Uniqueness of indecomposable idempotents in algebras with involution
Abstract
We prove uniqueness of a decomposition of 1 into indecomposable Hermitian idempotents in an order of a finite-dimensional $\mathbb{Q}$-algebra with positive involution, by generalising a result of Eichler on unique decomposition of lattices. We use this result to prove that polarised abelian varieties over any field admit a unique decomposition into indecomposable polarised abelian subvarieties, a result previously shown by Debarre over algebraically closed fields and Serre for arbitrary fields with different methods. We prove that an analogous uniqueness result holds true for arbitrary polarised integral Hodge structures, and derive a consequence for their automorphism groups.
Fields of Study & Subjects
Advanced Algebra and Logic
Advanced Topics in Algebra
Rings
Modules
and Algebras
Publication details
- Journal / source
- Asian Journal of Mathematics
- Publisher
- International Press of Boston
- Volume
- 30
- Issue
- 2
- Pages
- 133–145
- Publication date
- 1 January 2026
- Language
- EN