Abstract
The objective of this paper is to give a concrete interpretation of the dimension of the first Hochschild cohomology space of a cyclically oriented or tame cluster tilted algebra in terms of a numerical invariant arising from the potential.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Amiot, C., Labardini-Fragoso, D., Plamondon, P. -G.: Derived invariants for surface cut algebras ii: the punctured case. arXiv:1606.07364 (2016)
Assem, I., Brüstle, T., Charbonneau-Jodoin, G., Plamondon, P. -G.: Gentle algebras arising from surface triangulations. Algebra Number Theory 4 (2), 201–229 (2010)
Assem, I., Brüstle, T., Schiffler, R.: Cluster-tilted algebras as trivial extensions. Bull. Lond. Math Soc. 40(1), 151–162 (2008)
Assem, I., Bustamante, J. C., Dionne, J., Le Meur, P., Smith, D.: Representation theory of partial relation extensions. Coll. Math. 155(2), 157–186 (2019)
Assem, I., Bustamante, J. C., Igusa, K., Schiffler, R.: The first Hochschild cohomology group of a cluster tilted algebra revisited. Internat. J. Algebra Comput. 23(4), 729–744 (2013)
Assem, I., Gatica, M. A., Schiffler, R.: Hochschild cohomology of partial relation extensions. Comm. Algebra 46(12), 5273–5282 (2018)
Assem, I., Gatica, M. A., Schiffler, R., Taillefer, R.: Hochschild cohomology of relation extension algebras. J. Pure Appl. Algebra 220(7), 2471–2499 (2016)
Assem, I., Marcos, E., de la Peña, J.: The Simple Connectedness or a Tame Tilted Algebra. J. Algebra 237, 647–656 (2001)
Assem, I., Redondo, M. J.: The first Hochschild cohomology group of a Schurian cluster-tilted algebra. Manuscript. Math. 128(3), 373–388 (2009)
Assem, I., Redondo, M. J., Schiffler, R.: On the first Hochschild cohomology group of a cluster-tilted algebra. Algebr. Represent. Theory 18(6), 1547–1576 (2015)
Assem, I., Redondo, M. J., Schiffler, R.: On sequential walks
Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras 1 : Techniques of Representation Theory, vol. 65 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge. Techniques of representation theory (2006)
Assem, I., Skowroński, A.: On some classes of simply connected algebras. Proc. Lond. Math. Soc. (3) 56(3), 417–450 (1988)
Barot, M., Fernández, E., Platzeck, M. I., Pratti, N. I., Trepode, S.: From iterated tilted algebras to cluster-tilted algebras. Adv. Math. 223 (4), 1468–1494 (2010)
Barot, M., Geiss, C., Zelevinsky, A.: Cluster algebras of finite type and positive symmetrizable matrices. J. London Math. Soc. (2) 73(3), 545–564 (2006)
Barot, M., Trepode, S.: Cluster tilted algebras with a cyclically oriented quiver. Comm. Algebra 41(10), 3613–3628 (2013)
Bongartz, K.: Algebras and quadratic forms. J. London Math. Algebras Soc. (2) 28(3), 461–469 (1983)
Bongartz, K., Gabriel, P.: Covering spaces in representation-theory. Invent. Math. 65(3), 331–378 (1981/82)
Buan, A. B., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)
Buan, A. B., Marsh, R. J., Reiten, I.: Cluster-tilted algebras of finite representation type. J. Algebra 306(2), 412–431 (2006)
Buan, A. B., Marsh, R. J., Reiten, I.: Cluster-tilted algebras. Trans. Amer. Math. Soc. 359(1), 323–332 (2007). (electronic)
Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (an case). Trans. Amer. Math. Soc. 358(3), 1347–1364 (2006)
Chaparro, C., Schroll, S., Solotar, A.: On the lie algebra structure of the first hochschild cohomology of gentle algebras and brauer graph algebras. Journal of Algebra
David-Roesler, L., Schiffler, R.: Algebras from surfaces without punctures. J. Algebra 350, 218–244 (2012)
de la Peña, J. A., Saorín, M: On the first Hochschild cohomology group of an algebra. Manuscript. Math. 104(4), 431–442 (2001)
Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations. I. Mutations. Sel. Math. (N.S.) 14(1), 59–119 (2008)
Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math. 201(1), 83–146 (2008)
Happel, D.: Hochschild cohomology of finite-dimensional algebras, vol. 1404 of Lecture Notes in Math. Springer, Berlin, pp. 108–126 (1989)
Keller, B.: Deformed Calabi-Yau completions. J. Reine Angew. Math. 654, 125–180 (2011). With an appendix by Michel Van den Bergh
Labardini-Fragoso, D.: Quivers with potentials associated to triangulated surfaces. Proc. Lond. Math Soc. (3) 98(3), 797–839 (2009)
Le, M.P.: Topological invariants of piecewise hereditary algebras. Trans. Amer. Math. Soc. 363(4), 2143–2170 (2011)
Skowroński, A.: Simply connected algebras and Hochschild cohomologies [ MR1206961 (94e:16016)]. In: Representations of algebras (Ottawa, ON, 1992), vol. 14 of CMS Conf. Proc. Amer. Math. Soc., Providence, pp. 431–447 (1993)
Valdivieso-Diaz, Y.: Hochschild cohomology of jacobian algebras from unpunctured surfaces: A geometric computation. arXiv:1512.00738 (2015)
Acknowledgments
The first named author gratefully acknowledges partial support from NSERC of Canada. The third named author acknowledges partial support from ANPCyT, Argentina. The fourth named author is deeply grateful to Département de Mathématiques of the Université de Sherbrooke and Ibrahim Assem for supporting and providing warm and ideal working conditions during her stay at Sherbrooke, she is now a Royal Society Newton Fellow and part of this work was made under this fellowship.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Christof Geiss
Dedicated to José Antonio de la Peña for his 60th birthday
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Assem, I., Bustamante, J.C., Trepode, S. et al. From the Potential to the First Hochschild Cohomology Group of a Cluster Tilted Algebra. Algebr Represent Theor 24, 1191–1220 (2021). https://doi.org/10.1007/s10468-020-09985-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-020-09985-7