In order to do what it does, the SBStrips package includes several utility functions for use on quivers where each vertex has indegree and outdegree at most 2. (The existing term for such quivers is special biserial, abbreivated SB.) This class includes the 1-regular quivers: those where each vertex has indegree and outdegree exactly 1.
These quiver utility functions really build on the QPA package. We document them in this standalone chapter, alongside utilities for algebras presented by quivers.
The term quiver algebra means an object for which IsQuiverAlgebra (QPA: IsQuiverAlgebra) returns true.
‣ Is1RegQuiver( quiver ) | ( property ) |
Argument: quiver, a quiver
Returns: either true or false, depending on whether or not quiver is 1-regular.
‣ PathBySourceAndLength( vert, len ) | ( operation ) |
Arguments: vert, a vertex of a 1-regular quiver Q; len, a nonnegative integer.
Returns: the unique path in Q which has source vert and length len.
‣ PathByTargetAndLength( vert, len ) | ( operation ) |
Arguments: vert, a vertex of a 1-regular quiver Q; len, a nonnegative integer.
Returns: the unique path in Q which has target vert and length len.
‣ 1RegQuivIntAct( x, k ) | ( operation ) |
Arguments: x, which is either a vertex or an arrow of a 1-regular quiver; k, an integer.
Returns: the path x+k, as per the ℤ-action (see below).
Recall that a quiver is 1-regular iff the source and target functions s,t are bijections from the arrow set to the vertex set (in which case the inverse t^-1 is well-defined). The generator 1 ∈ ℤ acts as ``t^-1 then s'' on vertices and ``s then t^-1'' on arrows.
This operation figures out from x the quiver to which x belongs and applies 1RegQuivIntActionFunction (5.2-5) of tha quiver. For this reason, it is more user-friendly.
‣ 1RegQuivIntActionFunction( quiver ) | ( attribute ) |
Argument: quiver, a 1-regular quiver (as tested by Is1RegQuiver (5.2-1))
Returns: a single function f describing the ℤ-actions on the vertices and the arrows of quiver
Recall that a quiver is 1-regular iff the source and target functions s,t are bijections from the arrow set to the vertex set (in which case the inverse t^-1 is well-defined). The generator 1 ∈ ℤ acts as ``t^-1 then s'' on vertices and ``s then t^-1'' on arrows.
In practice you will probably want to use 1RegQuivIntAct (5.2-4), since it saves you having to remind SBStrips which quiver you intend to act on.
‣ Is2RegQuiver( quiver ) | ( property ) |
Argument: quiver, a quiver
Returns: either true or false, depending on whether or not quiver is 2-regular.
‣ 2RegAugmentationOfQuiver( ground_quiv ) | ( attribute ) |
Argument: ground_quiv, a sub2-regular quiver (as tested by IsSpecialBiserialQuiver (QPA: IsSpecialBiserialQuiver))
Returns: a 2-regular quiver of which ground_quiv may naturally be seen as a subquiver
If ground_quiv is itself sub-2-regular, then this attribute returns ground_quiv identically. If not, then this attribute constructs a brand new quiver object which has vertices and arrows having the same names as those of ground_quiv, but also has arrows with names augarr1, augarr2 and so on.
‣ Is2RegAugmentationOfQuiver( quiver ) | ( property ) |
Argument: quiver, a quiver
Returns: true if quiver was constructed by 5.3-2 or if quiver was an already 2-regular quiver, and false otherwise.
‣ OriginalSBQuiverOf2RegAugmentation( quiver ) | ( attribute ) |
Argument: quiver, a quiver
Returns: The sub-2-regular quiver of which quiver is the 2-regular augmentation.
Informally speaking, this attribute is the "inverse" to 2RegAugmentationOfQuiver.
‣ RetractionOf2RegAugmentation( quiver ) | ( attribute ) |
Argument: quiver, a quiver constructed using 2RegAugmentationOfQuiver
Returns: a function ret, which accepts paths in quiver as input and which outputs paths in OriginalSBQuiverOf2RegAugmentation( quiver ) 5.3-2.
One can identify OriginalSBQuiverOf2RegAugmentation( quiver ) with a subquiver of quiver. Some paths in quiver lie wholly in that subquiver, some do not. This function ret takes those that do to the corresponding path of OriginalSBQuiverOf2RegAugmentation( quiver ), and those that do not to the zero path of OriginalSBQuiverOf2RegAugmentation( quiver ).
What follows are minor additional utilities for QPA.
‣ String( path ) | ( method ) |
Argument: path, a path of length at least 2 in a quiver (see IsPath (QPA: IsPath) and LengthOfPath (QPA: LengthOfPath) for details)
Returns: a string describing path
Methods for String (Reference: String) already exist for vertices and arrows of a quiver; that is to say, paths of length 0 or 1. QPA forgets these for longer paths: at present, only the default answer "<object>" is returned.
A path in QPA is products of arrows. Accordingly, we write its string as a *-separated sequences of its constituent arrows. This is in-line with how paths are printed using ViewObj (Reference: ViewObj).
‣ ArrowsOfQuiverAlgebra( alg ) | ( operation ) |
Argument: alg, a quiver algebra
Returns: the residues of the arrows in the defining quiver of alg, listed together
‣ VerticesOfQuiverAlgebra( alg ) | ( operation ) |
Argument: alg, a quiver algebra
Returns: the residues of the vertices in the defining quiver of alg, listed together
‣ FieldOfQuiverAlgebra( alg ) | ( operation ) |
Argument: alg, a quiver algebra
Returns: the field of definition of alg
‣ DefiningQuiverOfQuiverAlgebra( alg ) | ( operation ) |
Argument: alg, a quiver algebra
Returns: the quiver of definition of alg
This single operation performs OriginalPathAlgebra (QPA: OriginalPathAlgebra) and then QuiverOfPathAlgebra (QPA: QuiverOfPathAlgebra)
‣ PathOneArrowLongerAtSource( path ) | ( attribute ) |
‣ PathOneArrowLongerAtTarget( path ) | ( attribute ) |
‣ PathOneArrowShorterAtSource( path ) | ( attribute ) |
‣ PathOneArrowShorterAtTarget( path ) | ( attribute ) |
Argument: path, a path
Returns: a path new_path which differs from path by one arrow in the appropriate way, or fail if no such arrow exists.
Both of the -Shorter- attributes require path to have length at least 1, as measured by LengthOfPath (QPA: LengthOfPath).
Both of the -Longer- attributes require there to exist a unique arrow to add. So, for example PathOneArrowLongerAtSource requires the source of path to have indegree exactly 1, as measured by InDegreeOfVertex (QPA: InDegreeOfVertex). This is always the situation with 1-regular quivers, where these operations are most intended to be used.
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