On the homology of almost Calabi-Yau algebras associated to SU(3) modular invariants

Untwisting a twisted Calabi-Yau algebra

Twisted Calabi-Yau algebras are a generalisation of Ginzburg's notion of Calabi-Yau algebras. Such algebras A come equipped with a modular automorphism σ\in Aut(A), the case σ= id being precisely the original class of Calabi-Yau algebras. Here we prove that every twisted Calabi-Yau algebra may be extended to a Calabi-Yau algebra. More precisely, we show that if A is a twisted Calabi-Yau algebra with modular automorphism σ, then the smash product algebras A \rtimes_σN and A \rtimes_σZ are Calabi-Yau.

Calabi-Yau algebras and superpotentials

We prove that complete d-Calabi-Yau algebras in the sense of Ginzburg are derived from superpotentials.

Pre-Calabi-Yau structures and moduli of representations

We establish a system of formal noncommutative calculus for differential forms and polyvector fields, which forms the foundations for the study of pre-Calabi-Yau categories. Using an explicit trace map, we show that any $n$-Calabi-Yau structure on a non-positively graded dg algebra $A$ induces a $(2-n)$-shifted symplectic structure on its derived moduli stack of representations; while any $n$-pre-Calabi-Yau structure on $A$ induces a $(2-n)$-shifted Poisson structure on this derived moduli stack.

On the structure of Calabi-Yau categories with a cluster tilting subcategory

We prove that for $d \geq 2$, an algebraic $d$-Calabi-Yau triangulated category endowed with a $d$-cluster tilting subcategory is the stable category of a DG category which is perfectly $(d+1)$-Calabi-Yau and carries a non degenerate $t$-structure whose heart has enough projectives.

Stable Calabi--Yau dimension of self-injective algebras of finite type

We give an equivalent definition of the stable Calabi--Yau dimension in terms of bimodule syzygies and so-called stably inner automorphisms. Using it, we complete the computation of the stable Calabi--Yau dimensions of the self-injective algebras of finite representation type which was started by K. Erdmann, A. Skowroński, J. Bia\lkowski and A. Dugas.

预-Calabi-Yau范畴的同伦转移定理与极小模型

Homotopy Transfer Theorem and minimal models for pre-Calabi-Yau categories

本文将预-Calabi-Yau代数的同伦性质推广到范畴情形，利用余代数技术直接证明了预-Calabi-Yau结构沿dg箭图拟同构的转移、极小模型的存在性以及拟同构的拟逆存在性。