The double centralizer theorem categoriﬁed is...?

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The double centralizer theorem categorified is...?

Or: Two different and yet similar answers Daniel Tubbenhauer

A ∼ = E nd _E _ndA _(M) (M)

Joint with Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz and Xiaoting Zhang

December 2020

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One version of the double centralizer theorem (DCT)

The DCT (Schur ∼1901+1927, Thrall∼1947, Morita∼1958).

LetAbe a self-injective, finite-dimensional algebra, andMbe a faithfulA-module.

Then there is a canonical algebra map

can:A→ End_EndA(M)(M), which is an isomorphism.

I Bad news. We can not create many new algebras out of (A,M). (Same for the categorified versions.)

I Good news. We can ^play AandB=EndA(M) against each other.

I Good news. There are plenty of ^examples which we know and like.

Question. What is a categorical analog of the DCT?

Mshould be aA-B-bimodule, soEndA(M) means right operators,

whileEndB(M) are left operators.

I will ignore this technicality.

Self-injective⇔projectives=injectives, faithful⇔only 0 acts as zero.

This is not the most general version, but I will stick to it for simplicity.

Daniel Tubbenhauer The double centralizer theorem categorified is...? December 2020 2 / 5

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One version of the double centralizer theorem (DCT)

The DCT (Schur ∼1901+1927, Thrall∼1947, Morita∼1958).

Then there is a canonical algebra map

can:A→ End_EndA(M)(M), which is an isomorphism.

I Bad news. We can not create many new algebras out of (A,M). (Same for the categorified versions.)

I Good news. We can ^play AandB=EndA(M) against each other.

I Good news. There are plenty of ^examples which we know and like.

Mshould be aA-B-bimodule, soEndA(M) means right operators,

whileEndB(M) are left operators. I will ignore this technicality. Self-injective⇔projectives=injectives,

faithful⇔only 0 acts as zero. This is not the most general version,

but I will stick to it for simplicity.

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Two potential answers.

Question. What is a categorical analog of the DCT?

Asemisimplecategori- fication of an algebra.

Anabeliancategorifi- cation of an algebra.

Anadditivecategori- fication of an algebra.

Asemisimplecategori- fication of a module.

Anabeliancategori- fication of a module.

Anadditivecategori- fication of a module.

Asemisimplecategori- fication of the DCT.

Anabeliancategori- fication of the DCT.

Anadditivecategori- fication of the DCT.

Goal. Explain both answers: first the abelian(easier), then the additive(harder).

Why only “two potential answers”?

Fun fact. Semisimple implies abelian, and is a special case of additive. In the middle the two outer ways coincide.

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Anabeliancategorifi- cation of an algebra.

Anadditivecategori- fication of an algebra.

Anabeliancategori- fication of a module.

Anadditivecategori- fication of a module.

Anabeliancategori- fication of the DCT.

Anadditivecategori- fication of the DCT.

Goal. Explain both answers: first the abelian(easier), then the additive(harder).

Why only “two potential answers”?

Fun fact. Semisimple implies abelian, and is a special case of additive.

In the middle the two outer ways coincide.

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Abelian DCT (Etingof–Ostrik ∼2003).

LetA be a finite, pivotal multitensor category and M a faithfulA-module. Then there is a canonical monoidal functor

can:A →End_End_A(M)(M), which is an equivalence.

Additive DCT (∼2020).

LetA be a monoidal fiat category, J a two-sided cell and M a simple transitive AJ-module with apex J. Then there is a canonical monoidal functor

can:AJ →End_End_A_J(M)(M),

which is an equivalence when restricted toadd(J) and corestricted to End^inj_E_nd

AJ(M)(M).

Do not worry: I will ^explain all the words! For now just note that the second statement already sounds more complicated.

These are not the most general versions, but I will stick to these for simplicity.

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One version of the double centralizer theorem (DCT) The DCT (Schur∼1901+1927, Thrall∼1947, Morita∼1958).

Then there is a canonical algebra map can:A→ EndEndA(M)(M), which is an isomorphism.

IBad news.We can not create many new algebras out of (A,M).(Same for the categorified versions.)

IGood news.We canplayAandB=EndA(M) against each other.

IGood news.There are plenty of^exampleswhich we know and like.

Mshould be aA-B-bimodule, soEndA(M) means right operators, whileEndB(M) are left operators. I will ignore this technicality. Self-injective⇔projectives=injectives,

faithful⇔only 0 acts as zero. This is not the most general version, but I will stick to it for simplicity.

Daniel Tubbenhauer The double centralizer theorem categorified is...? December 20202 / 5

Anabeliancategorifi-

cation of an algebra. Anadditivecategori-

fication of an algebra.

Anabeliancategori-

fication of a module. Anadditivecategori-

fication of a module.

Anabeliancategori-

fication of the DCT. Anadditivecategori-

fication of the DCT.

Goal.Explain both answers: first the abelian(easier), then the additive(harder).

Why only “two potential answers”? Fun fact.Semisimple implies abelian, and is a special case of additive.

Abelian DCT (Etingof–Ostrik∼2003).

LetAbe a finite, pivotal multitensor category and M a faithfulA-module. Then there is a canonical monoidal functor

can:A→EndEndA(M)(M), which is an equivalence.

LetAbe a monoidal fiat category,Ja two-sided cell and M a simple transitive AJ-module with apexJ. Then there is a canonical monoidal functor

can:AJ→EndEndAJ(M)(M), which is an equivalence when restricted toadd(J) and corestricted to End^injEndAJ(M)(M).

Do not worry: I will^explainall the words! For now just note that the second statement already sounds more complicated.

AknowsB, andBknowsA, right?

A-Mod'B-Mod

⇔

∃Mprogenerator such thatA∼=EndB(M)

⇔

∃Mprogenerator such thatB∼=EndA(M).

Back

Morita∼1958.

The DCT goes hand-in-hand with classical Morita-theory.

Schur∼1901+1927. The DCT goes hand-in-hand with classical Schur–Weyl duality.

Green∼1980. The DCT applies for Schur–Weyl in the non-semisimple case.

Soergel∼1990. The DCT applies in categoryO.

ExampleG=Z/2Z×Z/2Z(Klein four group).

IfKis not of characteristic 2,KGis semisimple and additive=abelian. So let us have a look at characteristic 2, where we haveKG∼=K[X,Y]/(X²,Y²) First, abelian:

IXandYhave to act as zero on each simple, soKGhas justKas a simple.

IKG-Modhas just one element.

Then additive:

IOnlyX²andY²have to act as zero on each indecomposable, and one can cook-up infinitely many,e.g.

•^X•^Y•^X•^Y•^X...^Y•^X• IKG-Modhas infinitely many elements.

Back

Theorem (Higman∼1953). Forchar(K) =p,KG-Modis... ...always a finite, pivotal multitensor category. ... monoidal fiat if and only if (p-|G|or thep-Sylow subgroup ofGis cyclic).

Example (G-Mod, ground fieldC).

ILetA=G-Mod, forGbeing a finite group. AsAis semisimple, abelian=additive. Simples are simpleG-modules.

IFor anyM,N∈A, we haveM⊗N∈A: g(m⊗n) =gm⊗gn for allg∈G,m∈M,n∈N. There is a trivial moduleC.

IThe regularA-module M :A→EndC(A):

M //

f

M⊗

f⊗

N //N⊗

.

IThe decategorification is the regularK0(A)-module.

Back

Semisimple example.

IA=Vect, and fix M = Vect^⊕n, which is faithful.

IB=EndVect(Vect^⊕n)∼=Matn×n(Vect) and EndMatn×n(Vect)(Vect^⊕n)∼=Vect.

Another semisimple example.

IA=VectG, and fix M = Vect, which is faithful.

IB=EndVectG(Vect)∼=G-ModandEndG-Mod(Vect)∼=VectG. An abelian example.

IA=H-Mod, and fix M = Vect, which is faithful.

IB=EndH-Mod(Vect)∼=H^?-ModandEndH?-Mod(Vect)∼=H-Mod.

Back Upshot

Morita equivalence (Etingof–Ostrik∼2003).

LetB=EndA(M) for M a faithful, exactA-module. Then A-mod'B-mod.

Example.

A=VectGandB=G-Modhave the “same” module categories, which is a very non-trivial fact.

Back An additive example

Exact⇔the unit acts as an exact functor.

If M is semisimple, then exactness is automatic.

Additive example (∼2020).

S=S(W,C) Soergel bimodules forWfinite, the coinvariant algebra and overC, Ja two-sided cell and CJthe cellSJ-module.

IAdditive DCT. We have can:SJ→EndEndSJ(CJ)(CJ), is an equivalence when restricted toadd(J) and corestricted to End^injEndAJ(CJ)(CJ).

I“Endomorphismensatz”. We have EndAJ(CJ)'AJ whereAJis the asymptotic category (semisimple!).

IMorita equivalence. We have SJ-stmod'AJ-stmod.

Back Sorry, this example is not self-contained.

But just to explain all the ingredients carefully is another talk.

This looks weaker than the abelian DCT, but this is what we can prove right now.

Anyway, let explain why it is weaker, which finally explains all words in the additive DCT.

To make CJfaithful, quotientSby “bigger stuff”

and getSJ. add(J): Since “lower stuff” still acts pretty much in an uncontrolable way,

restrict to only things inJ. injmeans injective endofunctors. In this case you could also consider projective endofunctors.

AJis the “degree zero part” ofSJ.

“AJis the crystal associated toSJ.”

stmodare simple transitive modules. The analogs of categories of simple modules downstairs.

There is still much to do...

Thanks for your attention!

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One version of the double centralizer theorem (DCT) The DCT (Schur∼1901+1927, Thrall∼1947, Morita∼1958).

Then there is a canonical algebra map can:A→ EndEndA(M)(M), which is an isomorphism.

IBad news.We can not create many new algebras out of (A,M).(Same for the categorified versions.)

IGood news.We canplayAandB=EndA(M) against each other.

IGood news.There are plenty of^exampleswhich we know and like.

Mshould be aA-B-bimodule, soEndA(M) means right operators, whileEndB(M) are left operators. I will ignore this technicality. Self-injective⇔projectives=injectives,

faithful⇔only 0 acts as zero. This is not the most general version, but I will stick to it for simplicity.

Anabeliancategorifi-

cation of an algebra. Anadditivecategori-

fication of an algebra.

Anabeliancategori-

fication of a module. Anadditivecategori-

fication of a module.

Anabeliancategori-

fication of the DCT. Anadditivecategori-

fication of the DCT.

Goal.Explain both answers: first the abelian(easier), then the additive(harder).

Why only “two potential answers”? Fun fact.Semisimple implies abelian, and is a special case of additive.

Abelian DCT (Etingof–Ostrik∼2003).

LetAbe a finite, pivotal multitensor category and M a faithfulA-module. Then there is a canonical monoidal functor

can:A→EndEndA(M)(M), which is an equivalence.

LetAbe a monoidal fiat category,Ja two-sided cell and M a simple transitive AJ-module with apexJ. Then there is a canonical monoidal functor

can:AJ→EndEndAJ(M)(M), which is an equivalence when restricted toadd(J) and corestricted to End^injEndAJ(M)(M).

Do not worry: I will^explainall the words! For now just note that the second statement already sounds more complicated.

A-Mod'B-Mod

⇔

Back

Morita∼1958.

Schur∼1901+1927. The DCT goes hand-in-hand with classical Schur–Weyl duality.

Green∼1980. The DCT applies for Schur–Weyl in the non-semisimple case.

Soergel∼1990. The DCT applies in categoryO.

ExampleG=Z/2Z×Z/2Z(Klein four group).

IfKis not of characteristic 2,KGis semisimple and additive=abelian. So let us have a look at characteristic 2, where we haveKG∼=K[X,Y]/(X²,Y²) First, abelian:

IXandYhave to act as zero on each simple, soKGhas justKas a simple.

IKG-Modhas just one element.

Then additive:

IOnlyX²andY²have to act as zero on each indecomposable, and one can cook-up infinitely many,e.g.

•^X•^Y•^X•^Y•^X...^Y•^X• IKG-Modhas infinitely many elements.

Back

Theorem (Higman∼1953). Forchar(K) =p,KG-Modis... ...always a finite, pivotal multitensor category. ... monoidal fiat if and only if (p-|G|or thep-Sylow subgroup ofGis cyclic).

ILetA=G-Mod, forGbeing a finite group. AsAis semisimple, abelian=additive. Simples are simpleG-modules.

IFor anyM,N∈A, we haveM⊗N∈A: g(m⊗n) =gm⊗gn for allg∈G,m∈M,n∈N. There is a trivial moduleC.

IThe regularA-module M :A→EndC(A):

M //

f

M⊗

f⊗

N //N⊗

.

IThe decategorification is the regularK0(A)-module.

Back

Semisimple example.

IA=Vect, and fix M = Vect^⊕n, which is faithful.

IB=EndVect(Vect^⊕n)∼=Matn×n(Vect) and EndMatn×n(Vect)(Vect^⊕n)∼=Vect.

IA=VectG, and fix M = Vect, which is faithful.

IB=EndVectG(Vect)∼=G-ModandEndG-Mod(Vect)∼=VectG. An abelian example.

IA=H-Mod, and fix M = Vect, which is faithful.

IB=EndH-Mod(Vect)∼=H^?-ModandEndH?-Mod(Vect)∼=H-Mod.

Back Upshot

Morita equivalence (Etingof–Ostrik∼2003).

LetB=EndA(M) for M a faithful, exactA-module. Then A-mod'B-mod.

Example.

A=VectGandB=G-Modhave the “same” module categories, which is a very non-trivial fact.

S=S(W,C) Soergel bimodules forWfinite, the coinvariant algebra and overC, Ja two-sided cell and CJthe cellSJ-module.

IAdditive DCT. We have can:SJ→EndEndSJ(CJ)(CJ), is an equivalence when restricted toadd(J) and corestricted to End^injEndAJ(CJ)(CJ).

I“Endomorphismensatz”. We have EndAJ(CJ)'AJ whereAJis the asymptotic category (semisimple!).

IMorita equivalence. We have SJ-stmod'AJ-stmod.

Back Sorry, this example is not self-contained.

To make CJfaithful, quotientSby “bigger stuff”

and getSJ. add(J): Since “lower stuff” still acts pretty much in an uncontrolable way,

restrict to only things inJ. injmeans injective endofunctors. In this case you could also consider projective endofunctors.

AJis the “degree zero part” ofSJ.

“AJis the crystal associated toSJ.”

stmodare simple transitive modules. The analogs of categories of simple modules downstairs.

There is still much to do...

Thanks for your attention!

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A knows B, andBknows A, right?

A-Mod'B-Mod

⇔

Back

Morita∼1958.

The DCT goes hand-in-hand with classical Morita-theory. Schur∼1901+1927.

The DCT goes hand-in-hand with classical Schur–Weyl duality. Green∼1980.

The DCT applies for Schur–Weyl in the non-semisimple case. Soergel∼1990.

The DCT applies in categoryO.

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A-Mod'B-Mod

⇔

Back

Morita∼1958.

Schur∼1901+1927.

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IfA⊂ End_K(M),B=EndA(M) andAis semisimple, then:

I A=EndB(M);

I Bis semisimple;

I As aA⊗B^op-module we have M∼= M

simples ofA,B

ALⁱ⊗LⁱB.

Back

Morita∼1958.

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IfA⊂ End_K(M),B=EndA(M) andAis semisimple, then:

I A=EndB(M);

I Bis semisimple;

I As aA⊗B^op-module we have M∼= M

simples ofA,B

ALⁱ⊗LⁱB.

Back

Morita∼1958.

Schur∼1901+1927.

The DCT goes hand-in-hand with classical Schur–Weyl duality.

Green∼1980.

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IfM=Aefore²=e,Mfaithful andB=EndA(Ae), then:

I B∼=eAe andA∼=End_eAe(Ae);

I TheB-simples are in bijection withA-simplesNsuch thatNe6= 0;

I Ais encoded in the (usually) much smaller algebraB.

Back

Morita∼1958.

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IfM=Aefore²=e,Mfaithful andB=EndA(Ae), then:

I B∼=eAe andA∼=End_eAe(Ae);

I TheB-simples are in bijection withA-simplesNsuch thatNe6= 0;

I Ais encoded in the (usually) much smaller algebraB.

Back

Morita∼1958.

The DCT goes hand-in-hand with classical Schur–Weyl duality.

Green∼1980.

The DCT applies for Schur–Weyl in the non-semisimple case.

Soergel∼1990.

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Example. (Looks silly, but is prototypical.) I A=K, and fixM=Kⁿ, which is faithful.

I B=End_K(Kⁿ)∼=Mat_n×n(K) and EndMat_n×n(K)(Kⁿ)∼=K. I M∼=K⊗Kⁿ, perfect matching of isotypic components.

Non-example. (Faithfulness missing.)

I A=K[X]/(X³), and fixM=K²,X 7→(^{0 1}_{0 0}), which is not-faithful.

I B=End_K[X]/(X³₎(K²)∼=K[X]/(X²) andEnd_K[X_]/(X²₎(K²)∼=K[X]/(X²).

I M∼=K²⊗Kas aK[X]/(X³)-module, M∼=K⊗K²as aK[X]/(X²)-module.

Non-example. (Self-injectivity missing.) I A= ^{K K}₀_K

, and fixM=K², which is faithful.

I B=EndK K

0K(K²)∼=KandEnd_K(K²)∼=Mat_2×2(K).

I M∼=K²⊗Kas a ^{K K}₀_K

-module,M∼=K⊗K² as aMat_2×2(K)-module.

Back More sophisticated

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Example (Schur ∼1901+1927, Green ∼1980).

I A=K[S_d], and fixM= (Kⁿ)^⊗d forn≥d, which is faithful.

I B=End_K[Sd] (Kⁿ)^⊗d∼=S(n,d) (Schur algebra) and End_S(n,d) (Kⁿ)^⊗d∼=K[S_d].

I K[S_d]∼=eS(n,d)e and theK[S_d]-simples are in bijection withS(n,d)-simples Nsuch thatNe6= 0.

Example (Soergel’s Struktursatz∼1990).

I Aa finite-dimensional algebra forO0(g_C). FixM=Ae, which is faithful for the right choice of idempotente_w₀ (the big projective).

I B=EndA(Aew0)∼=ew0Aew0 (Soergel’s Endomorphismensatz∼1990:

B=coinvariant algebra) andEndew0Aew0(Aew0)∼=A.

I Acan be recovered frome_w₀Ae_w₀, althoughAis much more complicated.

Explicitly, forg_C=sl₂one getse.g.

A= 1 ^a s

b /(a|b= 0), B∼=C{s,b|a}, As= ^a s

b

Back

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ExampleG =Z/2Z×Z/2Z (Klein four group).

IfKis not of characteristic 2,KG is semisimple and additive=abelian. So let us have a look at characteristic 2, where we haveKG ∼=K[X,Y]/(X²,Y²)

First, abelian:

I X andY have to act as zero on each simple, soKG has justKas a simple.

I KG-Modhas just one element.

Then additive:

I OnlyX²andY² have to act as zero on each indecomposable, and one can cook-up infinitely many,e.g.

• ^X • ^Y • ^X • ^Y • ^X ... ^Y • ^X • I KG-Modhas infinitely many elements.

Back

Theorem (Higman∼1953).

Forchar(K) =p,KG-Modis... ...always a finite, pivotal multitensor category.

... monoidal fiat if and only if (p-|G|or thep-Sylow subgroup ofG is cyclic).

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ExampleG =Z/2Z×Z/2Z (Klein four group).

IfKis not of characteristic 2,KG is semisimple and additive=abelian. So let us have a look at characteristic 2, where we haveKG ∼=K[X,Y]/(X²,Y²)

First, abelian:

I X andY have to act as zero on each simple, soKG has justKas a simple.

I KG-Modhas just one element.

Then additive:

I OnlyX²andY² have to act as zero on each indecomposable, and one can cook-up infinitely many,e.g.

• ^X • ^Y • ^X • ^Y • ^X ... ^Y • ^X • I KG-Modhas infinitely many elements.

Back

Theorem (Higman∼1953).

Forchar(K) =p,KG-Modis...

...always a finite, pivotal multitensor category.

... monoidal fiat if and only if (p-|G|or thep-Sylow subgroup ofG is cyclic).

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I LetA =G-Mod, forG being a finite group. AsA is semisimple, abelian=additive. Simples are simpleG-modules.

I For anyM,N∈A, we haveM⊗N∈A:

g(m⊗n) =gm⊗gn for allg ∈G,m∈M,n∈N. There is a trivial moduleC. I The regularA-module M :A →End_C(A):

M //

f

M⊗

f⊗

N //N⊗

.

I The decategorification is the regularK₀(A)-module.

Back

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I LetK ⊂G be a subgroup.

I K-Mod is aA-module, with action

Res^G_K⊗ : G-Mod→End_C K-Mod ,

M //

f

Res^G_K(M)⊗

Res^GK(f)⊗

N //Res^G_K(N)⊗ .

which is indeed an action becauseRes^G_K is a⊗-functor.

I The decategorifications areK₀(A)-modules.

Back

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Left partial preorder≥L on indecomposable objects by

F≥LG⇔there existsHsuch that Fis isomorphic to a direct summand ofHG.

Left cellsLare the equivalence classes with respect to≥L, on which≥Linduces a partial order. Similarly, right and two-sided, denoted byRandJ respectively.

CellA-modules associated toL are:

add {F|F≥LL}

/“kill ≥L-bigger stuff”.

Examples.

I Cells inA give⊗-ideals.

I IfA is semisimple, then FF^?andF^?Fboth contain the identity, so cell theory is trivial. The cellA-module is the regularA-module.

I For Soergel bimodules cells are Kazhdan–Lusztig cells and cell modules categorify Kazhdan–Lusztig cell modules.

I For categorified quantum groups you can push everything to cyclotomic KLR algebras, and cell modules categorify simple modules.

Back

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A finite, pivotal multitensor category A:

I Basics. A isK-linear and monoidal,⊗isK-bilinear. Moreover, A is abelian (this implies idempotent complete).

I Involution. A is pivotal,e.g. F^??∼=F.

I Finiteness. Hom-spaces are finite-dimensional, the number of simples is finite, finite length, enough projectives.

I Categorification. The abelian Grothendieck ring gives a finite-dimensional algebra with involution.

A monoidal fiat category A:

I Basics. A isK-linear and monoidal,⊗isK-bilinear. Moreover,A is additive and idempotent complete.

I Finiteness. Hom-spaces are finite-dimensional, the number of indecomposables is finite.

I Categorification. The additive Grothendieck ring gives a finite-dimensional algebra with involution.

Back Further

The crucial difference...

...is what we like to consider as “elements” of our theory: Abelian prefers simples,

additive prefers indecomposables.

This is a ^huge difference – for example in the fiat case there is simply no Schur’s lemma. Abelian examples.

H-ModforHa finite-dimensional Hopf algebra. (Think: KG,G finite.) Finite Serre quotients ofG-ModforG being a reductive group.

Abelian and additive examples.

H-Modfor Ha finite-dimensional, semisimple Hopf algebra. (Think: CG,G finite.) VectG forG gradedK-vector spaces,e.g.Vect=Vect1.

Additive examples.

H-ProjforHa finite-dimensional Hopf algebra. (Think: KG,G finite.) Finite quotients ofG-TiltforG being a reductive group.

Why I like the additive case.

All the example I know from my youth are not abelian, but only additive: Diagram categories, categorified quantum group

and their Schur quotients, Soergel bimodules, tilting module categoriesetc.

And these only fit into the fiat and not the tensor framework.

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Back Further

...is what we like to consider as “elements” of our theory:

Abelian prefers simples, additive prefers indecomposables.

This is a ^huge difference – for example in the fiat case there is simply no Schur’s lemma.

Abelian examples.

Additive examples.

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Back Further

This is a ^huge difference – for example in the fiat case there is simply no Schur’s lemma.

Abelian examples.

H-ModforHa finite-dimensional, semisimple Hopf algebra. (Think: CG,G finite.) VectG forG gradedK-vector spaces,e.g.Vect=Vect1.

Additive examples.

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Back Further

This is a ^huge difference – for example in the fiat case there is simply no Schur’s lemma. Abelian examples.

Additive examples.

All the example I know from my youth are not abelian, but only additive:

Diagram categories, categorified quantum group and their Schur quotients, Soergel bimodules,

tilting module categoriesetc.

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Abelian. An A-module M:

I Basics. M isK-linear and abelian. The action is a monoidal functor M :A →End_K,lex(M) (K-linear, left exactness).

I Categorification. The abelian Grothendieck group gives a finite-dimensional G0(A)-module.

Additive. AnA-module M:

I Basics. M isK-linear, additive and idempotent complete. The action is a monoidal functor M :A →End_K(M) (K-linear).

I Categorification. The additive Grothendieck group gives a finite-dimensional K₀(A)-module.

Back Further

Faithful⇔only 0 (the object) acts as zero (functor).

This already clarifies the abelian DCT.

Example.

Everything is constructed such that the regularA-moduleA exists.

Smarter version of the regularA-module are cellA-modules. ^What? But of course there are many ^more examples.

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Abelian. An A-module M:

I Basics. M isK-linear and abelian. The action is a monoidal functor M :A →End_K,lex(M) (K-linear, left exactness).

I Categorification. The abelian Grothendieck group gives a finite-dimensional G0(A)-module.

Additive. AnA-module M:

I Basics. M isK-linear, additive and idempotent complete. The action is a monoidal functor M :A →End_K(M) (K-linear).

I Categorification. The additive Grothendieck group gives a finite-dimensional K₀(A)-module.

Back Further

Faithful⇔only 0 (the object) acts as zero (functor). This already clarifies the abelian DCT.

Example.

Everything is constructed such that the regularA-moduleA exists.

Smarter version of the regularA-module are cellA-modules. ^What?

But of course there are many ^more examples.

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Semisimple example.

I A =Vect, and fix M = Vect^⊕n, which is faithful.

I B=End_Vect(Vect^⊕n)∼=Mat_n×n(Vect) and End_Mat_n×n(Vect)(Vect^⊕n)∼=Vect.

I A =Vect_G, and fix M = Vect, which is faithful.

I B=End_Vect_G(Vect)∼=G-ModandEndG-Mod(Vect)∼=VectG. An abelian example.

I A =H-Mod, and fix M = Vect, which is faithful.

I B=End_H-Mod(Vect)∼=H^?-ModandEndH^?-Mod(Vect)∼=H-Mod.

Back Upshot

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A knows B, andB knows A, right?

Morita equivalence (Etingof–Ostrik ∼2003).

LetB=End_A(M) for M a faithful, exactA-module. Then A-mod'B-mod.

Example.

A =Vect_G andB=G-Modhave the “same” module categories, which is a very non-trivial fact.

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S =S(W,C) Soergel bimodules forW finite, the coinvariant algebra and overC, J a two-sided cell and C_J the cellSJ-module.

I Additive DCT. We have

can:SJ →End_End_S_J(C_J)(C_J),

is an equivalence when restricted toadd(J) and corestricted to End^inj_E_nd

AJ(CJ)(C_J).

I “Endomorphismensatz”. We have

End_A_J(C_J)'AJ

where AJ is the asymptotic category (semisimple!).

I Morita equivalence. We have

SJ-stmod'AJ-stmod.

Back

Sorry, this example is not self-contained.

To make C_J faithful, quotientS by “bigger stuff”

and getSJ.

add(J): Since “lower stuff” still acts pretty much in an uncontrolable way, restrict to only things inJ.

injmeans injective endofunctors.

In this case you could also consider projective endofunctors. AJ is the “degree zero part” ofSJ.

“AJ is the crystal associated toSJ.”

stmodare simple transitive modules.

The analogs of categories of simple modules downstairs.

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AJ(CJ)(C_J).

End_A_J(C_J)'AJ

SJ-stmod'AJ-stmod.

Back

Sorry, this example is not self-contained. But just to explain all the ingredients carefully is another talk.

This looks weaker than the abelian DCT, but this is what we can prove right now. Anyway, let explain why it is weaker, which finally explains all words in the additive DCT.

and getSJ.

In this case you could also consider projective endofunctors.

AJ is the “degree zero part” ofSJ.

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AJ(CJ)(C_J).

End_A_J(C_J)'AJ

SJ-stmod'AJ-stmod.

Back

and getSJ.

In this case you could also consider projective endofunctors.

AJ is the “degree zero part” ofSJ.

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AJ(CJ)(C_J).

End_A_J(C_J)'AJ

SJ-stmod'AJ-stmod.

Back

and getSJ.

In this case you could also consider projective endofunctors. AJ is the “degree zero part” ofSJ.

The double centralizer theorem categoriﬁed is...?

The double centralizer theorem categorified is...?

A ∼ = E nd E ndA (M) (M)

A ∼ = E nd _E _ndA _(M) (M)