2018/03/13 by Shigeo Koshitani, Taro Sakurai
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Block (permutation group theory) #Characterization (materials science) #Codimension #Combinatorics #Commutator #Finite Group Theory Research #Jacobson radical #Lie conformal algebra #Mathematics #Pure mathematics #Ring (chemistry) #math.GR #math.RA #math.RT #msc:16E40 #msc:16G10 #msc:16P10 #msc:20C20
paper · pdf · doi:10.1007/s00013-019-01311-y
published in Archiv der Mathematik 113(1), 1-10 (Birkhäuser) · 9 pages
arxiv created 2018/03/13 · openalex created_date 2018/03/29 · openalex publication_date 2019/03/19 · arxiv updated 2019/07/01 · openalex updated_date 2026/08/05
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 nth Jacobson radical Radn(A). We prove that this is Morita invariant and give an upper bound for the codimension as well.