| Full text | |
| Author(s): |
Centrone, Lucio
;
Koshlukov, Plamen
;
Pereira, Kaue
Total Authors: 3
|
| Document type: | Journal article |
| Source: | Journal of Pure and Applied Algebra; v. 229, n. 6, p. 20-pg., 2025-06-01. |
| Abstract | |
Let A = A(1 )circle plus & ctdot;circle plus A(r) be a decomposition of the algebra A as a direct sum of vector subspaces. If for every choice of the indices 1 <= i(j )<= r there exist a(ij )is an element of A(ij) such that the product a(i1 )& ctdot;a(in )not equal 0, and for every 1 <= i, j <= r there is a constant beta(i, j) not equal 0 with a(i)a(j )= beta(i, j)a(j)a(i) for a(i )is an element of A(i), a(j )is an element of A(j), the above decomposition is regular. Bahturin and Regev raised the following conjecture: suppose that the regular decomposition comes from a group grading on A, and form the rxr matrix whose (i, j)th entry equals beta(i, j). Then this matrix is invertible if and only if the decomposition is minimal (that is one cannot get a regular decomposition of A by coarsening the decomposition). Aljadeff and David proved that the conjecture is true in the case the base field is of characteristic 0. We prove that the conjecture does not hold for algebras over fields of positive characteristic, by constructing algebras with minimal regular decompositions such that the associated matrix is singular.<br /> (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies. (AU) | |
| FAPESP's process: | 23/01673-0 - Graded tensor product and regular algebras |
| Grantee: | Kauê Orlando Pereira |
| Support Opportunities: | Scholarships in Brazil - Doctorate |
| FAPESP's process: | 18/23690-6 - Structures, representations, and applications of algebraic systems |
| Grantee: | Ivan Chestakov |
| Support Opportunities: | Research Projects - Thematic Grants |