Bocses

A Magma software package for computing with bocses.

By Julian Külshammer and Ulrich Thiel, 2017

Prerequisites

You need Magma version at least 2.19.

Downloading and running

You can download the latest version as a zip archive here. Then just run ./Bocses.sh.

Alternatively, from within a Magma session, run AttachSpec("Bocses.s.m");.

Usage

We illustrate the use of this software with two examples.

Example 1

We want to create the following bocs and its reduction:

//First, create the underlying quiver of the bocs.
//Edge labels are passed as list at the beginning in
//the same order as the successive list of tuples defining
//the edges. The edge labels can be used to define the
//differentials later.
B<a12,a23,a13,phi12,phi23,phi13>:=Bocs([1,2,3], \
[<1,2>,<2,3>, <1,3>,<1,2>,<2,3>,<1,3>] : Title:="Example1"); 

//Set non-zero edge degrees
B`EdgeDegrees["phi12"] :=1;
B`EdgeDegrees["phi23"] :=1;
B`EdgeDegrees["phi13"] :=1;

//Set non-zero edge levels
B`EdgeLevels["phi13"] :=2;
B`EdgeLevels["a13"] :=2;

//Set non-zero edge differentials
B`EdgeDifferentials["a13"]:=phi23*a12;

//Draw the bocs.
//This creates an HTML file in the directory "Output/Example 1"
//which is opened automatically. It contains an interactive graph
//like above.
Draw(B);

//Do reduction
Bred := Reduction(B);

//draw without adding edge labels
Draw(Bred : EdgeLabels:=false);

//Silently draw each reduction step and put in 
//output directory.
Bred := Reduction(B : DrawAll:=true);

The Draw command will create and open an HTML file in the Output directory containing the bocs as an interactive graph (vertices and edges can be moved with the mouse). The reduction of the bocs above looks as follows:

Example 2

In this example we do a reduction of the following bocs at a specific dimension vector:

The result is a whole list of bocses, each bocs representing one step in the reduction process.

B<a1,a2,a3,a4,a6,a5,a7,a8,b3,b4,b6,b5,b7,c4,c6,phi4,phi6,phi7,\
phi9,phi10,psi6,psi9> := Bocs([1,2,3,4,5,6,7,8,9,10,11],\
[<1,2>,<2,3>,<3,4>,<4,6>,<6,9>,<5,7>,<7,10>,<8,11>,<3,5>,\
<4,7>,<6,10>,<5,8>,<7,11>,<4,8>,<6,11>,<4,5>,<6,7>,<7,8>,\
<9,10>,<10,11>,<6,8>,<9,11>] : Title:="Example2");

B`EdgeDegrees["phi4"] := 1;
B`EdgeDegrees["phi6"] := 1;
B`EdgeDegrees["phi9"] := 1;
B`EdgeDegrees["phi7"] := 1;
B`EdgeDegrees["phi10"] := 1;
B`EdgeDegrees["psi6"] := 1;
B`EdgeDegrees["psi9"] := 1;
B`EdgeDegrees["phi10"] := 1;

B`EdgeLevels["b3"] := 2;
B`EdgeLevels["b4"] := 2;
B`EdgeLevels["b6"] := 2;
B`EdgeLevels["b5"] := 2;
B`EdgeLevels["b7"] := 2;
B`EdgeLevels["psi6"] := 2;
B`EdgeLevels["psi9"] := 2;
B`EdgeLevels["c4"] := 3;
B`EdgeLevels["c6"] := 3;

B`EdgeDifferentials["b3"] := phi4*a3;
B`EdgeDifferentials["b4"] := phi6*a4-a5*phi4;
B`EdgeDifferentials["b6"] := phi9*a6-a7*phi6;
B`EdgeDifferentials["b5"] := phi7*a5;
B`EdgeDifferentials["b7"] := phi10*a7-a8*phi7;
B`EdgeDifferentials["c4"] := phi7*b4-b5*phi4+psi6*a4;
B`EdgeDifferentials["c6"] := phi10*b6-b7*phi6+psi9*a6-a8*psi6;
B`EdgeDifferentials["psi6"] := phi7*phi6;
B`EdgeDifferentials["psi9"] := phi10*phi9;

Bred := Reduction(B, [ 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1 ]);