What is...2-representation theory?

(1)

What is...2-representation theory?

Or: Why do I care?

Daniel Tubbenhauer

Daniel Tubbenhauer What is...2-representation theory? September 2020 1 / 6

(2)

Representation theory is group theory in vector spaces LetCbe a group, an algebraetc.

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

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

withVbeing some vector space. (Called modules or representations.)

Basic question: Try to develop a reasonable theory of such actions.

Examples.

I Weyl∼1923++. The representation theory of (semi)simple Lie groups.

I Noether∼1928++. The representation theory of finite-dimensional algebras.

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.

(3)

2-representation theory is group theory in categories

LetCbe a reasonable 2-category.

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

M:C −→End(V),

withVbeing some finitary category. (Called 2-modules or 2-representations.)

Basic question: Try to develop a reasonable theory of such 2-actions.

Examples.

I Chuang–Rouquier & Khovanov–Lauda style. The 2-representation theory of (semi)simple Lie groups. Another time.

I Abelian ∼2000++ or additive∼2010++. The 2-representation theory of finite-dimensional algebras. Today.

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.

(4)

2-representation theory is group theory in categories

LetCbe a reasonable 2-category.

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

M:C −→End(V),

withVbeing some finitary category. (Called 2-modules or 2-representations.)

Basic question: Try to develop a reasonable theory of such 2-actions.

Examples.

I Chuang–Rouquier & Khovanov–Lauda style. The 2-representation theory of (semi)simple Lie groups. Another time.

I Abelian ∼2000++ or additive∼2010++. The 2-representation theory of finite-dimensional algebras. Today.

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.

(5)

Abelian vs. additive a.k.a. “What are the elements?”.

Finite tensor categories—the abelian world.

I Elements are simple objects. Finite means finitely many of these.

I What acts are finite multitensor categoriesC,i.e. finite abelian,K-linear, rigid (without duality all hope is lost) monoidal categories, with⊗:C ×C →C being bilinear.

I We act on finite abelian,K-linear categoriesV, with the 2-action

⊗:C ×V→Vbeing bilinear and biexact.

I The abelian Grothendieck groups are finite-dimensional algebras or finite-dimensional modules of such, respectively.

Examples.

I Finite-dimensional vector spaces, or any fusion category(fusion=finite tensor+semisimple).

I Modules of finite groups, or more generally, of finite-dimensional Hopf algebras.

I We see examples of 2-modules momentarily.

Finite additive means additive finitely many indecomposables finite-dimensional hom-spaces

Krull–Schmidt.

Abelian and additive run in parallel,

but additive is harder,e.g. no version of Schur’s 2-lemma.

Another point why additive is harder.

(6)

Fiat 2-categories—the additive world.

I Elements are indecomposable objects. Finite means finitely many of these.

I What acts are multifiat categoriesC,i.e. finite additive,K-linear, rigid (without duality all hope is lost) monoidal categories, with⊗:C ×C →C being bilinear.

I We act on finite additive,K-linear categoriesV, with the 2-action

⊗:C ×V→Vbeing bilinear.

I The additive Grothendieck groups are finite-dimensional algebras or finite-dimensional modules of such, respectively.

Examples.

I Finite-dimensional vector spaces, or any fusion category (fusion=fiat+semisimple).

I Modules of finite groups of finite representation type, or more generally, of finite-dimensional Hopf algebras of finite representation type.

I Projective/injective modules of finite groups of finite representation type, or more generally, of finite-dimensional Hopf algebras.

Krull–Schmidt.

(7)

Fiat 2-categories—the additive world.

I Elements are indecomposable objects. Finite means finitely many of these.

I What acts are multifiat categoriesC,i.e. finite additive,K-linear, rigid (without duality all hope is lost) monoidal categories, with⊗:C ×C →C being bilinear.

I We act on finite additive,K-linear categoriesV, with the 2-action

⊗:C ×V→Vbeing bilinear.

I The additive Grothendieck groups are finite-dimensional algebras or finite-dimensional modules of such, respectively.

Examples.

I Finite-dimensional vector spaces, or any fusion category (fusion=fiat+semisimple).

I Modules of finite groups of finite representation type, or more generally, of finite-dimensional Hopf algebras of finite representation type.

I Projective/injective modules of finite groups of finite representation type, or more generally, of finite-dimensional Hopf algebras.

Krull–Schmidt.

(8)

Take your favorite theorem and categorify it.

Some facts run in parallel, e.g.

the regular module M: C −→

End(V),a7→a·

the regular 2-module M:C −→

End(V),M7→M⊗ simples (no non-trivial C-stable

subspace) and Jordan–H¨older

2-simples (no non-trivialC-stable ideal) and 2-Jordan–H¨older double centralizer theorem, i.e.

C ∼= End_End_C_(V)(V) for V being faithful.

2-double centralizer theorem,i.e.

C ∼=End_E_nd_C_(V)(V) forVbeing 2-faithful. (Theorem 2020) Some do not, e.g.

Schur’s lemma, i.e. hom-spaces between simples are trivial

hom-spaces between 2-simples can be arbitrary complicated there are finitely many simples there can be∞many 2-simples

(9)

Example (Rep(G), ground fieldC).

I LetC =Rep(G), forG being a finite group.

I C is fusion: For anyM,N∈C, we haveM⊗N∈C: g(m⊗n) =gm⊗gn for allg ∈G,m∈M,n∈N. There is a trivial module1.

I The regular 2-moduleM:C →End(C):

M //

f

M⊗

f⊗

N //N⊗

.

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

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. ^Example

This is no coincidence.

Theorem (Etingof–Nikshych–Ostrik∼2004). IfC is fusion (fiat and semisimple),

then it has only finitely many 2-simples. This is false if one drops semisimplicity. Theorem (2020).

The non-semisimple, non-abelian Hecke category has only finitely many 2-simples.

(10)

I LetK ⊂G be a subgroup.

I Rep(K) is a 2-module ofRep(G), with 2-action

Res^G_K⊗ :Rep(G)→End Rep(K) ,

M //

f

Res^G_K(M)⊗

Res^G_K(f)⊗

N //Res^G_K(N)⊗ .

which is indeed a 2-action because Res^G_K is a⊗-functor.

I The decategorifications areN-modules.

⇔

(11)

I Letψ∈H²(K,C^∗). LetV(K, ψ) be the category of projectiveK-modules with Schur multiplierψ,i.e. a 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-module ofC =Rep(G):

Rep(G) V(K, ψ) ^Res

G KId

−−−−−−→Rep(K) V(K, ψ)−^⊗→V(K, ψ).

⇔

(12)

⊗: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. ^Example This is no coincidence.

(13)

⊗:V(K, φ)V(K, ψ)→V(K, φψ).

G KId

−−−−−−→Rep(K) V(K, ψ)−^⊗→V(K, ψ).

⇔

(14)

⊗:V(K, φ)V(K, ψ)→V(K, φψ).

G KId

−−−−−−→Rep(K) V(K, ψ)−^⊗→V(K, ψ).

⇔

Theorem (Etingof–Nikshych–Ostrik∼2004).

IfC is fusion (fiat and semisimple),

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

Theorem (2020).

(15)

Why2-representation theory?

Or: Representation theory of the 21th century!?

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer Why2-representation theory? April 20201 / 6

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

There is still much to do...

Thanks for your attention!

(16)

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

Daniel Tubbenhauer

April 2020

There is still much to do...

Thanks for your attention!

(17)

Example. Rep(Z/5Z) in characteristic 5.

B Indecomposables correspond to Jordan blocks ofF5[X]/(X⁵)∼=F5(Z/5Z):

Z₁!X 7→(0), Z₂!X 7→(^{0 1}_{0 0}), Z₃!X 7→_{0 1 0}

0 0 1 0 0 0

,

Z₄!X 7→

_{0 1 0 0}

0 0 1 0 0 0 0 1 0 0 0 0

, Z₅!X 7→

0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0

! .

⇒Rep(Z/5Z) has five elements as an additive category.

B OnlyZ₁is simple⇒Rep(Z/5Z) has only one element as an abelian category.

B OnlyZ₅is projective⇒Proj(Z/5Z) =Inj(Z/5Z) has one element as an additive category, andProj(Z/5Z) not abelian.

In characteristic6= 5 we haveRep(Z/5Z) =Proj(Z/5Z) =Inj(Z/5Z) and there is no difference between tensor (abelian) and fiat (additive).

Back

(18)

For example, forRep(S5) we have:

K 1 Z/2Z Z/3Z Z/4Z (Z/2Z)² Z/5Z S₃ Z/6Z D₄ D₅ A₄ D₆ GA(1,5) S₄ A₅ S₅

# 1 2 1 1 2 1 2 1 1 1 1 1 1 1 1 1

H² 1 1 1 1 Z/2Z 1 1 1 Z/2Z Z/2Z Z/2Z Z/2Z 1 Z/2Z Z/2Z Z/2Z

rk 1 2 3 4 4,1 5 3 6 5,2 4,2 4,3 6,3 5 5,3 5,4 7,5

Rep(S5)

This is completely different from their classical representation theory ofS₅.

Back