Skip to main content Accessibility help


  • ZHIHUA WANG (a1) (a2), LIBIN LI (a3) and YINHUO ZHANG (a4)


This paper deals with the Green ring $\mathcal{G}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$ with finitely many isomorphism classes of indecomposable objects over an algebraically closed field. The first part of this paper deals with the question of when the Green ring $\mathcal{G}(\mathcal{C})$ , or the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$ K over a field K, is Jacobson semisimple (namely, has zero Jacobson radical). It turns out that $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$ K is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero in K. For the Green ring $\mathcal{G}(\mathcal{C})$ itself, $\mathcal{G}(\mathcal{C})$ is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero. The second part of this paper focuses on the case where $\mathcal{C}=\text{Rep}(\mathbb {k}G)$ for a cyclic group G of order p over a field $\mathbb {k}$ of characteristic p. In this case, the Casimir number of $\mathcal{C}$ is computable and is shown to be 2p 2. This leads to a complete description of the Jacobson radical of the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$ K over any field K.



Hide All
1. Auslander, M., Reiten, I. and Smalø, S. O., Representation theory of Artin algebras, vol. 36 (Cambridge Studies in Advanced Mathematics, Cambridge, 1994).
2. Bakalov, B. and Kirillov, A. A., Lectures on tensor categories and modular functors, vol. 21, (AMS, Providence, 2001).
3. Benson, D. J., The Green ring of a finite group, J. Algebra 87 (1984), 290331.
4. Benson, D. J. and Carlson, J. F., Nilpotent elements in the Green ring, J. Algebra 104 (1986), 329350.
5. Bhargava, M. and Zieve, M. E., Factoring Dickson polynomials over finite fields, Finite Fields Appl. 5 (2) (1999), 103111.
6. Chen, H., The green ring of drinfeld double D(H 4), Algebras and Representation Theory 17 (5) (2014), 14571483.
7. Chen, H., Oystaeyen, F. V. and Zhang, Y., The green rings of taft algebras, Proc. Amer. Math. Soc. 142 (2014), 765775.
8. Chou, W. S., The factorization of Dickson polynomials over finite fields, Finite Fields Appl. 3 (1997), 8496.
9. Darpö, E. and Herschend, M., On the representation ring of the polynomial algebra over perfect field, Math. Z 265 (2011), 605–615.
10. Domokos, M. and Lenagan, T. H., Representation rings of quantum groups, J. Algebra 282 (2004), 103128.
11. Etingof, P., Gelaki, S., Nikshych, D. and Ostrik, V., Tensor categories, Mathematical Surveys and Monographs, vol. 205 (AMS, Providence, RI, 2015).
12. Etingof, P. and Ostrik, V., Finite tensor categories, Mosc. Math. J 4 (3) (2004), 627654.
13. Green, J. A., A transfer theorem for modular representations, J. Algebra 1 (1964), 7384.
14. Green, J. A., The modular representation algebra of a finite group, Ill. J. Math. 6 (4) (1962), 607619.
15. Higman, D. G., On orders in separable algebras, Canad. J. Math. 7 (1955), 509515.
16. Huang, H., Oystaeyen, F. V., Yang, Y. and Zhang, Y., The Green rings of pointed tensor categories of finite type, J. Pure Appl. Algebra 218 (2014), 333342.
17. Li, Y. and Hu, N., The Green rings of the 2-rank Taft algebra and its two relatives twisted, J. Algebra 410 (2014), 135.
18. Li, L. and Zhang, Y., The Green rings of the generalized Taft Hopf algebras, Contemp. Math. 585 (2013), 275288.
19. Liu, S., Auslander-Reiten theory in a Krull-Schmidt category, São Paulo J. Math. Sci. 4 (3) (2010), 425472.
20. Lorenz, M., Some applications of Frobenius algebras to Hopf algebras, Contemp. Math. 537 (2011), 269289.
21. McDonald, B. R., Finite rings with identity, vol. 28 (Marcel Dekker Incorporated, 1974).
22. Ringel, C. M., Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099 (Springer, Berlin Heidelberg, 1984).
23. Wang, Z., Li, L. and Zhang, Y., Green rings of pointed rank one Hopf algebras of nilpotent type, Algebras Represent. Theory 17 (6) (2014), 19011924.
24. Wang, Z., Li, L. and Zhang, Y., Green rings of pointed rank one Hopf algebras of non-nilpotent type, J. Algebra 449 (2016), 108137.
25. Witherspoon, S. J., The representation ring of the quantum double of a finite group, J. Algebra 179 (1996), 305329.
26. Zemanek, J., Nilpotent elements in representation rings, J. Algebra 19 (1971), 453469.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed