Abstract
This paper provides a method for the computation of Yoneda algebras for algebras of dihedral type. The Yoneda algebras for one infinite family of algebras of dihedral type (the family \(D(3\mathcal R)\) in K. Erdmann’s notation) are computed. The minimal projective resolutions of simple modules were calculated by an original computer program implemented by one of the authors in C++ language. The algorithm of the program is based on a diagrammatic method presented in this paper and inspired by that of D. Benson and J. Carlson.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Erdmann, K.: Blocks of Tame Representation Type and Related Algebras. Lecture Notes in Math. vol. 1428. Springer, Berlin Heidelberg New York (1990)
Generalov, A.I.: Cohomology of algebras of dihedral type, I. Zapiski Nauchn. Sem. POMI 265, 139–162 (1999). (in Russian); English transl. in: J. Math. Sci. 112(3), 4318–4331 (2002)
Balashov, O.I., Generalov, A.I.: The Yoneda algebras for some class of dihedral algebras. Vestn. St.-Peterbg. Univ., Ser. 1 3(15), 3–10 (1999). (in Russian); English transl. in: Vestnik St.-Petersburg. Univ. Math. 32(3), 1–8 (1999)
Balashov, O.I., Generalov, A.I.: Cohomology of algebras of dihedral type, II. Algebra Anal. 13(1), 3–25 (2001). (in Russian); English transl. in: St. Petersburg. Math. J. 13(1), 1–16 (2002)
Generalov, A.I.: Cohomology of algebras of semidihedral type, I. Algebra Anal. 13(4), 54–85 (2001). (in Russian); English transl. in: St. Petersburg. Math. J. 13(4), 549–573 (2002)
Antipov, M.A., Generalov, A.I.: Cohomology of algebras of semidihedral type, II. Zapiski nauchn. sem. POMI 289, 9–36 (2002). (in Russian); English transl. in: J. Math. Sci. 124(1), 4681–4697 (2004)
Generalov, A.I., Osiuk, E.A.: Cohomology of algebras of dihedral type, III. The family \(D{\left( {2{\user1{\mathcal{A}}}} \right)}\). Zapiski nauchn. sem. POMI 289, 113–133 (2002). (in Russian); English transl. in: J. Math. Sci. 124(1), 4741–4753 (2004)
Generalov, A.I.: Cohomology of algebras of dihedral type, IV. The family \(D{\left( {2{\user1{\mathcal{B}}}} \right)}\). Zapiski nauchn. sem. POMI 289, 76–89 (2002). (in Russian); English transl. in: J. Math. Sci. 124(1), 4719–4726 (2004)
Kosmatov, N.: Resolut, http://www.lifc.univ-fcomte.fr/~kosmatov/resolut/, 2003
Benson, D.J., Carlson, J.F.: Diagrammatic methods for modular representations and cohomology. Comm. Algebra 15(1–2), 53–121 (1987)
Alperin, J.L.: Diagrams for modules. J. Pure Appl. Algebra 16(2), 111–119 (1980)
Fuller, K.R.: Algebras from diagrams. J. Pure Appl. Algebra 48(1–2), 23–37 (1987)
Benson, D.J.: Representations and Cohomology, I. Basic Representation Theory of Finite Groups and Associative Algebras. Cambridge University Press, Cambridge (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the grant \(N^o\) 06-01-00200 of the Russian Foundation for Basic Research.
Rights and permissions
About this article
Cite this article
Generalov, A., Kosmatov, N. Projective Resolutions and Yoneda Algebras for Algebras of Dihedral Type. Algebr Represent Theor 10, 241–256 (2007). https://doi.org/10.1007/s10468-006-9012-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-006-9012-7