We study the oriented exchange graph EG(Gamma_N Q) ofreachable hearts in the finite-dimensional derived category D(Gamma_N Q) of the CY-N Ginzburg algebra Gamma_N Q associated to an acyclic quiver Q. We show that any such heart is induced from some heart in the bounded derived category D(Q) via some ‘Lagrangian immersion’ L :D(Q)->D(Gamma_N Q). We build on this to show that the quotient of EG(Gamma_N Q) by the Seidel–Thomas braid group is the exchange graph CEG_N−1(Q)of cluster tilting sets in the (higher) cluster category C_N−1(Q). As an application, we interpret Buan–Thomas’ coloured quiver for a cluster tilting set in terms of the Ext quiver of any corresponding heart in D(Gamma_N Q).

We give a proof of the parabolic/singular Koszul duality for the category O of affine Kac–Moodyalgebras. The main new tool is a relation between moment graphs and finite codimensional affine Schubert varieties. We apply this duality to q-Schur algebras and to cyclotomic rational double affine Heckealgebras.This yields a proof of a conjecture of Chuang–Miyachi relating the level-rank duality with the Ringel–Koszul duality of cyclotomic rational double affine Hecke algebras.

We study the functorially finite maximal rigid subcategories in 2-CY triangulated categories and their endomorphism algebras. Cluster tilting subcategories are obviously functorially finite and maximal rigid; we prove that the converse is true if the 2-CY triangulated categories admit a cluster tilting subcategory. As a generalization of a result of Keller and Reiten (2007) [KR], we prove that any functorially finite maximal rigid subcategory is Gorenstein with Gorenstein dimension at most 1. Similar as cluster tilting subcategory, one can mutate maximal rigid subcategories at any indecomposable object. If two maximal rigid objects are reachable via simple mutations, then their endomorphism algebras have the same representation type.

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Let O_c be the category of finite-length central-charge-c modules for the Virasoro Lie algebra whose composition factors are irreducible quotients of reducible Verma modules. Recently, it has been shown that O_c admits vertex algebraic tensor category structure for any c∈C. Here, we determine the structure of this tensor category when c=13−6p−6/p for an integer p>1. For such c, we prove that O_c is rigid, and we construct projective covers of irreducible modules in a natural tensor subcategory O_c^0. We then compute all tensor products involving irreducible modules and their projective covers. Using these tensor product formulas, we show that O_c has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine sl_2 at levels −2+p^±1. Next, as a straightforward consequence of the braided tensor category structure on O_c together with the theory of vertex operator algebra extensions, we rederive known results for triplet vertex operator algebras W(p), including rigidity, fusion rules, and construction of projective covers. Finally, we prove a recent conjecture of Negron that O_c^0 is braided tensor equivalent to the PSL(2,C)-equivariantization of the category of W(p)-modules.