Derived autoequivalences from periodic algebras

Abstract

We present a construction of autoequivalences of derived categories of symmetric algebras based on projective modules with periodic endomorphism algebras. This construction generalises autoequivalences previously constructed by Rouquier-Zimmermann and is related to the autoequivalences of Seidel-Thomas and Huybrechts-Thomas. We show that compositions and inverses of these equivalences are controlled by the resolutions of our endomorphism algebra and that each autoequivalence can be obtained by certain compositions of derived equivalences between algebras which are in general not Morita equivalent.

Lifts of longest elements to braid groups acting on derived categories

“…First we revise the construction of periodic twists from [Gra1]. The construction given here is the same but we will apply it in a more general situation: this greater generality will be important in our proofs.…”

“…The main result of [Gra1] states that if E is periodic and Y is a truncated resolution, so F is a shifted twisted copy of E , then Ψ P,f is an autoequivalence. We call such an autoequivalence a periodic twist and write it as Ψ P,Y , or just Ψ P when our truncated resolution is minimal.…”

“…Then using the lifting theorem developed in the previous section, we will show that lifts of longest elements from symmetric groups to braid groups act in the way suggested by the example at the end of [Gra1].…”

“…It was noted in [Gra1] that periodic twists on Γ 3 associated to the direct sum of the first two projectives, which have endomorphism algebra Γ 2 , are isomorphic to the composition F 1 F 2 F 1 of spherical twists. We will show that this pattern continues.…”

“…In [Gra1], a new construction of autoequivalences of derived categories of symmetric algebras, called periodic twists, was introduced, generalizing the construction of spherical twists. In Section 6 of that article it was noted that, given an action of B 3 on the derived category D b (Γ 3 ), the lift of w 0 3 to B 3 acts by a periodic twist.…”