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2-Verma modules

  • Grégoire Naisse EMAIL logo and Pedro Vaz

Abstract

We construct a categorification of parabolic Verma modules for symmetrizable Kac–Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also introduce a notion of dg-2-representation for quantum Kac–Moody algebras, and in particular of parabolic 2-Verma modules.

Funding statement: Grégoire Naisse is a Research Fellow of the Fonds de la Recherche Scientifique – FNRS, under Grant No. 1.A310.16, and he is grateful to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. Pedro Vaz was supported by the Fonds de la Recherche Scientifique – FNRS under Grant No. J.0135.16.

A Summary on the homotopy category of dg-categories and pretriangulated dg-categories

We gather some useful results on the homotopy category of dg-categories. References for this section are [21] and [42]. We also suggest [22] and [43] for nice surveys on the subject.

Our goal is to recall how to construct a category of dg-categories up to quasi-equivalence, so that the space of functors between two “triangulated categories” is “triangulated”.

A.1 Dg-categories

Recall the definition of a dg-category:

Definition A.1.

A dg-category𝒜 is a 𝕜-linear category such that

  1. Hom𝒜(X,Y) is a -graded 𝕜-vector space,

  2. the composition

    Hom𝒜(Y,Z)𝕜Hom𝒜(X,Y)--Hom𝒜(X,Z),

    preserves the -degree,

  3. there is a differential d:Hom𝒜(X,Y)iHom𝒜(X,Y)i-1 such that

    d2=0,d(fg)=dfg+(-1)|f|fdg.

Remark A.2.

We use a differential of degree -1 to match the conventions used in the rest of the paper.

Example A.3.

Any dg-algebra (A,d) can be seen as a dg-category 𝑩𝑨 with a single abstract object and Hom𝑩𝑨(,):=(A,d).

Example A.4.

Let 𝒞 be an abelian, Grothendieck, 𝕜-linear category. Consider the category C(𝒞) of complexes in 𝒞, and define Cdg(𝒞) as the category, where

  1. objects are complexes in 𝒞,

  2. hom-spaces are homogeneous maps of -graded modules,

  3. the differential d:HomCdg(𝒞)(X,Y)iHomCdg(𝒞)(X,Y)i-1 is given by

    df:=dYf-(-1)|f|fdX.

This data forms a dg-category.

Given a dg-category 𝒜, one defines

  1. the underlying category Z0(𝒜) as

    1. having the same objects as 𝒜,

    2. HomZ0(𝒜)(X,Y):=ker(Hom𝒜(X,Y)0𝑑Hom𝒜(X,Y)-1),

  2. the homotopy category H0(𝒜) (or [𝒜]) as

    1. having the same objects as 𝒜,

    2. HomH0(𝒜)(X,Y):=H0(Hom𝒜(X,Y),d).

Example A.5.

For 𝒞 as in Example A.4, we have

Z0(Cdg(𝒞))C(𝒞)andH0(Cdg(𝒞))Kom(𝒞)

the homotopy category of complexes in 𝒞.

A.2 Category of dg-categories

Definition A.6.

A dg-functor F:𝒜 is a functor between two dg-categories such that F(d𝒜f)=d(Ff). We write [F]:H0(𝒜)H0() for the induced functor.

We write dg-cat for the category of dg-categories, where objects are dg-categories and hom-spaces are given by dg-functors.

Let F,G:𝒜 be a pair of dg-functors between dg-categories. Then one defines 𝑜𝑚(F,G) as the -graded 𝕜-module of homogeneous natural transformations equipped with the differential induced by dHom(FX,GX) for all X𝒜. Then we put

Hom(F,G):=Z0(𝑜𝑚(F,G)).

Definition A.7.

A dg-functor 𝒜 is a quasi-equivalence if

  1. F:Hom𝒜(X,Y)Hom(FX,FY) is a quasi-isomorphism for all X,Y𝒜,

  2. [F]:H0(𝒜)H0() is essentially surjective (thus an equivalence).

One defines the dg-category 𝑜𝑚(𝒜,) of dg-functors between 𝒜 and as

  1. objects are dg-functors 𝒜,

  2. hom-spaces are Hom𝑜𝑚(𝒜,)(F,G):=𝑜𝑚(F,G).

There is also a notion of tensor product of dg-categories 𝒜 defined as

  1. objects are pairs XY for all X𝒜 and Y,

  2. hom-spaces are Hom𝒜(XY,XY):=Hom𝒜(X,X)𝕜Hom(Y,Y) with composition

    (fg)(fg):=(-1)|g||f|(ff)(gg),
  3. the differential is d(fg):=dfg+(-1)|f|fdg.

Then there is a bijection

Homdg-cat(𝒜,𝒞)Homdg-cat(𝒜,𝑜𝑚(,𝒞)).

This defines a symmetric closed monoidal structure on dg-cat. However, the tensor product of dg-categories does not preserve quasi-equivalences.

A.3 Dg-modules

Let 𝒜 be a dg-category. The opposite dg-category 𝒜op is given by

  1. same objects as in 𝒜,

  2. Hom𝒜op(X,Y):=Hom𝒜(Y,X),

  3. composition g𝒜opf:=(-1)|f||g|f𝒜g.

A left (resp. right) dg-moduleM over 𝒜 is a dg-functor

M:𝒜Cdg(𝕜)(resp. N:𝒜opCdg),

where Cdg(𝕜) is the dg-category of 𝕜-complexes. The dg-category of (right) dg-modules is 𝒜op-mod:=𝑜𝑚(𝒜op,Cdg(𝕜)). The category of (right) dg-modules is C(𝒜):=Z0(𝒜-mod), and it is an abelian category. The derived category 𝒟(𝒜) is the localization of Z0(𝒜op-mod) along quasi-isomorphisms.

Moreover, for any X𝒜 there is a right dg-module

X:=Hom𝒜(-,X).

One calls such dg-module representable. Any dg-module quasi-isomorphic to a representable dg-module is called quasi-representable. It yields a dg-enriched Yoneda embedding

𝒜𝒜op-mod.

Example A.8.

Let (A,d) be a dg-algebra. Then

Z0(𝑩𝑨)-mod(A,d)-modand𝒟(𝑩𝑨)𝒟(A,d).

The unique representable dg-module Hom𝑩𝑨(-,) is equivalent to the free module (A,d).

A.4 Model categories

We recall the basics of model category theory from [17]. Model category theory is a powerful tool to study localization of categories. For example, we can use it to compute hom-spaces in a derived category. We will mainly use it to describe the homotopy category of dg-categories up to quasi-equivalence.

Let M be a category with limits and colimits.

Definition A.9.

A model category on M is the data of three classes of morphisms

  1. the weak equivalences W,

  2. the fibrations Fib,

  3. the cofibrations Cof

satisfying

  1. for X𝑓Y𝑔ZM, if two out of three terms in {f,g,gf} are in W, then so is the third,

  2. stability along retracts:W, Fib and Cof are stable along retracts, that is if we have a commutative diagram

    and fW, Fib or Cof then so is g,

  3. factorization: any map X𝑓Y factorizes as pi, where pFib and iCofW or pFibW and iCof, and the factorization is functorial in f,

  4. lifting property: given a commutative square diagram

    with iCof and pFib, if either iW or pW, then there exists h:BX making the diagram commute.

We tend to think about fibrations as “nicely behaved surjections”, and cofibrations as “nicely behaved injections”.

The localization Ho(M):=W-1M of M along weak equivalences is called the homotopy category of M. It has a nice description in terms of homotopy classes of maps between fibrant and cofibrant objects.

Definition A.10.

If XCof, then we say X is cofibrant. If Y*Fib, then Y is fibrant.

One says that fg, that is f:XY is homotopy equivalent to g:XY, if there is a commutative diagram

where ij:XXC(X)Cof. One calls C(X) the cylinder object of X. When X is cofibrant and Y fibrant, then is an equivalence relation on HomM(X,Y). Moreover, we have

HomHo(M)(X,Y)HomM(X,Y)/

whenever X is cofibrant and Y fibrant. Note that any XM admits a cofibrant replacement QX since we have a commutative diagram

Similarly, any YM admits a fibrant replacement RY.

Let Mcf be the full subcategory of M given by objects that are both fibrant and cofibrant. Let Mcf/ be the quotient of Mcf by identifying maps that are homotopy equivalent. Then the localization functor MHo(M) restricts to Mcf, inducing an equivalence of categories

Mcf/Ho(M).

Example A.11.

Let C(𝕜) be the category of complexes of 𝕜-modules. It comes with a model category structure where W is the quasi-isomorphisms, Fib is the surjective maps, and Cof is given by the maps respecting the lifting property. All objects are fibrant and the cofibrant objects are essentially the complexes of projective 𝕜-modules. Then Ho(C(𝕜))𝒟(𝕜).

A model category on M is a C(𝕜)-model category if it is (strongly) enriched over C(𝕜), and the models are compatible (see [43, Section 3.1] for a precise definition). This definition means that we have

  1. a tensor product --:C(𝕜)×MM,

  2. an enriched dg-hom-space 𝑜𝑚M(X,Y)C(𝕜) for any X,YM compatible with the tensor product:

    HomM(EX,Y)HomC(𝕜)(E,𝑜𝑚M(X,Y)),
  3. Ho(M) is enriched over 𝒟(𝕜)Ho(C(𝕜)),

  4. a derived hom-functor

    𝑜𝑚M(X,Y):=𝑜𝑚M(QX,RY)𝒟(𝕜),

    where QX is a cofibrant replacement of X, and RY a fibrant replacement of Y,

  5. HomHo(M)(X,Y)H0(𝑜𝑚M(X,Y)).

Note that in particular for X,YMcf we have HomHo(M)(X,Y)H0(𝑜𝑚(X,Y)).

Example A.12.

Let 𝒜 be a dg-category. There is a C(𝕜)-model category on 𝒜-mod, where W is given by the quasi-isomorphisms, Fib are the surjective morphisms, and Cof is given by the maps respecting the lifting property. Then Ho(𝒜-mod)𝒟(𝒜).

Remark A.13.

In the C(𝕜)-model category 𝒜-mod, all objects are fibrant. Moreover, P is cofibrant if and only if for all surjective quasi-isomorphism f:LX (i.e. map in WFib) then there exists h:PL such that the following diagram commutes:

Note that, in a practical way, cofibrant dg-modules are quasi-isomorphic to direct summand of dg-modules admitting a (possibly infinite) exhaustive filtration where all the quotients are free dg-modules.

Definition A.14.

For M a C(𝕜)-model category, let M¯ (resp. Int(M)) be the dg-category with

  1. the same objects as M (resp. Mcf),

  2. HomM¯(X,Y):=𝑜𝑚M(X,Y).

Then we have H0(Int(M))Ho(M), and we say that Int(M) is a dg-enhancement of Ho(M).

Definition A.15.

We write

𝒟dg(𝒜):=Int(𝒜-mod)

for the dg-enhanced derived category of 𝒜.

Note that 𝒟dg(𝒜) is a dg-enhancement of 𝒟(𝒜) since we have H0(𝒟dg(𝒜))𝒟(𝒜).

Example A.16.

Let R be a 𝕜-algebra viewed as a dg-category with trivial differential. Then we have that 𝒟dg(R) is the dg-category of complexes of projective R-modules.

A.5 The model category of dg-categories

Let W be the collection of quasi-equivalences in dg-cat. Let Fib be the collection of dg-functors F:𝒜 in dg-cat such that

  1. FX,Y:Hom𝒜(X,Y)Hom(FX,FY) is surjective,

  2. for every isomorphism v:F(X)YH0() there exists an isomorphism

    u:XY0H0(𝒜)

    such that [F](u)=v.

This defines a model structure on dg-cat where everything is fibrant. One calls

Hqe:=Ho(dg-cat)

the homotopy category of dg-categories (up to quasi-equivalence).

How can we compute HomHqe(𝒜,)? It appears that constructing a cofibrant replacement for 𝒜 is in general a difficult problem. However, we can do the following:

  1. replace 𝒜 by a 𝕜-flat quasi-equivalent dg-category 𝒜: meaning it is such that

    Hom𝒜(X,Y)𝕜-

    preserves quasi-isomorphisms (e.g. when Hom𝒜(X,Y) is cofibrant in C(𝕜), i.e. a complex of projective 𝕜-modules),

  2. define Rep(𝒜,) as the subcategory of 𝒟(𝒜op) with FRep(𝒜,) if and only if for all X𝒜 there exists Y such that

    XLF𝒟()Y

    (in other words, F is a dg-bimodule sending representable 𝒜-modules to quasi-representable -modules),

  3. then

    HomHqe(𝒜,)Iso(Rep(𝒜,)),

    where Iso means the set of objects up to isomorphism.

Remark A.17.

Note that whenever 𝕜 is a field, all dg-categories are 𝕜-flat.

We refer to elements in Rep(𝒜,) as quasi-functors. Since a quasi-functor F:𝒜 induces a functor

[F]:H0(𝒜)H0(),

we can think of Rep(𝒜,) as the category of “representations up to homotopy” of 𝒜 in .

A.5.1 Closed monoidal structure

If 𝒜 is cofibrant, then -𝒜 preserves quasi-equivalences and one can define the bifunctor

-L-:Hqe×HqeHqe,𝒜L:=Q𝒜Q,

where Q𝒜 and Q are cofibrant replacements. Then, as proven by Toen [42], there exists an internal hom-functor 𝑜𝑚Hqe(-,-) such that

HomHqe(𝒜L,𝒞)HomHqe(𝒜,𝑜𝑚Hqe(,𝒞)).

Therefore, Hqe is a symmetric closed monoidal category.

Remark A.18.

Note that the internal hom can not simply be the derived hom functor (because tensor product of cofibrant dg-categories is not cofibrant in general).

Define the dg-category of quasi-functorsRepdg(𝒜,) as

  1. the objects in Rep(𝒜,)(𝒜op-mod)cf,

  2. the dg-homs 𝑜𝑚(X,Y) of Int(𝒜op-mod).

In other words, Repdg(𝒜,) is the full subcategory of quasi-functors in 𝒟dg(𝒜op), thus of cofibrant dg-bimodules that preserves quasi-representable modules. It is a dg-enhancement of Rep(𝒜,).

If 𝒜 is 𝕜-flat, then

𝑜𝑚Hqe(𝒜,)HqeRepdg(𝒜,).

Thus H0(𝑜𝑚Hqe(𝒜,))HomHqe(𝒜,).

Remark A.19.

If 𝕜 is a field of characteristic 0, then the dg-category 𝑜𝑚Hqe(𝒜,) is equivalent to the A-category of strictly unital A-functors [14].

Example A.20.

We have Repdg(𝒜,Int(C(𝕜)))Int(𝒜op-mod)𝒟dg(𝒜).

Recall that classical Morita theory says that for A and B being 𝕜-algebras, there is an equivalence

Homcop(A-mod,B-mod)Aop𝕜B-mod,

where Homcop is given by the functors that preserve coproducts.

Similarly, we put

Repdgcop(𝒟dg(𝒜),𝒟dg())

for the subcategory of Repdg(𝒟dg(𝒜),𝒟dg()) where FRepdgcop(𝒟dg(𝒜),𝒟dg()) if and only if [F]:𝒟(𝒜)𝒟() preserves coproducts.

Theorem A.21.

If A is 𝕜-flat, then we have

𝑜𝑚Hqecop(𝒟dg(𝒜),𝒟dg()):=Repdgcop(𝒟dg(𝒜),𝒟dg())Hqe𝒟dg(𝒜op).

Under the hypothesis of Theorem A.21, the internal composition of dg-quasifunctors preserving coproducts is given by taking a cofibrant replacement of the derived tensor product over 𝒜.

A.6 Pretriangulated dg-categories

Basically, a triangulated dg-category is a dg-category such that its homotopy category is canonically triangulated. But before being able to give a precise definition, we need to do a detour through Quillen exact categories, Frobenius categories and stable categories.

A.6.1 Frobenius structure on C(𝒜)

Recall that a Quillen exact category [37] is an additive category with a class of short exact sequences

0X𝑓Y𝑔Z0,

called conflations, which are pairs of ker-coker, where f is called an inflation and g a deflation, respecting some axioms:

  1. the identity is a deflation,

  2. the composition of deflations is a deflation,

  3. deflations (resp. inflations) are stable under base (resp. cobase) change.

A Frobenius category is a Quillen exact category having enough injectives and projectives, and where injectives coincide with projectives. The stable category 𝒞¯ of a Frobenius category 𝒞 is given by modding out the maps that factor through an injective/projective object. It carries a canonical triangulated structure, where

  1. the suspension functor S is obtained by taking the target of a conflation

    0XIXSX0,

    where IA is an injective hull of X, for all X𝒞,

  2. the distinguished triangles are equivalent to standard triangles

    X𝑓Y𝑔ZSX,

    obtained from conflations by the following commutative diagram:

Example A.22.

Let 𝒜 be a small dg-category. One can put a Frobenius structure on C(𝒜)(:=Z0(𝒜op-mod)) by using split short exact sequences as class of conflations. Then there is an equivalence C(𝒜)¯H0(𝒜op-mod), and the suspension functor coincides with the usual homological shift. Moreover, 𝒟(𝒜) inherits the triangulated structure from H0(𝒜-mod), where distinguished triangles are equivalent to distinguished triangles obtained from all short exact sequences in C(𝒜).

A.6.2 Pretriangulated dg-categories

Remark that for any dg-category 𝒜 there is a Yoneda functor

Z0(𝒜)C(𝒜),XHom𝒜(-,X).

Definition A.23.

A dg-category 𝒯 is pretriangulated if the image of the Yoneda functor is stable under translations and extensions (for the Quillen exact structure on C(𝒯) described in Example A.22).

This definition implies that

  1. Z0(𝒯) is a Frobenius subcategory of C(𝒯),

  2. H0(𝒯) inherits a triangulated structure, called canonical triangulated structure, from H0(𝒯-mod).

Example A.24.

Let 𝒜 be a dg-category. We have that 𝒟dg(𝒜) is pretriangulated with Z0(𝒟dg(𝒜))C(𝒜)cf. Moreover, the canonical triangulated structure of H0(𝒟dg(𝒜)) coindices with the usual on 𝒟(𝒜).

Then it is possible to show that

  1. any dg-category 𝒜 admits a pretriangulated hull pretr(𝒜) such that

    𝑜𝑚Hqe(𝒜,𝒯)𝑜𝑚Hqe(pretr(𝒜),𝒯)

    for all pretriangulated dg-category 𝒯,

  2. 𝑜𝑚Hqe(𝒜,𝒯) is pretriangulated whenever 𝒯 is pretriangulated,

  3. any dg-functor F:𝒯𝒯 between pretriangulated dg-categories induces a triangulated functor [F]:H0(𝒯)H0(𝒯).

For 𝒜 being 𝕜-flat, the pretriangulated structure of 𝑜𝑚Hqe(𝒟dg(𝒜),𝒟dg()) restricts to the one of 𝒟dg(𝒜op) (viewed as sub-dg-category). In particular, we obtain distinguished triangles of quasi-functors from short exact sequences of dg-bimodules.

Definition A.25.

For a morphism f:XYZ0(𝒯) in the underlying category of pretriangulated dg-category 𝒯, one calls mapping cone an object Cone(f)𝒯 such that

Cone(f)Cone(Xf¯Y)H0(𝒯-mod).

A.6.3 Dg-Morita equivalences

Definition A.26.

A dg-functor F:𝒜 is a dg-Morita equivalence if it induces an equivalence

𝑳F:𝒟(𝒜)𝒟(),XF(QX),

where QX is a cofibrant replacement of X.

Example A.27.

In particular, a quasi-equivalence is a dg-Morita equivalence and the functor that sends dg-categories to their pretriangulated hull 𝒜pretr(𝒜) is a dg-Morita equivalence.

Theorem A.28 ([41]).

There is a model structure dg-catmor on dg-cat, where the weak-equivalences are the dg-Morita equivalences and the fibrations are the same as before.

Definition A.29.

We say that 𝒯 is triangulated if it is fibrant in dg-catmor.

Equivalently, 𝒯 is triangulated if and only if the Yoneda functor induces an equivalence H0(𝒯-mod)𝒟c(𝒯) (i.e. every compact object is quasi-representable). Also equivalently, 𝒯 is triangulated if and only if 𝒯 is pretriangulated and H0(𝒯-mod) is idempotent complete.

In particular, any category admits a triangulated hull tr(𝒜) (i.e. fibrant replacement). It is given by

tr(𝒜):=𝒟dgc(𝒜),

the dg-category of compact objects in 𝒟dg(𝒜).

Example A.30.

Let R be a 𝕜-algebra viewed as a dg-category. Then 𝒟dgc(R) is the dg-category of perfect complexes, i.e. bounded complexes of finitely generated projective R-modules.

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Received: 2020-02-26
Revised: 2021-05-14
Published Online: 2021-10-16
Published in Print: 2022-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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