Maximal Green Sequences for Preprojective Algebras

Let \(S(n)\) be the category of invariant subspaces of nilpotent operators with nilpotency index at most \(n\). Such submodule categories have been studied already in 1934 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra \(\Pi _n\) of type \(\mathbb {A}_n\); the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schroer). We are going to discuss the connection between the submodule category \(\mathcal {S}(n)\) and the module category \(\hbox {mod}\;\Pi _{n-1}\) of the preprojective algebra \(\Pi _{n-1}\). Dense functors \(\mathcal {S}(n) \rightarrow \hbox {mod}\;\Pi _{n-1}\) are known to exist: one has been constructed quite a long time ago by Auslander and Reiten, recently another one by Li and Zhang. We will show that these two functors are full, dense, objective functors with index \(2n\), thus \(\hbox {mod}\;\Pi _{n-1}\) is obtained from \(\mathcal {S}(n)\) by factoring out an ideal which is generated by \(2n\) indecomposable objects. As a byproduct we also obtain new examples of ideals in triangulated categories, namely ideals \(\mathcal {I}\) in a triangulated category \(\mathcal {T}\) which are generated by an idempotent such that the factor category \(\mathcal {T}/\mathcal {I}\) is an abelian category.

Let Λ be a finite dimensional algebra with an action by a finite group G and A:=Λ*G the skew group algebra. One of our main results asserts that the canonical restriction-induction adjoint pair induced by the skew group algebra extension Λ⊂A induces a poset isomorphism between the poset of G-stable support τ-tilting modules over Λ and that of (modG)-stable support τ-tilting modules over A. We also establish a similar poset isomorphism between posets of appropriate classes of silting complexes over Λ and A. These two results generalize and unify the preceding results by Zhang–Huang, Breaz–Marcus–Modoi and the second and the third authors. Moreover, we give a practical condition under which τ-tilting finiteness and silting discreteness of Λ are inherited by A. As applications we study τ-tilting theory and silting theory of the (generalized) preprojective algebras and the folded mesh algebras. Among other things, we determine the posets of support τ-tilting modules and of silting complexes over preprojective algebra Π(𝕃 n ) of type 𝕃 n .

I-Sequences in the Category of Modules

We introduce and study a fermionisation procedure for the cohomological Hall algebra $\mathcal{H}_{\Pi_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson–Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel–Moore homology of the stack of representations of the $\unicode{x03BC}$ -deformed preprojective algebra introduced by Crawley–Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras and my earlier results on the Borel–Moore homology of the stack of representations of the undeformed preprojective algebra.

This paper is dedicated to the memory of Maurice Auslander, whose love of mathematics will always be an inspiration

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category 𝒪 associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category 𝒪 and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category 𝒪 for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra 𝔮(2).

Covering theory for linear categories with application to derived categories

Let H=kQ be a finite-dimensional connected wild hereditary path algebra, over some field k. Denote by H-reg the category of finite-dimensional regular H-modules, that is, the category of modules M with τ H − m ( τ H m M ) ≅ M for all integers m, where τH denotes the Auslander–Reiten translation. Call a filtration ( ∗ ) M = M 0 ⊃ M 1 ⊃ … ⊃ M r ⊃ M r + 1 = 0 of a regular H-module M a regular filtration if all subquotients Mi/Mi+1 are regular. Call a regular filtration (*) a regular composition series if it is strictly decreasing and has no proper refinement. A regular component C in the Auslander–Reiten quiver Γ (H) of H-mod is called filtration closed if, for each M∈ add C, the additive closure of C, and each regular filtration (*) of M, all the subquotients Mi/Mi+1 are also in add C. We show that most wild hereditary algebras have filtration-closed Auslander–Reiten components. Moreover, we deduce from this that there are also almost serial components, that is regular components C, such that any indecomposable X∈C has a unique regular composition series. This composition series coincides with the Auslander–Reiten filtration of X, given by the maximal chain of irreducible monos ending at X. 1991 Mathematics Subject Classification: 16G70, 16G20, 16G60, 16E30.

Wide Morita contexts and equivalences of comodule categories

Let A be a finite-dimensional algebra which is quasi-hereditary with re- spect to the poset (�,�), with standard modules �(�) for 2 �. Let F(�) be the category of A-modules which have filtrations where the quotients are standard modules. We determine some inductive results on the relative Auslander-Reiten quiver of F(�). Quasi-hereditary algebras were introduced by Cline, Parshall and Scott in the study of highest weight categories arising in the representation theory of semisimple Lie algebras and algebraic groups (2), and they were studied by Dlab and Ringel and others (4, 5). LetA be quasi-hereditary. Then the simple A-modules are labelled as L(�) forin some partially ordered set (�,≤). Of central importance are the standard modules, denoted by �(�), and the categoryF(�) of modules which have a filtration where the quotients are standard modules. In (12) it was proved that this category has relative Auslander-Reiten (AR) sequences. The aim of our paper is to present some inductive results on the relative Auslander-Reiten quiver ofF(�). For any subset of �, letF (�) be the category of modules inF(�) where all quotients are �(�) with �∈ . If � is an ideal in the poset (�,≤) then there is a subalgebra A of A which is quasi-hereditary with poset (, ≤). In this case, the categoryF (�) is equivalent toF(�A ). A main result in this paper relates, for a non-projective indecomposable mod- ule X inF (�), the two relative Auslander-Reiten sequences with cokernel X inF(�) and inF (�) (Theorem 2.5). By Ringel duality, there is an equivalent version for categories of modules filtered by costandard mod- ules. We describe this and some variations in Section 3. We continue with further results on relative Auslander-Reiten sequences. In particular, we determine certain relative Auslander-Reiten sequences where a projective- injective summand occurs; this generalizes the usual almost split sequence where a projective-injective module occurs.

Motivated by connections to the singlet vertex operator algebra in the $\mathfrak{g}=\mathfrak{sl}_2$ case, we study the unrolled restricted quantum group $\overline{U}_q^{ H}(\mathfrak{g})$ for any finite dimensional complex simple Lie algebra $\mathfrak{g}$ at arbitrary roots of unity with a focus on its category of weight modules. We show that the braid group action naturally extends to the unrolled quantum groups and that the category of weight modules is a generically semi-simple ribbon category (previously known only for odd roots) with trivial Müger center and self-dual projective modules. Bibliography: 44 titles.

The aim of this paper is to clarify the relation between the following objects: (a) rank 1 projective modules (ideals) over the first Weyl algebra A_1; (b) simple modules over deformed preprojective algebras $\Pi_{\lambda}(Q)$ introduced by Crawley-Boevey and Holland; and (c) simple modules over the rational Cherednik algebras $H_{0,c}(S_n)$ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on A-infinity modules over A_1 to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities $C^2/\Gamma$, where $\Gamma$ is a finite cyclic subgroup of $SL(2,C)$.

The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. Given this conjecture, I give a proof of the reduceness conjecture.

In this paper, we introduce generalized quiver varieties which include as special cases classical and cyclic quiver varieties. The geometry of generalized quiver varieties is governed by a finitely generated algebra $${{\mathcal {P}}}$$ : the algebra $${{\mathcal {P}}}$$ is self-injective if the quiver Q is of Dynkin type, and coincides with the preprojective algebra in the case of classical quiver varieties. We show that in the Dynkin case the strata of generalized quiver varieties are in bijection with the isomorphism classes of objects in $${\text {proj}}\,{{\mathcal {P}}}$$ , and that their degeneration order coincides with the Jensen–Su–Zimmermann’s degeneration order on the triangulated category $${\text {proj}}\,{{\mathcal {P}}}$$ . Furthermore, we prove that classical quiver varieties of type $$A_n$$ can be realized as moduli spaces of representations of an algebra $${{\mathcal {S}}}$$ .

From Supersymmetry to Quantum Commutativity