AR quiver calculator
A tool to deal with the Auslander-Reiten quiver of a module category or a triangualted category.
(The Auslander-Reiten quiver of mod k(D5), a torsion class of it, and Ext-projectives of it)
You can input your translation quiver by your mouse and keyboard, save and load your translation quiver, and import the AR quiver from String Applet.
Then this computes various things.
- Compute Hom and Ext^1
- For a module category, this computes all torsion(-free) classes, wide subcategories, ICE-closed subcategories (closed under Image, Cokernel, Extensions), IE-closed subcategories (closed under Images, Extensions), etc,
- and its Ext-projective (injective) objects.
- For a triangulated category, this computes a shift functor, and list all maximal Ext-orthogonal objects.
See GitHub Readme for details and examples.
Representation theory of algebra in Lean
I’m trying to formalize representation theory of algebra (especially Auslander-Reiten theory) using a proof assistant Lean. For those who don’t know a proof assistant, it’s like a computer game for proving theorems, so write a programming code which states and proves theorems.
What I have done so far in lean-noncommutative-ring is
- The left right symmetry of Jacobson radical of a ring (intersection of maximal left ideals coincides that of maximal right ideals), and a well-known characterization of elements in the Jacobson radical
- The left right symmetry of local ring (ring has a unique maximal left ideal iff it has a unique maximal right ideal), and a well-known characterization of local rings.
Obviously there are lots of things to play with Auslander-Reiten theory in Lean, so if you are interested in this game project, please contact me.
- Natural number game and its mirror. A very funny game for proving basic statements (associativity, commutativity, etc) from Peano’s axioms. Very gentle and instructive, so no prerequisite knowledge is needed.
- Formalising Mathematics, which explains basics of Lean from undergraduate materials like set theory and group theory
- Lean community
The lattice of torsion classes in SageMath
tors_lattice.py below enables us to compute and construct various objects from the lattice of torsion classes of a τ-tilting finite algebra in SageMath. Internally, this defines a class
FiniteTorsLattice, which is a subclass of a SageMath class for finite lattices:
Once you input the lattice of torsion classes (e.g. using my StringApplet-to-SageMath-converter below), this program can compute (or construct) various objects which naturally arise in the representation theory of algebras in SageMath, such as the lattice of wide subcategories, the lattice of ICE-closed subcategories, the simplicial complex of support τ-tilting modules, and the number of indecomposable Ext-projectives of each torsion class or a heart of each interval, and so on.
- GitHub Repository
- An introduction video based on the second part of my talk
- Slides used in the above video
- A Demo Notebook used in the above video, and its ipynb file
StringApplet to SageMath converter
This enables us to import the lattice of torsion classes in SageMath from String Applet. String Applet can compute the Hasse quiver of torsion classes for an inputted algebra (which should be representation-finite special biserial). This module makes a data which we can use to construct and study the lattice of torsion classes in SageMath (e.g. the kappa map below), from the TeX file exported by String Applet.
Kappa maps for lattices
This adds methods to a Sage class
FiniteLatticePoset which compute the (extended) kappa (dual) map. Using this, one can compute kappa maps for finite lattices in SageMath.
- GitHub Repository
- Notes for representation theorists: I strongly recommend you to read this if you are working with representation theory of algebras, not just a lattice theory: this explains how the kappa map arises in the rep-theory, and demonstrates how to use it to study bricks and torsion classes.