Idempotent words in finite-dimensional simple Jordan algebras
Abstract
We study finite left-nested Jordan products of idempotents in finite-dimensional simple Jordan algebras over a field 𝔽 of characteristic different from $2$. Over an algebraically closed field, the structure theorem reduces the problem to the scalar algebra, spin factors, the full matrix Jordan algebras Mn(𝔽)+, the symmetric matrix algebras Hn(𝔽), the symplectic Hermitian algebras H2n(𝔽,sp), and the split Albert algebra. We determine exactly which elements are representable: except in H2(𝔽), these are all nonscalar elements together with $0$ and $1$, whereas in H2(𝔽) the nonzero square-zero elements are also not representable. We also study the minimal word length of representable elements. For Mn(𝔽)+ over an arbitrary field, the maximal word length grows logarithmically in n, and for n=2 we determine the exact length of every nonscalar matrix. The same logarithmic growth holds for H2n(𝔽,sp). The spin-factor and symmetric-matrix results hold over quadratically closed fields, whereas the full matrix, symplectic Hermitian, and split Albert results hold over arbitrary fields of characteristic different from $2$.
Related
- Classification of Finite-Dimensional Simple Novikov Algebras and Their Irreducible Bimodules in Positive Characteristic
- Classification of Hom--Lie structures on finite-dimensional simple Lie algebras
- Singular solutions from simple Euclidean Jordan algebras
- Isomorphism classes and geometric classification of three-dimensional Jordan algebras and the length function
- simple cartan-free modules of finite rank over the super-virasoro algebras
BibTeX
@misc{gogic2026idempotent,
title = {{Idempotent words in finite-dimensional simple Jordan algebras}},
author = {Ilja Gogić and Mateo Tomašević},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23581},
eprint = {2609.23581},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23581},
}
RIS
TY - UNPB
TI - Idempotent words in finite-dimensional simple Jordan algebras
AU - Ilja Gogić
AU - Mateo Tomašević
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23581
UR - https://arxiv.org/abs/2609.23581
AB - We study finite left-nested Jordan products of idempotents in finite-dimensional simple Jordan algebras over a field 𝔽 of characteristic different from $2$. Over an algebraically closed field, the structure theorem reduces the problem to the scalar algebra, spin factors, the full matrix Jordan algebras M_n(𝔽)^+, the symmetric matrix algebras H_n(𝔽), the symplectic Hermitian algebras H_2n(𝔽,sp), and the split Albert algebra. We determine exactly which elements are representable: except in H_2(𝔽), these are all nonscalar elements together with $0$ and $1$, whereas in H_2(𝔽) the nonzero square-zero elements are also not representable. We also study the minimal word length of representable elements. For M_n(𝔽)^+ over an arbitrary field, the maximal word length grows logarithmically in n, and for n=2 we determine the exact length of every nonscalar matrix. The same logarithmic growth holds for H_2n(𝔽,sp). The spin-factor and symmetric-matrix results hold over quadratically closed fields, whereas the full matrix, symplectic Hermitian, and split Albert results hold over arbitrary fields of characteristic different from $2$.
ER -