2021/06/11 by Ralf Schiffler, Schiffler, Ralf, Khrystyna Serhiyenko +1 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2106.06496
openalex publication_date 2021/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we consider the class of 2-Calabi-Yau tilted algebras that are defined by a quiver with potential whose dual graph is a tree. We call these algebras dimer tree algebras because they can also be realized as quotients of dimer algebras on a disc. These algebras are wild in general. For every such algebra B, we construct a polygon S with a checkerboard pattern in its interior that gives rise to a category Diag(S). The indecomposable objects of Diag(S) are the 2-diagonals in S, and its morphisms are given by certain pivoting moves between the 2-diagonals. We conjecture that the category Diag(S) is equivalent to the stable syzygy category over the algebra B, such that the rotation of the polygon corresponds to the shift functor on the syzygies. In particular, the number of indecomposable syzygies is finite and the projective resolutions are periodic. We prove the conjecture in the special case where every chordless cycle in the quiver is of length three. As a consequence, we obtain an explicit description of the projective resolutions. Moreover, we show that the syzygy category is equivalent to the 2-cluster category of type \mathbbA, and we introduce a new derived invariant for the algebra B that can be read off easily from the quiver.