Actions of algebraic groups on DG categories provide a convenient, unifying framework in some parts of geometric representation theory, especially the representation theory of reductive Lie algebras. We extend this theory to loop groups and affine Lie algebras, extending previous work of Beraldo, Gaitsgory and the author.
Along the way, we introduce some subjects of independent interest: ind-coherent sheaves of infinite type indschemes, topological DG algebras, and weak actions of group indschemes on categories. We also present a new construction of semi-infinite cohomology for affine Lie algebras, based on a modular character for loop groups.
As an application of our methods, we establish an important technical result for Kac-Moody representations at the critical level, showing that the appropriate ("renormalized") derived category of representations admits a large class of symmetries coming from the adjoint action of the loop group and from the center of the enveloping algebra.
In the first part of this paper, we prove the existence of torsion free covers in the category of representations of quivers, $(Q,R-Mod)$, for a wide class of quivers included in the class of the so-called source injective representation quivers provided that any direct sum of torsion free and injective $R$-modules is injective. In the second part, we prove the existence of $\mathscr{F}_{cw}$-covers and $\mathscr{F}_{cw}^{\perp}$-envelopes for any quiver $Q$ and any ring $R$ with unity, where $\mathscr{F}_{cw}$ is the class of all "componentwise" flat representations of $Q$.
We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories of semimodules and semicontramodules. Certain (co)flatness and/or (co)projectivity conditions have to be imposed on the coring and semialgebra to make the module categories abelian (an…
Given a finite dimensional algebra $Λ$, we show that a frequently satisfied finiteness condition for the category ${\cal P}^{\infty}(Λ\rm{-mod})$ of all finitely generated (left) $Λ$-modules of finite projective dimension, namely contravariant finiteness of ${\cal P}^{\infty}(Λ\rm{-mod})$ in $Λ\rm{-mod}$, forces arbitrary modules of finite projective dimension to be direct limits of objects in ${\cal P}^{\infty}(Λ\rm{-mod})$. Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimension conjecture, that is, for the little finit…
We give an algebraic construction of standard modules (infinite dimensional modules categorifying the PBW basis of the underlying quantized enveloping algebra) for Khovanov-Lauda-Rouquier algebras in all finite types. This allows us to prove in an elementary way that these algebras satisfy the homological properties of an `affine quasi-hereditary algebra.' In simply-laced types these properties were established originally by Kato via a geometric approach. We also construct some Koszul-like projective resolutions of standard modules corresponding to multiplicity-free positive roots.
We describe the structure and homological properties of arbitrary generalized standard Auslander-Reiten components of artin algebras. In particular, we prove that for all but finitely many indecomposable modules in such components the Euler characteristic is defined and nonnegative. Further, we provide a handy criterion for an infinite Auslander-Reiten component of an artin algebra to be generalized standard. We solve also the long standing open problem concerning the structure of artin algebras admitting a~separating family of Auslander-Reiten components.
The ADO invariants are a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group $U_q(sl(2))$ at roots of unity. Ito showed that these invariants are sums of traces of quotients of homological representations of braid groups (truncated Lawrence representations). In this paper we show a direct homological formula for the ADO invariants, as sums of partial traces of Lawrence type representations, without further truncations.
A new homological dimension is introduced to measure the quality of resolutions of `singular' finite dimensional algebras (of infinite global dimension) by `regular' ones (of finite global dimension). Upper bounds are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological conjectures. Then invariance under stable equivalences is shown to hold, with some exceptions when there are nodes in case of additive equivalences, and without exceptions in case of triangulated equivalences. Stable equivalences of Morita type and derived equiva…
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of positive characteristic $p$. Under some restrictions on the size of $p$, the present paper establishes new results on the $G$-module structure of $\Ext^\bullet_{G_1}(V,W)$ when $V,W$ belong to several important classes of rational $G$-modules, and $G_1$ denotes the first Frobenius kernel of $G$. For example, it is proved that, if $L,L'$ are ($p$-regular) irreducible $G_1$-modules, then $\Ext^n_{G_1}(L,L')^{[-1]}$ has a good filtration with computable multiplicities. This and many other results depend on the entir…
We prove that thick category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra is extension full in the category of all modules. We also prove the weak Alexandru conjecture both for regular blocks of thick category $\mathcal{O}$ and the associated categories of Harish-Chandra bimodules, but disprove it for singular blocks.
Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the ring $Γ:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $Γ$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda ring $Y(e):={\rm Ext}^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is artinian of finite global dimension, then $A$ has finite global dimension if and only if so is $Γ$. We also investigate the situation where both $A,Γ$…
We show that, also within the class of representation-tame finite dimensional algebras $Λ$, the big left finitistic dimension of $Λ$ may be strictly larger than the little. In fact, the discrepancies $Findim Λ- findim Λ$ need not even be bounded for special biserial algebras which constitute one of the (otherwise) most thoroughly understood classes of tame algebras. More precisely: For every positive integer $r$, we construct a special biserial algebra $Λ$ with the property that $findim Λ= r + 1$, while $Findim Λ= 2r + 1$. In particular, there are infinite dimensional representations of $Λ$ wh…
We prove homological mirror symmetry for the universal centralizer $J_G$ (a.k.a Toda space), associated to any complex semisimple Lie group $G$. The A-side is a partially wrapped Fukaya category on $J_G$, and the B-side is the category of coherent sheaves on the categorical quotient of a dual maximal torus by the Weyl group action (with some modification if $G$ has a nontrivial center). This is the first and the main part of a two-part series, dealing with $G$ of adjoint type. The general case will be proved in the forthcoming second part [Jin2].
We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of $A_\infty$-structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an $n$-dimensional affine hypersurface of degree $n + 2$ and the double cover of the $n$-dimensional affine space br…
In this paper we consider several homological dimensions of crossed products $A _α ^σ G$, where $A$ is a left Noetherian ring and $G$ is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of $A ^σ _α G$ are classified: global dimension of $A ^σ _α G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^σ _α G$ coincides with that of $A$. A criterion for skew group rings to have finite global dimensions is deduced. Under the hy…
Modular functors, i.e. consistent systems of projective representations of mapping class groups of surfaces, have been constructed for non-semisimple modular categories already decades ago. Concepts from homological algebra have not been used in this construction although it is an obvious question how they should enter in the non-semisimple case. In the present paper, we elucidate the interplay between the structures from topological field theory and from homological algebra by constructing a homotopy coherent projective action of the mapping class group $\text{SL}(2,\mathbb{Z})$ of the torus …
Let $k$ be an algebraically closed field and $A$ be a (left and right) Noetherian associative $k$-algebra. Assume further that $A$ is either positively graded or semiperfect (this includes the class of finite dimensional $k$-algebras, and $k$-algebras that are finitely generated modules over a Noetherian central Henselian ring). Let $e$ be a primitive idempotent of $A$, which we assume is of degree $0$ if $A$ is positively graded. We consider the idempotent subalgebra $Γ= (1-e)A(1-e)$ and $S_e$ the simple right $A$-module $S_e = eA/e{\rm rad}A$, where ${\rm rad}A$ is the Jacobson radical of $A…
We prove that a finite dimensional algebra $A$ with representation-finite subcategory consisting of modules that are semi-Gorenstein-projective and $n$-th syzygy modules is left weakly Gorenstein. This generalises a theorem of Ringel and Zhang who proved the result in the case $n=1$. As an application we show that monomial algebras and endomorphism rings of modules over representation-finite algebras are weakly Gorenstein. We then give a new connection between the theory of dominant dimension and Gorenstein homological algebra for gendo-symmetric algebras. As an application, we will see that t…
We study rewriting properties of the column presentation of plactic monoid for any semisimple Lie algebra such as termination and confluence. Littelmann described this presentation using L-S paths generators. Thanks to the shapes of tableaux, we show that this presentation is finite and convergent. We obtain as a corollary that plactic monoids for any semisimple Lie algebra satisfy homological finiteness properties.
For a recollement of derived module categories of rings, we provide sufficient conditions to guarantee the additivity formula of higher algebraic K-groups of the rings involved, and establish a long Mayer-Vietoris exact sequence of higher algebraic K-groups for homological exact contexts introduced in the first paper of this series. Our results are then applied to recollements induced from homological ring epimorphisms and noncommutative localizations. Consequently, we get an infinitely long Mayer-Vietoris exact sequence of K-theory for Milnor squares, re-obtain a result of Karoubi on localiza…
This paper provides a homological algebraic foundation for generalizations of classical Hecke algebras introduced in math.QA/9805134. These new Hecke algebras are associated to triples of the form (A,B,e), where A is an associative algebra containing subalgebra B with character e. These algebras are connected with cohomology of associative algebras in the sense that for every left A-module V and right A-module W the Hecke algebra associated to triple (A,B,e) naturally acts in the B-cohomology and B-homology spaces of V and W, respectively. We also introduce the semi-infinite cohomology functor…
