= The sl webalgebra

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The sl

₃

web algebra

Daniel Tubbenhauer

Joint work with Marco Mackaay and Weiwei Pan

09.10.2012

=

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1 Introduction Categorification The rough idea

2 Kuperberg’s sl₃-webs Basic definitions

Representation theory of U_q(sl₃)

3 The sl₃ web algebra Basic definitions Thesl₃ web algebra KS

Its Grothendieck groupK₀(K_S)

4 Properties of the sl₃ web algebra Frobenius structure

The centerZ(KS) The algebra is cellular

Daniel Tubbenhauer Thesl₃web algebra 09.10.2012 2 / 59

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A well-known example

Noether, Hopf, Mayer

Let X be a reasonable finite-dimensional spaces. Then the homology groups H_k(X) are a categorification of the Betti numbers ofX and the singular chain complex (C,di) is categorification of the Euler characteristic of X.

Note the following common features of the two examples above.

The Betti numbers and the Euler characteristic can be seen as parts of “bigger, richer”structures.

In both categorifications it isvery easy to “decategorify”, i.e. by taking the dimension or the alternating sum of the dimensions.

Both notions arenot obvious, e.g. the first notion of “Betti numbers”

was in the year 1857 (B. Riemann!!) and the first notion of

“homology groups” was in the year 1925.

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The ladder of categories

...

forget 3-arrows

... ...

forget 2-arrows

2−categories = “arrows between arrows”

add 3-arrows

XX

forget arrows

1−categories = usual categories

add 2-arrows

ZZ

size

0−categories = sets, vector spaces

add arrows

ZZ

−1−categories = cardinals, numbers

internal structure

ZZ

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Categorification is ill-defined

Note that the notion of categorification is ill-defined. The rough idea is to replace set theoretical structures by category theoretical structures. So categorification could mean

An inverse process of somedecategorification, e.g.

Degroupoidification (Baez, Dolan, Trimble): a functor D: Span(Gpd)→Hilb.

Grothendieck groupC 7→K0(C) constructions (Khovanov, Lauda).

DimensionV 7→dim(V) constructions (homology groups).

more...

Common feature: decategorification iseasy, categorification ishard.

Reveals hidden structure.

Today we use decategorification = Grothendieck group.

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Webs and foams

If you live in a two-dimensional world, then it is easy to imagine a one-dimensional world, but hard to imagine a three-dimensional world!

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The rough idea

The classicalpicture.

sln-webs oo Intertwiners

//

Kauffman,Kuperberg,MOY❘❘❘❘❘

((

❘❘

❘

U_q(sl_n)-Tensors

Reshetikhin,Turaev❥❥❥❥❥

tt❥❥❥❥❥❥

sln-knot polynomials And its categorification.

sln-foams^oo ^??? ^//

Khovanov❘❘❘,Khovanov❘❘ −Rozansky

((

❘❘

❘

sln-string diagrams

Webster❥❥❥❥❥❥

tt

❥❥❥❥❥❥

sln-knot homologies

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Basic definitions

Example

Aweb is an oriented trivalent graph such that any vertex is either a sink or a source. Anyweb can be obtained by gluing and disjoint union of some basic webs.

The boundary of a web corresponds to a sign stringS, i.e. +, if the orientation is pointing in, and−otherwise. The sign string for the example is S = (+ +−+−−).

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Basic definitions

Definition(Kuperberg)

The web spaceW_S for a given sign stringS is WS =C(q){w |∂w =S}/IS, where I_S is generated by the relations

= [3]

= [2]

= +

a a

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Basic definitions

Example

Webs can becoloured with flows: Define a flowf on a webw to be an oriented subgraph that contains exactly two of the three edges incident to each trivalent vertex. The connected components are called the flow lines.

At the boundary, the flow lines can be represented by a state string J. By convention, at the i-th boundary edge, we setji = +1 if the flow line is oriented upward, j_i =−1 if the flow line is oriented downward and j_i = 0 there is no flow line. SoJ = (0,0,0,0,0,−1,1) in the example.

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Basic definitions

Given a web with a flow, denoted w_f attribute a weight to each trivalent vertex and each arc in wf. The total weight ofwf is by definition the sum of the weights at all trivalent vertices and arcs.

wt=0 wt=0 wt=0 wt=-1 wt=-1 wt=-1

wt=0 wt=0 wt=1 wt=1 wt=1

wt=0

wt=0 wt=-1 wt=-2 wt=0 wt=-1 wt=-2

The total weight from the example before is −3.

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Representation theory of U

_q

(sl

3

)

A sign string S = (s1, . . . ,s_n) corresponds to V_S =V_s₁⊗ · · · ⊗V_s_n,

where V₊ is the fundamental representation andV₋∼=V₊∧V₊ its dual.

Webs correspond to intertwiners.

Theorem(Kuperberg)

WS ∼=Hom(C(q),VS)∼=Inv(VS)

The basis ofW_S, denotedB_S, is called web basisof Inv(VS). From the relations before, it follows that the webs of BS arenon-elliptic webs, i.e.

without circles, digons or squares.

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Representation theory of U

_q

(sl

3

)

Theorem(Khovanov, Kuperberg)

A pair of a sign string S = (s1, . . . ,s_n) and a state string J = (j1, . . . ,j_n) correspond to the coefficients of the web basis relative to thestandard basis {e₋₁^± ,e₀^±,e₊₁^± }of V_±.

Example

weight =−1 weight =−3

The basis web wS has a decomposition

w_S =· · · −(q⁻¹+q⁻³)(e₀⁺⊗e₀⁻⊗e₀⁺⊗e₀⁻⊗e₀⁺⊗e₋₁⁺ ⊗e₊₁⁺ )± · · · .

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Please, fasten your seat belts!

Let’s categorify everything!

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sl

₃

-foams

A pre-foamis a cobordism with singular arcs between two webs. Pre-foam composition consists of placing one pre-foam ontop of the other. The orientation of the singular arcs is, by convention, as in the diagrams below (called the zipand theunziprespectively):

We allow pre-foams to havedots that can movefreelyabout the facet on which they belong, but we donot allow dot to cross singular arcs.

A foamis a formal C-linear combination of isotopy classes of pre-foams modulo the following relations.

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The foam relations ℓ = (3 D , NC , S , Θ)

= 0 (3D)

=− − − (NC)

= = 0, =−1 (S)

α β

δ =







1, (α, β, δ) = (1,2,0) or a cyclic permutation,

−1, (α, β, δ) = (2,1,0) or a cyclic permutation, 0, else.

(Θ)

The relationsℓ= (3D,NC,S,Θ) sufficeto evaluate any closed foam!

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Just to frighten you: more relations

From the relationsℓ followa lot of identities.

=− (Bamboo)

= − (RD)

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And more relations

= 0 (Bubble)

= − (DR)

=− − (SqR)

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And even more relations

+ + = 0

= 0

(Dot Migration)

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The sl

₃

-foam category

Let Foam₃ be thecategory of foams, i.e. objects are webs andmorphisms are foams between webs.

The category is gradedby theq−degree of a foamF q(F) =χ(∂F)−2χ(F) + 2d +b,

where d is the number of dots andb is the number of vertical boundary components.

Example

The q-degrees are 2, 1 and 0 respectively.

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Closed webs and foams

Definition

There is an involution^∗ on the webs.

w

w*

A closed webis defined by closing of two webs.

u v*

A closed foamis a foam from ∅to a closed web.

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Foam homology

Definition

The foam homologyof a closed web w is defined by F(w) =Foam₃(∅,w).

F(w) is a graded complex vector space, whose q-dimension can be computed by the Kuperberg bracket:

Dw ∐ E

= [3]hwi,

h i= [2]h i,

=

+D E

.

The relations above correspond to the decompositionofF(w) into direct summands.

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The sl

₃

web algebra

Definition(MPT)

Let S = (s₁, . . . ,s_n). The sl₃ web algebra K_S is defined by K_S = M

u,v∈BS

uK_v,

with

uKv :=F(u^∗v){n}.

Multiplication is defined as follows:

uK_v₁⊗v₂K_w →uK_w

is zero, if v₁ 6=v₂. Ifv₁ =v₂, use themultiplication foamm_v, e.g.

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The sl

₃

web algebra

v w*

v v*

w*

v

Proposition(MPT)

The multiplication is associative and unital. The multiplication foamm_v only depends on the isotopy type ofv and hasq-degree n. Hence, KS is a finite dimensional, unital and graded algebra.

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En(c)hanced sign strings

Definiton

An enhanced sign sequence is a sequenceS = (s₁, . . . ,sn) with s_i ∈ {◦,−,+,×}, for alli = 1, . . . ,n. The corresponding weight µ=µS ∈Λ(n,d) is given by the rules

µi =











0, ifsi =◦, 1, ifs_i = 1, 2, ifs_i =−1, 3, ifs_i =×.

Let Λ(n,d)₃⊂Λ(n,d) be the subset of weights with entries between 0 and 3. For any enhanced sign string S, we defineSb by deleting the entries equal to◦ or ×.

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En(c)hanced sign strings

Moreover for n=d = 3^k we define

WS =W_S_b and BS =B_b_S and W₍₃k) = M

µs∈Λ(n,n)3

WS

one the levelof webs and on thelevel of foams, we define K_S =K_S_b and W₍₃^k₎= M

µs∈Λ(n,n)₃

K_S−pmod_gr.

I will sketch in the following how we obtain one of our main results as a corollary.

Corollary(MPT)

K₀(W₍₃^k₎)⊗Z[q,q⁻¹]C(q)∼=W₍₃^k₎.

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Skew Howe-duality

The natural actions of GL_k andGLn on

Alt^p(C^k ⊗Cⁿ) = Λ^p(C^k⊗Cⁿ) are Howe dual (skew Howe duality).

This impliesthat

Inv_SL_k(Λ^p¹(C^k)⊗ · · · ⊗Λ^pⁿ(C^k))∼=W(p₁, . . . ,p_n), where W(p1, . . . ,p_n) denotes the (p1, . . . ,p_n)-weight space of the irreducible GLn-module W(k^ℓ), if n=k^ℓ.

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Quantum groups

Definition

The algebra U_q(gl_n) is generated byK₁^±1, . . . ,K_n^±1 and E_±1, . . . ,E_±(n−1) subject to alonglist of relations.

The algebra U_q(sln)⊂U_q(gl_n) is generated byK_iK_i⁻¹₊₁ andE_±i. Their idempotented completions U˙_q(sl_n) and ˙U_q(gl_n), are defined by adjoining idempotents 1λ for any weight λ∈Zⁿ (andλ∈Zⁿ⁻¹ for the special linear group) subject to a longlist of relations.

Note that the idempotented complete version are much easier, e.g. it is much easier to write down a nice basis.

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A finite-dimensional semi-simple quotient

Lemma(Doty, Giaquinto)

Theq-Schur algebra S_q(n,d) is generated by 1_λ, for λ∈Λ(n,d), andE_±1, for i = 1, . . . ,n−1, such that

1_λ1_λ =δ_λ,µ1_λ, X

λ∈Λ(n,d)

1λ = 1,

E_±11_λ = 1_λ±α_iE_±1, E_iE_−j−E_−jE_i =δ_i_,j X

λ∈Λ(n,d)

[λ_i −λ_i+1] 1_λ.

It is finite-dimensional and semi-simple. It is known that S (n,n)1 /(µ >(3^ℓ))∼=V .

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The action

We definedan action φof S_q(n,n) on W₍₃ℓ) by

1λ 7→

λ1 λ2 λn

E_±i1_λ 7→

λ₁ λi−1 λi λi+1

λi±1 λi+1∓1

λi+2 λn

We use the convention that vertical edges labeled 1 are oriented upwards, vertical edges labeled 2 are oriented downwards and edges labeled 0 or 3 are erased. The hard part was to show that this iswell-defined.

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Exempli gratia

E₊₁1₍₂₂₎ 7→

2 2

3 1

E₋₂E₊₁1₍₁₂₁₎7→

1 2 1

2 0 2

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An instance of skew Howe-duality

Lemma

The action φgives riseto an isomorphism φ:V₍₃ℓ) →W₍₃ℓ)

of Sq(n,n)-modules.

Note that their are categorifications of ˙U_q(sl_n) and ˙U_q(gl_n), denoted as U(sln) andU(gl_n), by Khovanov and Lauda.

The idea now is to categorifythe whole process!

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Some work was already done!

Theorem(Mackaay, Stoˇsi´c, Vaz)

Define, similar to the uncategorified story, a 2-category S(n,d). Let S˙(n,d) be the Karoubi envelope ofS(n,d). Then

K₀( ˙S(n,d))⊗Z[q,q⁻¹]C(q)∼=S_q(n,d).

The following was conjectured by Khovanov and Lauda in 2008. Note that V =R_λ−pmod_gr for λ∈Λ(n,n)⁺ (the algebra R_λ is a quotient of S(n,d) and is called Khovanov-Lauda-Rouquieralgebra).

Theorem(Brundan-Kleshchev, Lauda-Vazirani, Webster, Kang-Kashiwara,...)

As ˙U_q(sln) we have

K (V )⊗ C(q)∼=V .

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The action (categorified)

We definedan action φof Sq(n,n) on W₍₃^ℓ₎ by

On objects its the aforementioned actionφof S_q(n,n) onW₍₃ℓ). On morphisms we do it, like before, on the generators.

Note that this time everything gets (categorification is “richer”,

remember?) more complicated, i.e. their are eleven completely different generators instead of two, their are way morerelations to check and the pictures are two-dimensional now.

Lets me give two of the definitions for the generators and one example one has to check.

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The signs are important!

i+1,i,λ

7→

λi λi+1 λi+2

i,λ

7→(−1)^⌈^λ²ⁱ^⌉+⌊^λⁱ⁺¹² ^⌋

λi λ_i+1

But until everything is checked, we get the very nice result that this action is well-defined.

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It’s horrible!

1 2 1

2 2 0

1 3

3 =

1 2 1

2 2 0

3

•

−

1 2 1

2 2 0

3

• .

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It’s harvest time!

By Rouquier’s universality theorem, afterpulling back the categorical action, we get

Theorem(MPT)

Let V beany idempotent complete category, which allows an integrable graded categorical action by U(sln) (plus some extra conditions). Then there exists an equivalence of categorical U(sln)-representations

Φ : V₍₃^k₎→ W₍₃^k₎, and therefore to V.

Note that we are using the sl₃ web algebra to obtain the result forU(sln)!

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It’s harvest time!

Checking all the definitions, we see that we have a commuting square of isomorphisms (bijective isometries even). Hence, we finally get our hands on K₀.

V₍₃k) γ₍₃k)

//

φ

K₀(V₍₃^k₎)⊗Z[q,q⁻¹]C(q)

K₀(Φ)

W₍₃k) ψ //K₀(W₍₃^k₎)⊗Z[q,q⁻¹]C(q)

Corollary(MPT)

K₀(W₍₃^k₎)⊗Z[q,q⁻¹]C(q)∼=W₍₃k).

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More to harvest!

The result above leads to the following theorem.

Theorem(MPT)

The two algebras R₃ℓ and K₃ℓ are Morita equivalent.

Note that Morita invariant properties can be check in both algebras now.

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A trace form on the sl

₃

web algebra

Definition

Their is a natural trace formon the algebraK_S. We take, by definition, the trace form

tr:K_S →C

to be zero on uKv, when u 6=v ∈B_S. For anyv ∈B_S, we define tr:vK_v →C

by closing any foam f_v with 1v, e.g.

v*

v 1v

fv

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It’s Frobenius!

The trace is non-degenerated andsymmetric. Both can be seen

geometrical, e.g. the fact that tr(gf) =tr(fg) holds follows from slidingf around the closure until it appears on the other side of g, e.g.

g

f 1v

1u

= ^f

g 1u

1v

The non-degenerate trace form onK_S gives riseto a graded (KS,K_S)-bimodule isomorphism K_S^∨ ∼=K_S{−2n}, i.e. we have Theorem(MPT)

For any sign stringS of length n, the algebra K_S is a graded, symmetric Frobenius algebra of Gorenstein parameter 2n.

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Tableaux and flows

Let p_S be the number of positive entries andn_S the number of negative entries of S. By definition, we have that d =p_S+ 2n_S. Our key idea is to reduce everything to the case where n_S = 0. Fix any state stringJ of length n, wedefine a new state stringbJ of length d by the following algorithm:

1 Let ₀bJ be the empty string.

2 For 1≤i ≤n, let_ibJ be the result of concatenating j_i to _i−1bJ if µ_i = 1. If µ_i = 2 then

1 concatenate (1,0) toi−1bJ ifji = 1,

2 concatenate (0,−1) toi−1bJ ifji=−1,

3 concatenate (1,−1) toi−1bJ ifji= 0.

1 0

1

0 -1

-1

1 -1

0

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Tableaux and flows

Proposition(MPT)

There is a bijectionbetween Col^λ_µ and the set of state stringsJ such that there exists a w ∈B_S and a flow f onw which extendsJ. The bijection is given by an algorithm.

Example

The tableau on the left gives rise to the web with flow next to it.

1 0 -1

2 4 1 3 2

7 4

6 5

0 0 0 0 0 1 -1

+ + + +

-

+ -

1^st 2^nd 4^th

3^rd

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The center of K

_S

Let X_µ^λ be the (λ, µ)-Spaltenstein variety. Note that, ifn_s = 0, then X_µ^λ=X^λ, the latter being the Springer fiber associated toλ.

Let P =C[x₁, . . . ,xd]. If µis the composition associated toS, then letSµ

be the corresponding parabolic subgroup of the symmetric group S_d and therefore letP^µ:=P^S^µ ⊂P be the subring of polynomials which are invariant under Sµ.

For a specific idealI_µ^λ letR_µ^λ :=P^µ/I_µ^λ. Brundan and Ostrik proved that H^∗(X_µ^λ)∼=R_µ^λ.

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The center of K

_S

We showed that R_µ^λ actsonK_S and that (as graded complex algebras) R_µ^λ1⊂Z(K_S).

By a dimension argument (based on Morita equivalence) we get Theorem(MPT)

H^∗(X_µ^λ) is isomorphic (as graded algebras) toZ(K_S). The dimension of the center is #Col^λ_µ, i.e. the center is parametrised by flows on the boundary line.

Since one can say that X_µ^λ “generalises” Schubert calculus, we say that Z(KS) “categorifies” a part of the calculations with symmetric

polynomials.

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Two motivating examples

Example

Let A=M_n×n(R), i.e. the set of n×n-matrices over R. Set P={∗} and T(∗) ={1, . . . ,n}. The standard basis ofA, i.e. the e_ij-matrices, has a very special property, namely that the coefficients for multiplication with a matrix from the rightonly depend on the rowi and vice versafor

multiplication from the left. Moreover, for i(M) =M^t, we havei(eij) =eji. Example

Let A=R[x]/(xⁿ) andi=id. Then set P={0, . . . ,n−1} and

T(k) ={1}. Then the standard basisc₁₁^k =x^k has a very special property, namely that the coefficients for multiplication onlydepends on higher powers of x (moduloxⁿ).

The idea of Graham and Lehrer was to “interpolate”between the two

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Cellular algebras

Definition(Graham, Lehrer)

SupposeA is a free algebra overR of finite rank. A cell datumis an order quadruple (P,T,C,i), where (P,⊲) is theweight poset,T(λ) is a finite set for all λ∈P,iis an involutionand an injection

C: a

λ∈P

T(λ)× T(λ)→A, (s,t)7→c_st^λ,

such that the c_st^λ form aR-basis ofAwithi(c_st^λ) =c_ts^λ and for alla∈A c_st^λa= X

u∈T(λ)

rtu(a)c_st^λ (modA^⊲^λ).

The c_st^λ are called acellular basisof A(with respect to the involutioni).

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K

_S

is cellular!

Note that the whole notions of cellularity can be generalised to the concept of graded cellularity. As mentioned before, we know that the algebras K₍₃k) andR₍₃k) are Morita equivalent. Hu and Mathas showed that latter is a graded cellular algebra. Moreover, K¨onig and Xi showed that cellularity is (up to some technicalities with the involutions) an invariant under Morita equivalence. Hence, we have:

Theorem(MPT)

The algebra K_S is a finite dimensional, graded cellular and symmetric Frobenius algebra.

Note that we don’t have a cellular basis at the moment (the proof of the invariance of cellularity isnot constructive), but we have a good candidate!

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The world of algebras

division algebras simple algebras semisimple algebras

symmetric algebras weakly symmetric algebras

Frobenius algebras selfinjective algebras

cellular algebras

KS

Ter ra per ic ol osa

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There is stillmuch to do...

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Thanks for your attention!