2-representation theory of Soergel bimodules

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2-representation theory of Soergel bimodules

Or: Mind your groups Daniel Tubbenhauer

2 1

1 2

1

left cells

“left modules”

2 1

1 2

1

right cells

“right modules”

2 1

1 2

1

two-sided cells

“bimodules”

2 1

1 2

1

H-cells

“subalgebras”

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

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2-representation theory in a nutshell

2-moduleM i7→M(i)

category F7→M(F)

functor α7→ M(α)

nat. trafo

1-moduleM i7→ M(i)

vector space F7→M(F)

linear map

0-modulem i7→m(i)

number

categorical module

categorifies

Examples of2-categories.

Monoidal categories, module categoriesRep(G) of finite groupsG, module categories of Hopf algebras, fusion or modular tensor categories, Soergel bimodulesS, categorified quantum groups, categorified Heisenberg algebras.

Examples of2-representation of these. Categorical modules, functorial actions,

(co)algebra objects, conformal embeddings of affine Lie algebras,

the LLT algorithm, cyclotomic Hecke/KLR algebras, categorified (anti-)spherical module. Applications of2-representations.

Representation theory (classical and modular), link homology, combinatorics TQFTs, quantum physics, geometry.

Plan for today.

1) Give an overview of the main ideas of 2-representation theory. 2) Discuss the group-like exampleRep(G).

3) Discuss the semigroup-like exampleS.

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2-moduleM i7→M(i)

category F7→M(F)

nat. trafo

linear map

0-modulem i7→m(i)

number

categorical module

categorifies

Plan for today.

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2-moduleM i7→M(i)

category F7→M(F)

nat. trafo

linear map

0-modulem i7→m(i)

number

categorical module

categorifies

Examples of2-representation of these.

Categorical modules, functorial actions,

the LLT algorithm, cyclotomic Hecke/KLR algebras, categorified (anti-)spherical module.

Applications of2-representations.

Plan for today.

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2-moduleM i7→M(i)

category F7→M(F)

nat. trafo

linear map

0-modulem i7→m(i)

number

categorical module

categorifies

Examples of2-representation of these.

Categorical modules, functorial actions,

the LLT algorithm, cyclotomic Hecke/KLR algebras, categorified (anti-)spherical module.

Applications of2-representations.

Plan for today.

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2-moduleM i7→M(i)

category F7→M(F)

nat. trafo

linear map

0-modulem i7→m(i)

number

categorical module

categorifies

Plan for today.

1) Give an overview of the main ideas of 2-representation theory.

2) Discuss the group-like exampleRep(G).

3) Discuss the semigroup-like example S.

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Representation theory is group theory in vector spaces

LetCbe a finite-dimensional algebra.

Frobenius∼1895++, Burnside∼1900++, Noether∼1928++.

Representation theory is the ^useful? study of algebra actions M:C−→ End(V),

withVbeing some vector space. (Called modules or representations.) The “atoms” of such an action are called simple.

Maschke∼1899, Noether, Schreier∼1928. All modules are built out of simples (“Jordan–H¨older” filtration).

Basic question: Find the periodic table of simples.

Empirical fact.

Most of the fun happens already for monoidal categories (one-object 2-categories); I will stick to this case for the rest of the talk,

but what I am going to explain works for 2-categories.

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2-representation theory is group theory in categories

LetC be a finitary 2-category.

Etingof–Ostrik, Chuang–Rouquier, many others ∼2000++. 2-representation theory is the useful? study of actions of 2-categories:

M:C −→End(V),

withV being some finitary category. (Called 2-modules or 2-representations.) The “atoms” of such an action are called 2-simple (“simple transitive”).

Mazorchuk–Miemietz∼2014. All 2-modules are built out of 2-simples (“weak 2-Jordan–H¨older filtration”).

Basic question: Find the periodic table of 2-simples.

Empirical fact.

Most of the fun happens already for monoidal categories (one-object 2-categories); I will stick to this case for the rest of the talk,

but what I am going to explain works for 2-categories.

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2-representation theory is group theory in categories

LetC be a finitary 2-category.

Etingof–Ostrik, Chuang–Rouquier, many others ∼2000++. 2-representation theory is the useful? study of actions of 2-categories:

M:C −→End(V),

withV being some finitary category. (Called 2-modules or 2-representations.) The “atoms” of such an action are called 2-simple (“simple transitive”).

Mazorchuk–Miemietz∼2014. All 2-modules are built out of 2-simples (“weak 2-Jordan–H¨older filtration”).

Basic question: Find the periodic table of 2-simples.

Empirical fact.

Most of the fun happens already for monoidal categories (one-object 2-categories);

I will stick to this case for the rest of the talk, but what I am going to explain works for 2-categories.

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A category V is called finitary if its equivalent toC-pMod. In particular:

I It has finitely many indecomposable objectsM_j (up to∼=).

I It has finite-dimensional hom-spaces.

I Its Grothendieck group [V] = [V]_Z⊗ZCis finite-dimensional.

A finitary, monoidal category C can thus be seen as a categorification of a finite-dimensional algebra.

Its indecomposable objectsCi give a distinguished basis of [C].

A finitary 2-representation of C: I A choice of a finitary categoryV. I (Nice) endofunctorsM(Ci) acting onV. I [M(C_i)] giveN-matrices acting on [V].

The atoms (decat).

ACmodule is called simple if it has noC-stable ideals.

The atoms (cat).

AC 2-module is called 2-simple if it has noC-stable⊗-ideals.

Dictionary.

cat finitary finitary+monoidal fiat functors decat vector space algebra self-injective matrices

Instead of studyingCand its action via matrices, studyC-pModand its action via functors.

Example (decat).

C=C= 1 acts on any vector space viaλ· . It has only one simpleV=C.

Example (cat).

C =Vec=Rep(1) acts on any finitary category viaC⊗C

It has only one 2-simpleV=Vec.

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The atoms (decat).

The atoms (cat).

Dictionary.

Example (decat).

Example (cat).

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The atoms (decat).

The atoms (cat).

Dictionary.

Example (decat).

Example (cat).

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The atoms (decat).

The atoms (cat).

Dictionary.

Example (decat).

Example (cat).

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An algebra A= (A, µ, ι) inC:

µ=

A

A A

, ι=

1 A

, = , = = .

Its (right) modules (M, δ):

δ=

M M

A

, = , = .

Example. Algebras inVec are algebras; modules are modules.

Example. Algebras inRep(G) and their modules ^Click .

The category of (right)A-modulesMod_C(A_M) is a leftC 2-representation.

Theorem (spread over several papers). Completeness. For every 2-simpleM there exists a simple algebra objectAM in(a quotient of)C (fiat)

such thatM∼=Mod_C(A_M). Non-redundancy. M ∼=N if and only if AM andAN are Morita–Takeuchi equivalent.

Example.

Simple algebra objects inVecare simple algebras.

Up to Morita–Takeuchi equivalence these are justC; andMod_Vec(C)∼=Vec. The above theorem is a vast generalization of this.

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µ=

A

A A

, ι=

1 A

, = , = = .

δ=

M M

A

, = , = .

Example. Algebras inRep(G) and their modules ^Click . The category of (right)A-modulesMod_C(A_M)

is a leftC 2-representation.

Theorem (spread over several papers). Completeness. For every 2-simpleM there exists a simple algebra objectAM in(a quotient of)C (fiat)

such thatM∼=Mod_C(A_M). Non-redundancy. M ∼=N if and only if AM andAN are Morita–Takeuchi equivalent.

Example.

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µ=

A

A A

, ι=

1 A

, = , = = .

δ=

M M

A

, = , = .

Theorem (spread over several papers).

Completeness. For every 2-simpleM there exists a simple algebra objectAM in(a quotient of)C (fiat)

such thatM∼=Mod_C(A_M).

Non-redundancy. M ∼=N if and only if AM andAN are Morita–Takeuchi equivalent.

Example.

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µ=

A

A A

, ι=

1 A

, = , = = .

δ=

M M

A

, = , = .

Theorem (spread over several papers).

Completeness. For every 2-simpleM there exists a simple algebra objectAM in(a quotient of)C (fiat)

such thatM∼=Mod_C(A_M).

Non-redundancy. M ∼=N if and only if AM andAN are Morita–Takeuchi equivalent.

Example.

Up to Morita–Takeuchi equivalence these are justC; andMod_Vec(C)∼=Vec.

The above theorem is a vast generalization of this.

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Example (Rep(G)).

I LetC =Rep(G) (G a finite group).

I C is monoidal and finitary (and fiat). For anyM,N∈C, we haveM⊗N∈C: g(m⊗n) =gm⊗gn

for allg ∈G,m∈M,n∈N. There is a trivial representation1.

I The regular 2-representationM:C →End(C):

M //

f

M⊗

f⊗

N //N⊗

.

I The decategorification is aN-representation, the regular representation.

I The associated algebra object isA_M =1∈C.

Theorem (folklore?).

Completeness. All 2-simples ofRep(G) are of the formV(K, ψ). Non-redundancy. We haveV(K, ψ)∼=V(K⁰, ψ⁰)

⇔

the subgroups are conjugate orψ⁰=ψ^g, whereψ^g(k,l) =ψ(gkg⁻¹,glg⁻¹). Note thatRep(G) has only finitely many 2-simples.

This is no coincidence.

Theorem (Etingof–Nikshych–Ostrik ∼2004); the group-like case. IfC is fusion (fiat and semisimple),

then it has only finitely many 2-simples. This is false if one drops the semisimplicity. ^Example

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Example (Rep(G)).

I LetK ⊂G be a subgroup.

I Rep(K) is a 2-representation of Rep(G), with action Res^G_K⊗ :Rep(G)→End(Rep(K)) which is indeed a 2-action because Res^G_K is a⊗-functor.

I The decategorifications areN-representations.

I The associated algebra object isA_M =Ind^G_K(1K)∈C.

⇔

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Example (Rep(G)).

I Letψ∈H²(K,C^∗). LetV(K, ψ) be the category of projectiveK-modules with Schur multiplierψ,i.e. vector spacesVwithρ: K → End(V) such that

ρ(g)ρ(h) =ψ(g,h)ρ(gh), for allg,h∈K. I Note thatV(K,1) =Rep(K) and

⊗:V(K, φ)V(K, ψ)→ V(K, φψ).

I V(K, ψ) is also a 2-representation ofC =Rep(G):

Rep(G) V(K, ψ) ^Res

G KId

−−−−−−→ Rep(K) V(K, ψ)−→ V^⊗ (K, ψ).

I The decategorifications areN-representations. ^Example

I The associated algebra object isA_M =Ind^G_K(1K)∈C, but withψ-twisted multiplication.

⇔

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Example (Rep(G)).

⊗:V(K, φ)V(K, ψ)→ V(K, φψ).

G KId

−−−−−−→ Rep(K) V(K, ψ)−→ V^⊗ (K, ψ).

Completeness. All 2-simples ofRep(G) are of the formV(K, ψ).

Non-redundancy. We haveV(K, ψ)∼=V(K⁰, ψ⁰)

⇔

the subgroups are conjugate orψ⁰=ψ^g, whereψ^g(k,l) =ψ(gkg⁻¹,glg⁻¹).

Note thatRep(G) has only finitely many 2-simples. This is no coincidence.

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Example (Rep(G)).

⊗:V(K, φ)V(K, ψ)→ V(K, φψ).

G KId

−−−−−−→ Rep(K) V(K, ψ)−→ V^⊗ (K, ψ).

⇔

Note thatRep(G) has only finitely many 2-simples.

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Example (Rep(G)).

⊗:V(K, φ)V(K, ψ)→ V(K, φψ).

G KId

−−−−−−→ Rep(K) V(K, ψ)−→ V^⊗ (K, ψ).

⇔

Note thatRep(G) has only finitely many 2-simples.

Theorem (Etingof–Nikshych–Ostrik∼2004); the group-like case.

IfC is fusion (fiat and semisimple), then it has only finitely many 2-simples.

This is false if one drops the semisimplicity. ^Example

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Clifford, Munn, Ponizovski˘ı, Green∼1942++. ^Semigroups

Write X≤LYifYis a direct summand ofZXforZ∈C,i.e. Y⊂⊕ZX. X∼LYif X≤LYandY≤LX. ∼L partitions C into left cells L. Similarly for rightR, two-sided cellsJ or 2-modules.

An apex is a maximal two-sided cell not annihilating a 2-module.

Fact (Chan–Mazorchuk∼2016). Any 2-simple has a unique apex.

Mackaay–Mazorchuk–Miemietz–Zhang∼2018. For any fiat 2-category C (semigroup-like) there exists a fiat 2-subcategoryAH (almost group-like) such that

(2-simples ofC with apexJ

)

one-to-one

←−−−−→

(2-simples of AH

with apexH ⊂ J )

Catch. In generalAHis not fusion.

Example (group-like).

Fusion categories,e.g. Rep(G), have only one cell. AH is everything. Example (semigroup-like).

LetRep(G,K) forKbeing of prime characteristic. The projectives form a two-sided cell. AH can be complicated.

Example (Kazhdan–Lusztig∼1979, Soergel∼1990). Soergel bimodulesS(Sn) for the symmetric group have cells coming from the Robinson–Schensted correspondence.

AH has one indecomposable object, but is not fusion. Example (Taft algebraT2).

T2-Modhas two cells – the lowest cell containing the trivial representation; the biggest containing the projectives.

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)

one-to-one

←−−−−→

(2-simples of AH

with apexH ⊂ J )

Fusion categories,e.g. Rep(G), have only one cell. AH is everything.

Example (semigroup-like).

LetRep(G,K) forKbeing of prime characteristic. The projectives form a two-sided cell. AH can be complicated.

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)

one-to-one

←−−−−→

(2-simples of AH

with apexH ⊂ J )

LetRep(G,K) forKbeing of prime characteristic.

The projectives form a two-sided cell. AH can be complicated.

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)

one-to-one

←−−−−→

(2-simples of AH

with apexH ⊂ J )

Example (Kazhdan–Lusztig∼1979, Soergel∼1990).

Soergel bimodulesS(Sn) for the symmetric group have cells coming from the Robinson–Schensted correspondence.

AH has one indecomposable object, but is not fusion.

Example (Taft algebraT2).

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)

one-to-one

←−−−−→

(2-simples of AH

with apexH ⊂ J )

Example (Kazhdan–Lusztig∼1979, Soergel∼1990).

Soergel bimodulesS(Sn) for the symmetric group have cells coming from the Robinson–Schensted correspondence.

AH has one indecomposable object, but is not fusion.

Example (Taft algebraT2).

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Let Γ be a Coxeter graph.

Artin ∼1925, Tits ∼1961++. The Artin–Tits group and its Coxeter group quotient are given by generators-relations:

AT =hb_i | · · ·b_ib_jb_i

| {z }

mijfactors

=· · ·b_jb_ib_j

| {z }

mij factors

i

W =hs_i |s²= 1,· · ·s_is_js_i

| {z }

mijfactors

=· · ·s_js_is_j

| {z }

mijfactors

i .

Generalize classical braid groups, or ^generalize polyhedron groups, respectively.

His the quotient ofZ[v,v⁻¹]AT by the quadratic relations,e.g.

− = (v−v⁻¹) .

Fact (Kazhdan–Lusztig∼1979, Soergel–Elias–Williamson∼1990,2012). H has a distinguished basis, called the ^{KL basis} , which is a decategorification of

Question. What can one say about simples ofHusing KL cells?

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Let Γ be a Coxeter graph.

Artin ∼1925, Tits ∼1961++. The Artin–Tits group and its Coxeter group quotient are given by generators-relations:

AT =hb_i | · · ·b_ib_jb_i

| {z }

mijfactors

=· · ·b_jb_ib_j

| {z }

mij factors

i

W =hs_i |s²= 1,· · ·s_is_js_i

| {z }

mijfactors

=· · ·s_js_is_j

| {z }

mijfactors

i .

Generalize classical braid groups, or ^generalize polyhedron groups, respectively.

His the quotient ofZ[v,v⁻¹]AT by the quadratic relations,e.g.

− = (v−v⁻¹) .

Fact (Kazhdan–Lusztig∼1979, Soergel–Elias–Williamson∼1990,2012). H has a distinguished basis, called the ^{KL basis} , which is a decategorification of

Question. What can one say about simples ofHusing KL cells?

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Example (type B₂).

W =hs,t|s²=t²= 1,tsts=ststi. Number of elements: 8. Number of cells: 3, named 0 (lowest) to 2 (biggest).

Cell order:

0 1 2

Size of the cells:

cell 0 1 2 size 1 6 1

Cell structure:

s,sts st ts t,tst

1

w0

number of elements

−−−−−−−−−−−→ ²₁ ¹₂

1

Example (SAGE). 1·1 = 1. Example (SAGE). cs·cs=(1+bigger powers)cs. csts·cs=(1+bigger powers)csts.

csts·csts=(1+bigger powers)cs+higher cell elements. csts·ctst=(bigger powers)cst+ higher cell elements.

Example (SAGE). cw0·cw0=(1+bigger powers)cw0.

Fact (Lusztig∼1984++). For any Coxeter groupW there is a well-defined function

a:W →N

which is constant on two-sided cells.

Big example

Idea (Lusztig ∼1984).

Ignore everything except the leading coefficient of the classical KL basis shifted bya(two-sided cell).

Those shifted versions are what I denote bycw.

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Cell order:

0 1 2

Size of the cells:

cell 0 1 2 size 1 6 1

Cell structure:

s,sts st ts t,tst

1

w0

number of elements

−−−−−−−−−−−→ ²₁ ¹₂

1

1 Example (SAGE).

1·1 = 1.

Example (SAGE). cs·cs=(1+bigger powers)cs. csts·cs=(1+bigger powers)csts.

a:W →N

Big example

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Cell order:

0 1 2

Size of the cells:

cell 0 1 2 size 1 6 1

Cell structure:

s,sts st ts t,tst

1

w0

number of elements

−−−−−−−−−−−→ ²₁ ¹₂

1

1 Example (SAGE).

1·1 = 1.

Example (SAGE).

cs·cs=(1+bigger powers)cs. csts·cs=(1+bigger powers)csts.

csts·csts=(1+bigger powers)cs+higher cell elements.

csts·ctst=(bigger powers)cst+ higher cell elements.

a:W →N

Big example

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Cell order:

0 1 2

Size of the cells:

cell 0 1 2 size 1 6 1

Cell structure:

s,sts st ts t,tst

1

w0

number of elements

−−−−−−−−−−−→ ²₁ ¹₂

1

1 Example (SAGE).

1·1 = 1.

Example (SAGE).

cs·cs=(1+bigger powers)cs. csts·cs=(1+bigger powers)csts.

csts·csts=(1+bigger powers)cs+higher cell elements.

csts·ctst=(bigger powers)cst+ higher cell elements.

Example (SAGE).

cw₀·cw₀=(1+bigger powers)cw₀.

a:W →N

Big example

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Cell order:

0 1 2

Size of the cells:

cell 0 1 2 size 1 6 1

Cell structure:

s,sts st ts t,tst

1

w0

number of elements

−−−−−−−−−−−→ ²₁ ¹₂

1

Fact (Lusztig∼1984++).

For any Coxeter groupW there is a well-defined function

a:W →N

Big example

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Cell order:

0 1 2

Size of the cells:

cell 0 1 2 size 1 6 1

Cell structure:

s,sts st ts t,tst

1

w0

number of elements

−−−−−−−−−−−→ ²₁ ¹₂

1

For any Coxeter groupW there is a well-defined function

a:W →N

Big example

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The asymptotic limit A0(W) ofHv(W) is defined as follows.

As a freeZ-module:

A₀(W) =L

J Z{a_w|w ∈ J }. vs. H_v(W) =Z[v,v⁻¹]{c_w |w ∈W}.

Multiplication.

a_xa_y =P

z∈J γ_x,y^z a_z. vs. c_xc_y =P

z∈J v^a(z)h_x,y^z c_z+ bigger friends.

where γ_x^z_,y ∈Nis the leading coefficient ofh^z_x,y ∈N[v,v⁻¹].

Example (typeB2).

The multiplication tables (empty entries are 0 and [2] = 1 +v²) in 1:

as asts ast at atst ats

as as asts ast

asts asts as ast

ats ats ats at+atst

at at atst ats

atst atst at ats

ast ast ast as+asts

cs csts cst ct ctst cts

cs [2]cs [2]csts [2]cst cst cst+cw0 cs+csts

csts [2]csts [2]cs+ [2]²cw₀ [2]cst+ [2]cw₀ cs+csts cs+ [2]²cw₀ cs+csts+ [2]cw₀

cts [2]cts [2]cts+ [2]cw0 [2]ct+ [2]ctst ct+ctst ct+ctst+ [2]cw0 2cts+cw0

ct cts cts+cw0 ct+ctst [2]ct [2]ctst [2]cts

ctst ct+ctst ct+ [2]²cw0 ct+ctst+ [2]cw0 [2]ctst [2]ct+ [2]²cw0 [2]cts+ [2]cw0

cst cs+csts cs+csts+ [2]cw0 2cst+cw0 [2]cst [2]cst+ [2]cw0 [2]cs+ [2]csts

(Note the “subalgebras”.) The asymptotic algebra is much simpler!

Big example

Fact (Lusztig∼1984++). A0(W) =L

JA^J₀(W) with theaw basis and all its summandsA^J₀(W) =Z{aw|w ∈ J }

are multifusion algebras. (Group-like.)

Multifusion algebras = decategorifications of multifusion categories. Surprising fact 1 (Lusztig∼1984++).

It seems one throws almost away everything, but: There is an explicit embedding

Hv(W),→A0(W)⊗ZZ[v,v⁻¹]

which is an isomorphism after scalar extension toC(v). Surprising fact 2 –H-cell-theorem (Lusztig∼1984++).

There is an explicit one-to-one correspondence

{simples ofHv(W) with apexJ }←−−−−→ {simples of^one-to-one A^H₀ (W)}.

Example

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As a freeZ-module:

A₀(W) =L

Multiplication.

a_xa_y =P

Example (typeB2).

as as asts ast

asts asts as ast

ats ats ats at+atst

at at atst ats

atst atst at ats

ast ast ast as+asts

Fact (Lusztig∼1984++). A0(W) =L

JA^J₀(W) with theaw basis and all its summandsA^J₀(W) =Z{aw|w ∈ J }

Multifusion algebras = decategorifications of multifusion categories. Surprising fact 1 (Lusztig∼1984++).

It seems one throws almost away everything, but: There is an explicit embedding

Hv(W),→A0(W)⊗ZZ[v,v⁻¹]

Example

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As a freeZ-module:

A₀(W) =L

Multiplication.

a_xa_y =P

Example (typeB2).

as as asts ast

asts asts as ast

ats ats ats at+atst

at at atst ats

atst atst at ats

ast ast ast as+asts

Big example

A0(W) =L

JA^J₀(W) with theaw basis and all its summandsA^J₀(W) =Z{aw|w∈ J }

Multifusion algebras = decategorifications of multifusion categories.

Surprising fact 1 (Lusztig∼1984++). It seems one throws almost away everything, but:

There is an explicit embedding Hv(W),→A0(W)⊗ZZ[v,v⁻¹]

Example