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Algebras with bounded multiplicities of the central cocharacters
Abstract
Let A be an associative PI-algebra over a field of characteristic zero. The central polynomials of A give rise to a sequence {χnz(A)}n≥ 1 of Sn-characters, called its central cocharacters. A natural problem is to understand which algebraic properties of A are imposed by bounds on the multiplicities occurring in the decompositions of these cocharacters into irreducible characters. We prove that these multiplicities are bounded by one if and only if there exist scalars (α,β)≠(0,0) such that α[x1,x2]x1+βx1[x1,x2] takes values in the center of A. This provides a central analogue of the Ananin-Kemer theorem for ordinary cocharacters. We also characterize the algebras whose central cocharacter multiplicities are uniformly bounded.
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BibTeX
@misc{cota2026algebras,
title = {{Algebras with bounded multiplicities of the central cocharacters}},
author = {Wesley Q. Cota and Luiz H. S. Matos and Breno O. Souza},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28885},
eprint = {2610.28885},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28885},
}
RIS
TY - UNPB
TI - Algebras with bounded multiplicities of the central cocharacters
AU - Wesley Q. Cota
AU - Luiz H. S. Matos
AU - Breno O. Souza
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28885
UR - https://arxiv.org/abs/2610.28885
AB - Let A be an associative PI-algebra over a field of characteristic zero. The central polynomials of A give rise to a sequence {χ_n^z(A)}_n≥ 1 of S_n-characters, called its central cocharacters. A natural problem is to understand which algebraic properties of A are imposed by bounds on the multiplicities occurring in the decompositions of these cocharacters into irreducible characters. We prove that these multiplicities are bounded by one if and only if there exist scalars (α,β)≠(0,0) such that α[x_1,x_2]x_1+βx_1[x_1,x_2] takes values in the center of A. This provides a central analogue of the Ananin-Kemer theorem for ordinary cocharacters. We also characterize the algebras whose central cocharacter multiplicities are uniformly bounded.
ER -