## Thorsten Holm

### Books

 [2] Algebras and Representation Theory. (with K. Erdmann) Introductory textbook. Springer Undergraduate Mathematics Series (SUMS), Springer (2018) [1] Triangulated Categories (eds. T. Holm, P. Jørgensen, R. Rouquier) London Mathematical Society Lecture Notes Series, No. 375, Cambridge University Press (2010)

### Snapshot (for general audience)

 [1] Friezes and tilings. appeared in Snapshots of modern mathematics from Oberwolfach (2015); part of the IMAGINARY open mathematics platform

### Articles

 [57] Frieze patterns with coefficients. (with M. Cuntz, P. Jørgensen) Preprint (2019), 22 pages. arXiv:1909.02332 [56] A p-angulated generalisation of Conway and Coxeter's theorem on frieze patterns. (with P. Jørgensen) Int. Math. Res. Not., Vol. 2020, No.1, 71-90. DOI:10.1093/imrn/rny020 [55] Cluster tilting subcategories and torsion pairs in Igusa-Todorov cluster categories of Dynkin type A∞. (with S. Gratz, P. Jørgensen) Math. Z. 292 (2019), 33-56 . DOI:10.1007/s00209-018-2117-y [54] Frieze patterns over integers and other subsets of the complex numbers. (with M. Cuntz) J. Comb. Algebra 3 (2019), 153-188. DOI:10.4171/JCA/29 [53] All SL2-tilings come from infinite triangulations. (with C. Bessenrodt, P. Jørgensen) Adv. Math. 315 (2017), 194-245. DOI:10.1016/j.aim.2017.05.019 An announcement (3 pages, written for the Oberwolfach workshop on Cluster Algebras and Related Topics, December 2013) can be found here . [52] Generalized friezes and a modified Caldero-Chapoton map depending on a rigid object, II. (with P. Jørgensen) Bull. Sci. Math. 140 (2016), 112-131. DOI:10.1016/j.bulsci.2015.05.001 [51] Generalized friezes and a modified Caldero-Chapoton map depending on a rigid object. (with P. Jørgensen) Nagoya Math. J. 218 (2015), 101-124. DOI: 10.1215/00277630-2891495 [50] Cluster tilting vs. weak cluster tilting in Dynkin type A infinity. (with P. Jørgensen) Forum Math. 27 (2015), 1117-1137. DOI: 10.1515/forum-2012-0093 [49] Towards derived equivalence classification of the cluster-tilted algebras of Dynkin type D. (with J. Bastian, S. Ladkani) J. Algebra 410 (2014), 277-332. DOI: 10.1016/j.jalgebra.2014.03.034 [48] Torsion pairs in cluster tubes. (with P. Jørgensen, M. Rubey) J. Algebraic Combin. 39 (2014), 587-605. DOI: 10.1007/s10801-013-0457-6 [47] Generalized frieze pattern determinants and higher angulations of polygons. (with C. Bessenrodt, P. Jørgensen) J. Combin. Theory Ser. A 123 (2014), 30-42. DOI: 10.1016/j.jcta.2013.11.003 [46] Ptolemy diagrams and torsion pairs in the cluster categories of Dynkin type D. (with P. Jørgensen, M. Rubey) Adv. in Appl. Math. 51 (2013), 583-605. DOI: 10.1016/j.aam.2013.07.005 [45] SL2 tilings and triangulations of the strip. (with P. Jørgensen) J. Combin. Theory Ser. A 120 (2013), 1817-1834. DOI: 10.1016/j.jcta.2013.07.001 [44] Realising higher cluster categories of Dynkin type as stable module categories. (with P. Jørgensen) Quart. J. Math. 64 (2013), 409-435. DOI: 10.1093/qmath/has013 [43] Derived equivalence classification of cluster-tilted algebras of Dynkin type E. (with J. Bastian, S. Ladkani) Algebr. Represent. Theory 16 (2013), 527-551. DOI: 10.1007/s10468-011-9318-y For previous versions, see arXiv:0906.3422. Supplementary material. [42] Sparseness of t-structures and negative Calabi-Yau dimension in triangulated categories generated by a spherical object. (with P. Jørgensen, D. Yang) Bull. Lond. Math. Soc. 45 (2013), 120-130. DOI: 10.1112/blms/bds072 [41] On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon. (with P. Jørgensen) Math. Z. 270 (2012), 277-295. DOI: 10.1007/s00209-010-0797-z [40] Deformed preprojective algebras of type L: Külshammer ideals and derived equivalences. (with A. Zimmermann) J. Algebra 346 (2011), 116-146. DOI: 10.1016/j.jalgebra.2011.08.024 [39] Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type An. (with P. Jørgensen, M. Rubey) J. Algebraic Combin. 34 (2011), 507-523. DOI: 10.1007/s10801-011-0280-x [38] Classification of torsion pairs in cluster categories of Dynkin type. Oberwolfach Reports 8 (2011), 555-558. [37] Derived equivalence classification of symmetric algebras of polynomial growth. (with A. Skowronski) Glasgow Math. J. 53 (2011), 277-291. DOI: 10.1017/S0017089510000698 [36] Triangulated categories: definitions, properties and examples. (with P. Jørgensen) In: Triangulated Categories (eds. T. Holm, P. Jørgensen, R. Rouquier), London Mathematical Society Lecture Notes Series (No. 375), Cambridge University Press (2010), 1-51. [35] On the relation between cluster and classical tilting. (with P. Jørgensen) J. Pure Appl. Algebra 214 (2010), 1523-1533. DOI: 10.1016/j.jpaa.2009.11.012 [34] Generalized Reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type. (with A. Zimmermann) J. Algebra 320, no.9 (2008), 3425-3437. DOI: 10.1016/j.jalgebra.2008.07.026 [33] Maximal n-orthogonal modules for selfinjective algebras. (with K. Erdmann) Proc. Amer. Math. Soc. 136, no.9 (2008), 3069-3078. DOI: 10.1090/S0002-9939-08-09297-6 [32] Weighted locally gentle quivers and Cartan matrices. (with C. Bessenrodt) J. Pure Appl. Algebra 212 (2008), 204-221. DOI: 10.1016/j.jpaa.2007.05.004 [31] Blocks with a quaternion defect group over a 2-adic ring: the case $Ã$4. (with R. Kessar, M. Linckelmann) Glasgow Math. J. 49 (2007), 29-43. DOI: 10.1017/S0017089507003394 [30] Generalized Reynolds ideals for non-symmetric algebras. (with C. Bessenrodt, A. Zimmermann) J. Algebra 312 (2007), no.2, 985-994. DOI: 10.1016/j.jalgebra.2007.02.028 [29] Derived equivalence classification of nonstandard selfinjective algebras of domestic type. (with R. Bocian, A. Skowronski) Comm. Algebra 35 (2007), no.2, 515-526. DOI: 10.1080/00927870601052521 [28] q-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras. (with C. Bessenrodt) Pacific J. Math. 229 (2007), No.1, 25-48. DOI: 10.2140/pjm.2007.229.25 [27] A local conjecture on Brauer character degrees of finite groups. (with W. Willems) Trans. Amer. Math. Soc. 359 (2007), no.2, 591-603. DOI: 10.1090/S0002-9947-06-03888-8 [26] Derived equivalence classification of one-parametric self-injective algebras. (with R. Bocian, A. Skowronski) J. Pure Appl. Algebra 207, no.3 (2006), 491-536. DOI: 10.1016/j.jpaa.2005.10.015 [25] Derived equivalence classification of symmetric algebras of domestic type. (with A. Skowronski) J. Math. Soc. Japan 58 no.4 (2006), 1133-1149. [24] The representation dimension of $k\left[x,y\right]/\left(x2, yn\right).$ (with W. Hu) J. Algebra 301 no.2 (2006), 791-802. DOI: 10.1016/j.jalgebra.2005.11.037 [23] Cartan determinants for gentle algebras. Arch. Math. 85 (2005), 233-239. DOI: 10.1007/s00013-005-1344-8 [22] The representation dimension of Schur algebras: the tame case. Quart. J. Math. 55 (2004), 477-490. DOI: 10.1093/qmath/hah014 [21] Radical embeddings and representation dimension. (with K. Erdmann, O. Iyama and J. Schröer) Adv. Math. 185 (2004), 159-177. DOI: 10.1016/S0001-8708(03)00169-5 [20] Derived equivalence classification of weakly symmetric algebras of Euclidean type. (with R. Bocian, A. Skowronski) J. Pure Appl. Algebra 191 (2004), 43-74. DOI: 10.1016/j.jpaa.2003.12.003 [19] The representation dimension of domestic weakly symmetric algebras. (with R. Bocian, A. Skowronski) Cent. Eur. J. Math. 2, No.1 (2004), 67-75. [18] Hochschild cohomology of tame blocks. J. Algebra 271 No.2 (2004), 798-826. DOI: 10.1016/j.jalgebra.2003.09.030 [17] Representation dimension of some tame blocks of finite groups. Algebra Colloquium 10:3 (2003), 275-284. [16] On nonstandard tame selfinjective algebras having only periodic modules. (with J. Bialkowski, A. Skowronski) Colloquium Math. 97 (2003), 33-47. [15] Derived equivalences for tame weakly symmetric algebras having only periodic modules. (with J. Bialkowski, A. Skowronski) J. Algebra 269 No.2 (2003), 652-668. DOI: 10.1016/S0021-8693(03)00368-5 [14] Twisted bimodules and Hochschild cohomology for selfinjective algebras of class An, II. (with K. Erdmann and N. Snashall) Algebras and Representation Theory 5 (2002), 457-482. DOI: 10.1023/A:1020551906728 [13] Blocks of Tame Representation Type and Related Algebras: Derived Equivalences and Hochschild Cohomology. Habilitationsschrift (2001), Otto-von-Guericke-Universität Magdeburg. (137 pages) [12] Hochschild Cohomology Rings of Algebras k[X]/(f). Beiträge zur Algebra und Geometrie 41 (2000), No.1, 291-301 [11] Twisted bimodules and Hochschild cohomology for selfinjective algebras of class An. (with K. Erdmann) Forum Math. 11 (1999), no. 2, 177-201. [10] Derived Equivalence Classification of Algebras of Dihedral, Semidihedral and Quaternion Type. J. Algebra 211 (1999), 159-205. DOI: 10.1006/jabr.1998.7544 [9] Derived Categories, Derived Equivalences and Representation Theory. In: Linckelmann, M. (ed.), Proceedings of the summer school on representation theory of algebras, finite and reductive groups, Cluj-Napoca, Romania, September 15-25, 1997. Cluj-Napoca: ``Babes Bolyai" University, Faculty of Mathematics and Computer Science, 33-66 (1998) [8] Hochschild Cohomology of Brauer Tree Algebras. Comm. Algebra 26 (11), 3625-3646 (1998). DOI: 10.1080/00927879808826363 [7] Derived Equivalent Tame Blocks. J. Algebra 194 (1997), no.1, 178-200. DOI: 10.1006/jabr.1996.7007 [6] Derived Equivalences and Hochschild Cohomology for Blocks with Quaternion Defect Groups. In: Darstellungstheorietage Jena 1996, Sitzungsber. Math.-Naturwiss. Kl., 7, Akad. Gemein. Wiss. Erfurt, Erfurt, 1996, 75-86. [5] Hochschild Cohomology of the Integral Group Ring of a Cyclic Group and Related Algebras. Archiv der Mathematik, 67 (No.5), 360-366 (1996). DOI: 10.1007/BF01189095 [4] The Hochschild Cohomology Ring of a Modular Group Algebra: The Commutative Case. Comm. Algebra, 24(6), 1957-1969 (1996). DOI: 10.1080/00927879608825682 [3] The even Hochschild Cohomology Ring of a Block with Cyclic Defect Group. Journal of Algebra 178, 317-341 (1995). DOI: 10.1006/jabr.1995.1352 [2] Hochschild-Kohomologie von Blöcken mit zyklischer Defektgruppe. Vorlesungen aus dem Fachbereich Mathematik der Universität GH Essen. Heft 22 (1994).

Didactics  [1] Der Euklidische Algorithmus - warum nicht in der Schule? (with W. Willems) Mathematische Unterrichtspraxis, Heft 4/1999, S.34-41.

### Unpublished Notes

• Representation dimension for group algebras - a very brief survey, 3 pages, Comments welcome!
Notes based on a problem session at the Workshop The Representation Dimension of Artin Algebras, Bielefeld, 1-4 May 2008
• Algebren, Darstellungen und Homologische Algebra.
Manuskript (2003). Sehr vorläufige Version!
(Ausarbeitung von Kolloquiumsvorträgen für ein allgemeines mathematisches Publikum.)
• Notes on Donovan's Conjecture for Blocks of Tame Representation Type.
Manuscript (2001)
Abstract: We summarize in this note the results on Donovan's conjecture for tame blocks. In particular, we point out that the conjecture holds for blocks with dihedral and semidihedral defect groups, but that the conjecture is still open for blocks with generalized quaternion defect group. The results follow from Erdmann's work on the classification of tame blocks up to Morita equivalence. But for several cases there are no complete proofs in the literature and the aim of these notes is to provide the necessary calculations.
• Broué's conjecture for the non-principal 3-block of the Higman-Sims group.
Manuscript (1997).
• Tilting complexes for selfinjective algebras of class D_n.
Manuscript (1997).
• Derivations of group rings over commutative rings.
Manuscript (1995).

### Theses

• Blocks of Tame Representation Type and Related Algebras: Derived Equivalences and Hochschild Cohomology.
Habilitation thesis, Otto-von-Guericke-Universität Magdeburg, 2001
• Hochschild-Kohomologie von Blöcken mit zyklischer Defektgruppe.
Doctoral thesis (supervisor: Gerhard O. Michler), University of Essen, 1994.
• Über einen Zusammenhang zwischen der geometrischen Invariante Σ1 und Darstellungen von Gruppen.
Diploma thesis (supervisor: Robert Bieri), University of Frankfurt, 1991.

### Book Reviews

• F. Padberg: Zahlentheorie und Arithmetik.
Mathematische Unterrichtspraxis, Heft 1/2000, S.40-41.
• P. Göthner: Elemente der Algebra.
Mathematische Unterrichtspraxis, Heft 2/99, S.43-44.

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Thorsten Holm