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Archiv der Mathematik

, Volume 113, Issue 1, pp 1–10 | Cite as

On theorems of Brauer-Nesbitt and Brandt for characterizations of small block algebras

  • Shigeo Koshitani
  • Taro SakuraiEmail author
Article
  • 19 Downloads

Abstract

In 1941, Brauer-Nesbitt established a characterization of a block with trivial defect group as a block B with \(k(B) = 1\) where k(B) is the number of irreducible ordinary characters of B. In 1982, Brandt established a characterization of a block with defect group of order two as a block B with \(k(B) = 2\). These correspond to the cases when the block is Morita equivalent to the one-dimensional algebra and to the non-semisimple two-dimensional algebra, respectively. In this paper, we redefine k(A) to be the codimension of the commutator subspace K(A) of a finite-dimensional algebra A and prove analogous statements for arbitrary (not necessarily symmetric) finite-dimensional algebras. This is achieved by extending the Okuyama refinement of the Brandt result to this setting. To this end, we study the codimension of the sum of the commutator subspace K(A) and the nth power \({\text {Rad}}^n(A)\) of the Jacobson radical \({\text {Rad}}(A)\). We prove that this is Morita invariant and give an upper bound for the codimension as well.

Keywords

Codimension Commutator subspace Finite-dimensional algebra Morita invariant Morita equivalence 

Mathematics Subject Classification

Primary 16G10 Secondary 16P10 16E40 20C20 

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Notes

Acknowledgements

We would like to thank the referee for useful comments and advice on the earlier version of the manuscript. We wish to thank Yoshihiro Otokita for helpful discussions. This is a part of the PhD thesis of the second author. After the first version of this paper had appeared on the arXiv, Burkhard Külshammer informed us that Theorems 1.2 and 1.3 in this version already had appeared in the unpublished Ph.D. thesis by Marlene Chlebowitz [6]. We thank Burkhard Külshammer very much.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Center for Frontier ScienceChiba UniversityChibaJapan
  2. 2.Department of Mathematics and Informatics, Graduate School of ScienceChiba UniversityChiba-shiJapan

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