formed by the paths of the quiver of the representation

Then to specify the spaces and maps associated to the quiver there are three possible options. The first is the 'values' option, where the next two arguments give the data to be assigned. The following can either be the next two entries in the argument list or they can be passed by keyword. If the argument list is long enough the keywords are ignored; the keywords are only checked in the event that the argument list does not have enough entries after P.

  • spaces – dict (default: empty); a dictionary associating to each vertex a free module over the base ring \(k\). Not all vertices must be specified; unspecified vertices are automatically set to \(k^0\). Keys of the dictionary that don’t correspond to vertices are ignored.
  • maps - dict (default: empty); a dictionary associating to each edge a map whose domain and codomain are the spaces associated to the initial and terminal vertex of the edge respectively. Not all edges must be specified; unspecified edges are automatically set to the zero map. Keys of the dictionary that don’t correspond to edges are ignored.

The second option is the paths option which creates a module by generating a right ideal from a list of paths. Thus the basis elements of this module correspond to paths of the quiver and the maps are given by right multiplication by the corresponding edge. As above this can be passed either as the next entry in the argument list or as a keyword. The keyword is only checked if there is no entry in the argument list after Q.

  • basis - list; a nonempty list of paths in the quiver Q. Entries that do not represent valid paths are ignored and duplicate paths are deleted. There must be at least one valid path in the list or a ValueError is raised. The closure of this list under right multiplication forms the basis of the resulting representation.

The third option is the dual paths option which creates the dual of a left ideal in the quiver algebra. Thus the basis elements of this module correspond to paths of the quiver and the maps are given by deleting the corresponding edge from the start of the path (the edge map is zero on a path if that edge is not the initial edge of the path). As above this can be passed either as the next entry in the argument list or as a keyword.

  • basis – list; a nonempty list of paths in the quiver Q. Entries that do not represent valid paths are ignored and duplicate paths are deleted. There must be at least one valid path in the list or a ValueError is raised. The closure of this list under left multiplication of edges forms the basis of the resulting representation.

Using the second and third options requires that the following keyword be passed to the constructor. This must be passed as a keyword.

  • option - string (default: None), either 'values' or 'paths' or 'dual paths'. None is equivalent to 'values'.

OUTPUT:

  • QuiverRep

EXAMPLES:

sage: Q1 = DiGraph({1:{2:['a']}}).path_semigroup()

When the option keyword is not supplied the constructor uses the 'values' option and expects the spaces and maps to be specified. If no maps or spaces are given the zero module is created:

sage: M = Q1.representation(GF(5))
sage: M.is_zero()
True

The simple modules, indecomposable projectives, and indecomposable injectives are examples of quiver representations:

sage: S = Q1.S(GF(3), 1)
sage: I = Q1.I(QQ, 2)
sage: P = Q1.P(GF(3), 1)

Various standard submodules can be computed, such as radicals and socles. We can also form quotients and test for certain attributes such as semisimplicity:

sage: R = P.radical()
sage: R.is_zero()
False
sage: (P/R).is_simple()
True
sage: P == R
False

With the option 'paths' the input data should be a list of QuiverPaths or things that QuiverPaths can be constructed from. The resulting module is the submodule generated by these paths in the quiver algebra, when considered as a right module over itself:

sage: P1 = Q1.representation(QQ, [[(1, 1)]], option='paths')
sage: P1.dimension()
2

In the following example, the 3rd and 4th paths are actually the same, so the duplicate is removed:

sage: N = Q1.representation(QQ, [[(1, 1)], [(2, 2)], [(1, 1), (1, 2, 'a'), (2, 2)], [(1, 2, 'a')]], option='paths')
sage: N.dimension()
3

The dimension at each vertex equals the number of paths in the closed basis whose terminal point is that vertex:

sage: Q2 = DiGraph({1:{2:['a'], 3:['b', 'c']}, 2:{3:['d']}}).path_semigroup()
sage: M = Q2.representation(QQ, [[(2, 2)], [(1, 2, 'a')]], option='paths')
sage: M.dimension_vector()
(0, 2, 2)
sage: N = Q2.representation(QQ, [[(2, 2)], [(1, 2, 'a'), (2, 3, 'd')]], option='paths')
sage: N.dimension_vector()
(0, 1, 2)
create_key(k, P, *args, **kwds)

Return a key for the specified module.

The key is a tuple. The first and second entries are the base ring k and the partial semigroup P formed by the paths of a quiver. The third entry is the option and the remaining entries depend on that option. If the option is 'values' and the quiver has \(n\) vertices then the next \(n\) entries are the vector spaces to be assigned to those vertices. After that are the matrices of the maps assigned to edges, listed in the same order that Q.edges() uses. If the option is 'paths' or 'dual paths' then the next entry is a tuple containing a sorted list of the paths that form a basis of the quiver.

INPUT:

See the class documentation.

OUTPUT:

  • tuple

EXAMPLES:

sage: P = DiGraph({1:{2:['a']}}).path_semigroup()
sage: from sage.quivers.representation import QuiverRep
sage: QuiverRep.create_key(GF(5), P)
(Finite Field of size 5,
 Partial semigroup formed by the directed paths of Multi-digraph on 2 vertices,
 'values',
 Vector space of dimension 0 over Finite Field of size 5,
 Vector space of dimension 0 over Finite Field of size 5,
 [])
create_object(version, key, **extra_args)

Create a QuiverRep_generic or QuiverRep_with_path_basis object from the key.

The key is a tuple. The first and second entries are the base ring k and the quiver Q. The third entry is the 'option' and the remaining entries depend on that option. If the option is 'values' and the quiver has \(n\) vertices then the next \(n\) entries are the vector spaces to be assigned to those vertices. After that are the matrices of the maps assigned to edges, listed in the same order that Q.edges() uses. If the option is 'paths' or 'dual paths' then the next entry is a tuple containing a sorted list of the paths that form a basis of the quiver.

INPUT:

  • version – the version of sage, this is currently ignored
  • key – tuple

OUTPUT:

EXAMPLES:

sage: Q = DiGraph({1:{2:['a']}}).path_semigroup()
sage: from sage.quivers.representation import QuiverRep
sage: key = QuiverRep.create_key(GF(5), Q)
sage: QuiverRep.create_object(Q.version(), key)
Representation with dimension vector (0, 0)
class sage.quivers.representation.QuiverRep_generic(k, P, spaces, maps)

Bases: sage.misc.fast_methods.WithEqualityById, sage.modules.module.Module

A generic quiver representation.

This class should not be called by the user.

Call QuiverRep with option='values' (which is the default) instead.

INPUT:

  • k – ring, the base ring of the representation
  • P – the path semigroup of the quiver \(Q\) of the representation
  • spaces – dict (default: empty), a dictionary associating to each vertex a free module over the base ring \(k\). Not all vertices need to be specified, unspecified vertices are automatically set to \(k^0\). Keys of the dictionary that don’t correspond to vertices are ignored.
  • maps – dict (default: empty), a dictionary associating to each edge a map whose domain and codomain are the spaces associated to the initial and terminal vertex of the edge respectively. Not all edges need to be specified, unspecified edges are automatically set to the zero map. Keys of the dictionary that don’t correspond to edges are ignored.

OUTPUT:

  • QuiverRep

EXAMPLES:

sage: Q = DiGraph({1:{3:['a']}, 2:{3:['b']}}).path_semigroup()
sage: spaces = {1: QQ^2, 2: QQ^3, 3: QQ^2}
sage: maps = {(1, 3, 'a'): (QQ^2).Hom(QQ^2).identity(), (2, 3, 'b'): [[1, 0], [0, 0], [0, 0]]}
sage: M = Q.representation(QQ, spaces, maps)
sage: Q = DiGraph({1:{2:['a']}}).path_semigroup()
sage: P = Q.P(GF(3), 1)
sage: I = Q.I(QQ, 1)
sage: P.an_element() in P
True
sage: I.an_element() in P
False

TESTS:

sage: TestSuite(M).run()
sage: TestSuite(P).run()
sage: TestSuite(I).run()
sage: TestSuite(Q.S(ZZ,2)).run()
AR_translate()

Return the Auslander-Reiten translate of self.

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: M = Q.representation(GF(3), {1: 1, 2: 1}, {(1, 2, 'a'): 1})
sage: tauM = M.AR_translate()
sage: tauM
Representation with dimension vector (1, 1)
sage: tauM.get_map((1, 2, 'a')).matrix()
[1]
sage: tauM.get_map((1, 2, 'b')).matrix()
[0]

The module M above is its own AR translate. This is not always true:

sage: Q2 = DiGraph({3:{1:['b']}, 5:{3:['a']}}).path_semigroup()
sage: Q2.S(QQ, 5).AR_translate()
Representation with dimension vector (0, 1, 0)
Element

alias of QuiverRepElement

Hom(codomain)

Return the hom space from self to codomain.

For more information see the QuiverHomSpace documentation.

TESTS:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: Q.S(QQ, 2).Hom(Q.P(QQ, 1))
Dimension 2 QuiverHomSpace
actor()

Return the quiver path algebra acting on this representation.

OUTPUT:

  • a quiver path algebra

EXAMPLES:

sage: Q = DiGraph({1:{2:['a']}}).path_semigroup()
sage: M = Q.representation(GF(5))
sage: M.base_ring()
Finite Field of size 5
sage: M.actor()
Path algebra of Multi-digraph on 2 vertices over Finite Field of size 5
algebraic_dual(basis=False)

Compute the algebraic dual \(Hom_Q(M, kQ)\) of the module \(M\) = self.

INPUT:

  • basis – bool; if False, then only the module is returned. If True, then a tuple is returned. The first element is the QuiverRep and the second element is a dictionary which associates to each vertex a list. The elements of this list are the homomorphisms which correspond to the basis elements of that vertex in the module.

OUTPUT:

  • QuiverRep or tuple

Note

Here \(kQ\) is the path algebra considered as a right module over itself. If \(e\) is an edge of the quiver \(Q\) then we let \((fe)(m) = ef(m)\). This gives \(Hom_Q(M, kQ)\) a module structure over the opposite quiver Q.reverse().

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b'], 3: ['c', 'd']}, 2:{3:['e']}}).path_semigroup()
sage: Q.free_module(GF(7)).algebraic_dual().dimension_vector()
(7, 2, 1)
an_element()

Return an element of self.

OUTPUT:

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: M = Q.P(QQ, 1)
sage: M.an_element()
Element of quiver representation
coordinates(vector)

Return the coordinates when vector is expressed in terms of the gens.

INPUT:

OUTPUT:

  • list, the coefficients when the vector is expressed as a linear combination of the generators of the module

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: M = Q.P(QQ, 1)
sage: x, y, z = M.gens('xyz')
sage: M.coordinates(x - y + z)
[1, -1, 1]
sage: M.coordinates(M.an_element())
[1, 1, 0]
sage: M.an_element() == x + y
True
dimension(vertex=None)

Return the dimension of the space associated to the given vertex vertex.

INPUT:

  • vertex – integer or None (default: None), the given vertex

OUTPUT:

  • integer, the dimension over the base ring of the space associated to the given vertex. If vertex=None then the dimension over the base ring of the module is returned

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: P = Q.P(GF(2), 1)
sage: P.dimension(1)
1
sage: P.dimension(2)
2

The total dimension of the module is the sum of the dimensions at each vertex:

sage: P.dimension()
3
dimension_vector()

Return the dimension vector of the representation.

OUTPUT:

  • tuple

Note

The order of the entries in the tuple matches the order given by calling the vertices() method on the quiver.

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: P = Q.P(GF(2), 1)
sage: P.dimension_vector()
(1, 2)

Each coordinate of the dimension vector is the dimension of the space associated to that coordinate:

sage: P.get_space(2).dimension()
2
direct_sum(modules, return_maps=False)

Return the direct sum of self with the given modules modules.

The modules must be modules over the same quiver and base ring.

INPUT:

  • modulesQuiverRep or list of QuiverRep‘s
  • return_maps – Boolean (default: False); if False, then the output is a single QuiverRep object which is the direct sum of self with the given module or modules. If True, then the output is a list [sum, iota, pi]. The first entry sum is the direct sum of self with the given module or modules. Both iota and pi are lists of QuiverRepHoms with one entry for each summand; iota[i] is the inclusion map and pi[i] is the projection map of the \(i\)-th summand. The summands are ordered as given with self being the zeroth summand.

OUTPUT:

  • QuiverRep or tuple

EXAMPLES:

sage: Q = DiGraph({1:{2:['a'], 3:['b']}, 2:{4:['c']}, 3:{4:['d']}}).path_semigroup()
sage: P1 = Q.P(QQ, 1)
sage: P2 = Q.P(QQ, 2)
sage: S = P1.direct_sum(P2)
sage: P1.dimension_vector()
(1, 1, 1, 2)
sage: P2.dimension_vector()
(0, 1, 0, 1)
sage: S.dimension_vector()
(1, 2, 1, 3)
sage: S, iota, pi = P1.direct_sum(P2, return_maps=True)
sage: iota[0].domain() is P1
True
sage: iota[1].domain() is P2
True
sage: pi[0].codomain() is P1
True
sage: pi[1].codomain() is P2
True
sage: iota[0].codomain() is S
True
sage: iota[1].codomain() is S
True
sage: pi[0].domain() is S
True
sage: pi[1].domain() is S
True
sage: iota[0].get_matrix(4)
[1 0 0]
[0 1 0]
sage: pi[0].get_matrix(4)
[1 0]
[0 1]
[0 0]
sage: P1prime = S/iota[1].image()
sage: f = P1prime.coerce_map_from(S)
sage: g = f*iota[0]
sage: g.is_isomorphism()
True
gens(names='v')

Return a list of generators of self as a \(k\)-module.

INPUT:

  • names – an iterable variable of length equal to the number of generators, or a string (default: 'v'); gives the names of the generators either by giving a name to each generator or by giving a name to which an index will be appended

OUTPUT:

  • list of QuiverRepElement objects, the linear generators of the module (over the base ring)

Note

The generators are ordered first by vertex and then by the order given by the gens() method of the space associated to that vertex.

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: M = Q.P(QQ, 1)
sage: M.gens()
[v_0, v_1, v_2]

If a string is given then it is used as the name of each generator, with the index of the generator appended in order to differentiate them:

sage: M.gens('generator')
[generator_0, generator_1, generator_2]

If a list or other iterable variable is given then each generator is named using the appropriate entry. The length of the variable must equal the number of generators (the dimension of the module):

sage: M.gens(['w', 'x', 'y', 'z'])
Traceback (most recent call last):
...
TypeError: can only concatenate list (not "str") to list
sage: M.gens(['x', 'y', 'z'])
[x, y, z]

Strings are iterable, so if the length of the string is equal to the number of generators then the characters of the string will be used as the names:

sage: M.gens('xyz')
[x, y, z]
get_map(edge)

Return the map associated to the given edge edge.

INPUT:

  • edge – tuple of the form (initial vertex, terminal vertex, label) specifying the edge whose map is returned

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: Q.P(ZZ, 1).get_map((1, 2, 'a'))
Free module morphism defined by the matrix
[1 0]
Domain: Ambient free module of rank 1 over the principal ideal domain Integer Ring
Codomain: Ambient free module of rank 2 over the principal ideal domain Integer Ring
get_space(vertex)

Return the module associated to the given vertex vertex.

INPUT:

  • vertex – integer, a vertex of the quiver of the module

EXAMPLES:

sage: Q = DiGraph({1:{2:['a'], 3:['b']}}).path_semigroup()
sage: Q.P(QQ, 1).get_space(1)
Vector space of dimension 1 over Rational Field
is_semisimple()

Test whether the representation is semisimple.

OUTPUT:

  • bool

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: M = Q.P(QQ, 1)
sage: (M/M.radical()).is_semisimple()
True
is_simple()

Test whether the representation is simple.

OUTPUT:

  • bool

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: Q.P(RR, 1).is_simple()
False
sage: Q.S(RR, 1).is_simple()
True
is_zero()

Test whether the representation is zero.

OUTPUT:

  • bool

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: M = Q.representation(ZZ)
sage: N = Q.representation(ZZ, {1: 1})
sage: M
Representation with dimension vector (0, 0)
sage: N
Representation with dimension vector (1, 0)
sage: M.is_zero()
True
sage: N.is_zero()
False
linear_combination_of_basis(coordinates)

Return the linear combination of the basis for self given by coordinates.

INPUT:

  • coordinates – list; a list whose length is the dimension of self. The \(i\)-th element of this list defines the coefficient of the \(i\)-th basis vector in the linear combination.

OUTPUT:

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: M = Q.P(QQ, 1)
sage: x, y, z = M.gens('xyz')
sage: e = x - y + z
sage: M.coordinates(e)
[1, -1, 1]
sage: M.linear_combination_of_basis([1, -1, 1]) == e
True
linear_dual()

Compute the linear dual \(Hom_k(M, k)\) of the module \(M =\) self over the base ring \(k\).

OUTPUT:

  • QuiverRep, the dual representation

Note

If \(e\) is an edge of the quiver \(Q\) then we let \((fe)(m) = f(me)\). This gives \(Hom_k(M, k)\) a module structure over the opposite quiver Q.reverse().

EXAMPLES:

sage: Q = DiGraph({1:{2:['a','b']}, 2:{3:['c']}}).path_semigroup()
sage: M = Q.P(QQ, 1)
sage: M.linear_dual()
Representation with dimension vector (1, 2, 2)
sage: M.linear_dual().quiver() is Q.reverse().quiver()
True
projective_cover(return_maps=False)

Return the projective cover of self.

EXAMPLES:

sage: Q = DiGraph({1:{2:['a','b']}, 2:{3:['c','d']}}).path_semigroup()
sage: S1 = Q.S(GF(3), 1)
sage: f1 = S1.projective_cover()
sage: f1.is_surjective()
True
sage: f1._domain
Representation with dimension vector (1, 2, 4)
sage: Q.P(GF(3), 1)
Representation with dimension vector (1, 2, 4)
quiver()

Return the quiver of the representation.

OUTPUT:

EXAMPLES:

sage: Q = DiGraph({1:{2:['a']}}).path_semigroup()
sage: M = Q.representation(GF(5))
sage: M.quiver() is Q.quiver()
True
quotient(sub, check=True)

Return the quotient of self by the submodule sub.

INPUT:

  • subQuiverRep; this must be a submodule of self, meaning the space associated to each vertex \(v\) of sub is a subspace of the space associated to \(v\) in self and the map associated to each edge \(e\) of sub is the restriction of the map associated to \(e\) in self
  • check – bool; if True then sub is checked to verify that it is indeed a submodule of self and an error is raised if it is not

OUTPUT:

  • QuiverRep, the quotient module self / sub

Note

This function returns only a QuiverRep object quot. The projection map from self to quot can be obtained by calling quot.coerce_map_from(self).

EXAMPLES:

sage: Q = DiGraph({1:{2:[‘a’,’b’]}, 2:{3:[‘c’]}}).path_semigroup() sage: M = Q.I(GF(3), 3) sage: N = Q.S(GF(3), 3) sage: M.quotient(N) Representation with dimension vector (2, 1, 0) sage: M.quotient(M.radical()) Representation with dimension vector (2, 0, 0) sage: M.quotient(M) Representation with dimension vector (0, 0, 0)
radical()

Return the Jacobson radical of self.

OUTPUT:

  • QuiverRep, the Jacobson radical

EXAMPLES:

sage: Q = DiGraph({1:{2:['a','b']}, 2:{3:['c']}}).path_semigroup()
sage: M = Q.P(QQ, 1)
sage: M.radical()
Representation with dimension vector (0, 2, 2)
right_edge_action(element, path)

Return the result of element*path.

INPUT:

  • elementQuiverRepElement, an element of self
  • pathQuiverPath or list of tuples

OUTPUT:

  • QuiverRepElement, the result of element*path when path is considered an element of the path algebra of the quiver

EXAMPLES:

sage: Q = DiGraph({1:{2:['a','b']}, 2:{3:['c']}}).path_semigroup()
sage: M = Q.P(QQ, 1)
sage: v = M.an_element()
sage: v.support()
[1, 2, 3]
sage: M.right_edge_action(v, (1, 1)).support()
[1]
sage: M.right_edge_action(v, [(1, 1)]).support()
[1]
sage: M.right_edge_action(v, [(1, 1), (1, 2, 'a')]).support()
[2]
sage: M.right_edge_action(v, (1, 2, 'a')) == M.right_edge_action(v, [(1, 1), (1, 2, 'a'), (2, 2)])
True
socle()

The socle of self.

OUTPUT:

  • QuiverRep, the socle

EXAMPLES:

sage: Q = DiGraph({1:{2:['a','b']}, 2:{3:['c']}}).path_semigroup()
sage: M = Q.P(QQ, 1)
sage: M.socle()
Representation with dimension vector (0, 0, 2)
submodule(elements=[], spaces=None)

Returns the submodule generated by the data.

INPUT:

  • elements – a collection of QuiverRepElements (default: empty list), each should be an element of self
  • spaces – dictionary (default: empty), this dictionary should contain entries of the form {v: S} where \(v\) is a vertex of the quiver and \(S\) is a subspace of the vector space associated to \(v\)

OUTPUT:

  • QuiverRep, the smallest subrepresentation of self containing the given elements and the given subspaces

Note

This function returns only a QuiverRep object sub. The inclusion map of sub into M = self can be obtained by calling M.coerce_map_from(sub).

EXAMPLES:

sage: Q = DiGraph({1:{3:['a']}, 2:{3:['b']}}).path_semigroup()
sage: spaces = {1: QQ^2, 2: QQ^3, 3: QQ^2}
sage: maps = {(1, 3, 'a'): (QQ^2).Hom(QQ^2).identity(), (2, 3, 'b'): [[1, 0], [0, 0], [0, 0]]}
sage: M = Q.representation(QQ, spaces, maps)
sage: v = M.an_element()
sage: M.submodule([v])
Representation with dimension vector (1, 1, 1)

The smallest submodule containing the vector space at vertex 1 also contains the entire vector space associated to vertex 3 because there is an isomorphism associated to the edge (1, 3, 'a'):

sage: M.submodule(spaces={1: QQ^2})
Representation with dimension vector (2, 0, 2)

The smallest submodule containing the vector space at vertex 2 also contains the image of the rank 1 homomorphism associated to the edge (2, 3, 'b'):

sage: M.submodule(spaces={2: QQ^3})
Representation with dimension vector (0, 3, 1)

As v is not already contained in this submodule, adding it as a generator yields a larger submodule:

sage: v.support()
[1, 2, 3]
sage: M.submodule([v], {2: QQ^3})
Representation with dimension vector (1, 3, 1)

Giving no generating data yields the zero submodule:

sage: M.submodule().is_zero()
True

If the given data generates all of M then the result is M:

sage: M.submodule(M.gens()) is M
True
support()

Return the support of self as a list.

OUTPUT:

  • list, the vertices of the representation that have nonzero spaces associated to them

EXAMPLES:

sage: Q = DiGraph({1:{2:['a']}, 3:{2:['b'], 4:['c']}}).path_semigroup()
sage: M = Q.P(QQ, 3)
sage: M
Representation with dimension vector (0, 1, 1, 1)
sage: M.support()
[2, 3, 4]
top()

Return the top of self.

OUTPUT:

  • QuiverRep, the quotient of self by its radical

EXAMPLES:

sage: Q = DiGraph({1:{2:['a','b']}, 2:{3:['c']}}).path_semigroup()
sage: M = Q.P(QQ, 1)
sage: M.top()
Representation with dimension vector (1, 0, 0)
sage: M.top() == M/M.radical()
True
transpose()

Return the transpose of self.

The transpose, \(\mbox{Tr} M\), of a module \(M\) is defined as follows. Let \(p: P_1 \to P_2\) be the second map in a minimal projective presentation \(P_1 \to P_2 \to M \to 0\) of \(M\). If \(p^t\) is the algebraic dual of \(p\) then define \(\mbox{Tr} M = \mbox{coker} p^t\).

OUTPUT:

  • QuiverRep

EXAMPLES:

sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: M = Q.representation(GF(3), {1: 1, 2: 1}, {(1, 2, 'a'): 1})
sage: M.transpose()
Representation with dimension vector (1, 1)
zero_submodule()

Return the zero submodule of self.

OUTPUT:

  • QuiverRep, the zero submodule of self.

EXAMPLES:

sage: Q = DiGraph({1:{2:['a','b']}, 2:{3:['c']}}).path_semigroup()
sage: M = Q.P(QQ, 1)
sage: M.zero_submodule()
Representation with dimension vector (0, 0, 0)
sage: M.zero_submodule().is_zero()
True
class sage.quivers.representation.QuiverRep_with_dual_path_basis(k, P, basis)

Bases: sage.quivers.representation.QuiverRep_generic

The basis of the module must be closed under left deletion of an edge; that is, deleting any edge from the beginning of any path in the basis must result in a path also contained in the basis of the module.

INPUT:

  • k – ring; the base ring of the representation
  • P – the path semigroup of the quiver \(Q\) of the representation
  • basis – list (default: empty); should be a list of paths (also lists) in the quiver \(Q\). Entries that do not represent valid paths are ignored and duplicate paths are deleted. The closure of this list under left deletion forms the basis of the resulting representation.
class sage.quivers.representation.QuiverRep_with_path_basis(k, P, basis)

Bases: sage.quivers.representation.QuiverRep_generic

The basis of the module must be closed under right multiplication by an edge; that is, appending any edge to the end of any path in the basis must result in either an invalid path or a valid path also contained in the basis of the module.

INPUT:

  • k – ring, the base ring of the representation
  • P – the path semigroup of the quiver \(Q\) of the representation
  • basis – list (default: empty); should be a list of paths (also lists) in the quiver \(Q\). Entries that do not represent valid paths are ignored and duplicate paths are deleted. The closure of this list under right multiplication forms the basis of the resulting representation.
is_left_module()

Test whether the basis is closed under left multiplication.

EXAMPLES:

sage: Q1 = DiGraph({1:{2:['a']}}).path_semigroup()
sage: P2 = Q1.representation(QQ, [[(2, 2)]], option='paths')
sage: P2.is_left_module()
False

The supplied basis is not closed under left multiplication, but it’s not closed under right multiplication either. When the closure under right multiplication is taken the result is also closed under left multiplication and therefore produces a left module structure:

sage: kQ = Q1.representation(QQ, [[(1, 1)], [(2, 2)]], option='paths')
sage: kQ.is_left_module()
True

Taking the right closure of a left closed set produces another left closed set:

sage: Q2 = DiGraph({1:{2:['a'], 3:['b', 'c']}, 2:{3:['d']}}).path_semigroup()
sage: M = Q2.representation(QQ, [[(2, 2)], [(1, 2, 'a')]], option='paths')
sage: M.is_left_module()
True

Note that the second path is length 2, so even though the edge (1, 2, ‘a’) appears in the input the path [(1, 2, ‘a’)] is not in the right closure:

sage: N = Q2.representation(QQ, [[(2, 2)], [(1, 2, 'a'), (2, 3, 'd')]], option='paths')
sage: N.is_left_module()
False

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