We implement braid group actions on the homotopy category of complexes of projective modules over the zigzag algebra of any simple undirected graph, that is an undirected graph without multiple edges.

We now explain what the various files do. References for the mathematical content in this code can be found in the subsection titled “References”.

Constructs the zigzag algebra of a (doubled) quiver by running ZigZagAlgebra(ct), where the argument ct can be any of the following.

1. An abstract symmetric Cartan matrix, which is a Sage matrix whose diagonal entries are 2, and off-diagonal entries are either 0 or negative integers.
2. A named CartanType in Sage, which could be something like CartanType(['A', 3]) for type A3, or CartanType(['D', 5]) for type D5, or CartanType(['A', 3, 1]) for type affine \(A_3\).
3. A string shorthand for a named CartanType, such as ~ ‘A3’ ~ or ~ ‘D5’ ~.
4. An object of class CartanMatrix.

At the moment, we do not support Dynkin diagrams with multiple edges. This means that we only support Cartan matrices whose off-diagonal entries are either -1 or 0.

Constructs elements of a zigzag algebra. For now mostly a stub that inherits from the Sage class FiniteDimensionalAlgebraElement.

Constructs indecomposable projective left modules over a zigzag algebra.

Constructs complexes of projective modules over a possibly non-commutative but finite-dimensional algebra over a field. These should be thought of as objects of the homotopy category of complexes. In particular, we implement a method for Gaussian elimination of null-homotopic complexes.

Implements spherical twists by the standard projective modules over the zigzag algebra.

Implements Harder–Narasimhan filtrations of spherical objects under a given Bridgeland stability condition (coming soon).

Here are some references.

1. Zigzag algebras originate in the paper A category for the adjoint representation by Huerfano and Khovanov (arXiv:0002060). The relevant content is in Section 3.
2. The categorical braid group action implemented here originates in the paper Braid group actions on derived categories of coherent sheaves by Seidel and Thomas.
3. The more general quiver version of this action was developed in the paper Quivers, Floer cohomology, and braid group actions by Khovanov and Seidel.
4. We now use a signed version of the zigzag algebra, as described in Section 6 (specifically, Definition 6.4) of the paper A Thurston compactification of the space of stability conditions by the authors and Anthony Licata (arXiv:2011.07908).
5. An accessible introduction to braid group actions on these quiver categories can be found in Section 1 of the article Notes on link homology by Asaeda and Khovanov (arXiv:0804.1279). Note, however, that the braid actions are described here in terms of tensoring by complexes of bimodules, and this construction is somewhat special to zigzag algebras of quivers. In the code, we choose to implement braid actions using the more general setup of spherical twists, namely by constructing cones of universal evaluation maps. This construction is described, e.g., in the paper of Seidel and Thomas mentioned above.
6. Gaussian elimination of complexes that are obviously null-homotopic is explained, e.g., in Lemma 4.2 of the paper Fast Khovanov homology computations by Dror Bar-Natan (arXiv:0606318).

We are greatly indebted to Anthony Licata, with whom this project began, and who is a long-time collaborator of the authors on the mathematics behind this code. His remarkable facility with calculating these braid actions was both extremely inspiring and extremely valuable — we mostly started writing this code because we were too slow to keep up with him! We are grateful to Hoel Queffelec for countless useful conversations and for suggesting critical bugfixes and improvements.