[Paper Review] The geometry of Brauer graph algebras and cluster mutations

Bethany Marsh, Sibylle Schroll|arXiv (Cornell University)|Sep 17, 2013

Algebraic structures and combinatorial models30 references5 citations

TL;DR

This paper establishes a geometric correspondence between compact oriented marked surfaces and Brauer graph algebras up to derived equivalence, using ribbon graphs and Kauer moves. It shows that diagonal flips in m-angulations of a disc correspond to Whitehead moves in dual (m−1)-ary trees, and constructs explicit tilting complexes realizing derived equivalences, while also providing counterexamples where dual graph mutations do not preserve derived equivalence in non-simply connected surfaces.

ABSTRACT

In this paper we establish a connection between ribbon graphs and Brauer graphs. As a result, we show that a compact oriented surface with marked points gives rise to a unique Brauer graph algebra up to derived equivalence. In the case of a disc with marked points we show that a dual construction in terms of dual graphs exists. The rotation of a diagonal in an m-angulation gives rise to a Whitehead move in the dual graph, and we explicitly construct a tilting complex on the related Brauer graph algebras reflecting this geometrical move.

Motivation & Objective

• To establish a geometric correspondence between compact oriented marked surfaces and Brauer graph algebras up to derived equivalence.
• To show that Kauer moves on Brauer graphs correspond to cluster mutations in surface-based cluster algebras.
• To construct explicit two-term tilting complexes that realize derived equivalences induced by geometric moves in m-angulations of a disc.
• To demonstrate that derived equivalence is not preserved in general for dual graphs under Kauer moves, especially in non-simply connected surfaces.
• To clarify the relationship between Brauer graph algebras, Brauer tree algebras, and cluster categories via geometric and homological methods.

Proposed method

• Uses ribbon graph theory to associate a cyclic ordering of edges around vertices to Brauer graphs without exceptional vertices.
• Applies Kauer moves to Brauer graphs and proves they induce derived equivalences, generalizing known results.
• Constructs a dual graph for m-angulations of a disc, showing it forms an (m−1)-ary tree with Whitehead moves corresponding to mutation.
• Employs tilting complexes to explicitly realize derived equivalences between Brauer graph algebras induced by geometric moves.
• Applies Sylvester’s law of inertia and Cartan matrix eigenvalue analysis to test derived equivalence, using characteristic polynomials and determinant comparisons.
• Provides counterexamples using triangulations of a sphere and a punctured disc to show derived equivalence fails for dual graphs under Kauer moves.

Experimental results

Research questions

• RQ1 How can Brauer graph algebras be associated with compact oriented marked surfaces up to derived equivalence?
• RQ2 Do Kauer moves on Brauer graphs correspond to cluster mutations in surface-based cluster algebras?
• RQ3 Can explicit tilting complexes be constructed to realize derived equivalences induced by geometric moves in m-angulations of a disc?
• RQ4 Under what conditions does a Kauer move on a dual graph preserve derived equivalence of the associated Brauer graph algebras?
• RQ5 What role do Cartan matrices and eigenvalue signatures play in distinguishing derived equivalence classes of Brauer graph algebras?

Key findings

• A compact oriented marked surface gives rise to a unique Brauer graph algebra up to derived equivalence, via triangulation or m-angulation.
• In the case of a disc with m-angulations, a mutation of a diagonal corresponds to a Whitehead move on the dual (m−1)-ary tree.
• An explicit two-term tilting complex is constructed that realizes the derived equivalence induced by such a move.
• For the dual graph of a triangulation of a disc, the Kauer move corresponds to the associativity rule in cluster categories.
• Counterexamples show that the Brauer graph algebras of dual graphs are not always derived equivalent after a Kauer move, as evidenced by differing numbers of zero eigenvalues in Cartan matrices.
• The Cartan matrices of the dual graph algebras in the counterexample have characteristic polynomials with different degrees of zero eigenvalues: 5 for the original, 4 for the mutated algebra, proving non-equivalence via Corollary 5.3.