SELFINJECTIVE ALGEBRAS: FINITE AND TAME TYPE

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1 SELFINJECTIVE ALGEBRAS: FINITE AND TAME TYPE Andrzej Skowroński (Querétaro, August 2004) ( skowron/selfinjective2004.pdf)

2 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

3 1. PRELIMINARIES K A mod A moda algebraically closed field finite dimensional K-algebra category of finite dimensional right A-modules stable category of mod A (modulo projectives) D b (mod A) derived category of bounded complexes over mod A (triangulated category) Γ A Γ s A Auslander-Reiten quiver of A τ A = D Tr, τ A =TrD Auslander-Reiten translations stable Auslander-Reiten quiver of A A, B finite dimensional K-algebras A and B are: Morita equivalent if mod A = mod B stably equivalent if moda = modb derived equivalent if D b (mod A) = D b (mod B) (as triangulated categories) 1

4 A finite dimensional K-algebra A tame: d 1 M1,...,M nd K[x]-A-bimodules such that M i free left K[x]-modules of finite rank all but finitely many isoclasses of indecomposable right A-modules of dimension d are of the form K[x]/(x λ) K[x] M i,1 i n d, λ K µ A (d) = least number on K[x]-A-bimodules satisfying the above condition for d A tame = { finite discrete ind d A = set } { µa (d) one-parameter families } A is not tame ==== Drozd representation theory of A comprises the representation theories of all finite dimensional K-algebras 2

5 Hierarchy of tame algebras: A of finite type d 1 µ A (d) =0 A domestic m 1 A polynomial growth m 1 µ A(d) m d 1 µ A(d) d m d 1 A tame d 1 µ A (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 by the Morita and stable equivalences (Krause, Krause-Zwara) Open problem: A polynomial growth A linear growth ( µ A(d) md) m 1 d 1 3

6 A finite dimensional K-algebra D = Hom K (,K) standard duality of mod A A is selfinjective if A A = D(A)A (projective A-modules are injective) A is symmetric if A A A = A D(A) A A basic algebra, then A selfinjective A Frobenius A is Frobenius (symmetric) if there is a nondegenerate (symmetric) K-bilinear associative form (, ) :A A K (a, bc) =(ab, c), a, b, c, K A is weakly symmetric if top P = soc P for any indecomposable projective A-module P Symmetric weakly symmetric selfinjective Frobenius 4

7 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)/(x p 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)/(x p x [p] χ(x) p 1,x L) of restricted Lie algebras (L, [p]), for linear forms χ : L K, dim K U(L, χ) = p dim K L, u(l, 0) = u(l) (are Frobenius algebras: Farnsteiner-Strade) (3) Finite dimensional Hopf algebras (are Frobenius algebras: Larson-Sweedler) 5

8 (4) Hochschild extension algebras 0 D(A) T A 0 (equivalence classes form the Hochschild cohomology group H 2 (A, D(A))) In particular, we have the trivial extension 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, then we have in mod A an Auslander-Reiten sequence of the form 0 rad P (rad P/soc P ) P P/soc P 0 Hence Γ s A is obtainded from Γ A by removing the indecomposable projective modules and the arrows attached to them A selfinjective Nakayama ==== soc A A = soca = soc A A A, B selfinjective algebras A and B are socle equivalent if A/ soc A = B/ soc B 6

9 PROBLEM. Determine the Morita equivalence classes of the tame finite dimensional selfinjective algebras For selfinjective algebras, Morita equivalence derived equivalence Rickard ==== 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 selfinjective algebras 7

10 We may assume algebra = basic, connected, finite dimensional K-algebra A algebra A = KQ/I Q = Q A Gabriel quiver of A, I admissible ideal in the path algebra KQ of Q tame (basic) selfinjective algebras mod A = rep K (Q, I) ( standard algebras admit simply connected Galois coverings nonstandard algebras Representation theory of tame standard selfinjective algebras can be reduced to the representation theory of tame algebras of finite global dimension (tame simply connected algebras with nonnegaive Euler forms) 8 )

11 A connected K-category R is locally bounded if: distinct objects of R are nonisomorphic R(x, x) is a local algebra x obr x obr y obr (dim K R(x, y)+dim K R(y, x)) < R = KQ/I, Q locally finite connected quiver, I admissible ideal of the path category KQ mod R category of finitely generated contravariant functors R mod K mod R =rep K (Q, I) R bounded (has finitely many objects) R = R(x, y) finite dimensional basic x,y obr connected K-algebra We will identify a bounded K-category R with the associated finite dimensional algebra R 9

12 R locally bounded K-category G group of K-linear automorphisms of R G is admissible if G acts freely on the objects of R and has finitely many orbits R/G orbit (bounded) category objects: G-orbits of objects of R (R/G)(a, b) = (f yx) (x,y) a b R(x, y) g f yx = f g(y),g(x) F : R R/G canonical Galois covering ob(r) x Fx = G x ob(r/g) x obr a ob(r/g) Fy=a Fy=a F induces isomorphisms R(x, y) R(y, x) (R/G)(Fx,a), (R/G)(a, F x) g G,x a,y b 10

13 The group G acts also on mod R mod R M gm = Mg 1 mod R We have also the push-down functor (Bongartz-Gabriel) F λ :modr mod R/G M mod R, a ob(r/g) (F λ M)(a) = x a M(x) Assume G is torsion-free. an injection (Gabriel) G-orbits of isoclasses of indecomposable modules in mod R F λ Then F λ induces isoclasses of indecomposable modules in mod R/G R is locally support-finite if for any x obr supp(m) is a bounded category M ind R M(x) 0 R locally support-finite Dowbor-Skowroński ==== is dense F λ Then Γ R/G = ΓR /G (Gabriel) 11

14 R selfinjective locally bounded K-category G admissible group of automorphisms of R R/G basic connected finite dimensional selfinjective K-algebra B algebra 1 B = e e n e 1,...,e n orthogonal primitive idempotents of B B repetitive category of B (selfinjective locally bounded K-category) objects: e m,i,m Z, 1 i n B(e m,i,e r,j )= e j Be i,r= m D(e i Be j ),r= m +1 0, otherwise e j Be i =Hom B (e i B, e j B), D(e i Be j )=e j D(B)e i (m,i) Z {1,...,n} B(,e r,j )(e m,i )=e j B D(Be j ) 12

15 ν B : B B Nakayama automorphism of B ν B(e m,i )=e m+1,i for all m, i Z {1,...,n} (ν B) admissible group of automorphisms of B F B : B B/(ν B) =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(e m,i )=e p,j for some p m and j {1,...,n} ϕ is rigid if for each pair (m, i) Z {1,...,n} exists j {1,...,n} such that ϕ(e m,i )=e m,j ϕ is strictly positive if it is positive but not rigid Note that ν B is strictly positive. 13

16 Assume B is triangular (Q B has no oriented cycles) Then B is triangular B is the full bounded subcategory of B given by the objects e 0,i, 1 i n Let i be a sink of Q B B S i + B reflection of B at i S + i B the full subcategory of B given by the objects e 0,j,1 j n, j i, ande 1,i = ν B(e 0,i ). σ + i Q B = Q S + i B reflection of Q B at i Observe that B = S + i B 14

17 Reflection sequence of sinks of Q B : a sequence i 1,...,i t of vertices of Q B such that i s is a sink of σ + i s 1...σ + i 1 Q B for 1 s t. Two triangular algebras B and C are said to be reflection equivalent if C = S + i t...s + i 1 B for a reflection sequence of sinks i 1,...,i t of Q B. B, C reflection equivalent triangular algebras B = Ĉ 15

18 1 α 1 2 α 2 3 n 1 α n 1 n B = K B = K /În Î n generated by all compositions of n + 1 consecutive arrows in ν B : B B Nakayama automorphism, ν B(r, i) =(r +1,i) (r, i) Z {1,...,n} ϕ : B B, ϕ n = ν B N n m = B/(ϕ m )=KC m /J m,n m 1 α m 2 α m 1 m 2 m αm 1... C m... α J m,n generated by all compositions of n + 1 consecutive arrows in C m Nm n Nakayama algebra, Nn = T (B) Nm n symmetric m n ϕ m is a root of ν B α 2 :. (2,n) α 2,n (1, 1) α 1,1 (1, 2) α 1,2. (1,n 1) α 1,n 1 (1,n) α 1,n (0, 1) α 0,1 (0, 2) α 0,2. (0,n 1) α 0,n 1 (0,n) α 0,n ( 1, 1) α 1,1 ( 1, 2) α 1,2. 16

19 1 2 α 1 α 2 α B = K B = K /În : Î n generated by β m,i α m,i β m,j α m,j, α m,i β m 1,j, m Z, i, j {1, 2, 3}, i j. ν B : B B Nakayama ν B(m, i) =(m +1,i) ϱ : B B ϱ = {((m, 1), (m, 3))} σ : B B σ = {((m, 1), (m, 2), (m, 3))}.. β 1,1 (2, 0) β 1,2 (1, 1) (1, 2) α 1,1 β 0,1 (1, 0). β 1,3 α 1,2 α 1,3 β 0,2 (0, 1) (0, 2) α 0,1 β 1,1 (0, 0) β 0,3 α 0,2 α 0,3 β 1,3 β 1,2 ( 1, 1) ( 1, 2). α 1,1 α 1,2 (1, 3) (0, 3) ( 1, 3) α 1,3 ( 1, 0).. A = T (B), A = B/(ϱν B), A = B/(σν B) 17

20 A = KQ/I, A = KQ/I, A = KQ/I, 2 Q : 1 α 1 β 2 β 1 0 α 2 α 3 I = β 1 α 1 β 2 α 2,β 2 α 2 β 3 α 3,α 1 β 2,α 1 β 3,α 2 β 1,α 2 β 3,α 3 β 1,α 3 β 2 I = β 1 α 1 β 2 α 2,β 2 α 2 β 3 α 3,α 1 β 1,α 1 β 2,α 2 β 1,α 2 β 3,α 3 β 2,α 3 β 3 I = β 1 α 1 β 2 α 2,β 2 α 2 β 3 α 3,α 1 β 1,α 1 β 2,α 2 β 2,α 2 β 3,α 3 β 1,α 3 β 3 β 3 3 Γ A : Γ A : P 0 P 1 P 1 /S 1 P 2 P 1 /S 1 P 2 /S 2 P 2 /S 2 P 3 /S 3 P 3 /S 3 P 3 /S 1 P 3 P 0 P 3 P 1 /S 3 P 2 P 3 /S 1 P 2 /S 2 P 2 /S 2 P 1 /S 3 P 1 Γ A : P 0 P 2 P 1 /S 3 P 3 P 2 /S 1 P 2 /S 1 P 3 /S 2 P 1 /S 3 P 3 /S 2 P 1 18

21 Λ = KQ/I locally bounded K-category (Q, I) Π 1 (Q, I) fundamental group I is generated by elements (relations) ofthe path category KQ of the form ϱ = λ 1 u 1 + λ 2 u λ m u m,m 1, λ 1,λ 2,...,λ m K \{0}, u 1,u 2,...,u m are parallel paths in Q x u 1 u 2. y u m m(i) a set of minimal relations generating the ideal I Π 1 (Q, I) =Π 1 (Q, x 0 )/N (Q, m(i),x 0 ) Π 1 (Q, x 0 ) fundamental group of the quiver Q at a fixed vertex x 0 of Q N(Q, I, x 0 ) normal subgroup of Π 1 (Q, x 0 ) generated by the homotopy classes I [wuv 1 w 1 ] x 0 w x walk u y u v = u r,v = u s ϱ = λ i u i m(i) 19

22 WemayassociatetoΛ=KQ/I a universal Galois covering F : Λ =K Q/Ĩ Λ/G =Λ=KQ/I with group G =Π 1 (Q, I). (Green, Martinez de la Peña) W topological universal cover Q with base point x 0 Π 1 (Q, x 0 )actsonw Π 1 (Q, m(i),x 0 )actsonw Q = W/N(Q, m(i),x 0 ) orbit quiver Π 1 (Q, I) actson Q and induces a map p : Q Q of quivers Ĩ generated by liftings of minimal generators (from m(i)) of I to K Q 20

23 Λ=K[x, y]/(x 2,y 2 ) Λ=KQ/I Q : α β I = α 2,β 2,αβ βα Π 1 (Q, I) =Z Z, Λ =K Q/Ĩ is given by α β... α β... α α β α β α β β Ĩ is generated by all paths α 2, β 2, αβ βα in Q Note that Λ = T (K ), where : α Kronecker quiver β and Λ =K Q/Î is given by Q :... α α... β β Î = all α 2,β 2,αβ βα Λ = R for R = Λ 21

24 R locally bounded K-category R is simply connected (Assem-Skowroński) if, for any presentation R = KQ/I of R as a bound quiver category, Q = Q R has no oriented cycles (R is triangular) 1(Q, I) istrivial 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

25 Brauer tree algebras Brauer tree: a finite connected tree T = T m S together with a circular ordering of the edges converging at each vertex one exceptional vertex S with multiplicity m 1 We draw T in a plane such that the edges converging at any vertex have the clockwise order Brauer tree T Brauer quiver Q T : the vertices of Q T are the edges of T there is an arrow i j in Q T j is the consecutive edge of i in the circular ordering of the edges converging at a vertex of T Q T has the following structure: Q T is a union of oriented cycles corresponding to the vertices of T Every vertex of Q T belongs to exactly two cycles The cycles of Q T are divided into two camps: α-camps and β-camps such that two cycles of Q T having nontrivial intersection belong to different camps. We assume that the cycle of Q T corresponding to the exceptional vertex S of T is an α-cycle. 23

26 i vertex of Q T i i α i α(i) the arrow in α-camp of Q T starting at i β i β(i) the arrow in β-camp of Q T starting at i α 2 (i)... α α(i) α(i) β(i) ββ(i) β2 (i) α 2 (i) αα 2 (i) α 1 (i) α i α α 1 (i) i β i β β 1 (i) β 1 (i) β β 2 (i) β 2 (i)... A i = α i α α(i)...α α 1 (i) B i = β i β β(i)...β β 1 (i) T = TS m A(T )=A(T S m)=kq TS m/im S Brauer tree algebra IS m ideal in KQ TS m generated by elements : β β 1 (i) α i and α α 1 (i) β i A m i B i if the α-cycle passing through i is exceptional A i B i if the α-cycle passing through i is not exceptional for all vertices i of Q T m. S 24

27 Example. Let T = T 6 S is of the form S m =6 Q T = Q T 6 S is of the form β 4 β 5 4 α 4 5 α 3 α 5 3 β 2 β 3 α α 6 β 1 β 6 1 α 1 IS 6 generated by α 1 β 1, β 6 α 1, β 1 α 2, α 2 β 2, β 2 α 3, α 5 β 3, α 3 β 4, β 4 α 4, α 4 β 5, β 5 α 5, α 6 β 6, β 3 α 6, α 6 1 β 1β 2 β 3 β 6, α 2 β 2 β 3 β 6 β 1, α 3 α 4 α 5 β 3 β 6 β 1 β 2, α 4 α 5 α 3 β 4, α 5 α 3 α 4 β 5, α 6 β 6 β 1 β 2 β 3 25

28 Example. Let T = T m S be the star Q T = Q T m S e S e 1 is of the form β e 1 e 1 e α e 1 β 1 β e α e α β 2 α 2.. symmetric Nakayama alge- A(TS m bra )=N em e In general, T = TS m a Brauer tree the Brauer tree algebra A(T ) is (special) biserial and of finite type 26

29 Λ selfinjective algebra Λ is biserial if the heart H(P )=radp/soc P 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 conditions: (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 (Skowroński-Waschbüsch, 1983) 27

30 char K = p>0 G finite group, p G KG = B 0 B 1 B r, B 0,B 1,...,B r 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-subgroup then the blocks B 0,B 1,...,B r are Morita equivalent to symmetric Nakayama algebras (application of Clifford s theorem) In general, let B be a block of KG B D = D B defect group of B Dp-subgroup of G mod B X X Y KD KG, for some Y mod KD B of finite type D B is cyclic 28

31 Theorem (Dade-Janusz-Kupisch, ). Let B be a block of a group algebra KG with cyclic defect group D B. Then B is Morita equivalent to a Brauer tree algebra A(TS m). (Here me +1=p n if D B = p n and B has e simple modules) Remark. Most of the Brauer tree algebras A(T m s ) are not Morita equivalent to blocks of group algebras (Feit, 1984). Theorem (Gabriel-Riedtmann (1979), Rickard (1989)). Let A be a selfinjective algebra. 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 group algebras) with cyclic defect groups D B and D B, one gets: B and B are derived equivalent D B = DB (solution of Broué s conjecture in the cyclic defect case) 29

32 2. SELFINJECTIVE ALGEBRAS OF POLYNOMIAL GROWTH Selfinjective algebras of finite type Domestic selfinjective algebras of infinite type Nondomestic selfinjective algebras of polynomial growth Selfinjective algebras of Dynkin type: B tilted of Dynkin type B/G, Selfinjective algebras of Euclidean type: B tilted of Euclidean type B/G, Selfinjective algebras of tubular type: B tubular B/G, 30

33 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 Ā (standard form of A) ofpolynomial growth such that dim K A =dim K Ā, A and Ā are socle equivalent, Ā is a degeneration of A. (3) If A is nonstandard domestic then char K = 2. (4) If A is nonstandard of polynomial growth then char K =2or 3. COROLLARY. Every selfinjective algebra of polynomial growth is of linear growth. 31

34 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 ν B of B. PROBLEM. Let A and A be stably equivalent 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-Skowroński, 2004) weakly symmetric algebras of polynomial growth (Bia lkowski-holm-skowroński, 2003) 32

35 SELFINJECTIVE ALGEBRAS OF FINITE TYPE Selfinjective algebras of Dynkin type {A m, D m, E 6, E 7, E 8 } Dynkin graph a Dynkin quiver with underlying graph H = K the path algebra of T mod H tilting H-module: Ext 1 H (T,T)=0 T = T 1 T n, n = 0 T 1,...,T n indecomposable pairwise nonisomorphic B =End H (T ) tilted algebra of type gl. dim B 2 B is of finite-type The Auslander-Reiten quiver Γ B of B is of the form Dynkin section 33

36 Selfinjective algebra of Dynkin type : algebra of the form B/G, whereb is a tilted algebra of a Dynkin type andg is an admissible 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

37 For tilted algebras B and B of Dynkin type, we have B = B Hughes-Waschbüsch ==== B and B are reflection equivalent For r, s 1, let Λ(r, s) =KQ r,s /I r,s, Q r,s the quiver r 2 1 β α 1 σ γ I r,s generated by αβ γσ. 2 s Λ(r, s) tilted algebra of type D r+s+2 Λ(r, s) andλ(r,s ) are reflection equivalent r + s = r + s 35

38 A = B/G selfinjective algebra of Dynkin type Γ B is of the form ( projective-injective vertices) and Γ s B is of the form Z (=Z ) τ m B where m An = n, m Dn =2n 3, m E6 = 11, m E7 = 17, m E8 = 29. In fact, we have ν B = τ m on mod B: B B =End H (T ), H = K mod T (B) = mod T (H) (Tachikawa-Wakamatsu) ind T (H) = 2 ind H (Tachikawa, Yamagata) 36

39 37

40 C the set of vertices of Γ s B = Z given by the radicals of the indecomposable projective B-modules (configuration of Z ) Then Γ B = Z C completion of Z by for all c C. c c τ c In particular, the configuration C is stable under the action of ν B = τ m on Γ B s B = Z Moreover, the admissible automorphism group G of B is infinite cyclic: G acts on Γ B, and hence also on Γ s B = Z As an automorphism group of the translation quiver Z, G =(τ r ϱ)wherer 1andthe automorphism ϱ fixes at least one vertex of Z. Note that ϱ is of order 1, 2, or 3. 38

41 We have the canonical Galois covering F : B B/G = A with G infinite cyclic. Moreover, B is locally support-finite (even locally representationfinite). Hence, the push-down functor F λ :mod B mod B/G =moda is dense, anda is of finite type. In particular Γ A = Γ B/G and Γ s A =Γs B/G = Z /G cylinder Möbius strip ZD 4 /(τ 5r (1, 2, 3)) 39

42 Theorem (Riedtmann, Waschbüsch, ). 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 Γ s A = Z /G and Γ A = 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-Waschbüsch)) 40

43 A selfinjective algebra of finite type 1 A = e e n, e 1,...,e n orthogonal primitive idempotents A standard A ( standard (Zürich) ) A regular (Berlin) ind A = K(ΓA ) e iae j cyclic mesh category 1 i,j n left e i Ae i -module and cyclic right e j Ae j -module nonstandard (classified by Riedtmann) nonregular (classified by Waschbüsch) 41

44 T = T S = TS 2 Brauer tree with at least two edges and the extreme vertex S of multiplicity 2 B 3 B S S... r r 1 B r B r 1 Then the Brauer quiver Q T = Q T 2 S form α 1 Q Br 1 Q Br 1 r 1... r j +1 β r Q B2 β 1 cycle S Q Bj+1 β j 2 j 1 3 Q B3... j Q Bj β j 1 Q Bj+1 is of the For each edge i of T (vertex i of Q T )wehave the cycles A i and B i around i Define B j = β j...β r α 1 β 1...β i 1, j 1,j S 0 42

45 For each λ K, define the algebra D(T S,λ)=KQ T /I(T S,λ) where I(T S,λ)istheidealofKQ T generated by β β 1 (i) α i and α α 1 (i) β i, i (Q T ) 0 \{1}, A 2 1 = B 1, A i B i, i (Q T ) 0 \ S 0, A j B j, j S 0 \{1}, β r β 1 λβ r α 1 β 1. Proposition. (1) D(T S,λ), λ K, are weakly symmetric algebras of finite type. (2) For λ, µ K \{0}, D(T S,λ) = D(T S,µ). (3) D(T S, 0) = D(T S, 1) char K 2. (4) D(T S, 0) and D(T S, 1) are socle equivalent. (5) D(T S, 0) = B/(ϕ), for an exceptional tilted algebra B of Dynkin type D 3m and a 3- root ϕ of ν B. (6) For char K =2, D(T S, 1) is nonstandard and degenerates to D(T S, 0). 43

46 Example. T = T 2 S of the form S 1 S 2 3 Q T = Q T 2 S of the form α 1 1 β 2 β 1 2 α 3 α 2 3 β 3 D(T S, 0) = KQ T /I(T S, 0) I(T S, 0) generated by D(T S, 1) = KQ T /I(T S, 0) I(T S, 1) generated by β 1 α 2, α 3 β 2 β 3 α 3, α 2 β 3 α 2 1 β 1β 2 α 2 α 3 β 2 α 1 β 1 α 3 α 2 β 3 β 2 β 1 1(Q T,I(T S, 0)) = Z β 1 α 2, α 3 β 2 β 3 α 3, α 2 β 3 α 2 1 β 1β 2 α 2 α 3 β 2 α 1 β 1 α 3 α 2 β 3 β 2 β 1 β 2 α 1 β 1 1(Q T,I(T S, 1)) trivial 44

47 Q : α 1 α 1 β 1 β 2 β 1 β 2 α 3 α 2 α 3 I generated by α 2 1 β 1β 2, β 2 β 1, α 3 β 2 B = KQ/I tilted algebra of type D 9 = D 3 3 B = K Q/Î is of the form Q : α 1 α 1 α 1 α 1... β 1 α2 β 2 α 3 β 1 α2 β 2 α 3 β 1 α2 β 2 α 3 β 1 α2 β 2.. Î generated by all α 2 1 β 1β 2, β 2 β 1, α 2 α 3 β 2 α 1 β 1, β 1 α 2, α 3 β 2 ϕ=shift up by one ϕ 3 = ν B D(T S, 0) = B/(ϕ) For a Brauer tree T and extreme vertex S of T, we put D(T S )=D(T S, 0) and D(T S ) 45 = D(T S, 1)

48 Theorem (Riedtmann, Waschbüsch,...). 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 ν B. (4) A is isomorphic to one of the algebras (a) T (B), B tilted of Dynkin type. (b) A(TS m), T S m Brauer tree, S exceptional of multiplicity m 2. (c) D(T S ), T Brauer tree, S extreme exceptional. Remark. The Brauer tree algebras A(T )= A(TS 1 ) are exactly the trivial extensions T (B) of tilted algebras of types A n Theorem (Riedtmann (1983), Waschbüsch (1981)). Let A be a selfinjective algebra over K. TFAE: (1) A is nonstandard of finite type, (2) A = D(T S ), T Brauer tree, S extreme exceptional, and char K =2. 46

49 Standard selfinjective algebras of finite type B tilted algebra of Dynkin type B is exceptional: there is a reflection sequence i 1,...,i t of sinks in Q t with t<rk K 0 (B) = 0 such that B = S i + t...s i + B 1 B exceptional there is a strictly positive automorphism ϕ of B such that rk K 0 ( B/(ϕ)) < rk K 0 (T (B)) = rk K 0 (B) A(TS m ) Brauer tree algebra of multiplicity m 2 D(T S ) B(TS m ) exceptional tilted algebra of type A n A(TS m) = B(T S m)/(ϕ) ϕ m = ν B(T m S ) B (T S ) exceptional tilted algebra of type D 3m D(T S ) = B (T S )/(ϕ) ϕ 3 = ν B (T S ) 47

50 Proposition. Let B be a tilted algebra of Dynkin type. TFAE: (1) B is exceptional. (2) There is an automorphism ϕ of B with ϕ m = ν B for some m 2 (ϕ proper root of ν B). (3) B = B(T m S ), m 2, or B = B (T S ). Remark. There are no exceptional tilted algebras of Dynkin types E 6, E 7, E 8 (Bretscher- Läser-Riedtmann (1981)) There is a description (by bound quivers and relations) of all (iterated) tilted algebras of type A n (Happel- Ringel (1981), Assem-Happel (1981)) (iterated) tilted algebras of type D n (Conti (1986), Assem-Skowroński (1989), Keller (1991)) The numbers of reflection classes of tilted algebras of types E 6, E 7, E 8,are: E 6 : 22 E 7 : 143 E 8 :

51 Theorem. Let A be a standard selfinjective algebra of finite type. Then A is isomorphic to an algebra of one of the forms: B/(ν r B), r 1, B tilted of type {A n, D n, E 6, E 7, E 8 } B/(ϱν r B), r 1, B tilted of type {A 2p+1, D n, E 6 }, ϱ automorphism of order 2 B/(σν r B), r 1, B = K, =, σ automorphism of order 3 B/(ϕ r ), r 1, B = B(TS m ) tilted of type =A n, ϕm-root ν B(T m S ), B/(ϕ r ), r 1, B = B (T S ) tilted of type =D 3m, ϕ 3-root ν B (T S ), There is also a complete classification of the derived and stable equivalence classes of 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

52 ZÜRICH SCHOOL APPROACH A selfinjective algebra of finite type Assume A is basic, connected, A K Γ A finite, connected, translation quiver p : Γ A Γ A / A =Γ A universal Galois covering of translation quivers, A = 1(Γ A ) fundamental group of Γ A, Γ A simply connected ( 1( Γ A )istrivial) A = 1(Γ s A ), so we have also Galois covering p : Γ s A Γ s A / A =Γ s A Theorem (Riedtmann, 1977). Γ s A = Z for a Dynkin graph {A n, D n, E 6, E 7, E 8 }. In particular, Γ s A = Z / A,and A is infinite cyclic. C A set of vertices representing the radicals of indecomposable projective A-modules (configuration of Γ s A ) Γ A =(Γ s A ) C A =(Z / A) CA C A = p 1 (C A ) configuration of Γ s A = Z Γ A =(Γ s A ) CA =(Z ) CA 50

53 K(Γ A )=KΓ A /I A mesh-category of Γ A I A generated by the meshes x y 1 y 1. y r τ x K( Γ A )=K Γ A /ĨA mesh-category of Γ A à the full subcategory of K( Γ A ) given by the projective vertices à locally bounded K-category ind A (respectively, ind Ã) full subcategory of mod A (respectively, mod Ã) formed by a complete set of indecomposable modules Then ind à = K( Γ A ). A is called standard if ind A = K(Γ A ) Then we have Galois coverings ind à = K( Γ A ) K( Γ A )/ A = K(Γ A )=inda F : à Ã/ A = A where à is simply connected and A infinite cyclic group 51

54 In fact à = B for a tilted algebra B of Dynkin type (a simple application of tilting theory) Hence, for the selfinjective algebras of finite type, the both concepts of standardness coincide The configuration C = C A of Γ = Γ s A = Z is a combinatorial configuration of Z : (a) For any vertex x of Γ there exists a vertex c C such that Hom K(Γ) (x, c) 0; (b) For any vertices c, d C we have Hom K(Γ) (c, d) = 0, if c d and Hom K(Γ) (c, c) = K;. (P indecomposable projective, Ω (rad P )= à P/rad P ) Moreover, the configuration C = C A is τ m - stable Ch. Riedtmann classified (1977) all τ m - stable combinatorial configurations of the stable translation quivers Z, in case = E 6, E 7 or E 8 together with F. Jenni by computer (E 6 : 22, E 7 : 143, E 8 : 598 isoclasses of configurations) Moreover, Bretscher-Läser-Riedtmann (1981) proved that for any τ m -stable configuration C of Z and an admissible group of Z stabilizing C,wehave Z C / = ΓA for a selfinjective algebra A of finite type 52

55 In the fact, the main result of their paper says: the configurations of Z correspond bijectively to the isomorphism classes of square -free tilting modules over K (iso- classess of basic tilted algebras of Dynkin types ) This is not correct because there are nonisomorphic tilted algebras of Dynkin types having isomorphic repetitive categories! The correct result is: the configurations of Z correspond bijectively to the reflection equivalence classes of tilted algebras of Dynkin types A selfinjective of finite type Riedtmann ==== exists a well-behaved covering functor there F : K( Γ A )=indã ind A The functor is the canonical Galois covering functor K( Γ A ) K( Γ A )/ A = K(Γ A ) (hence A is standard), except char K =2andΓ A = ZD 3m / A, A =(τ 2m 1 ), and there exist nonstandard selfinjective algebras of finite type 53

56 BERLIN SCHOOL APPROACH A selfinjective algebra of finite type Assume A is basic, connected, A K 1=e 1 + e e n, e 1,e 2,...,e n orthogonal primitive idempotents Jans (1957) A finite type ==== A has finite ideal lattice for all i, j {1,...,n} we have e i Ae i is serial e i Ae j is cyclic left e i Ae i -module or cyclic right e j Ae j -module A is called regular if all bimodules e i Ae j are cyclic as left e i Ae i -modules and cyclic as right e j Ae j -modules A Stamm-algebra A (Kupisch, 1965) M(A) = {e i (rad t A)e j 1 i, j n, t 0} semigroup The nonzero elements of M(A) form a K- basis M 0 of the algebra A = K M 0,withmultiplication 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

57 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 selfinjective of Dynkin type Waschbüsch (1981) classified all nonregular selfinjective algebras of finite type, by the (modified) Brauer tree algebras with extreme exceptional vertex. They coincides with the nonstandard selfinjective algebras of finite type, described by Riedtmann (1983). Hence A regular A standard A nonregular A nonstandard Theorem (Hughes-Waschbüsch, 1983). For an algebra A, T (A) is of finite type T (A) = T (B) for a tilted algebra B of Dynkin type 55

58 DOMESTIC SELFINJECTIVE ALGEBRAS OF INFINITE TYPE {Ãn, D n, Ẽ6, Ẽ7, Ẽ8} Euclidean graph Selfinjective algebra of Euclidean type : algebra of the form B/G, whereb is a tilted algebra of an Euclidean type andg is an admissible group of automorphisms of B A = B/G selfinjective algebra of Euclidean type F : B B/G = A canonical Galois covering with the group G In fact, G is infinite cyclic For B tilted of type { D n, Ẽ6, Ẽ7, Ẽ8}, B is simply connected For B tilted of type = Ãn, B is not simply connected, but A = B/G admits a simply connected Galois covering F : R R/H = A with H = Z Z Hence, A = B/G is a standard selfinjective algebra 56

59 Theorem (Skowroński, 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 Γ s A admits a component Z for an Euclidean graph. 57

60 A = B/G selfinjective algebra of Euclidean type Γ B is of the form (Y r C r ): m Z Y s i = Z, Y 1 C 1 Y 0 C 0 Y 1 C s i = P 1 (K)-families of stable tubes (of the same tubular type), ν B(Y i )=Y i+2, ν B(C i )=C i+2, Then G is infinite cyclic generated by a strictly positive automorphism of B Moreover, B is locally support-finite F λ :mod B mod B/G =moda is dense Γ A =Γ B/G =Γ B/G 58

61 Hence, Γ A is of the form (for some r 1and X i = F λ (Y i ), T i = F λ (C i ), 0 i<r) T X r 1 0 T 0 X r 1 X 1 T r 2 T 1 X r 2 X 2 X s i = Z, T s i P 1 (K)-families of stable tubes (of the same tubular type) Then A is called r-parametric 59

62 Euclidean algebra = tubular (branch) extension 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 Ãp+q 1, D r+2, Ẽ6, Ẽ7, orẽ8, having a complete slice in the preinjective component. B Euclidean algebra (of type ) gl. dim B 2 B is domestic of infinite type (one-parametric) The Auslander-Reiten quiver Γ B of B is of the form P T Q 60

63 Proposition (Assem-Nehring-Skowroński, 1989). Let B be a tilted algebra of Euclidean type. Then there exists a reflection sequence of sinks i 1,i 2,...,i m in Q B such that B = S i + m...s i + S + 2 i B is an Euclidean algebra 1 of type. In particular, B = B. Proposition (Assem-Nehring-Skowroński, 1989). Let B, B be Euclidean algebras. TFAE: (1) B = B. (2) T (B) = T ( B ). (3) B = S + i r...s + i 2 S + i 1 B for a reflection sequence of sinks i 1,i 2,...,i r in Q B, r rk K 0 (B). In fact, at most two Euclidean algebras may have isomorphic repetitive categories 61

64 An Euclidean algebra B is exceptional if there exists a reflection sequence of sinks in Q B i 1,i 2,...,i t such that t<rk K 0 (B) and B = S + i t S + i 2 S + i 1 B. Proposition (Skowroński, 1989). Let B be an Euclidean algebra. Then B is exceptional there exists an automorphism ϕ of B such that ϕ d = ϱν B for some d 2 and a rigid automorphism ϱ of B. Moreover, then d =2. Theorem (Skowroński, 1989). Every selfinjective algebra of Euclidean type is one of the forms: (1) B/ ( σν k B),whereB is an Euclidean algebra, σ is a rigid automorphism of B, and k is a positive integer. (2) B/ ( µϕ 2k+1), where B is an exceptional Euclidean algebra, µ is a rigid automorphism of B, ϕ is an automorphism of B such that ϕ 2 = ϱν B for a rigid automorphism ϱ of B, andk is a positive integer. 62

65 PROBLEM. Describe the exceptional Euclidean algebras and their repetitive algebras Theorem (Lenzing-Skowroński, 1999). There are no exceptional Euclidean algebras of types Ẽ6, Ẽ7, Ẽ8. Theorem (Bocian-Skowroński, 2003). Let B be an Euclidean algebra. TFAE: (1) B is an exceptional algebra. (2) There is an automorphism ϕ of B with ϕ 2 = ν B. (3) B is reflection equivalent to an exceptional Euclidean algebra of one of the forms: B(T,v 1,v 2 ), B (T ) (type Ãn). Θ (i) (l, m, B), 0 i 8 (type D n ). 63

66 B exceptional Euclidean algebra oneparametric (weakly) symmetric selfinjective algebra B/(ϕ), ϕ 2 = ν B. B(T,v 1,v 2 ) Λ(T,v 1,v 2 ) B (T ) Λ (T ) Θ 0 (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,v 1,v 2 ) 64

67 T = Brauer tree with two (different) distinguished vertices v 1 and v 2 Λ(T,v 1,v 2 )=KQ T /I(T,v 1,v 2 ) one-parametric symmetric algebra of Euclidean type Ãm. Example. Let T be the Brauer tree v 1 v Then Q T is of the form α 2 α 6 6 β 5 β 6 5 α 3 β 2 2 β 1 α α 1 α 8 8 β 7 7 α 7 β 3 β 4 4 β 8 β 9 9 α 4 α 9 65

68 and the ideal I (T,v 1,v 2 ) in KQ T generated by α 1 β 1, β 1 α 2, α 2 β 2, β 2 α 3, α 7 β 3, β 3 α 4, α 4 β 4, β 4 α 1, α 3 β 5, β 5 α 6, α 6 β 6, β 6 α 5, α 5 β 7, β 9 α 7, α 8 β 8, β 7 α 8, α 9 β 9, β 8 α 9, α 2 1 β 1β 2 β 3 β 4, α 2 β 2 β 3 β 4 β 1, α 4 β 4 β 1 β 2 β 3, (α 3 α 5 α 7 ) 2 β 3 β 4 β 1 β 2, (α 5 α 7 α 3 ) 2 β 5 β 6, (α 7 α 3 α 5 ) 2 β 7 β 8 β 9, α 6 β 6 β 5, α 8 β 8 β 9 β 7, α 9 β 9 β 7 β 8. T = Brauer graph with exactly one cycle, having moreover an odd number of edges. Λ (T ) =KQ T /I (T ) one-parametric symmetric algebra of Euclidean type Ãm. 66

69 Example. Let T be the Brauer graph Then Q T is the quiver α 2 α 6 6 γ 5 5 α 3 β 2 2 β 1 γ 6 γ 8 α α 1 8 β 7 7 α 7 β 3 β 4 4 β 8 β 9 9 α 4 α 9 67

70 and the ideal I (T ) in KQ T is generated by: α 1 β 1, β 1 α 2, α 2 β 2, β 2 α 3, α 7 β 3, β 3 α 4, α 4 β 4, β 4 α 1, α 3 γ 5, γ 5 α 6, α 5 β 7, β 9 α 7, α 9 β 9, β 8 α 9, β 7 γ 8, γ 8 α 5, γ 6 β 8, α 6 γ 6, α 1 β 1 β 2 β 3 β 4, α 2 β 2 β 3 β 4 β 1, α 3 α 5 α 7 β 3 β 4 β 1 β 2, α 4 β 4 β 1 β 2 β 3, α 5 α 7 α 3 γ 5 γ 6 γ 8, α 6 γ 6 γ 8 γ 5, α 7 α 3 α 5 β 7 β 8 β 9, γ 8 γ 5 γ 6 β 8 β 9 β 7, α 9 β 9 β 7 β 8. 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 Euclidean type D n. 68

71 Example. Let T be the Brauer graph with the distinguished vertex v v a =5 1 6 b =4 3 2 Then Q (0) T is the quiver γ 4 1 γ 1 β 1 6 β 6 α 6 α 5 δ β 2 δ 4 γ 3 γ 2 3 β 3 69

72 and the ideal I (0) (T,v) in KQ (0) T is generated by: α 6 β 6, β 1 α 6, γ 1 β 2, γ 2 β 3, γ 4 β 1, β 2 γ 2, β 3 γ 3, β 6 γ 1, α 5 δ 5, δ 4 α 5, α 6 β 6 β 1, α 5 δ 5 γ 4 γ 1 γ 2 γ 3 δ 4, β 1 β 6 γ 1 γ 2 γ 3 δ 4 δ 5 γ 4, β 2 γ 2 γ 3 δ 4 δ 5 γ 4 γ 1, β 3 γ 3 δ 4 δ 5 γ 4 γ 1 γ 2, γ 3 γ 4, δ 5 γ 4 γ 1 γ 2 γ 3 δ 5 δ 4 δ 5, γ 4 γ 1 γ 2 γ 3 δ 4 δ 4 δ 5 δ 4. Note that α 5 (δ 5 δ 4 ) 2 I (0) (T,v). T = Brauer graph with one distinguished vertex 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) converge in a common vertex v 1 (respectively, v 2, v 3 ). 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, andthe edge c is a direct successor of the edge a, in the cyclic orders of edges at the vertices v 1, v 2 and v 3, respectively Γ (1) (T,v) = KQ (1) T /I(1) (T,v) one-parametric symmetric algebra of Euclidean type D n. 70

73 Example. Let T be the Brauer graph with the distinguished vertex v b =6 c =4 a = v Then Q (1) T is the quiver α 8 α α 9 β 7 β 8 β 6 β 9 6 β 4 4 α 4 1 β 1 γ 5 γ 6 α 3 α 1 β 5 5 γ 3 3 α 2 2 β 2 71

74 and the ideal I (1) (T,v) in KQ (1) T is generated by: β 1 α 1, β 2 α 2, β 6 α 7, β 7 α 8, β 8 α 9, β 9 α 4, α 2 γ 3, γ 3 β 5, γ 5 β 6, α 1 β 2, α 4 β 1, α 7 β 7, α 8 β 8, α 9 β 9, β 5 γ 5, α 2 α 3 α 4 α 1 β 2, α 7 β 7 β 8 β 9 β 4 β 6, α 8 β 8 β 9 β 4 β 6 β 7, α 9 β 9 β 4 β 6 β 7 β 8, β 5 γ 5 γ 6 γ 3, α 1 α 2 α 3 α 4 β1 2, γ 5γ 6 α 3, α 2 α 3 β 4, β 9 β 4 γ 6, β 6 β 7 β 8 β 9 γ 6 α 3, γ 3 γ 5 α 3 β 4, α 4 α 1 α 2 β 4 γ 6. Note that α 4 α 1 α 2 α 3 β 4 β 6 β 7 β 8 β 9,α 3 α 4 α 1 α 2 γ 3 γ 5 γ 6,β 6 β 7 β 8 β 9 β 4 γ 6 γ 3 γ 5 I (1) (T,v). T = Brauer tree with two (different) distinguished vertices v 1 and v 2 such that v 1 is the end of exactly one edge Γ (2) (T,v 1,v 2 ) = KQ (2) T /I(2) (T,v 1,v 2 ) oneparametric symmetric algebra of Euclidean type D n. 72

75 Example. Let T be the Brauer tree with two distinguished vertices v 1 and v b =5 a =4 e =1 2 c =3 v 2 Then Q (2) T is the quiver v 1 β 8 α 6 α β 6 α 1 β 1 α 5 α 2 α α 4 4 α β 3 3 β 7 β 5 γ 2 β 4 9 γ β 2 γ 3 73

76 and the ideal I (2) (T,v 1,v 2 ) in KQ (2) T is generated by: α 1 β 2, α 2 β 3, α 3 β 4, α 4 β 5, α 5 β 6, α 6 β 1, α 7 β 7, α 8 β 8, β 1 α 1, β 2 α 2, β 3 α 3, β 4 α 4, β 5 α 7, β 6 α 8, β 7 α 5, β 8 α 6, α 2 α 3 α 4 α 5 α 6 α 1 β 2, α 3 α 4 α 5 α 6 α 1 α 2 β 3, α 4 α 5 α 6 α 1 α 2 α 3 β 4, α 5 α 6 α 1 α 2 α 3 α 4 β 5 β 7, α 6 α 1 α 2 α 3 α 4 α 5 β 6 β 8, α 7 β 7 β 5, α 8 β 8 β 6, α 1 α 2 α 3 α 4 α 5 α 6 β1 2, γ 2β 5, β 3 γ 1, γ 1 γ 3, γ 3 γ 2, γ 2 α 5 α 6 α 1 α 2 α 3, α 4 α 5 α 6 α 1 α 2 γ 1, α 3 α 4 γ 1 γ 2, γ 2 α 5 α 6 α 1 α 2 γ 1 γ 3. 74

77 Weakly symmetric algebras of Euclidean type C A =(dim K Hom A (P i,p j )) Cartan matrix of A, P 1,P 2,...,P n complete family of pairwise nonisomorphic indecomposable projective A- modules Theorem (Bocian-Skowroński, 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

78 Theorem (Bocian-Skowroński, 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), whereb is an (exceptional) Euclidean algebra and ϕ is a square root of the Nakayama automorphism ν B of B. (v) A is isomorphic to an algebra of the form Λ (T,v 1,v 2 ), Λ (T ), Γ (0) (T,v), Γ (1) (T,v), or Γ (2) (T,v 1,v 2 ). Theorem. A local algebra A is a selfinjective (weakly symmetric) algebra of Euclidean type if and only if A is isomorphic to an algebra K x, y / ( x 2,y 2,xy λyx ),forλ K\{0}. 76

79 T = Brauer graph with exactly one loop which is also its direct successor B r B S.. B r r 1 B r 1 The Brauer quiver Q T is of the form α 1 Q Br 1 Q Br 1 r 1 S... r j +1 β r Q B2 β 1 Q Bj+1 β j 2 j 1 3 Q B3... j Q Bj β j 1 Q Bj+1 A i,b i, i T, cycles in Q T around i B j = β j...β r α 1 β 1...β i 1, j 1,j S 0 77

80 Ω (T ) = KQ T /Ī (T ) one-parametric biserial symmetric algebra, where Ī (T ) is the ideal of KQ T generated by β β 1 (i) α i and α α 1 (i) β i, i (Q T ) 0 \{1}, β r β 1, A i B i, i (Q T ) 0 \ S 0, A j B j, j S 0 \{1}, A 2 1 A 1B 1, A 1 B 1 B 1 A 1, (A 1 = α 1 ) Theorem (Bocian-Skowroński, 2004). Let Λ be a basic connected selfinjective K-algebra. Then Λ is socle equivalent to a selfinjective 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 isomorphic to an algebra of the form Ω (T ). 78

81 Example. Let T be the Brauer graph 4 3 T 2 1 α 2 2 β 1 Q T β 4 4 α 4 α 3 β β 3 α 1 Ω (T ) = KQ T /I (T ), wherei (T ) is the ideal in KQ T generated by: β 1 α 2, α 2 β 2, β 2 α 3, α 4 β 3, α 3 β 4, β 4 α 4, β 3 β 1, α 2 β 2 β 3 α 1 β 1, α 3 α 4 β 3 α 1 β 1 β 2, α 2 1 α 1β 1 β 2 β 3, α 1 β 1 β 2 β 3 β 1 β 2 β 3 α 1, α 4 α 3 β 4. 79

82 Nonstandard algebras Proposition (Bocian-Skowroński, 2004). Let T be a Brauer graph such that Λ (T ) and Ω (T ) are defined. Then (1) dim K Ω (T )=dim K Λ (T ). (2) Ω (T ) = Λ (T ) char K 2. (3) char K =2 Ω (T ) is nonstandard. (4) Ω (T ) and Λ (T ) are socle equivalent. (5) Λ (T ) is a degeneration of Ω (T ). Theorem (Skowroński, 2004). Let A be a selfinjective algebra. TFAE: (1) A is nonstandard domestic of infinite type. (2) char K =2and A = Ω (T ) for a Brauer graph T with one loop. 80

83 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. dim B =2 rk K 0 (B) = 6, 8, 9, or 10 B is nondomestic of polynomial growth The Auslander-Reiten quiver Γ B of B is of the form P T 0 q Q + T q T Q 81

84 Selfinjective algebra of tubular type: algebra of the form B/G, whereb is a tubular algebra and G is an admissible group of automorphisms 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, because B is simply connected (property of all tubular algebras) Hence, A = B/G is a standard selfinjective algebra Theorem (Skowroński, 1989, 2002). Let A be a selfinjective algebra. TFAE: (1) A is selfinjective of tubular type. (2) A is standard nondomestic of polynomial growth. (3) A is standard tame and Γ s A of (stable) tubes. consists only 82

85 A = B/G selfinjective algebra of tubular type Γ B is of the form q Q 1 C q C0 0 q Q 0 C q C 1 1 C s i, i Z, P 1(K)-families of stable tubes q Q i 1 i =Q (i 1,i) C q P 1 (K)-family of stable tubes Then G is infinite cyclic generated by a strictly positive automorphism of B Moreover, B is locally support-finite F λ :mod B mod B/G =moda is dense Γ A =Γ B/G =Γ B/G 83

86 Hence, Γ A is of the form (for some r 0and T q = F λ (C q )) T 0 = T r q Q r 1 T q r q Q 0 1 T q T r 1 T 1 q Q r 2 T q r 1 q Q 1 2 T q 84

87 Proposition (Nehring-Skowroński, 1989). Let B, B be tubular algebras. TFAE: (1) B = B. (2) T (B) = T (B ). (3) B = S + i r...s + i r B for a reflection sequence of sinks i 1,...,i r in Q B, r rk K 0 (B). A tubular algebra B is exceptional if there exists a reflection sequence i 1,...,i t of sinks in Q B such that t<rk K 0 B and B = S + i t S + i 1 B. Proposition (Skowroński, 1989). Let B be a tubular algebra. Then B is exceptional there exists an automorphism ϕ of B such that ϕ d = ϱν B for some d 2 and a rigid automorphism ϱ of B. 85

88 Theorem (Skowroński, 1989). Every selfinjective algebra of tubular type is one of the forms: (1) B/(σν k B), whereb is a tubular algebra, σ is a rigid automorphism of B, andk is a positive integer. (2) B/(µϕ k ),whereb is an exceptional tubular algebra, µ is a rigid automorphism of B, ϕ is an automorphism of B such that ϕ d = ϱν B for some d 2 andarigidautomorphism ϱ of B, and k is a positive integer with d k. 86

89 PROBLEM. Describe the exceptional tubular algebras and their repetitive algebras. Consider the following family of bound quiver algebras (where a dotted line means that the sum of paths indicated by this line is zero if it indicates exactly three parallel paths, the commutativity of paths if it indicates exactly two parallel paths, and the zero path if it indicates only one path): B φ γ α 4 1 ψ σ 2 3 β φγα = φσβ ψγα = λψσβ B 1 (λ) λ K \{0, 1} B 4 3 ξ α 5 6 η ζ 3 4 σ γ ω 7 β 7 1 ξα = ηγ, 2 ζα = ωγ ξσ = ηβ, ζσ = λωβ B 2 (λ) λ K \{0, 1} B 5 87

90 B B B B B B B B B 14 88

91 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 B 1 (λ) or B 2 (λ), forsomeλ K\{0, 1} (Skowroński, 1989). (ii) B is exceptional of tubular type (3, 3, 3) if and only if B is isomorphic to B 3, B 4 B 5, B 6, B 7,or B 8 (Bia lkowski Skowroński, 2002). (iii) B is exceptional of tubular type (2, 4, 4) if and only if B is isomorphic to B 9, B 10 B 11, B 12,or B 13 (Bia lkowski Skowroński, 2002). (iv) B is exceptional of tubular type (2, 3, 6) if and only if B is isomorphic to B 14 (Lenzing Skowroński, 2000). 89

92 Example B 6 S + 1 B S 2 + S+ 1 B 6 = B6 op S 4 + S+ 3 S+ 2 S+ 1 B 6 = B 6 Hence, B 6 is exceptional (of tubular type (3, 3, 3)). 90

93 Weakly symmetric algebras of tubular type Theorem (Bia lkowski-skowroński, 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

94 Theorem (Bia lkowski-skowroński, 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), whereb is a tubular algebra and ϕ is a proper root of the Nakayama automorphism ν B of B. (iii) A is isomorphic to one of the bound quiver algebras. α γ σ β αγα = ασβ βγα = λβσβ γαγ = σβγ γασ = λσβσ A 1 (λ) λ K \{0, 1} α σ γ α 2 = σγ λβ 2 = γσ γα = βγ σβ = ασ A 2 (λ) λ K \{0, 1} β δ γ α β ε ξ βα + δγ + εξ =0 αβ =0, ξε =0 γδ =0 A 3 δ γ α β ε ξ βα + δγ + εξ =0 αβ =0, γε =0 ξδ =0 A 4 α γ β α 2 = γβ βαγ =0 A 5 92

95 α γ β α 3 = γβ βγ =0 βα 2 =0 α 2 γ =0 α β δ γ βα = δγ γδ = εξ αδε =0 ξγβ =0 ε ξ σ δ ξβα =0 α δαβ =0 β γ βαγ =0 ξ αβσ =0 αβα = σξ, ξγ =0 βαβ = γδ, δσ =0 A 6 A 7 A 8 δ α ε γ βσ δα = εβ, γε = βσ, ασβ =0 εγδ =0,σγεγ =0 A 9 α γ δ β δβδ = αγ γβα =0, β(δβ) 3 =0 A 12 α β ξ δ γ ξαβ = ξδγξ αβδ = δγξδ βα =0, (γξδ) 2 γ =0 A 10 β γ α α 2 = γβ, βδ =0, γβ =0 σγ =0, αδ =0, σα =0 α 3 = δσ A 13 δ σ β α ξ γ γαβ = γξγ αβξ = ξγξ βα =0, δγ =0 ζ δ ξζ =0, (γξ) 2 = ζδ A 11 α β βα = δγδγ αδγδ =0 γδγβ =0 αβ =0 A 14 δ γ α γ δ β σ γβα =0, α 2 = δβ βδ =0,ασ =0,αδ = σγ A 15 α σ γ δ β αβγ =0, α 2 = βδ δβ =0,σα =0,δα = γσ A 16 93

96 Corollary. For an algebra A the following conditions 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 A 1 (λ), A 2 (λ), λ K \ {0, 1}, A 3 (if char K =2), or A i, 4 i 16. (A 3 is the preprojective algebra of type D 4 ) Corollary. Let A be a weakly symmetric algebra of tubular type with nonsingular Cartan matrix. Then A has at most four simple modules and the stable Auslander-Reiten quiver of A consists of tubes of rank 4. 94

97 Socle equivalences Theorem (Bia lkowski-skowroński, 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 isomorphic to one of the bound quiver algebras α γ β α 2 = γβ βαγ = βα 2 γ βαγβ =0 γβαγ =0 Λ 1 α γ β α 2 γ =0, βα 2 =0 γβγ =0, βγβ =0 βγ = βαγ α 3 = γβ Λ 2 (iii) K is of characteristic 2 and Λ is isomorphic to one of the bound quiver algebras 95

98 α σ γ α 4 =0, γα 2 =0, α 2 σ =0 α 2 = σγ + α 3, λβ 2 = γσ γα = βγ, σβ = ασ Λ 3 (λ) λ K \{0, 1} β α γ δ β δβδ = αγ, (βδ) 3 β =0 γβαγ =0, αγβα =0 γβα = γβδβα Λ 4 β γ α δ σ α 2 = γβ, α 3 = δσ, βδ =0 σγ =0, αδ =0, σα =0 γβγ =0, βγβ =0, βγ = βαγ Λ 5 α δ β γ αδγδ =0, γδγβ =0 αβα =0, βαβ =0 αβ = αδγβ βα = δγδγ Λ 6 α γ δ β σ βδ = βαδ, ασ =0,αδ = σγ γβα =0,α 2 = δβ, γβδ =0 βδβ =0, δβδ =0 Λ 7 α σ γ δ β δβ = δαβ, σα =0,δα= γσ αβγ =0,α 2 = βδ, δβγ =0 βδβ =0, δβδ =0 Λ 8 δ γ α β ε ξ βα + δγ + εξ =0 γδ =0, ξε =0, αβα =0 βαβ =0, αβ = αδγβ Λ 9 η γ µ δ α ξ σ β µβ =0, αη =0, βα = δγ ξσ = ηµ, σδ = γξ + σδσδ δσδσ =0, ξγξγ =0 Λ 10 96

99 Nonstandard algebras char tub. type nonstandard algebras standard algebras α 2 = γβ βαγ = βα 2 γ βαγβ =0 γβαγ =0 3 (3, 3, 3) Λ 1 2 (2, 2, 2, 2) α α γ β 2 1 γ β 2 1 α 2 γ =0, βα 2 =0 γβγ =0, βγβ =0 βγ = βαγ α 3 = γβ Λ 2 σ α 1 2 γ β α 4 =0, γα 2 =0, α 2 σ =0 α 2 = σγ + α 3, λβ 2 = γσ γα = βγ, σβ = ασ Λ 3 (λ) λ K \{0, 1} 3 δ γ α 4 ε 1 β ξ 2 βα + δγ + εξ =0 γδ =0, ξε =0, αβα =0 βαβ =0, αβ = αδγβ (3, 3, 3) Λ 9 η µ 5 γ δ α ξ σ β 4 µβ =0, αη =0, βα = δγ ξσ = ηµ, σδ = γξ + σδσδ δσδσ =0, ξγξγ =0 (2, 3, 6) Λ 10 α α α γ β 2 1 α 2 = γβ βαγ =0 A 5 γ β 2 1 α 3 = γβ βγ =0 βα 2 =0 α 2 γ =0 A 6 σ γ 1 2 α 2 = σγ λβ 2 = γσ γα = βγ σβ = ασ A 2 (λ) λ K \{0, 1} β 3 δ γ α 4 ε 1 β ξ 2 βα + δγ + εξ =0 αβ =0, ξε =0 γδ =0 A 3 η µ 5 γ δ α ξ σ β 4 µβ =0, αη =0, σδ = γξ βα = δγ, ξσ = ηµ A 29 97

100 char tub. type nonstandard algebras standard algebras α γ δ β δβδ = αγ, (βδ) 3 β =0 γβαγ =0, αγβα =0 γβα = γβδβα 2 (2, 4, 4) Λ β γ α δ σ α 2 = γβ, α 3 = δσ, βδ =0 σγ =0, αδ =0, ασ =0 γβγ =0, βγβ =0, βγ = βαγ Λ 5 α β δ γ αδγδ =0, γδγβ =0 αβα =0, βαβ =0 αβ = αδγβ βα = δγδγ Λ 6 1 α γ δ 2 3 β δβδ = αγ γβα =0, (βδ) 3 β =0 A 12 β γ α δ σ α 2 = γβ, βδ =0, βγ =0 σγ =0, αδ =0, σα =0 α 3 = δσ A 13 α β δ γ βα = δγδγ αδγδ =0 γδγβ =0 αβ =0 A 14 α γ δ β σ βδ = βαδ, ασ =0, αδ = σγ γβα =0, α 2 = δβ, γβδ =0 βδβ =0, δβδ =0 Λ 7 α γ δ β σ γβα =0, α 2 = δβ βδ =0, ασ =0, αδ = σγ A 15 σ 3 γ α δ 2 1 β δβ = δαβ, σα =0, δα = γσ αβγ =0, α 2 = βδ, δβγ =0 βδβ =0, δβδ =0 Λ 8 α γ δ β σ αβγ =0, α 2 = βδ δβ =0, σα =0, δα = γσ A 16 98