selfinjective algebras: finite and tame typeskowron/selfinjective2004.pdfproblem. determine the...
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SELFINJECTIVE ALGEBRAS:
FINITE AND TAME TYPE
Andrzej Skowronski
(Queretaro, August 2004)
(http://www.mat.uni.torun.pl/˜skowron/Selfinjective2004.pdf)
1. PRELIMINARIES
2. SELFINJECTIVE ALGEBRAS OF
POLYNOMIAL GROWTH
• Selfinjective algebras of finite type
• Domestic selfinjective algebras of
infinite type
• Nondomestic selfinjective algebras of
polynomial growth
3. TAME SYMMETRIC ALGEBRAS WITH
PERIODIC MODULES
4. TAME STANDARD SELFINJECTIVE
ALGEBRAS
0
1. PRELIMINARIES
K algebraically closed field
A finite dimensional K-algebra
modA category of finite dimensionalright A-modules
modA stable category of modA(modulo projectives)
Db(modA) derived category of boundedcomplexes over modA(triangulated category)
ΓA Auslander-Reiten quiver of AτA = DTr, τ−A = TrDAuslander-Reiten translations
ΓsA stable Auslander-Reiten quiverof A
A, B finite dimensional K-algebras ⇒A and B are:
Morita equivalent if modA ∼= modB
stably equivalent if modA ∼= modB
derived equivalent if Db(modA) ∼= Db(modB)
(as triangulated categories)
1
A finite dimensional K-algebra
A tame: ∀d≥1∃M1,...,MndK[x]-A-bimodules
such that
• Mi free left K[x]-modules of finite rank
• all but finitely many isoclasses of inde-
composable right A-modules of dimen-
sion d are of the form K[x]/(x− λ)⊗K[x]
Mi, 1 ≤ i ≤ nd, λ ∈ K
µA(d) = least number on K[x]-A-bimodules
satisfying the above condition for d
A tame =⇒
indd A =
finite dis-crete set
⋃µA(d) one-para-meter families
A is not tameDrozd
====⇒ representation theory
of A comprises the representation theories of
all finite dimensional K-algebras
2
Hierarchy of tame algebras:
A of finite type ⇐⇒ ∀d≥1
µA(d) = 0
A domestic ⇐⇒ ∃m≥1
∀d≥1
µA(d) ≤ m
A polynomial growth ⇐⇒ ∃m≥1
∀d≥1
µA(d) ≤ dm
A tame ⇐⇒ ∀d≥1
µA(d) <∞
Examples:
• hereditary algebras of Dynkin type
• hereditary algebras of Euclidean type
• tubular algebras
• tame generalized canonical algebras
Hierarchy of the tame algebras is preserved bythe Morita and stable equivalences (Krause,Krause-Zwara)
Open problem: A polynomial growth ⇒A linear growth ( ∃
m≥1∀d≥1
µA(d) ≤ md)
3
A finite dimensional K-algebra
D = HomK(−,K) standard duality of modA
A is selfinjective if AA∼= D(A)A
(projective A-modules are injective)
A is symmetric if AAA∼= AD(A)A
A basic algebra, then
A selfinjective ⇐⇒ A Frobenius
A is Frobenius (symmetric) if there is a
nondegenerate (symmetric) K-bilinear asso-
ciative form (−,−) : A×A → K
(a, bc) = (ab, c), a, b, c,∈ K
A is weakly symmetric if topP ∼= socP for
any indecomposable projective A-module P
Symmetric ⇒ weakly symmetric ⇒ selfinjective
⇑Frobenius
4
Examples of selfinjective algebras:
(1) Group algebras KG of finite groups G,
more generally, blocks of group algebras
(symmetric algebras)
(2) Restricted enveloping algebras
u(L) = U(L)/(xp − x[p], x ∈ L)
of restricted Lie algebras (L, [p]) in
characteristic p > 0, or more generally,
reduced enveloping algebras
u(L, χ) = U(L)/(xp−x[p]−χ(x)p ·1, x ∈ L)
of restricted Lie algebras (L, [p]), for
linear forms χ : L → K, dimK U(L, χ) =
pdimK L, u(L,0) = u(L) (are Frobenius
algebras: Farnsteiner-Strade)
(3) Finite dimensional Hopf algebras (are
Frobenius algebras: Larson-Sweedler)
5
(4) Hochschild extension algebras
0 → D(A) → T → A→ 0
(equivalence classes form the Hochschildcohomology group H2(A,D(A)))In particular, we have the trivial exten-sion T(A) = A D(A) of A by D(A)T(A) = A ⊕ D(A) as K-vector space(a, f) · (b, g) = (ab, ag+ fb)a, b ∈ A, f, g ∈ D(A).T(A) are symmetric algebras
A selfinjective algebra
P indecomposable projective A-module, thenwe have in modA an Auslander-Reitensequence of the form
0 → radP → (radP/ socP)⊕P → P/ socP → 0
Hence ΓsA is obtainded from ΓA by removingthe indecomposable projective modules andthe arrows attached to them
A selfinjectiveNakayama====⇒ socAA = socA =
socAAA, B selfinjective algebras
A and B are socle equivalent if
A/ socA ∼= B/ socB
6
PROBLEM. Determine the Morita
equivalence classes of the tame finite
dimensional selfinjective algebras
For selfinjective algebras,
Morita
equivalence⇒ derived
equivalenceRickard====⇒ stable
equivalence
In particular, the hierarchy of tame algebras
is also preserved by the derived equivalences
Hence, we have the related
PROBLEM′. Determine the derived
(respectively, stable) equivalence classes
of the tame finite dimensional selfinjec-
tive algebras
7
We may assume
algebra = basic, connected, finite dimen-
sional K-algebra
A algebra ⇒ A ∼= KQ/I
Q = QA Gabriel quiver of A, I admissibleideal in the path algebra KQ of Q
modA ∼= repK(Q, I)
tame (basic)
selfinjective
algebras
standard algebras(admit simply connected
Galois coverings
)
nonstandard algebras
Representation theory of tame standardselfinjective algebras can be reduced to therepresentation theory of tame algebras offinite global dimension (tame simply connec-ted algebras with nonnegaive Euler forms)
8
A connected K-category R is locally bounded
if:
• distinct objects of R are nonisomorphic
• ∀x∈obR
R(x, x) is a local algebra
• ∀x∈obR
∑y∈obR
(dimK R(x, y) + dimK R(y, x)) <∞
⇒ R ∼= KQ/I, Q locally finite connected quiver,
I admissible ideal of the path category KQ
modR category of finitely generated
contravariant functors R → modK
modR = repK(Q, I)
R bounded (has finitely many objects) ⇒⊕R =
⊕x,y∈obR
R(x, y) finite dimensional basic
connected K-algebra
We will identify a bounded K-category
R with the associated finite dimensional
algebra ⊕R9
R locally bounded K-category
G group of K-linear automorphisms of R
G is admissible if G acts freely on the objectsof R and has finitely many orbits
R/G orbit (bounded) category
objects: G-orbits of objects of R
(R/G)(a, b) =(fyx) ∈ ∏(x,y)∈a×b
R(x, y)∣∣∣ g · fyx = fg(y),g(x) ∀
g∈G,x∈a,y∈b
F : R → R/G canonical Galois covering
ob(R) x → Fx = G · x ∈ ob(R/G)
∀x∈obR
∀a∈ob(R/G)
F induces isomorphisms
⊕Fy=a
R(x, y)∼−→ (R/G)(Fx, a),
⊕Fy=a
R(y, x)∼−→ (R/G)(a, Fx)
10
The group G acts also on modR
modR M → gM = Mg−1 ∈ modR
We have also the push-down functor(Bongartz-Gabriel)
Fλ : modR −→ modR/G
M ∈ modR, a ∈ ob(R/G) ⇒ (FλM)(a) =⊕x∈a
M(x)
Assume G is torsion-free. Then Fλ inducesan injection (Gabriel)
G-orbits ofisoclasses of
indecomposablemodules in modR
Fλ
isoclasses of
indecomposablemodules inmodR/G
R is locally support-finite if for any x ∈ obR⋃M∈indRM(x) =0
supp(M) is a bounded category
R locally support-finiteDowbor-Skowronski
====⇒ Fλis dense
Then ΓR/G∼= ΓR/G (Gabriel)
11
R selfinjective locally bounded K-category
G admissible group of automorphisms of R
⇒ R/G basic connected finite dimensional
selfinjective K-algebra
B algebra
1B = e1 + · · · + en
e1, . . . , en orthogonal primitive
idempotents of B
B repetitive category of B
(selfinjective locally bounded K-category)
objects: em,i,m ∈ Z,1 ≤ i ≤ n
B(em,i, er,j) =
ejBei , r = m
D(eiBej) , r = m+ 10 , otherwise
ejBei = HomB(eiB, ejB), D(eiBej) = ejD(B)ei⊕(m,i)∈Z×1,...,n
B(−, er,j)(em,i) = ejB⊕D(Bej)
12
νB
: B → B Nakayama automorphism of B
νB(em,i) = em+1,i for all m, i ∈ Z × 1, . . . , n
(νB) admissible group of automorphisms of
B
FB : B → B/(νB) = T(B) Galois covering
ϕ automorphism of the K-category of B
ϕ is positive if for each pair (m, i) ∈ Z ×1, . . . , n we have ϕ(em,i) = ep,j for some
p ≥ m and j ∈ 1, . . . , n
ϕ is rigid if for each pair (m, i) ∈ Z×1, . . . , nexists j ∈ 1, . . . , n such that ϕ(em,i)=em,j
ϕ is strictly positive if it is positive but not
rigid
Note that νB
is strictly positive.
13
Assume B is triangular (QB has no oriented
cycles)
Then B is triangular
B is the full bounded subcategory of B given
by the objects
e0,i,1 ≤ i ≤ n
Let i be a sink of QBB → S+
i B reflection of B at i
S+i B the full subcategory of B given by the
objects
e0,j, 1 ≤ j ≤ n, j = i, and e1,i = νB(e0,i).
σ+i QB = Q
S+i B
reflection of QB at i
Observe that B ∼= S+i B
14
Reflection sequence of sinks of QB: a se-
quence i1, . . . , it of vertices of QB such that
is is a sink of σ+is−1
. . . σ+i1QB for 1 ≤ s ≤ t.
Two triangular algebras B and C are said to
be reflection equivalent if C ∼= S+it. . . S+
i1B
for a reflection sequence of sinks i1, . . . , it of
QB.
B, C reflection equivalent triangular algebras
⇒ B ∼= C
15
∆ 1α1→ 2
α2→ 3 → · · · → n− 1αn−1−→ n
B = K∆
B = K∆/In
In generated by all compositions
of n+ 1 consecutive arrows in
∆νB
: B → B Nakayama automor-
phism, νB(r, i) = (r+ 1, i)
(r, i) ∈ Z × 1, . . . , nϕ : B → B, ϕn = ν
BNnm = B/(ϕm) = KCm/Jm,n
Cm
m αm 1α1
m− 1
αm−1
2
α2
m− 2
αm−2
3
. . .
. . .
Jm,n generated by all composi-
tions of n + 1 consecutive
arrows in CmNnm Nakayama algebra, Nn
n = T(B)
Nnm symmetric ⇐⇒ m | n ⇐⇒
ϕm is a root of νB
∆ :
...
(2, n)α2,n
(1,1)α1,1
(1,2)α1,2...
(1, n− 1)α1,n−1
(1, n)α1,n
(0,1)α0,1
(0,2)α0,2...
(0, n− 1)α0,n−1
(0, n)α0,n
(−1,1)α−1,1
(−1,2)α−1,2...
16
∆1α1
2α2
3α3
0
B = K∆
B = K∆/In
In generated by
βm,iαm,i − βm,jαm,j,
αm,iβm−1,j,
m ∈ Z, i, j ∈ 1,2,3, i = j.
νB
: B → B Nakayama
νB(m, i) = (m+ 1, i)
: B → B
= ((m,1), (m,3))σ : B → B
σ = ((m,1), (m,2), (m,3))
∆ :
...
...
...
(2,0)β1,1
β1,2
β1,3
(1,1)
α1,1
(1,2)
α1,2
(1,3)
α1,3
(1,0)β0,1
β0,2
β0,3
(0,1)
α0,1
(0,2)
α0,2
(0,3)
α0,3
(0,0)β−1,1
β−1,2
β−1,3
(−1,1)
α−1,1
(−1,2)
α−1,2
(−1,3)
α−1,3
(−1,0)
... ... ...
A = T(B), A′ = B/(νB), A′′ = B/(σν
B)
17
A ∼= KQ/I, A′ ∼= KQ/I ′, A′′ ∼= KQ/I ′′,
Q :
2α2
0
β1
β2
β3
1
α1
3α3
I = 〈β1α1−β2α2, β2α2−β3α3, α1β2, α1β3, α2β1, α2β3, α3β1, α3β2〉I ′ = 〈β1α1−β2α2, β2α2−β3α3, α1β1, α1β2, α2β1, α2β3, α3β2, α3β3〉I ′′ = 〈β1α1−β2α2, β2α2−β3α3, α1β1, α1β2, α2β2, α2β3, α3β1, α3β3〉
ΓA :
P0
P1
P1/S1
•
•
•
•
P2
P1/S1
P2/S2 •
• •
• •
• •
•
•
P2/S2
P3/S3
• •
• •
P3/S3
P3
ΓA′ :
P0
P3
P1/S3
•
•
•
•
P2
P3/S1
P2/S2 •
• •
• •
• •
•
•
P2/S2
P3/S1
• •
• •
P1/S3
P1
ΓA′′ :
P0
P2
P1/S3
•
•
•
•
P3
P2/S1
P2/S1 •
• •
• •
• •
•
•
P3/S2
P3/S2
• •
• •
P1/S3
P1
18
Λ = KQ/I locally bounded K-category
(Q, I) → Π1(Q, I) fundamental group
I is generated by elements (relations) of the
path category KQ of the form
= λ1u1 + λ2u2 + · · · + λmum,m ≥ 1,
λ1, λ2, . . . , λm ∈ K \ 0, u1, u2, . . . , um areparallel paths in Q
x
u1
!" #$ %& %& '( )*
u2
um
)* '( %& %& #$ !"
... y
m(I) a set of minimal relations generatingthe ideal I
Π1(Q, I) = Π1(Q, x0)/N(Q,m(I), x0)
Π1(Q, x0) fundamental group of the quiver Q
at a fixed vertex x0 of Q
N(Q, I, x0) normal subgroup of Π1(Q, x0)generated by the homotopy classes I [wuv−1w−1]
x0 wwalk
x
u+,
!" %& '(
-.
v-.
'( %& !"
+,y u = ur, v = us
=∑λiui ∈ m(I)
19
We may associate to Λ = KQ/I a universal
Galois covering
F : Λ = KQ/I → Λ/G = Λ = KQ/I
with group G = Π1(Q, I).
(Green, Martinez – de la Pena)
W topological universal cover Q with base
point x0
Π1(Q, x0) acts on W ⇒Π1(Q,m(I), x0) acts on W
Q = W/N(Q,m(I), x0) orbit quiver
Π1(Q, I) acts on Q and induces a map
p : Q→ Q of quivers
I generated by liftings of minimal generators
(from m(I)) of I to KQ
20
Λ = K[x, y]/(x2, y2) ⇒ Λ = KQ/I
Q : •α β I = 〈α2, β2, αβ − βα〉
Π1(Q, I) = Z ⊕ Z, Λ = KQ/I is given by
...
...
...
. . . • α
β
• α
β
•
β
. . .
. . . • α
β
• α
β
•
β
. . .
. . . • α
• α
•
. . .
... ... ...
I is generated by all paths α2, β2, αβ− βα in Q
Note that Λ = T(K∆), where
∆ : • α
β• Kronecker quiver
and Λ = KQ/I is given by
Q : . . . • α
β• α
β•
. . .
I = 〈all α2, β2, αβ − βα〉Λ = R for R = Λ
21
R locally bounded K-category
R is simply connected (Assem-Skowronski)
if, for any presentation R ∼= KQ/I of R as a
bound quiver category,
• Q = QR has no oriented cycles
(R is triangular)
• ∏1(Q, I) is trivial
R is simply connected ⇐⇒ R is triangular
and has no proper Galois coverings
An algebra A is called standard if there exists
a Galois covering R −→ R/G = A such that
• R is a simply connected locally bounded
K-category
• G is an admissible group of automorphisms
of R
22
Brauer tree algebras
Brauer tree: a finite connected tree T = TmStogether with• a circular ordering of the edges converg-
ing at each vertex• one exceptional vertex S with multiplicitym ≥ 1
We draw T in a plane such that the edgesconverging at any vertex have the clockwiseorder
Brauer tree T → Brauer quiver QT :• the vertices of QT are the edges of T• there is an arrow i → j in QT ⇐⇒ j
is the consecutive edge of i in the circu-lar ordering of the edges converging at avertex of T
QT has the following structure:• QT is a union of oriented cycles corre-
sponding to the vertices of T• Every vertex of QT belongs to exactly two
cycles
The cycles of QT are divided into two camps:α-camps and β-camps such that two cyclesof QT having nontrivial intersection belong todifferent camps. We assume that the cycle ofQT corresponding to the exceptional vertex Sof T is an α-cycle. 23
i vertex of QT
iαi−→ α(i) the arrow in α-camp of QT
starting at i
iβi−→ β(i) the arrow in β-camp of QT
starting at i
α2(i)
α(i)
αα(i) β(i)ββ(i) β2(i)
.
.
.
.
.
.
i
αi
βi
!!
α−2(i)αα−2(i)
α−1(i)
αα−1(i)
!!
β−1(i)
ββ−1(i)
β−2(i)ββ−2(i)
Ai = αiαα(i) . . . αα−1(i) Bi = βiββ(i) . . . ββ−1(i)
T = TmS → A(T) = A(TmS ) = KQTmS/ImS
Brauer tree algebra
ImS ideal in KQTmSgenerated by elements :
• ββ−1(i)αi and αα−1(i)βi
• Ami −Bi if the α-cycle passing through i
is exceptional• Ai −Bi if the α-cycle passing through i
is not exceptionalfor all vertices i of QTmS
.
24
Example. Let T = T6S is of the form
•5
•6
•4
•3
•2
1 • S
m = 6
•
QT = QT6S
is of the form
4β4
α4
2 α2
β2
""
3
α3
β3 ##
1 α1
β1
5β5
α5
!!
6α6
!!
β6
I6S generated by
α1β1, β6α1, β1α2, α2β2, β2α3, α5β3, α3β4,
β4α4, α4β5, β5α5, α6β6, β3α6,
α61−β1β2β3β6, α2−β2β3β6β1, α3α4α5−β3β6β1β2,α4α5α3 − β4, α5α3α4 − β5, α6 − β6β1β2β3
25
Example. Let T = TmS be the star
•e−1 •
e
...........• S •1
•2
•3
QT = QTmSis of the form
eβe
αe 1 β1
α1
e− 1βe−1
αe−1
2 β2
α2
..............................
.............
A(TmS ) = Neme symmetric Nakayama alge-
bra
In general,T = TmS a Brauer tree ⇒ the Brauer tree al-gebra A(T) is (special) biserial and of finitetype
26
Λ selfinjective algebra
Λ is biserial if the heart H(P) = radP/ socP
of every indecomposable projective Λ-
module P is a direct sum of at most two
serial modules
Λ is special biserial if Λ ∼= KQ/I, where
the bound quiver (Q, I) satisfies the con-
ditions:
(SB1) Each vertex of Q is the starting and
end point of at most two arrows
(SB2) For any arrow α of Q, there is at
most one arrow β and one arrow γ
of Q such that αβ, γα /∈ I.
Λ of finite type, then
Λ biserial ⇐⇒ Λ special biserial
(Skowronski-Waschbusch, 1983)
27
charK = p > 0
G finite group, p∣∣∣|G|
KG = B0 ×B1 × · · · ×Br,B0, B1, . . . , Br connected algebras
(blocks of KG)
KG is of
finite type
Higman (1954)⇐====⇒ Sylow p-subgroups
of G are cyclic
If G admits a normal cyclic Sylow p-subgroupthen the blocks B0, B1, . . . , Br are Moritaequivalent to symmetric Nakayama algebras(application of Clifford’s theorem)
In general, let B be a block of KG
B → D = DB defect group of B
D p-subgroup of G
modB X ⇒ X|Y ⊗KD KG, for some Y ∈modKD
B of finite type ⇐⇒ DB is cyclic
28
Theorem (Dade-Janusz-Kupisch,1966-1969).Let B be a block of a group algebra KG withcyclic defect group DB. Then B is Moritaequivalent to a Brauer tree algebra A(TmS ).
(Here me+ 1 = pn if | DB |= pn and B has esimple modules)
Remark. Most of the Brauer tree algebrasA(Tms ) are not Morita equivalent to blocksof group algebras (Feit, 1984).
Theorem (Gabriel-Riedtmann (1979),Rickard (1989)). Let A be a selfinjectivealgebra. TFAE:(1) A is Morita equivalent to a Brauer tree
algebra.(2) A is stably equivalent to a symmetric
Nakayama algebra.(3) A is derived equivalent to a symmetric
Nakayama algebra.
In particular, for blocks B and B′ (of groupalgebras) with cyclic defect groups DB andDB′, one gets:
B and B′ are derived equivalent ⇐⇒ DB∼= DB′
(solution of Broue’s conjecture in the cyclicdefect case)
29
2. SELFINJECTIVE ALGEBRAS OF
POLYNOMIAL GROWTH
• Selfinjective algebras of finite type
• Domestic selfinjective algebras of infinite
type
• Nondomestic selfinjective algebras of poly-
nomial growth
Selfinjective algebras of Dynkin type: B/G,
B tilted of Dynkin type
Selfinjective algebras of Euclidean type: B/G,
B tilted of Euclidean type
Selfinjective algebras of tubular type: B/G,
B tubular
30
THEOREM (2004). Let A be a nonsimple
selfinjective algebra over K. Then
(1) A is standard of polynomial growth ⇐⇒A is of Dynkin, Euclidean, or tubular type.
(2) A is of polynomial growth ⇐⇒ there
exists a unique standard selfinjective
algebra A (standard form of A) of poly-
nomial growth such that
• dimK A = dimK A,
• A and A are socle equivalent,
• A is a degeneration of A.
(3) If A is nonstandard domestic then charK =
2.
(4) If A is nonstandard of polynomial growth
then charK = 2 or 3.
COROLLARY. Every selfinjective algebra of
polynomial growth is of linear growth.
31
THEOREM (2003). Let A be a nonlocal
selfinjective algebra over K. Then
A is standard weakly symmetric of polynomial
growth ⇐⇒ A = B/(ϕ), B tilted of Dynkin
type, tilted of Euclidean type, or tubular, and
ϕ is a root of the Nakayama automorphism
νB
of B.
PROBLEM. Let A and A′ be stably equiv-
alent selfinjective algebras of polynomial
growth. Are A and A′ derived equivalent?
Confirmed for:
• selfinjective algebras of finite type (Asashiba,
1999)
• weakly symmetric domestic algebras (Bocian-
Holm-Skowronski, 2004)
• weakly symmetric algebras of polynomial
growth (Bialkowski-Holm-Skowronski,
2003)
32
SELFINJECTIVE ALGEBRAS OF FINITETYPE
Selfinjective algebras of Dynkin type
∆ ∈ Am,Dm,E6,E7,E8 Dynkin graph
∆ a Dynkin quiver with underlying graph ∆
H = K ∆ the path algebra of ∆
T ∈ modH tilting H-module:Ext1H(T, T) = 0T = T1 ⊕ · · · ⊕ Tn, n = |∆0|T1, . . . , Tn indecomposable pairwise
nonisomorphic
B = EndH(T) tilted algebra of type ∆• gl.dimB ≤ 2• B is of finite-type• The Auslander-Reiten quiver ΓB of B is
of the form
Dynkin section ∆33
Selfinjective algebra of Dynkin type ∆:
algebra of the form B/G, where B is a tilted
algebra of a Dynkin type ∆ and G is an ad-
missible group of automorphisms of B
A = B/G selfinjective algebra of Dynkin type
⇒
F : B −→ B/G
canonical Galois covering with the group G
In fact, G is infinite cyclic
Moreover, B is simply connected, because
B is simply connected (property of all tilted
algebras of Dynkin types)
Hence, A = B/G is a standard selfinjective
algebra
34
For tilted algebras B and B′ of Dynkin type,
we have
B ∼= B′ Hughes-Waschbusch⇐====⇒ B and B′ are re-
flection equivalent
For r, s ≥ 1, let Λ(r, s) = KQr,s/Ir,s,
Qr,s the quiver
•β$$
r · · · 2 1 1
α
γ$$
2 · · · s
•σ
Ir,s generated by αβ − γσ.
Λ(r, s) tilted algebra of type Dr+s+2
Λ(r, s) and Λ(r′, s′) are reflection equivalent
⇐⇒ r+ s = r′ + s′
35
A = B/G selfinjective algebra of Dynkin type
∆ ⇒ ΓB
is of the form
∗∗
∗ ∗
∗∗
∗ ∗
∗∗
∗ ∗
∗∗
∗ ∗
(∗ projective-injective vertices) and ΓsB
is of
the form Z∆ (= Z ∆)
∆ τ−m∆
B∆
where mAn = n, mDn = 2n − 3, mE6= 11,
mE7= 17, mE8
= 29.
In fact, we have νB
= τ−m∆
Bon modB:
B = EndH(T), H = K ∆ ⇒modT(B) ∼= modT(H)
(Tachikawa-Wakamatsu)
| indT(H) |∼= 2 | indH |(Tachikawa, Yamagata)
36
37
C the set of vertices of ΓsB
= Z∆ given by the
radicals of the indecomposable projective
B-modules (configuration of Z∆)
Then ΓB
= Z∆C completion of Z∆ by
c∗
c
%%
τ−cfor all c ∈ C .
In particular, the configuration C is stableunder the action of ν
B= τ
−m∆
Bon Γs
B= Z∆
Moreover, the admissible automorphism groupG of B is infinite cyclic:
G acts on ΓB, and hence also on Γs
B= Z∆
As an automorphism group of the translationquiver Z∆, G = (τ−r) where r ≥ 1 and theautomorphism fixes at least one vertex ofZ∆.
Note that is of order 1, 2, or 3.
38
We have the canonical Galois covering
F : B → B/G = A
with G infinite cyclic. Moreover, B is locally
support-finite (even locally representation-
finite).
Hence, the push-down functor
Fλ : mod B → mod B/G = modA
is dense, and A is of finite type. In particular
ΓA∼= Γ
B/G and ΓsA = Γs
B/G = Z∆/G
cylinder
Mobius strip
•
&&
• •
&&
•
''!!!!!
&&
• •
''!!!!!
&&
•
"""
###
· · ·
((###
))"""• •
%%$$$$$
%%$$$$$ •
''!!!!! • •
''!!!!! ZD4/(τ
−5r(1,2,3))
39
Theorem (Riedtmann, Waschbusch, . . .
1983). Let A be a nonsimple algebra. TFAE
(1) A is standard selfinjective of finite type.
(2) A is selfinjective of Dynkin type.
The fundamental result:
Theorem (Riedtmann, 1977). Let A be a
selfinjective algebra of finite type. Then
ΓsA = Z∆/G and ΓA = Z∆C /G
for a Dynkin graph ∆, an infinite cyclic group
G of automorphisms of Z∆, and a G-stable
configuration C of Z∆.
(The mesh-category K(Z∆C ) is isomorphic
to ind B for a tilted algebra B of type ∆
(Hughes-Waschbusch))
40
A selfinjective algebra of finite type
1A = e1 + · · · + en,
e1, . . . , en orthogonal primitive
idempotents
*
+" %%%%%
%%%%%%
%%%%%%%%%%% *
#+&&&
&&&&
&&&&
&&&&
&&&&
&&&
A standard
A standard (Zurich)(indA ∼= K(ΓA)mesh category
) A regular (Berlin)∀
1≤i,j≤neiAej cyclic
left eiAei-moduleand cyclic rightejAej-module
nonstandard
(classified by Riedtmann)
⇐⇒ nonregular
(classified by Waschbusch)
41
T = TS = T2S Brauer tree with at least two
edges and the extreme vertex S of multiplic-ity 2
•B3
3•
B2
2
...........S 1
S′
•Br
r
• Br−1
r−1
Then the Brauer quiver QT = QT2S
is of the
form
cycle S′
r − 1
QBr−1
,,''''''
''. . .
rQBr
βr--((((((((
j + 1
QBj+1..
1β1//))))))))
α1
j
βj00*******
11
βj−1(((((((
QBj
2QB2++
++++++
+ j − 1QBj+13QB3
. . .
For each edge i of T (vertex i of QT ) we havethe cycles Ai and Bi around i
Define B′j = βj . . . βrα1β1 . . . βi−1, j = 1, j ∈
S′0
42
For each λ ∈ K, define the algebra
D(TS, λ) = KQT/I(TS, λ)
where I(TS, λ) is the ideal of KQT generated
by
• ββ−1(i)αi and αα−1(i)βi, i ∈ (QT )0 \ 1,
• A21 = B1,
• Ai −Bi, i ∈ (QT )0 \ S′0,
• Aj −B′j, j ∈ S′
0 \ 1,
• βrβ1 − λβrα1β1.
Proposition. (1) D(TS, λ), λ ∈ K, are weakly
symmetric algebras of finite type.
(2) For λ, µ ∈ K \ 0, D(TS, λ)∼= D(TS, µ).
(3) D(TS,0)∼= D(TS,1) ⇐⇒ charK = 2.
(4) D(TS,0) and D(TS,1) are socle equiva-
lent.
(5) D(TS,0) = B/(ϕ), for an exceptional tilted
algebra B of Dynkin type D3m and a 3-
root ϕ of νB.
(6) For charK = 2, D(TS,1) is nonstandard
and degenerates to D(TS,0).
43
Example. T = T2S of the form
S 1S′ 2 • 3 •
QT = QT2S
of the form
1α1
β12
β2
α23
α3 β3
D(TS,0) = KQT/I(TS,0)
I(TS,0) generated by
β1α2, α3β2β3α3, α2β3α21 − β1β2
α2α3 − β2α1β1α3α2 − β3β2β1
∏1(QT , I(TS,0))
∼= Z
D(TS,1) = KQT/I(TS,0)
I(TS,1) generated by
β1α2, α3β2β3α3, α2β3α21 − β1β2
α2α3 − β2α1β1α3α2 − β3
β2β1 − β2α1β1
∏1(QT , I(TS,1)) trivial
44
Q :
•
α3
22,,,,,,,,,,,,,,,,,,,,,,,,
•
α1
β1
33---
----
----
----
--•
β244.................
α2
55///
////
////
////
////
////
/
•
α322,,,,,,,,,,,,,,,,,,,,,,,,
•
α1
β1
33---
----
----
----
--•
β244.................
•
• •
I generated by
α21 − β1β2, β2β1,
α3β2
B = KQ/I tilted algebra of type D9 = D3·3B = KQ/I is of the form
Q :
... ... ...•
α1
β1
33---
----
----
----
--•
β244................. α2
55///
////
////
////
////
////
/
66000
0000
•α3
22,,,,,,,,,,,,,,,,,,,,,,,,
•
α1
β1
33---
----
----
----
--•
β244................. α2
55///
////
////
////
////
////
/
•α3
22,,,,,,,,,,,,,,,,,,,,,,,,
•
α1
β1
33---
----
----
----
--•
β244................. α2
55///
////
////
////
////
////
/
•α3
22,,,,,,,,,,,,,,,,,,,,,,,,
•
α1
β1
33---
----
----
----
--•
β244................. α2
55111
1111
1111
1111
1111
1111
1
•
2222222222222222
• •
... ... •
I generated by all
α21 − β1β2, β2β1,
α2α3 − β2α1β1,
β1α2, α3β2
ϕ=shift up by one
ϕ3 = νB
D(TS,0)∼= B/(ϕ)
For a Brauer tree T and extreme vertex S of
T , we put
D(TS) = D(TS,0) and D(TS)′ = D(TS,1)
45
Theorem (Riedtmann, Waschbusch, . . . ).
Let A be a standard selfinjective algebra. TFAE:
(1) A is symmetric of finite type.
(2) A is weakly symmetric of finite type.
(3) A ∼= B/(ϕ), B tilted of Dynkin type, ϕ
root of the Nakayama automorphism νB.
(4) A is isomorphic to one of the algebras
(a) T(B), B tilted of Dynkin type.
(b) A(TmS ), TmS Brauer tree, S excep-
tional of multiplicity m ≥ 2.
(c) D(TS), T Brauer tree, S extreme ex-
ceptional.
Remark. The Brauer tree algebras A(T) =
A(T1S ) are exactly the trivial extensions T(B)
of tilted algebras of types An
Theorem (Riedtmann (1983), Waschbusch
(1981)). Let A be a selfinjective algebra over
K. TFAE:
(1) A is nonstandard of finite type,
(2) A ∼= D(TS)′, T Brauer tree, S extreme
exceptional, and charK = 2.
46
Standard selfinjective algebras of finite
type
B tilted algebra of Dynkin type ∆
B is exceptional: there is a reflection se-
quence i1, . . . , it of sinks in Qt with t < rkK0(B) =
|∆0| such that B ∼= S+it. . . S+
i1B
B exceptional ⇐⇒ there is a strictly positive
automorphism ϕ of B such that rkK0(B/(ϕ)) <
rkK0(T(B)) = rkK0(B)
A(TmS ) Brauer tree
algebra of multi-
plicity m ≥ 2
B(TmS ) exceptional til-
ted algebra of type AnA(TmS ) ∼= B(TmS )/(ϕ)
ϕm = νB(TmS )
D(TS) B∗(TS) exceptional til-
ted algebra of type D3m
D(TS)∼= B∗(TS)/(ϕ)
ϕ3 = νB∗(TS)
47
Proposition. Let B be a tilted algebra ofDynkin type. TFAE:
(1) B is exceptional.(2) There is an automorphism ϕ of B with
ϕm = νB
for some m ≥ 2 (ϕ proper rootof ν
B).
(3) B ∼= B(TmS ), m ≥ 2, or B ∼= B∗(TS).
Remark. There are no exceptional tilted al-gebras of Dynkin types E6, E7, E8 (Bretscher-Laser-Riedtmann (1981))
There is a description (by bound quivers andrelations) of all
• (iterated) tilted algebras of type An (Happel-Ringel (1981), Assem-Happel (1981))
• (iterated) tilted algebras of type Dn (Conti(1986), Assem-Skowronski (1989), Keller(1991))
The numbers of reflection classes of tiltedalgebras of types E6, E7, E8, are:E6: 22E7: 143E8: 598
48
Theorem. Let A be a standard selfinjectivealgebra of finite type. Then A is isomorphicto an algebra of one of the forms:
• B/(νrB), r ≥ 1, B tilted of type ∆ ∈
An,Dn,E6,E7,E8• B/(νr
B), r ≥ 1, B tilted of type ∆ ∈
A2p+1,Dn,E6, automorphism of order2
• B/(σνrB), r ≥ 1, B = K∆, ∆ =
•##
•
•""• , σ
automorphism of order 3
• B/(ϕr), r ≥ 1, B = B(TmS ) tilted of type∆ = An, ϕ m-root ν
B(TmS ),
• B/(ϕr), r ≥ 1, B = B∗(TS) tilted of type∆ = D3m, ϕ 3-root ν
B∗(TS),
There is also a complete classification ofthe derived and stable equivalence classesof the selfinjective algebras of finite type(Asashiba, 1999)
A, A′ selfinjective algebras of finite type ⇒A and A′ are
stably equivalent⇐⇒ A and A′ are
derived equivalent
49
ZURICH SCHOOL APPROACH
A selfinjective algebra of finite type
Assume A is basic, connected, A K
ΓA finite, connected, translation quiver
p : ΓA −→ ΓA/∏A = ΓA
universal Galois covering of translation quiv-ers,
∏A
∼= ∏1(ΓA) fundamental group of ΓA,
ΓA simply connected (∏
1(ΓA) is trivial)∏A
∼= ∏1(Γ
sA), so we have also Galois covering
p : ΓsA −→ ΓsA/∏A = ΓsA
Theorem (Riedtmann, 1977). ΓsA∼= Z∆
for a Dynkin graph ∆ ∈ An,Dn,E6,E7,E8.In particular, ΓsA = Z∆/
∏A, and
∏A is infinite
cyclic.
CA set of vertices representing the radicals
of indecomposable projective A-modules
(configuration of ΓsA)
ΓA = (ΓsA)CA= (Z∆/
∏A)CA
CA = p−1(CA) configuration of ΓsA∼= Z∆
ΓA = (ΓsA)CA
= (Z∆)CA
50
K(ΓA) = KΓA/IA mesh-category of ΓAIA generated by the meshes
y1
y17733333
33333333
x
44444444444444885555555555555
9+66666
666666
666 τ−x...yr
:977777777777777
K(ΓA) = KΓA/IA mesh-category of ΓA
A the full subcategory of K(ΓA) given by the
projective vertices
A locally bounded K-category
indA (respectively, ind A) full subcategory ofmodA (respectively, mod A) formed by a com-plete set of indecomposable modules
Then ind A ∼= K(ΓA).
A is called standard if indA ∼= K(ΓA)
Then we have Galois coverings
ind A = K(ΓA) → K(ΓA)/∏A = K(ΓA) = indA
F : A → A/∏A = A
where A is simply connected and∏A infinite
cyclic group
51
In fact A ∼= B for a tilted algebra B of Dynkintype ∆ (a simple application of tilting theory)
Hence, for the selfinjective algebras of finitetype, the both concepts of standardness co-incide
The configuration C = CA of Γ = ΓsA = Z∆is a combinatorial configuration of Z∆:(a) For any vertex x of Γ there exists a vertex
c ∈ C such that HomK(Γ)(x, c) = 0;(b) For any vertices c, d ∈ C we have
HomK(Γ)(c, d) = 0, if c = d andHomK(Γ)(c, c)
∼= K;.(P indecomposable projective, Ω−
A(radP) =
P/ radP)
Moreover, the configuration C = CA is τm∆-stable
Ch. Riedtmann classified (1977) all τm∆-stable combinatorial configurations of the sta-ble translation quivers Z∆, in case ∆ = E6,E7or E8 together with F. Jenni by computer(E6 : 22, E7 : 143, E8 : 598 isoclasses of con-figurations)
Moreover, Bretscher-Laser-Riedtmann (1981)proved that for any τm∆-stable configurationC of Z∆ and an admissible group
∏of Z∆
stabilizing C , we have
Z∆C /∏ ∼= ΓA
for a selfinjective algebra A of finite type52
In the fact, the main result of their pa-per says: the configurations of Z∆ corre-spond bijectively to the isomorphism classesof square -free tilting modules over K ∆ (iso-classess of basic tilted algebras of Dynkintypes ∆)
This is not correct because there are noniso-morphic tilted algebras of Dynkin types hav-ing isomorphic repetitive categories!
The correct result is: the configurationsof Z∆ correspond bijectively to the reflec-tion equivalence classes of tilted algebras ofDynkin types ∆
A selfinjective of finite typeRiedtmann====⇒ there
exists a well-behaved covering functor
F : K(ΓA) = ind A −→ indA
The functor is the canonical Galois coveringfunctor
K(ΓA) −→ K(ΓA)/∏A = K(ΓA)
(hence A is standard), exceptcharK = 2 and ΓA = ZD3m/
∏A,
∏A = (τ2m−1),
and there exist nonstandard selfinjective al-gebras of finite type
53
BERLIN SCHOOL APPROACH
A selfinjective algebra of finite type
Assume A is basic, connected, A K
1 = e1 + e2 + · · · + en,e1, e2, . . . , en orthogonal primitive idempotents
A finite typeJans (1957)====⇒ A has finite ideal
lattice ⇒ for all i, j ∈ 1, . . . , n we have• eiAei is serial• eiAej is cyclic left eiAei-module or cyclic
right ejAej-module
A is called regular if all bimodules eiAej are
cyclic as left eiAei-modules and cyclic as
right ejAej-modulesA → Stamm-algebra A′ (Kupisch, 1965)
M(A) = ei(radt A)ej | 1 ≤ i, j ≤ n, t ≥ 0semigroup
The nonzero elements of M(A) form a K-basis M0 of the algebra A′ = K ·M0, with mul-tiplication induced from the semigroup M(A)
A′ is a selfinjective algebra
A and A′ have isomorphic ideal lattices
A is weakly symmetric ⇐⇒ A′ is symmetric
54
Moreover, we have• A regular ⇒ A ∼= A′• A nonregular ⇒ A symmetric
(Kupisch (1978), Kupisch-Scherzler (1981))
A ∼= A′ ⇒ A has a nice canonical multiplicative
Cartan K-basis⇒ A standard algebra ⇒ A ∼= B/G selfinjec-tive of Dynkin type
Waschbusch (1981) classified all nonregu-lar selfinjective algebras of finite type, by the(modified) Brauer tree algebras with extremeexceptional vertex. They coincides with thenonstandard selfinjective algebras of finite type,described by Riedtmann (1983).
Hence
A regular ⇐⇒ A standard
A nonregular ⇐⇒ A nonstandard
Theorem (Hughes-Waschbusch, 1983). Foran algebra A,T(A) is of finite type ⇐⇒ T(A) ∼= T(B) fora tilted algebra B of Dynkin type
55
DOMESTIC SELFINJECTIVE ALGEBRASOF INFINITE TYPE
∆ ∈ An, Dn, E6, E7, E8 Euclidean graph
Selfinjective algebra of Euclidean type ∆:algebra of the form B/G, where B is a tiltedalgebra of an Euclidean type ∆ and G is anadmissible group of automorphisms of B
A = B/G selfinjective algebra of Euclideantype ⇒
F : B −→ B/G = A
canonical Galois covering with the group G
In fact, G is infinite cyclic
For B tilted of type ∆ ∈ Dn, E6, E7, E8, Bis simply connected
For B tilted of type ∆ = An, B is not simplyconnected, but A = B/G admits a simplyconnected Galois covering
F ′ : R −→ R/H = A
with H ∼= Z ⊕ Z
Hence, A = B/G is a standard selfinjectivealgebra
56
Theorem (Skowronski, 1989, 2003). Let
A be a selfinjective algebra. TFAE:
(1) A is selfinjective of Euclidean type.
(2) A is standard domestic of infinite type.
(3) A is standard, tame and ΓsA admits a
component Z∆ for an Euclidean graph
∆.
57
A = B/G selfinjective algebra of Euclidean
type ∆ ⇒ ΓB
is of the form∨
m∈Z(Yr ∨ Cr):
∗∗
∗∗∗ ∗
∗∗ ∗∗ ∗
∗Y0C−1 C0Y−1 Y1
Ysi = Z∆,
C si = P1(K)-families of stable tubes (of the
same tubular type),
νB(Yi) = Yi+2, νB(Ci) = Ci+2,
Then G is infinite cyclic generated by a strictly
positive automorphism of B
Moreover, B is locally support-finite ⇒Fλ : mod B → mod B/G = modA is dense ⇒
ΓA = ΓB/G
= ΓB/G
58
Hence, ΓA is of the form (for some r ≥ 1 and
Xi = Fλ(Yi), Ti = Fλ(Ci), 0 ≤ i < r)
∗∗∗
∗∗
∗ ∗
∗∗∗ ∗
∗∗ ∗∗
∗∗
∗∗ ∗
∗∗∗
X0Tr−1 T0Xr−1 X1
Tr−2 T1
Xr−2 X2
X si = Z∆,
T si P1(K)-families of stable tubes (of the
same tubular type)
Then A is called r-parametric
59
Euclidean algebra = tubular (branch) ex-
tension of a tame concealed algebra of one
the tubular types (p, q), 1 p q, (2,2, r),
r 2, (2,3,3), (2,3,4), or (2,3,5)
= representation-infinite tilted algebra of an
Euclidean type Ap+q−1, Dr+2, E6, E7, or E8,
having a complete slice in the preinjective
component.
B Euclidean algebra (of type ∆) ⇒
• gl.dimB ≤ 2
• B is domestic of infinite type (one-parametric)
• The Auslander-Reiten quiver ΓB of B is
of the form
∆
P T Q
60
Proposition (Assem-Nehring-Skowronski,
1989). Let B be a tilted algebra of Euclidean
type ∆. Then there exists a reflection se-
quence of sinks i1, i2, . . . , im in QB such that
B′ = S+im. . . S+
i2S+i1B is an Euclidean algebra
of type ∆. In particular, B ∼= B′.
Proposition (Assem-Nehring-Skowronski,
1989). Let B, B′ be Euclidean algebras.
TFAE:
(1) B ∼= B′.
(2) T (B) ∼= T(B′).
(3) B′ ∼= S+ir. . . S+
i2S+i1B for a reflection se-
quence of sinks i1, i2, . . . , ir in QB, r ≤rkK0 (B).
In fact, at most two Euclidean algebras may
have isomorphic repetitive categories
61
An Euclidean algebra B is exceptional if thereexists a reflection sequence of sinks in QBi1, i2, . . . , it such that t < rkK0 (B) and B ∼=S+it
· · ·S+i2S+i1B.
Proposition (Skowronski, 1989). Let B bean Euclidean algebra. ThenB is exceptional ⇐⇒ there exists an auto-
morphism ϕ of B such that
ϕd = νB
for some d 2 and
a rigid automorphism of B.Moreover, then d = 2.
Theorem (Skowronski, 1989). Every self-injective algebra of Euclidean type is one ofthe forms:
(1) B/(σνk
B
), where B is an Euclidean alge-
bra, σ is a rigid automorphism of B, andk is a positive integer.
(2) B/(µϕ2k+1
), where B is an exceptional
Euclidean algebra, µ is a rigid automor-phism of B, ϕ is an automorphism of Bsuch that ϕ2 = ν
Bfor a rigid automor-
phism of B, and k is a positive integer.
62
PROBLEM. Describe the exceptional
Euclidean algebras and their repetitive
algebras
Theorem (Lenzing-Skowronski, 1999).
There are no exceptional Euclidean algebras
of types E6, E7, E8.
Theorem (Bocian-Skowronski, 2003). Let
B be an Euclidean algebra. TFAE:
(1) B is an exceptional algebra.
(2) There is an automorphism ϕ of B with
ϕ2 = νB.
(3) B is reflection equivalent to an excep-
tional Euclidean algebra of one of the
forms:
• B(T, v1, v2), B′(T) (type An).
• Θ(i)(l,m,B), 0 ≤ i ≤ 8 (type Dn).
63
B exceptional Euclidean algebra one-
parametric (weakly) symmetric selfinjec-
tive algebra B/(ϕ), ϕ2 = νB.
B(T, v1, v2) Λ(T, v1, v2)
B′(T) Λ′(T)
Θ0(l,m,B) Γ(0)(T, v)
Θ(1)(l,m,B)
Θ(2)(l,m,B)
Θ(3)(l,m,B)
Θ(4)(l,m,B)
Γ(1)(T, v)
Θ(5)(l,m,B)
Θ(6)(l,m,B)
Θ(7)(l,m,B)
Θ(8)(l,m,B)
Γ(2)(T, v1, v2)
64
T = Brauer tree with two (different) distin-guished vertices v1 and v2⇒ Λ(T, v1, v2)=KQT/I(T, v1, v2)one-parametric symmetric algebra of Eu-
clidean type Am.
Example. Let T be the Brauer tree
1
4
3
25
6
79
8
v1 v2
Then QT is of the form
1
2
4
3
5
7
6
9
8
α1
α2
α3
α4
α5
α6
α7
α8
α9
β1β2
β3 β4
β5
β6
β7
β8 β9
65
and the ideal I (T, v1, v2) in KQT generated
by α1β1, β1α2, α2β2, β2α3, α7β3, β3α4, α4β4,
β4α1, α3β5, β5α6, α6β6, β6α5, α5β7, β9α7,
α8β8, β7α8, α9β9, β8α9, α21 − β1β2β3β4, α2 −
β2β3β4β1, α4−β4β1β2β3, (α3α5α7)2−β3β4β1β2,
(α5α7α3)2 − β5β6, (α7α3α5)
2 − β7β8β9, α6 −β6β5, α8 − β8β9β7, α9 − β9β7β8.
T = Brauer graph with exactly one cycle,
having moreover an odd number of edges.
⇒ Λ′(T)=KQT/I′(T) one-parametric
symmetric algebra of Euclidean type Am.
66
Example. Let T be the Brauer graph
1
4
3
2
7
9
5
8
6
Then QT is the quiver
1
2
4
3
5
7
6
9
8
α1
α2
α3
α4
α5
α6
α7
γ8
α9
β1β2
β3 β4
γ5
γ6
β7
β8 β9
67
and the ideal I ′ (T) in KQT is generated by:
α1β1, β1α2, α2β2, β2α3, α7β3, β3α4, α4β4,
β4α1, α3γ5, γ5α6, α5β7, β9α7, α9β9, β8α9,
β7γ8, γ8α5, γ6β8, α6γ6, α1 − β1β2β3β4, α2 −β2β3β4β1, α3α5α7−β3β4β1β2, α4−β4β1β2β3,α5α7α3−γ5γ6γ8, α6−γ6γ8γ5, α7α3α5−β7β8β9,γ8γ5γ6 − β8β9β7, α9 − β9β7β8.
T = Brauer graph with exactly one loop b
having the unique vertex denoted by u, and
one distinguished vertex v different from u
such that v is the end of exactly one edge a,
and the loop b and the edge a converge in
a common vertex u. Moreover, we assume
that the edge a is a direct successor of the
loop b, and the loop b is a direct successor
of the edge a but the loop b is not a direct
successor of itself in the cyclic order of edges
at the vertex u of the graph T
⇒ Γ(0)(T, v) = KQ(0)T /I(0)(T, v)
one-parametric symmetric algebra of Eu-
clidean type Dn.
68
Example. Let T be the Brauer graph with
the distinguished vertex v
1 6a = 5
b = 4 3 2
v
Then Q(0)T is the quiver
6
1
γ4
γ3γ2
γ1
α5
α6
4
5
3
2
δ4
δ5β2
β3
β1
β6
69
and the ideal I(0)(T, v) in KQ(0)T is generated
by: α6β6, β1α6, γ1β2, γ2β3, γ4β1, β2γ2, β3γ3,
β6γ1, α5δ5, δ4α5, α6−β6β1, α5−δ5γ4γ1γ2γ3δ4,β1β6 − γ1γ2γ3δ4δ5γ4, β2 − γ2γ3δ4δ5γ4γ1, β3 −γ3δ4δ5γ4γ1γ2, γ3γ4, δ5γ4γ1γ2γ3 − δ5δ4δ5,
γ4γ1γ2γ3δ4 − δ4δ5δ4.
Note that α5 − (δ5δ4)2 ∈ I(0)(T, v).
T = Brauer graph with one distinguished ver-
tex v and exactly one cycle having three edges
denoted by a, b and c. Assume that the edges
a and b (respectively, b and c, a and c) con-
verge in a common vertex v1 (respectively,
v2, v3). Moreover, we assume that the edge
a is a direct successor of the edge b, the edge
b is a direct successor of the edge c, and the
edge c is a direct successor of the edge a, in
the cyclic orders of edges at the vertices v1,
v2 and v3, respectively
⇒ Γ(1)(T, v) = KQ(1)T /I(1)(T, v)
one-parametric symmetric algebra of Eu-
clidean type Dn.
70
Example. Let T be the Brauer graph with
the distinguished vertex v
1
c = 4
8
a = 3
b = 6
7 9
5 2
v
Then Q(1)T is the quiver
1
2
97 8
3
4
α3
α8
α7 α9
γ6
γ3
6
5
β5
β1
α1
β9
β2
α2
α4
β6
β7 β8
β4
γ5
71
and the ideal I(1)(T, v) in KQ(1)T is gener-
ated by: β1α1, β2α2, β6α7, β7α8, β8α9, β9α4,
α2γ3, γ3β5, γ5β6, α1β2, α4β1, α7β7, α8β8,
α9β9, β5γ5, α2α3α4α1 − β2, α7 − β7β8β9β4β6,
α8−β8β9β4β6β7, α9−β9β4β6β7β8, β5−γ5γ6γ3,α1α2α3α4 − β2
1, γ5γ6α3, α2α3β4, β9β4γ6,
β6β7β8β9− γ6α3, γ3γ5−α3β4, α4α1α2−β4γ6.
Note that α4α1α2α3−β4β6β7β8β9, α3α4α1α2−γ3γ5γ6, β6β7β8β9β4 − γ6γ3γ5 ∈ I(1)(T, v).
T = Brauer tree with two (different) distin-
guished vertices v1 and v2 such that v1 is the
end of exactly one edge
⇒ Γ(2)(T, v1, v2) = KQ(2)T /I(2)(T, v1, v2) one-
parametric symmetric algebra of Euclidean
type Dn.
72
Example. Let T be the Brauer tree with two
distinguished vertices v1 and v2
e = 1
6
b = 5
8
a = 4c = 3
2
7v2
v1
Then Q(2)T is the quiver
1
α6
α5
α4 α3
α2
α8
α1
γ1γ2
6
8
5 4
3
2
9
7
β4
β5
β7
γ3
β2
α7 β3
β1
β8
β6
73
and the ideal I(2)(T, v1, v2) in KQ(2)T is gen-
erated by: α1β2, α2β3, α3β4, α4β5, α5β6,
α6β1, α7β7, α8β8, β1α1, β2α2, β3α3, β4α4,
β5α7, β6α8, β7α5, β8α6, α2α3α4α5α6α1 − β2,
α3α4α5α6α1α2 − β3, α4α5α6α1α2α3 − β4,
α5α6α1α2α3α4−β5β7, α6α1α2α3α4α5−β6β8,α7−β7β5, α8−β8β6, α1α2α3α4α5α6−β2
1, γ2β5,
β3γ1, γ1γ3, γ3γ2, γ2α5α6α1α2α3, α4α5α6α1α2γ1,
α3α4 − γ1γ2, γ2α5α6α1α2γ1 − γ3.
74
Weakly symmetric algebras
of Euclidean type
CA = (dimK HomA(Pi, Pj)) Cartan matrix of
A, P1, P2, . . . , Pn complete family of pairwise
nonisomorphic indecomposable projective A-
modules
Theorem (Bocian-Skowronski, 2003). Let
A be an algebra. TFAE:
(i) A is weakly symmetric of Euclidean type
and has singular Cartan matrix.
(ii) A is symmetric of Euclidean type and
has singular Cartan matrix.
(iii) A is two-parametric weakly symmetric of
Euclidean type.
(iv) A is isomorphic to the trivial extension
T(B) of an Euclidean algebra B.
75
Theorem (Bocian-Skowronski, 2003). Let
A be a nonlocal algebra. TFAE:
(i) A is weakly symmetric of Euclidean type
and has nonsingular Cartan matrix.
(ii) A is symmetric of Euclidean type and
has nonsingular Cartan matrix.
(iii) A is one-parametric weakly symmetric of
Euclidean type.
(iv) A ∼= B/ (ϕ), where B is an (exceptional)
Euclidean algebra and ϕ is a square root
of the Nakayama automorphism νB
of B.
(v) A is isomorphic to an algebra of the form
Λ(T, v1, v2), Λ′ (T), Γ(0)(T, v), Γ(1)(T, v),
or Γ(2)(T, v1, v2).
Theorem. A local algebra A is a selfinjec-
tive (weakly symmetric) algebra of Euclidean
type if and only if A is isomorphic to an alge-
bra K〈x, y〉/(x2, y2, xy−λyx
), for λ ∈ K\ 0.
76
T = Brauer graph with exactly one loop which
is also its direct successor
•B2
2))))))))))
•B3
3((((((((((
.........• S′1
•Br
r(((((((((( • Br−1
r−1))))))))))
The Brauer quiver QT is of the form
S′
r − 1
QBr−1
,,''''''
''. . .
rQBr
βr--((((((((
j + 1
QBj+1..
1β1//))))))))
α1
j
βj00*******
11
βj−1(((((((
QBj
2QB2++
++++++
+ j − 1QBj+13QB3
. . .
Ai,Bi, i ∈ T , cycles in QT around i
B′j = βj . . . βrα1β1 . . . βi−1, j = 1, j ∈ S′
0
77
Ω′(T) = KQT/I′(T) one-parametric biserial
symmetric algebra, where I ′(T) is the ideal
of KQT generated by
• ββ−1(i)αi and αα−1(i)βi, i ∈ (QT )0 \ 1,
• βrβ1,
• Ai −Bi, i ∈ (QT )0 \ S′0,
• Aj −B′j, j ∈ S′
0 \ 1,
• A21 − A1B1, A1B1 −B1A1,
(A1 = α1)
Theorem (Bocian-Skowronski, 2004). Let
Λ be a basic connected selfinjectiveK-algebra.
Then Λ is socle equivalent to a selfinjec-
tive algebra of Euclidean type if and only
if exactly one of the following cases holds:
(i) Λ is selfinjective of Euclidean type,
(ii) K is of characteristic 2 and Λ is isomor-
phic to an algebra of the form Ω′(T).
78
Example. Let T be the Brauer graph
1
3
2
4
T
1
2
3
4
α2
α1
β1
β2
β3α4
α3
β4
QT
Ω′(T) = KQT/I′(T), where I
′(T) is the ideal
in KQT generated by: β1α2, α2β2, β2α3, α4β3,
α3β4, β4α4, β3β1, α2 − β2β3α1β1, α3α4 −β3α1β1β2, α
21−α1β1β2β3, α1β1β2β3−β1β2β3α1,
α4α3 − β4.
79
Nonstandard algebras
Proposition (Bocian-Skowronski, 2004).
Let T be a Brauer graph such that Λ′(T) and
Ω′(T) are defined. Then
(1) dimK Ω′(T) = dimK Λ′(T).
(2) Ω′(T) ∼= Λ′(T) ⇐⇒ charK = 2.
(3) charK = 2 ⇒ Ω′(T) is nonstandard.
(4) Ω′(T) and Λ′(T) are socle equivalent.
(5) Λ′(T) is a degeneration of Ω′(T).
Theorem (Skowronski, 2004). Let A be a
selfinjective algebra. TFAE:
(1) A is nonstandard domestic of infinite type.
(2) charK = 2 and A ∼= Ω′(T) for a Brauer
graph T with one loop.
80
NONDOMESTIC SELFINJECTIVE
ALGEBRAS OF POLYNOMIAL GROWTH
B tubular algebra (in the sense of Ringel)
= tubular (branch) extension of a tame
concealed algebra of one of tubular types
(2,2,2,2), (3,3,3), (2,4,4), or (2,3,6).
B tubular =⇒• gl.dimB = 2
• rkK0(B) = 6, 8, 9, or 10
• B is nondomestic of polynomial growth
• The Auslander-Reiten quiver ΓB of B is
of the form
P T0∨q∈Q+ Tq T∞ Q
81
Selfinjective algebra of tubular type:algebra of the form B/G, where B is a tubu-lar algebra and G is an admissible group ofautomorphisms of B
A = B/G selfinjective algebra of tubular type⇒
F : B −→ B/G = A
canonical Galois covering with group G
In fact, G is infinite cyclic
Moreover, B is simply connected, becauseB is simply connected (property of all tubularalgebras)
Hence, A = B/G is a standard selfinjectivealgebra
Theorem (Skowronski, 1989, 2002). LetA be a selfinjective algebra. TFAE:
(1) A is selfinjective of tubular type.
(2) A is standard nondomestic of polynomialgrowth.
(3) A is standard tame and ΓsA consists onlyof (stable) tubes.
82
A = B/G selfinjective algebra of tubular type
⇒ ΓB
is of the form
∗∗ ∗
∨q∈Q−1
0Cq C0
∨q∈Q0
1Cq C1
C si , i ∈ Z, P1(K)-families of stable tubes
∀q∈Qi−1
i =Q∩(i−1,i)Cq P1(K)-family of stable tubes
Then G is infinite cyclic generated by a strictly
positive automorphism of B
Moreover, B is locally support-finite ⇒Fλ : mod B → mod B/G = modA is dense ⇒
ΓA = ΓB/G
= ΓB/G
83
Hence, ΓA is of the form (for some r ≥ 0 and
Tq = Fλ(Cq))
∗
∗∗
∗
T0 = Tr
∨q∈Qr−1
rTq
∨q∈Q0
1Tq
Tr−1 T1
∨q∈Qr−2
r−1Tq
∨q∈Q1
2Tq
84
Proposition (Nehring-Skowronski, 1989).
Let B, B′ be tubular algebras. TFAE:
(1) B ∼= B′.
(2) T(B) ∼= T(B′).
(3) B′ ∼= S+ir. . . S+
irB for a reflection sequence
of sinks i1, . . . , ir in QB, r ≤ rkK0(B).
A tubular algebra B is exceptional if there
exists a reflection sequence i1, . . . , it of sinks
in QB such that t < rkK0B and B ∼= S+it
· · ·S+i1B.
Proposition (Skowronski, 1989). Let B be
a tubular algebra. Then
B is exceptional ⇐⇒ there exists an auto-
morphism ϕ of B such that
ϕd = νB
for some d ≥ 2 and
a rigid automorphism of B.
85
Theorem (Skowronski, 1989). Every self-
injective algebra of tubular type is one of the
forms:
(1) B/(σνkB), where B is a tubular algebra, σ
is a rigid automorphism of B, and k is a
positive integer.
(2) B/(µϕk), where B is an exceptional tubu-
lar algebra, µ is a rigid automorphism of
B, ϕ is an automorphism of B such that
ϕd = νB
for some d ≥ 2 and a rigid au-
tomorphism of B, and k is a positive
integer with d | k.
86
PROBLEM. Describe the exceptional
tubular algebras and their repetitive
algebras.
Consider the following family of bound quiveralgebras (where a dotted line means that thesum of paths indicated by this line is zero ifit indicates exactly three parallel paths, thecommutativity of paths if it indicates exactlytwo parallel paths, and the zero path if itindicates only one path):
φγα = φσβψγα = λψσβB1(λ)
λ ∈ K \ 0,1
α β
γ σ
φ ψ
1
2 3
4
5 6
ξα = ηγ, ζα = ωγξσ = ηβ, ζσ = λωβ
B2(λ)λ ∈ K \ 0,1
α βγσ
η ζξ ω
1 2
3 4
5 6
B31
2 3 4
5
876
B41
3
5
4
2
6
8 7
B51
3
54
76
2
8
87
B6
1
4
6
3
2
5
78
B7
3
12
45
76
8
B8
3
1
4
5
7
6
8
2
B9
34
1
5
2
67
98
B10
5
2 31
6
7
4
8 9
B11
5
1
7
34
6
2
98
B12
1
7
6
2
5
9
3
4
8
B13
1
7
6
2
5
9
3
4
8
B141 2
3 4 5
6 7
8 9 10
88
Theorem. Let B be a tubular algebra. Then
the following equivalences hold:
(i) B is exceptional of tubular type (2,2,2,2)
if and only if B is isomorphic to B1(λ) or
B2(λ), for some λ ∈ K\0,1 (Skowronski,
1989).
(ii) B is exceptional of tubular type (3,3,3)
if and only if B is isomorphic to B3, B4
B5, B6, B7, or B8 (Bialkowski–Skowron-
ski, 2002).
(iii) B is exceptional of tubular type (2,4,4)
if and only if B is isomorphic to B9, B10
B11, B12, or B13 (Bialkowski–Skowron-
ski, 2002).
(iv) B is exceptional of tubular type (2,3,6)
if and only if B is isomorphic to B14
(Lenzing–Skowronski, 2000).
89
Example.
8
7
8
9:
;<
=.
>
?@AB
6
5
3
4
1 2
1′
CCC
CCC
CC
CC
CCC
CC
CC
C
8
""
7
6
""
##
5
3 4
2B6 S+
1 B6
1′
2′
BA@?
%D
E
,
(FG
H
8
""
##
7
""
6
""
##
5
""
3 4
3′
##
I
JKCL"
M&
6 N O P
4′
##
1′
##
2′
""
8
;:
QQQ
QQQQ
Q 7
""
6 5S+2 S
+1 B6
∼= Bop6 S+
4 S+3 S
+2 S
+1 B6
∼= B6
Hence, B6 is exceptional (of tubular type (3,3,3)).
90
Weakly symmetric algebrasof tubular type
Theorem (Bialkowski-Skowronski, 2003).
Let A be an algebra. TFAE:
(i) A is weakly symmetric of tubular type
and has singular Cartan matrix.
(ii) A is symmetric of tubular type and has
singular Cartan matrix.
(iii) A is isomorphic to the trivial extension
T(B) of a tubular algebra B.
91
Theorem (Bialkowski-Skowronski, 2003).
Let A be an algebra. TFAE:
(i) A is weakly symmetric of tubular type
and has nonsingular Cartan matrix.
(ii) A is isomorphic to an algebra of the form
B/(ϕ), where B is a tubular algebra and
ϕ is a proper root of the Nakayama au-
tomorphism νB
of B.
(iii) A is isomorphic to one of the bound
quiver algebras.
A1(λ)λ ∈ K \ 0,1
αγα = ασββγα = λβσβγαγ = σβγγασ = λσβσ
αγ
σβ
A2(λ)λ ∈ K \ 0,1
α2 = σγλβ2 = γσγα = βγσβ = ασ
α βσγ
A3
βα+ δγ + εξ = 0αβ = 0, ξε = 0
γδ = 0
αβ
δ γ
εξ
A4
βα+ δγ + εξ = 0αβ = 0, γε = 0
ξδ = 0
αβ
δ γ
εξ
A5
α2 = γββαγ = 0
αγ
β
92
A6
α3 = γβ
βγ = 0
βα2 = 0
α2γ = 0
αγ
β
A7
βα = δγ
γδ = εξ
αδε = 0
ξγβ = 0
αβ
δγ
εξ
A8
αβα = σξ, ξγ = 0
βαβ = γδ, δσ = 0
ξβα = 0
δαβ = 0
βαγ = 0
αβσ = 0
αβσ
ξ
γ
δ
A9
δα = εβ, γε = βσ, ασβ = 0
εγδ = 0, σγεγ = 0
αβσ
γε
δ
A10
ξαβ = ξδγξαβδ = δγξδ
βα = 0, (γξδ)2γ = 0
α β
δγξ
A11
γαβ = γξγ
αβξ = ξγξ
βα = 0, δγ = 0
ξζ = 0, (γξ)2 = ζδ
βα
ξγ
ζδ
A12
δβδ = αγ
γβα = 0, β(δβ)3 = 0
α
β
γδ
A13
α2 = γβ, βδ = 0, γβ = 0
σγ = 0, αδ = 0, σα = 0
α3 = δσ
αβγ
δσ
A14
βα = δγδγ
αδγδ = 0
γδγβ = 0
αβ = 0
αβ
δγ
A15
γβα = 0, α2 = δβ
βδ = 0, ασ = 0, αδ = σγ
αβ
γδ
σ
A16
αβγ = 0, α2 = βδ
δβ = 0, σα = 0, δα = γσ
αβ
γδ
σ
93
Corollary. For an algebra A the following con-
ditions are equivalent:
(i) A is symmetric of tubular type and has
nonsingular Cartan matrix,
(ii) A is isomorphic to one of the bound
quiver algebras A1(λ), A2(λ), λ ∈ K \0,1, A3 (if charK = 2), or Ai, 4 ≤ i ≤16.
(A3 is the preprojective algebra of type D4)
Corollary. Let A be a weakly symmetric al-
gebra of tubular type with nonsingular Car-
tan matrix. Then A has at most four sim-
ple modules and the stable Auslander-Reiten
quiver of A consists of tubes of rank ≤ 4.
94
Socle equivalences
Theorem (Bialkowski-Skowronski, 2003).
Let Λ be a selfinjective K-algebra. Then Λ is
socle equivalent to a selfinjective algebra of
tubular type if and only if exactly one of the
following cases holds:
(i) Λ is of tubular type,
(ii) K is of characteristic 3 and Λ is isomor-
phic to one of the bound quiver alge-
bras
Λ1
α2 = γββαγ = βα2γβαγβ = 0γβαγ = 0
αγβ
Λ2
α2γ = 0, βα2 = 0γβγ = 0, βγβ = 0
βγ = βαγα3 = γβ
αγβ
(iii) K is of characteristic 2 and Λ is isomor-
phic to one of the bound quiver algebras
95
Λ3(λ)λ ∈ K \ 0,1
α4 = 0, γα2 = 0, α2σ = 0α2 = σγ + α3, λβ2 = γσ
γα = βγ, σβ = ασ
α βσγ
Λ4
δβδ = αγ, (βδ)3β = 0γβαγ = 0, αγβα = 0
γβα = γβδβα
α
β
γδ
Λ5
α2 = γβ, α3 = δσ, βδ = 0σγ = 0, αδ = 0, σα = 0
γβγ = 0, βγβ = 0, βγ = βαγ
αβγ
δσ
Λ6
αδγδ = 0, γδγβ = 0αβα = 0, βαβ = 0
αβ = αδγβ
βα = δγδγ
αβ
δγ
Λ7
βδ = βαδ, ασ = 0, αδ = σγγβα = 0, α2 = δβ, γβδ = 0βδβ = 0, δβδ = 0
α β
γδ
σ
Λ8
δβ = δαβ, σα = 0, δα = γσαβγ = 0, α2 = βδ, δβγ = 0βδβ = 0, δβδ = 0
α β
γδ
σ
Λ9
βα+ δγ + εξ = 0γδ = 0, ξε = 0, αβα = 0βαβ = 0, αβ = αδγβ
αβ
δ γεξ
Λ10
µβ = 0, αη = 0, βα = δγ
ξσ = ηµ, σδ = γξ + σδσδ
δσδσ = 0, ξγξγ = 0
γξ
δσα β
η µ
96
Nonstandard algebras
char tub. type nonstandard algebras standard algebras
3 (3,3,3) Λ1
α2 = γββαγ = βα2γβαγβ = 0γβαγ = 0
αγ
β2 1
A5
α2 = γββαγ = 0
αγ
β2 1
Λ2
α2γ = 0, βα2 = 0γβγ = 0, βγβ = 0
βγ = βαγ
α3 = γβ
αγ
β2 1
A6
α3 = γββγ = 0βα2 = 0α2γ = 0
αγ
β2 1
2 (2,2,2,2)Λ3(λ)
λ ∈ K \ 0,1
α4 = 0, γα2 = 0, α2σ = 0α2 = σγ + α3, λβ2 = γσ
γα = βγ, σβ = ασ
α βσγ1 2
A2(λ)λ ∈ K \ 0,1
α2 = σγλβ2 = γσγα = βγσβ = ασ
α βσγ1 2
(3,3,3) Λ9
βα+ δγ + εξ = 0γδ = 0, ξε = 0, αβα = 0βαβ = 0, αβ = αδγβ
αβ
δ γ
εξ
4
3
1 2
A3
βα+ δγ + εξ = 0αβ = 0, ξε = 0
γδ = 0
αβ
δ γ
εξ
4
3
1 2
(2,3,6) Λ10
µβ = 0, αη = 0, βα = δγξσ = ηµ, σδ = γξ + σδσδδσδσ = 0, ξγξγ = 0
γξ
δσα β
η µ2 1 3
5
4
A29
µβ = 0, αη = 0, σδ = γξβα = δγ, ξσ = ηµ
γξ
δσα β
η µ2 1 3
5
4
97
char tub. type nonstandard algebras standard algebras
2 (2,4,4) Λ4
δβδ = αγ, (βδ)3β = 0γβαγ = 0, αγβα = 0
γβα = γβδβα
α
β
γδ
1
2 3
A12
δβδ = αγ
γβα = 0, (βδ)3β = 0
α
β
γδ
1
2 3
Λ5
α2 = γβ, α3 = δσ, βδ = 0σγ = 0, αδ = 0, ασ = 0
γβγ = 0, βγβ = 0, βγ = βαγ
αβγ
δσ
1 2 3
A13
α2 = γβ, βδ = 0, βγ = 0σγ = 0, αδ = 0, σα = 0
α3 = δσ
αβγ
δσ
1 2 3
Λ6
αδγδ = 0, γδγβ = 0αβα = 0, βαβ = 0
αβ = αδγββα = δγδγ
αβ
δγ1 2 3
A14
βα = δγδγαδγδ = 0γδγβ = 0αβ = 0
αβ
δγ1 2 3
Λ7
βδ = βαδ, ασ = 0, αδ = σγγβα = 0, α2 = δβ, γβδ = 0βδβ = 0, δβδ = 0
αβ
γδ
σ3
2 1
A15
γβα = 0, α2 = δββδ = 0, ασ = 0, αδ = σγ
αβ
γδ
σ3
2 1
Λ8
δβ = δαβ, σα = 0, δα = γσαβγ = 0, α2 = βδ, δβγ = 0βδβ = 0, δβδ = 0
αβ
γδ
σ3
2 1
A16
αβγ = 0, α2 = βδδβ = 0, σα = 0, δα = γσ
αβ
γδ
σ3
2 1
98
Theorem (Bialkowski-Skowronski, 2003).
Let Λ be a nonstandard algebra from the left
column and A the corresponding standard al-
gebra from the right column (of the above
table). Then
(1) dimK Λ = dimK A,
(2) Λ A,
(3) Λ and A are socle equivalent,
(4) A is a degeneration of Λ.
Theorem (Skowronski, 2004). Let A be a
selfinjective K-algebra. TFAE:
(1) A is nonstandard nondomestic of polyno-
mial growth.
(2) A is isomorphic to one of the algebras
• Λ1 or Λ2, for K of characteristic 3,
• Λ3(λ), Λ4, Λ5, Λ6, Λ7, Λ8, Λ9, or Λ10,
for K of characteristic 2.
99
Example.
Q
γ
B5
α
βγ
α
βγ
α
βγ
α
βγ
α
βγ
α
βγ
$
I =
⟨all α3 − γβ,
βγ, βα2, α2γ
⟩
R = KQ/I
R = B5
B5 tubular algebra
of type (3,3,3)
gl.dimB5 = 2
Q
α
β
γI =
⟨α3 − γβ, βγ, βα2, α2γ
⟩
A6 = KQ/I = R/Z is a nondomestic selfin-jective algebra of polynomial growth
100
Λ2 = KQ/I(1), I(1) =⟨α3−γβ, βγ−βαγ, βα2, α2γ
⟩,
A6, Λ2 selfinjective algebras of dimension 11
A6∼= Λ2 ⇐⇒ charK = 3
charK = 3 ⇒ Λ2 is nonstandard
A6/ socA6∼= Λ2/ socΛ2
Λ(t) = KQ/I(t), I(t) =⟨α3−γβ, βγ−tβαγ, βα2, α2γ
⟩,
t ∈ KΛ(t) ∼= Λ(1) = Λ2 for t ∈ K \ 0
A6 = Λ(0) = limt→0
Λ(t), A6 ∈ GL11(K)Λ2
A6 is a degeneration of Λ2 (Λ2 is a deformationof A6) ∗ ∗
T (1) = T (0) =∨
λ∈P1(K)T (0)λ
T (q) =∨
λ∈P1(K)T qλ
ΓA6= ΓΛ2
q ∈ Q ∩ (0,1)
101
3. TAME SYMMETRIC ALGEBRASWITH PERIODIC MODULES
A selfinjective K-algebra
ΩA Heller’s syzygy operator
M finite dimensional A-module
P(M) projective cover of M
0 −→ ΩA(M) −→ PA(M) −→ M −→ 0
M is ΩA-periodic if ΩnA(M) ∼= M for some
n ≥ 1
A symmetric ⇒ τA = DTr = Ω2A ⇒
(M is ΩA-periodic ⇐⇒ M is τA-periodic)
PROBLEM. Determine the Morita equiv-
alence classes of the (tame) finite dimen-
sional selfinjective algebras A whose all
indecomposable nonprojective finite di-
mensional modules are ΩA-periodic.
May assume A is basic and connected
102
charK = p > 0
G finite group
B block of the group algebra KG
DB defect group of B (p-subgroup of G)
B is of finite type ⇐⇒ DB cyclic
B is tame of
infinite type⇐⇒ p = 2 and DB is dihedral,
semidihedral, quaternion
The tame blocks B of infinite type belong
to the families of algebras of dihedral type,
semidihedral type and quaternion type,
classified completely by Erdmann (1988),
by quivers (with at most 3 vertices) and
relations.
103
For a block B of a group algebra KG, we
have:
B is of infinite type and
all indecomposable nonpro-
jective finite dimensional B-
modules are ΩB-periodic
⇐⇒
p = 2
and DB is a
quaternion
group
An algebra A is of quaternion type if:
• A is symmetric, connected, tame of infi-
nite type
• The indecomposable nonprojective finite
dimensional A-modules are ΩA-periodic
of period dividing 4
• The Cartan matrix of A is nonsingular
104
Theorem (Erdmann, 1988). Let A be an
algebra of quaternion type. Then A is Morita
equivalent to one of the bound quiver alge-
bras:
•α β
α2 = (βα)k−1β, β2 = (αβ)k−1α(αβ)k = (βα)k, (αβ)kα = 0k ≥ 2
•α β
charK = 2α2 = (βα)k−1β + c(αβ)k
β2 = (αβ)k−1α+ d(αβ)k
(αβ)k = (βα)k, (αβ)kα = 0(βα)kβ = 0k ≥ 2, c, d ∈ K, (c, d) = (0,0)
•α β
•γ
γβγ = (γαβ)k−1γαβγβ = (αβγ)k−1αβα2 = (βγα)k−1βγ + c(βγα)k
α2β = 0k ≥ 2, c ∈ K
•α β
•γ
η
γβ = ηs−1, βη = (αβγ)k−1αβηγ = (γαβ)k−1γαα2 = a(βγα)k−1βγ + c(βγα)k
α2β = 0, γα2 = 0k ≥ 1, s ≥ 3, a ∈ K∗, c ∈ K
•α β
•γ
η
αβ = βη, ηγ = γα, βγ = α2
γβ = η2 + aηs−1 + cηs
αs+1 = 0, ηs+1 = 0γαs−1 = 0, αs−1β = 0s ≥ 4, a ∈ K∗, c ∈ K
•α β
•γ
η
αβ = βη, ηγ = γα, βγ = α2
γβ = aηt−1 + cηt
α4 = 0, ηt+1 = 0, γα2 = 0α2β = 0t ≥ 3, a ∈ K∗, c ∈ K(t = 3 ⇒ a = 1, t > 3 ⇒ a = 1)
•β
•γ
δ •η
βδη = (βγ)k−1βδηγ = (γβ)k−1γηγβ = d(ηδ)s−1ηγβδ = d(δη)s−1δβδηδ = 0, ηγβγ = 0k, s ≥ 2, d ∈ K∗
(k = s = 2 ⇒ d = 1, else d = 1)
•β
•γ
δ •η
βγβ = (βδηγ)k−1βδηγβγ = (δηγβ)k−1δηγηδη = (ηγβδ)k−1ηγβδηδ = (γβδη)k−1γβδβγβδ = 0, ηδηγ = 0k ≥ 2
105
•α β
•γ
δ •η
βγ = αs−1
αβ = (βδηγ)k−1βδηγα = (δηγβ)k−1δηγηδη = (ηγβδ)k−1ηγβδηδ = (γβδη)k−1γβδα2β = 0, βδηδ = 0k ≥ 1, s ≥ 3
•β
•γ
δ •η
β = 0, γ = 0, η2 = 02δ = 0δη−γβ = s−1, η = (ηδ)k−1ηδ = (δη)k−1δ, (βγ)k−1βδ = 0(ηδ)k−1ηγ = 0k ≥ 2, s ≥ 3
•α β
•γ
δ •η
ξ
βγ = αs−1
γα = (δηγβ)k−1δηγαβ = (βδηγ)k−1βδηηδ = ξt−1
δξ = (γβδη)k−1γβδξη = (ηγβδ)k−1ηγβα2β = 0, δηδ = 0k ≥ 1, s, t ≥ 3
•β
κ33R
RRRR
RRRR
R •γ
δ44SSSSSSSSSS
•λ
<;RRRRRRRRRR
η=<SSSSSSSSSS
βδ = (κλ)a−1κηγ = (λκ)a−1λδλ = (γβ)b−1γκη = (βγ)b−1βλβ = (ηδ)c−1η, γκ = (δη)c−1δγβδ = 0, δηγ = 0, λκη = 0a, b, c ≥ 1 (at most one equal 1)
These algebras are of quaternion type:
tameness by the degeneration argument
(Geiss), and the Ω-period dividing 4 by the
derived equivalence classification (Holm) and
the explicit constructions of the bimodule
resolutions for the derived representatives
(Erdmann)
106
THEOREM (Erdmann-Skowronski, 2004).
Let A be a symmetric algebra. TFAE:
(1) A is tame with all indecomposable non-
projective finite dimensional modules are
ΩA-periodic.
(2) A is isomorphic to an algebra of one of
the forms:
• socle deformation of a symmetric
algebra of Dynkin type;
• socle deformation of a symmetric
algebra of tubular type;
• algebra of quaternion type.
107
Corollary. Let A be a tame symmetric alge-
bra with all indecomposable nonprojective fi-
nite dimensional modules ΩA-periodic. Then
(1) The Cartan matrix CA of A is singular if
and only if A is isomorphic to the trivial
extension T(B) of a tubular algebra B.
(2) If A is of infinite type with nonsingular
Cartan matrix CA then A has at most 4
simple modules and the stable Auslander-
Reiten quiver ΓsA of A consists of tubes
of rank at most 4.
(3) If A is of infinite type then A has at most
10 simple modules.
CT(B) = −(ΦB − In)CB, n = rkK0(B),
ΦB = CtBC−1B Coxeter matrix of B
B tubular algebra ⇒ 1 is an eigenvalue of ΦB⇒ detCT(B) = 0.
108
4. TAME STANDARDSELFINJECTIVE ALGEBRAS
B algebra, gl.dimB <∞
P1, P2, . . . , Pn complete set of indecompos-
able projective B-modules
CB =(dimK HomB(Pi, Pj)
)Cartan matrix of B
CB is Z-invertible, K0(B) = Zn
χB : K0(B) → Z Euler form of B
χB(x) = xC−tB xt for x ∈ K0(B)
M ∈ modB, [M ] ∈ K0(B)Ringel
====⇒
χB([M ]) =∞∑i=0
(−1)i dimK ExtiB(M,M)
109
THEOREM (Skowronski, 2002). Let B
be a simply connected algebra. TFAE:
(1) χB is nonnegative.
(2) T(B) is tame.
(3) Db(modB) is tame.
(4) B is tilting-cotilting equivalent to a tame
simply connected generalized canonical
algebra.
B triangular algebra with T(B)
(resp. Db(modB)) tame, then
B is simply connected ⇐⇒ HH1(B) = 0
110
Tame simply connected generalized
canonical algebras
(1) K∆(An), K∆(Dn), K∆(E6), K∆(E7),
K∆(E8)
∆(An) • • • · · · • • n ≥ 1 vertices
∆(Dn) •""
• •
•
• · · · • •
n ≥ 4 vertices
∆(E6) •""
•
• •
•
•
∆(E7) •""
•
• •
•
• •
∆(E8) •""
•
• •
•
• • •
111
(2) C(p, q, r), 2 ≤ p ≤ q ≤ r, 1p + 1
q + 1r ≥ 1
•α1
•α2 · · · • •αp−1
• •β1 •β2 · · · • •βq−1 •
αp βq
γq
•γ1
•γ2 · · · • •γr−1
bounded by
αp . . . α2α1 + βq . . . β2β1 + γr . . . γ2γ1 = 0
1p+
1q+
1r > 1 =⇒ (p, q, r) = (2,2, r), r ≥ 2,
(2,3,3), (2,3,4),
(2,3,5)1p+
1q+
1r = 1 =⇒ (p, q, r) = (3,3,3), (2,4,4),
(2,3,6)
112
(3) Cλ = C(2,2,2,2, λ), λ ∈ K,
•
α1
>========================
•β1
• •
α2
?>TTTTTTTTTTTTTTTTTTTTTTT
β2..
γ2
δ2
>========================
•γ1
..
•
δ1
?>TTTTTTTTTTTTTTTTTTTTTTT
bounded by α2α1 + β2β1 + γ2γ1 = 0,
α2α1 + λβ2β1 + δ2δ1 = 0.
C(2,2, r), r ≥ 2, C(2,3,3), C(2,3,4), C(2,3,5)
domestic canonical algebras of Euclidean
types Dr+2, E6, E7, E8
C(3,3,3), C(2,4,4), C(2,3,6), Cλ, λ ∈ K \ 0,1,tubular canonical algebras of tubular types
(3,3,3), (2,4,4), (2,3,6), (2,2,2,2)
113
(4) Λ(m,n), 2 ≤ m ≤ n,
•α1
@?UUUUUUUU
UUUUUUUU
UUUUUUUU
UUUUUUUU
UUUUUUUU
UUUUU
•β1A@VVVVVVVVVV
VVVVVVVVVVVVVVVVVVVV
VVVVVVVVVVVVVV
• •γn
--<<<<<<<<<
β2BAWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW
α2
CBXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
•γ1..YYYYYYY •γ2
η2DCEEEEEEEEEEE
· · · •444
444•γm−1
ηm−1DCEEEEEEEEEEE
· · ·,,444
444
EDYYYYYY
•γm
δm""
•γm+1 · · · • •γn−1
•δ1
FE////////////////
•δ2
ξ2
GF
· · · •YYYYYY •
δm−1
ξm−1
GF
bound by α2α1 + β2β1 + γn . . . γ2γ1 = 0,
γ2γ1 = η2δ1, δ2δ1 = ξ2γ1, γi+1γi = ηi+1ξi,
δi+1δi = ξi+1ηi, ξi+1γi = δi+1ξi, ηi+1δi =
γi+1ηi, i ∈ 2, . . . ,m− 2, γmγm−1 = δmξm−1,
δmδm−1 = γmηm−1.
Λ(2, n), n ≥ 2, canonical pg-critical algebras
Λ(2,2) ∼= C0∼= C1
114
THEOREM (Skowronski, 2003). Let A be
a nonsimple standard selfinjective algebra. Then
A is tame ⇐⇒ A ∼= B/G, where
(1) B is a simply connected locally bounded
K-category of the form B =⋃n≥0
Bn for a
chain
B0 ⊆ B1 ⊆ · · · ⊆ Bn ⊆ Bn+1 ⊆ · · ·
of simply connected bounded K-catego-
ries with nonnegative Euler forms, Bn
convex in Bn+1 for any n ≥ 0;
(2) G is a torsion-free admissible group of
automorphisms of B.
115
Theorem (Pogorzaly-Skowronski, 1991).
Let A be a selfinjective algebra of infinitetype. TFAE:
(1) A is standard and biserial.
(2) A is special biserial.
(3) A ∼= R/G, where R is a selfinjective sim-ply connected locally bounded K-categorywhose every full bounded subcategory isof finite type and G is an admissible group
of automorphisms of R.
(4) A ∼= B/G, where B is a simply connectedlocally bounded K-category of the formB =
⋃n≥0
Bn, for a chain
B0 ⊆ B1 ⊆ · · · ⊆ Bn ⊆ Bn+1 ⊆ · · ·of iterated tilted algebras of Dynkin typesAm, Bn convex in Bn+1 for any n ≥ 0,and G is an admissible torsion-free groupof automorphisms of B.
Special biserial algebras are tame: Wald-
Waschbusch (1985), Dowbor-Skowronski
(1987), Butler-Ringel (1987)
116
THEOREM (Skowronski, 2003). Let A be
a tame standard selfinjective algebra. Then
the connected components of the stable
Auslander-Reiten quiver ΓsA of A are of the
forms:
• Z∆/G, ∆ Dynkin graph
• Z∆, ∆ Euclidean graph
• ZA∞/(τn), n ≥ 1, stable tubes
• ZA∞∞
• ZD∞
117
PROBLEM. Does the class of all tame
selfinjective algebras over algebraically
closed fields forms an open Z-scheme in
any dimension d?
PROBLEM. Are the tame selfinjective
algebras of (Auslander’s) representation
dimension at most 3?
PROBLEM. Are the stable module
categories modA of the tame selfinjective
algebras A of (Rouquier’s) dimension at
most 1?
PROBLEM. Determine the Morita
equivalence classes of the tame blocks
of finite dimensional Hopf algebras
118