Chinese Annals of Mathematics, Series B

, Volume 36, Issue 6, pp 1027–1042 | Cite as

On the first Hochschild cohomology of admissible algebras

  • Fang LiEmail author
  • Dezhan Tan


The aim of this paper is to investigate the first Hochschild cohomology of admissible algebras which can be regarded as a generalization of basic algebras. For this purpose, the authors study differential operators on an admissible algebra. Firstly, differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the k-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field k of characteristic 0.


Quiver Admissible algebra Differential operators Cohomology 

2000 MR Subject Classification

16E40 16G20 16W25 17B70 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ames, G., Cagliero, L. and Tirao, P., Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras, J. Algebra, 322, 2009, 1466–1497.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    Assem, I., Simson, D. and Skowronski, A., Elements of the Representation Theory of Associative Algebras Vol I: Techniques of Representation Theory, London Mathematical Society Student Texts, 65, Cambridge University Press, Cambridge, 2006.Google Scholar
  3. [3]
    Auslander, M., Reiten, I. and Smalø, S. O., Representation Theory of Artin Algebra, Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
  4. [4]
    Bardzell, M. J., The alternating syzygy behavior of monomial algebras, J. Algebra, 188(1), 1997, 69–89.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Bollobas, B., Modern Graph Theory, Graduate Texts in Mathematics, 184, Springer-Verlag, New York, 1998.Google Scholar
  6. [6]
    Cibils, C., Rigidity of truncated quiver algebras, Adv. Math., 79, 1990, 18–42.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Cibils, C., Rigid monomial algebras, Math. Ann., 289, 1991, 95–109.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [[8]
    Crawley-Boevey, W. W., Lectures on Representations of Quivers. ˜pmtwc/quivlecs.pdfGoogle Scholar
  9. [9]
    Gross, J. L. and Tucker, T. W., Topological Graph Theory, John Wiley and Sons, New York, 1987.zbMATHGoogle Scholar
  10. [[10]
    Guo, L. and Li, F., Structure of Hochschild cohomology of path algebras and differential formulation of Euler’s polyhedron formula. arxiv:1010.1980v2Google Scholar
  11. [11]
    Happel, D., Hochschild cohomology of finite dimensional algebras, Lecture Notes in Math., 1404, 1989, 108–126.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Locateli, A. C., Hochschild cohomology of truncated quiver algebras, Comm. Algebra, 27, 1999, 645–664.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    Pena, J. A. and Saorin, M., On the first Hochschild cohomology group of an algebra, Manscripta Math., 104, 2001, 431–442.zbMATHCrossRefGoogle Scholar
  14. [14]
    Sanchez-Flores, S., The Lie module structure on the Hochschild cohomology groups of monomial algebras with radical square zero, J. Algebra, 320, 2008, 4249–4269.zbMATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Sanchez-Flores, S., On the semisimplicity of the outer derivations of monomial algebras, Communications in Algebra, 39, 2011, 3410–3434.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [[16]
    Sardanashvily, G., Differential operators on a Lie and graded Lie algebras. arxiv: 1004.0058v1[math-ph]Google Scholar
  17. [17]
    Strametz, C., The Lie algebra structure on the first Hochschild cohomology group of a monomial algebra, J. Algebra Appl., 5(3), 2006, 245–270.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    Xu, Y., Han, Y. and Jiang, W., Hochschild cohomology of truncated quiver algebras, Science in China, Series A: Mathematics, 50(5), 2007, 727–736.zbMATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    Zhang, P., Hochschild cohomology of truncated basic cycle, Science in China, Series A, 40(12), 1997, 1272–1278.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhouChina
  2. 2.College of Mathematics and Information ScienceShangqiu Normal UniversityShangqiu, HenanChina

Personalised recommendations