A Cluster tilting module for a representation-infinite block of a group algebra
AA CLUSTER TILTING MODULE FOR A REPRESENTATION-INFINITE BLOCKOF A GROUP ALGEBRA
BERNHARD B ¨OHMLER AND REN´E MARCZINZIK
Abstract.
Let G = SL (2 ,
5) be the special linear group of 2 × K of characteristic two of thegroup algebra KG has a 3-cluster tilting module. This gives the first example of a representation-infiniteblock of a group algebra having a cluster tilting module and answers a question by Erdmann and Holm. Introduction
We assume that algebras are finite dimensional over a field K . For n ≥
1, an A -module M is calledan n -cluster tilting module if it satisfies:add( M ) = { X ∈ mod − A | Ext iA ( M, X ) = 0 for 1 ≤ i ≤ n − } = { X ∈ mod − A | Ext iA ( X, M ) = 0 for 1 ≤ i ≤ n − } . We remark that in some references like [EH08] an n -cluster tilting module is called an maximal ( n − n -cluster tilting modules was introduced by Iyama in [Iya07a] and[Iya07b]. It has found several important applications, for instance in the theory of cluster algebras,see [GLS06]. Cluster tilting modules are especially important for selfinjective algebras where recentmethods allow to construct many examples related to other structures in algebra and combinatorics, seefor example [DI20] and [CDIM20]. One of the most important classes of selfinjective algebras are groupalgebras. This leads to the following natural question: Question.
When does a block of a group algebra have a cluster tilting module?
In [EH08], Erdmann and Holm showed that selfinjective algebras with a cluster tilting module havecomplexity at most one (recently it was shown in [MV21] that this result does not hold for non-selfinjectivealgebras). Erdmann and Holm used this result in [EH08, Section 5.3] to show that a block of a groupalgebra can only have a cluster tilting module when it is representation-finite or Morita equivalent to analgebra of quaternion type.Every representation-finite block of a group algebra is derived equivalent to a symmetric Nakayamaalgebra and recently a complete classification for the existence of cluster tilting modules was obtained in[DI20, Section 5] for selfinjective Nakayama algebras. For algebras of quaternion type (which are always ofinfinite representation type) it is however unknown whether they can have cluster tilting modules and thiswas posed in [EH08, Section 5.3] as an open question. There is no universal method to construct clustertilting modules or to show that they do not exist. The fact that the classification of indecomposablemodules for algebras of quaternion type is still not known makes the search and verification of clustertilting modules especially hard. We remark that the existence of an n -cluster tilting module is a Moritainvariant. In this article we show that the algebra of quaternion type Q (3 A ) has a 3-cluster tiltingmodule. This algebra is Morita equivalent to the principal block of the group algebra of SL (2 ,
5) over asplitting field of characteristic two. Our main result is as follows:
Theorem.
Let A be the principal block of the group algebra KG for G = SL (2 , over a splitting fieldof characteristic two. Then A has a -cluster tilting module. Date : January 26, 2021.2010
Mathematics Subject Classification.
Primary 16G10, 16E10.
Key words and phrases.
Cluster tilting modules, algebras of quaternion type, group algebras. a r X i v : . [ m a t h . R T ] J a n BERNHARD B ¨OHMLER AND REN´E MARCZINZIK
This gives the first example of a cluster tilting module of a representation-infinite block of a groupalgebra and it gives a positive answer to the question of Erdmann and Holm about the existence of clustertilting modules for algebras of quaternion type. We remark that we found the 3-cluster tilting module inthe main result by experimenting with the GAP-package [QPA16]. The proof also uses the calculation ofquiver and relations for an endomorphism ring which was obtained with the aid of the computer.1.
An example of a 3-cluster tilting module for the algebra of quaternion type Q (2 A ) We assume that all algebras are finite dimensional over a field K and all modules are finite dimensionalright modules unless stated otherwise. J will denote the Jacobson radical of an algebra and D =Hom K ( − , K ) the natural duality. We assume that the reader is familiar with the basics of representationtheory and homological algebra of finite dimensional algebras and refer for example to the textbook[SY11].The global dimension gldim A of an algebra A is defined as the supremum of all projective dimensionsof the simple A -modules. It is well known that the global dimension of A coincides with the globaldimension of the opposite algebra A op , see for example [W95, Exercise 4.1.1]. The dominant dimension domdim A of A is defined as the minimal n such that I n is not projective (or infinite if no such n exists),where 0 → A → I → I → · · · is a minimal injective coresolution of the regular A -module A . The dominant dimension of A coincideswith the dominant dimension of the opposite algebra A op , see [M68, Theorem 4].We will also need the following lemma on the behaviour of the global and dominant dimension underextensions of the ground field. For a field extension F of K , we denote by A F := A ⊗ K F the F -algebrawhich is obtained from A by the field extension. Lemma 1.1.
Let A be a finite dimensional algebra over the field K and let F be a field extension of K .(1) domdim A = domdim A F .(2) If A/J is separable (where J denotes the Jacobson radical of A ), then gldim A = gldim A F .Proof. (1) See [M68, Lemma 5].(2) See [ERZ57, Corollary 18]. (cid:3) Recall that a module M is a generator of mod − A when every indecomposable projective A -moduleis a direct summand of M and M is a cogenerator of mod − A when every indecomposable injective A -module is a direct summand of M . Theorem 1.2.
Let A be a non-semisimple connected finite dimensional algebra with an A -module M that is a generator and cogenerator of mod − A . Then M is an n -cluster tilting module if and only if B := End A ( M ) is a higher Auslander algebra of global dimension n + 1 , that is B has global dimensionequal to n + 1 and dominant dimension equal to n + 1 .Proof. See [Iya08, Theorem 2.6] for an elementary proof. (cid:3)
We refer to [E90, Section VII] for the precise definition of algebras of quaternion type, which arise inthe study of blocks of group algebras with quaternion defect groups. The tables starting at page 303 of[E90] give quiver and relations of algebras of quaternion type. In this article, we only need to know thealgebra of quaternion type Q (3 A ) that we describe next. Let K be a field of characterstic two. Let A = KQ/I be the following quiver algebra where Q is given by • • • by nd CLUSTER TILTING MODULE FOR A REPRESENTATION-INFINITE BLOCK OF A GROUP ALGEBRA 3 and the relations are given by I = (cid:104) byb − bdnybdn, yby − dnybdny, ndn − nybdnyb, dnd − ybdnybd, bybd, ndny (cid:105) . This is the algebra of quaternion type Q (3 A ) and this algebra is Morita equivalent to the principalblock of the the group algebra F G where G = SL (2 ,
5) is the special linear group of 2 × F is a splitting field of characteristic two, see for example page 110 of[H01] and section 7 of [E88]. The algebra A is symmetric of period 4 and dim K ( A ) = 36. The dimensionvectors of the indecomposable projective A -modules P , P and P are respectively given by [4 , , , [4 , , , , A -modules:(1) Let M = e A/nA , which has dimension vector [0 , , M = e A/nybdnyA , which has dimension vector [1 , , M = e A/nyA , which has dimension vector [0 , , M = e A/yA , which has dimension vector [1 , , M := A ⊕ M ⊕ M ⊕ M ⊕ M . Note that every indecomposable summand of M has simple top.We fix A and M as above for the rest of this article. We show that M is a 3-cluster tilting module. Theorem 1.3.
Let A be the algebra of quaternion type Q (3 A ) over a field F with characteristic two.Then M is a 3-cluster tilting module.Proof. Clearly M is a generator and cogenerator of mod − A . We show that B := End A ( M ) has globaldimension 4 and dominant dimension 4. Then, M is a 3-cluster tilting module by Theorem 1.2. Firstassume that K has two elements. The following QPA program calculates quiver and relations of B op overthe field with two elements and shows that B op has global dimension and dominant dimension equal to4. We remark that GAP applies functions from the right. Thus, it calculates the opposite algebra of theendomorphism ring of M . LoadPackage("qpa");k:=2;F:=GF(2);Q:=Quiver(3,[[1,2,"b"],[2,3,"d"],[2,1,"y"],[3,2,"n"]]);kQ:=PathAlgebra(F,Q);AssignGeneratorVariables(kQ);rel:=[b*y*b-(b*d*n*y)^(k-1)*b*d*n,y*b*y-(d*n*y*b)^(k-1)*d*n*y,n*d*n-(n*y*b*d)^(k-1)*n*y*b,d*n*d-(y*b*d*n)^(k-1)*y*b*d,b*y*b*d,n*d*n*y];A:=kQ/rel; B:=Basis(A);U:=Elements(B);Display(U);n:=Size(B);UU:=[];for i in [4..n] do Append(UU,[U[i]]);od;t1:=UU[4];M1:=RightAlgebraModuleToPathAlgebraMatModule(RightAlgebraModule(A, \*, RightIdeal(A,[t1])));N1:=CoKernel(InjectiveEnvelope(M1));M1:=N1;t2:=UU[33];M2:=RightAlgebraModuleToPathAlgebraMatModule(RightAlgebraModule(A, \*, RightIdeal(A,[t2])));N2:=CoKernel(InjectiveEnvelope(M2));M2:=N2;t3:=UU[10];M3:=RightAlgebraModuleToPathAlgebraMatModule(RightAlgebraModule(A, \*, RightIdeal(A,[t3])));N3:=CoKernel(InjectiveEnvelope(M3));M3:=N3;t4:=UU[3];M4:=RightAlgebraModuleToPathAlgebraMatModule(RightAlgebraModule(A, \*, RightIdeal(A,[t4])));N4:=CoKernel(InjectiveEnvelope(M4));M4:=N4;N:=DirectSumOfQPAModules([N1,N2,N3,N4]);projA:=IndecProjectiveModules(A);RegA:=DirectSumOfQPAModules(projA);M:=DirectSumOfQPAModules([RegA,N]);B:=EndOfModuleAsQuiverAlgebra(M)[3];QQ:=QuiverOfPathAlgebra(B);Display(QQ);rel:=RelatorsOfFpAlgebra(B);gd:=GlobalDimensionOfAlgebra(B,33);dd:=DominantDimensionOfAlgebra(B,33);
We observe that B op = K ˆ Q/ ˆ I is a quiver algebra where ˆ Q is given by BERNHARD B ¨OHMLER AND REN´E MARCZINZIK • • • • • • • α α α α α α α α α α α α α α and the relations are given byˆ I = (cid:104) α α , α α , α α + α α , α α , α α + α α , α α , α α , α α + α α α , α α α , α α α ,α α α , α α + α α α , α α α , α α α , α α α , α α α , α α + α α α ,α α α + α α α , α α α + α α α , α α + α α α , α α α + α α α ,α α α + α α α , α α α α , α α α α , α α α α , α α + α α α α ,α α α + α α α α , α α + α α α α , α α α α , α α α α , α α α α + α α α α ,α α α α , α α α α , α α α + α α α α , α α α α , α α α + α α α α ,α α + α α + α α α α α α , α α + α α α α α α (cid:105) . With B op also B has global dimension and dominant dimension equal to 4 and thus M is a 3-clustertilting module. Now let F be an arbitrary field with characteristic two, which is an extension of the field K with two elements. We haveEnd A F ( M ⊗ K F ) ∼ = End A ( M ) ⊗ K F ∼ = B F , which has also dominant and global dimension equal to 4. This follows from Lemma 1.1 and the factthat B/J is separable, since B is a quiver algebra. Thus M ⊗ K F is also a 3-cluster tilting module of A F . (cid:3) We remark that it took the supercomputer ”nenepapa” from the TU Kaiserslautern 105 hours tocompute the endomorphism ring of M . The data of this supercomputer are as follows. Compute-ServerLinux (Gentoo): Dell PowerEdge R730, 2x Intel Xeon E5-2697AV4 2.6 GHz, Turbo 3.60 GHz, 40 MBSmartCache, 32 Cores, 64 Threads, 768 GB RAM.As remarked earlier, the principal block of the group algebra KG for G = SL (2 ,
5) over a split-ting field K of characteristic two is Morita equivalent to the algebra of quaternion type Q (3 A ) . As acorollary of the previous Theorem we obtain our main result: Corollary 1.4.
Let G = SL (2 , and K be a field of characteristic two that is a splitting field for KG .Then the principal block of KG has a -cluster tilting module. Note that not every algebra of quaternion type has a cluster tilting module. In fact, the group algebra KG of the quaternions G of order 8 over a field K with characteristic two has no cluster tilting modules,since it is representation-infinite and we have Ext KG ( M, M ) (cid:54) = 0 for every non-projective KG -module M by a result of Tachikawa, see [T73, Theorem 8.6]. CLUSTER TILTING MODULE FOR A REPRESENTATION-INFINITE BLOCK OF A GROUP ALGEBRA 5
Acknowledgements
We thank Karin Erdmann for having informed us in private communication that she has also founda 3-cluster tilting module for another algebra of quaternion type which is not a block of a group algebra.We thank Thorsten Holm for providing a reference to his habilitation thesis. Bernhard B¨ohmler gratefullyacknowledges funding by the DFG (SFB/TRR 195). Ren´e Marczinzik gratefully acknowledges fundingby the DFG (with project number 428999796). We profited from the use of the GAP-package [QPA16].
References [CDIM20] Aaron Chan, Erik Darp¨o, Osamu Iyama, and Ren´e Marczinzik. Periodic trivial extension algebras and fractionallyCalabi-Yau algebras. https://arxiv.org/abs/2012.11927 .[DI20] Erik Darp¨o and Osamu Iyama. d-representation-finite self-injective algebras.
Advances in Mathematics , Volume 362,2020.[ERZ57] Samuel Eilenberg, Alex Rosenberg, and Daniel Zelinsky. On the dimension of modules and algebras VIII.
NagoyaMath. J.
Volume 12 , 1957, 71-93.[E90] Karin Erdmann. Blocks of tame representation type and related algebras.
Lecture Notes in Mathematics
Mathematische Annalen
Volume 281, pages 561-582, 1988.[EH08] Karin Erdmann and Thorsten Holm. Maximal n-orthogonal modules for selfinjective algebras.
Proc. Amer. Math.Soc. , 136(9):3069–3078, 2008.[GLS06] Christof Geiß, Bernard Leclerc, and Jan Schr¨oer. Rigid modules over preprojective algebras.
Inventiones mathe-maticae , 165(3):589–632, Sep 2006.[H01] Thorsten Holm. Blocks of Tame Representation Type and Related Algebras: Derived Equivalences and HochschildCohomology. Habilitation thesis 2001, .[Iya07a] Osamu Iyama. Auslander correspondence.
Advances in Mathematics , 210(1):51 – 82, 2007.[Iya07b] Osamu Iyama. Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories.
Advances inMathematics , 210(1):22 – 50, 2007.[Iya08] Osamu Iyama. Auslander-Reiten theory revisited.
Trends in representation theory of algebras and related topics.Proceedings of the 12th international conference on representations of algebras and workshop
EMS Series of CongressReports, 349-397 , 2008.[MV21] Rene Marczinzik, Leartis Vaso. Existence of a 2-cluster tilting module does not imply finite complexity. https://arxiv.org/abs/2101.05671 .[M68] Bruno M¨uller. The Classification of Algebras by Dominant Dimension.
Canadian Journal of Mathematics , Volume20 , 1968 , 398 - 409.[QPA16] The QPA-team.
QPA - Quivers, path algebras and representations - a GAP package, Version 1.25 , 2016.[SY11] Andrzej Skowro´nski and Kunio Yamagata.
Frobenius algebras I . EMS Textbooks in Mathematics. European Math-ematical Society (EMS), Z¨urich, 2011. Basic representation theory.[T73] Hiroyuki Tachikawa. Quasi-Frobenius Rings and Generalizations. Lecture Notes in Mathematics, Volume 351, 1973.[W95] Charles Weibel. An Introduction to Homological Algebra
Cambridge Studies in Advanced Mathematics Book 38 ,Cambridge University Press 1995.
FB Mathematik, TU Kaiserslautern, Gottlieb-Daimler-Str. 48, 67653 Kaiserslautern, Germany
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