No abstract uploaded!

Let $A$ be a finite dimensional k-algebra standardly stratified for a partial order $\leqslant$ and $\Delta$ be the direct sum of all standard modules. In this paper we study the extension algebra $E = Ext^{\ast}_A (Δ,Δ)$ of standard modules, characterize the stratification property of $E$ for $\geqslant$ and $\geqslant ^{op}$, and obtain a sufficient condition for $E$ to be a generalized Koszul algebra (in a sense which we define).

We show that the category of finite-length generalized modules for the singlet vertex algebra M(p), p∈Z_+, is equal to the category O_M(p) of C_1-cofinite M(p)-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since O_M(p) includes the uncountably many typical M(p)-modules, which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical M(p)-modules. We also introduce a tensor subcategory O_M(p)^T, graded by an algebraic torus T, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of sl_2 at a 2pth root of unity. We compute all tensor products involving simple and projective M(p)-modules, and we prove that both tensor categories O_M(p) and O_M(p)^T are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras W(p) are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.

We study the fundamental groups of the exchange graphs for the bounded derived category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category D(Gamma_N Q)of the Calabi–Yau-N Ginzburg algebra associated to Q. In the case of D(Q), we prove that its space of stability conditions (in the sense of Bridgeland) is simply connected. As an application, we show that the Donaldson–Thomas type invariant associated to Q can be calculated as a quantum dilogarithm function on its exchange graph. In the case of D(Gamma_N Q), we show that the faithfulness of the Seidel–Thomas braid group action (which is known for Q of type A or N=2) implies the simply connectedness of the space of stability conditions.

For each integer we describe the space of stability conditions on the derived category of the n-dimensional Ginzburg algebra associated to the A2 quiver. The form of our results points to a close relationship between these spaces and the Frobenius-Saito structure on the unfolding space of the A2 singularity.