Mutation of frozen Jacobian algebras

Abstract

We survey results on mutations of Jacobian algebras, while simultaneously extending them to the more general setup of frozen Jacobian algebras, which arise naturally from dimer models with boundary and in the context of the additive categorification of cluster algebras with frozen variables via Frobenius categories. As an application, we show that the mutation of cluster-tilting objects in various such categorifications, such as the Grassmannian cluster categories of Jensen-King-Su, is compatible with Fomin-Zelevinsky mutation of quivers. We also describe an extension of this combinatorial mutation rule allowing for arrows between frozen vertices, which the quivers arising from categorifications and dimer models typically have. Definition 2.1. A quiver is a tuple Q = (Q 0 , Q 1 , h, t), where Q 0 and Q 1 are sets, and h, t : Q 1 → Q 0 are functions. Graphically, we think of the elements of Q 0 as vertices and those of Q 1 as arrows, so that each α ∈ Q 1 is realised as an arrow α : tα → hα. We call Q finite if Q 0 and Q 1 are finite sets.Definition 2.2. Let Q be a quiver. A quiver F = (F 0 , F 1 , h , t ) is a subquiver of Q if it is a quiver such that F 0 ⊆ Q 0 , F 1 ⊆ Q 1 and the functions h and t are the restrictions of h and t to F 1 . We say F is a full subquiver if F 1 = {α ∈ Q 1 : hα, tα ∈ F 0 }, so that a full subquiver of Q is completely determined by its set of vertices.Definition 2.3. An ice quiver is a pair (Q, F ), where Q is a quiver, and F is a (not necessarily full) subquiver of Q.

Self-Injective Jacobian Algebras from Postnikov Diagrams

“…applying the local operation depicted in . This mutation rule also coincides with mutation of ice quivers with potential presented in [Pre18]. If we restrict our attention to the quiver Q, this difference disappears (since arrows between frozen vertices are not arrows in Q).…”

Ginzburg algebras of triangulated surfaces and perverse schobers

“…We thank Bernhard Keller for informing us about an alternative approach to Theorem 3. A result of Yilin Wu [Wu21] extends the argument from [Kel12, Section 7.6] to relative Ginzburg algebras, showing that the mutations of ice quivers with potential of [Pre20] induce derived equivalences between the associated relative Ginzburg algebras. Theorem 3 may then be recovered by additionally extending the results of [LF09] relating flips of the ideal triangulation and mutations of quivers with potentials to ice quivers.…”

Ginzburg algebras of triangulated surfaces and perverse schobers

“…The result assumes that the ice quiver which is mutated at a vertex v has no 2-cycles incident to v. Most cases of Theorem 3 would thus follow if one extends the result of [LF09] relating flips of the ideal triangulation and mutations of quivers with potentials to ice quivers. A discussion of mutations of ice quivers with potential can be found in [Pre20].…”