[Paper Review] Perverse equivalences, BB-tilting, mutations and applications

Sefi Ladkani|arXiv (Cornell University)|Jan 26, 2010

Algebraic structures and combinatorial models38 references27 citations

TL;DR

This paper establishes a deep connection between BB-tilting and perverse derived equivalences for finite-dimensional algebras, introducing algebra mutations via tilting complexes associated with quiver vertices. It proves that mutations at non-projective-injective vertices in 2-Calabi-Yau categories yield derived equivalent endomorphism algebras, generalizing cluster-tilting theory and showing that mutation-induced derived equivalences align with Fomin-Zelevinsky matrix mutations.

ABSTRACT

We relate the notions of BB-tilting and perverse derived equivalence at a vertex. Based on these notions, we define mutations of algebras, leading to derived equivalent ones. We present applications to endomorphism algebras of cluster-tilting objects in 2-Calabi-Yau categories and to algebras of global dimension at most 2.

Motivation & Objective

• To unify BB-tilting and perverse derived equivalences in the context of finite-dimensional algebras.
• To define algebra mutations via tilting complexes at quiver vertices, yielding derived equivalent algebras.
• To establish that mutations of cluster-tilting objects in 2-Calabi-Yau categories induce derived equivalences of their endomorphism algebras.
• To show that the Grothendieck group transformation under mutation matches Fomin-Zelevinsky matrix mutation.
• To clarify conditions under which quiver mutations correspond to derived equivalences of algebras, especially in 2-CY settings.

Proposed method

• Define two complexes, $T^{-}_{k}$ and $T^{+}_{k}$, associated with each vertex $k$ without loops in the quiver, which induce perverse derived equivalences.
• Prove that BB-tilting at a vertex $k$ corresponds precisely to the perverse equivalence induced by $T^{-}_{k}$, under homological conditions.
• Introduce three mutation operations: negative ($\mu^{-}_{k}$), positive ($\mu^{+}_{k}$), and BB-mutation ($\mu^{\mathrm{BB}}_{k}$), all producing derived equivalent algebras.
• Show that the Grothendieck group transformation induced by mutation matches the matrix mutation of the skew-symmetric quiver matrix.
• Use approximation sequences (exchange sequences) in Hom-finite, idempotent-split, Frobenius or triangulated 2-CY categories to define mutations.
• Apply results to endomorphism algebras of cluster-tilting objects in 2-CY categories, proving that mutations of objects induce mutations of their algebras.

Experimental results

Research questions

• RQ1 How are BB-tilting and perverse derived equivalences related at a vertex in a quiver with relations?
• RQ2 Under what conditions does a perverse equivalence induced by $T^{-}_{k}$ arise from BB-tilting?
• RQ3 When do mutations of algebras via $T^{-}_{k}$ or $T^{+}_{k}$ yield derived equivalent algebras?
• RQ4 How do mutations of cluster-tilting objects in 2-Calabi-Yau categories relate to mutations of their endomorphism algebras?
• RQ5 To what extent do quiver mutations (in the sense of Fomin-Zelevinsky) correspond to derived equivalences of the associated algebras?

Key findings

• BB-tilting at a vertex $k$ is equivalent to the perverse derived equivalence induced by the complex $T^{-}_{k}$, provided the homological conditions are satisfied.
• The negative mutation $\mu^{-}_{k}(A)$ and the BB-mutation $\mu^{\mathrm{BB}}_{k}(A)$ coincide when both are defined, and both yield derived equivalent algebras.
• In Hom-finite, idempotent-split Frobenius 2-CY categories, all BB-, negative, and positive mutations exist and coincide with the endomorphism algebra of the mutated cluster-tilting object.
• For triangulated 2-CY categories, neighboring 2-CY-tilted algebras are near-Morita equivalent but not necessarily derived equivalent, though many such pairs are derived equivalent.
• The Grothendieck group transformation induced by algebra mutation matches the matrix mutation of the skew-symmetric quiver matrix, as defined by Fomin and Zelevinsky.
• There exist quiver mutations (e.g., at vertices 2, 4, 5 in Example 6.9) that do not correspond to derived equivalences of the associated algebras, even when the cluster categories are Hom-finite.