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On $3$-generated axial algebras of Jordan type $\frac{1}{2}$
- Source :
- Communications in Mathematics, Volume 33 (2025), Issue 3 (Special issue: European Non-Associative Algebra Seminar) (October 7, 2024) cm:13307
- Publication Year :
- 2023
-
Abstract
- Axial algebras of Jordan type $\eta$ are a special type of commutative non-associative algebras. They are generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-\eta)$, where $\eta$ is a fixed value that is not equal to $0$ or $1$. These algebras have restrictive multiplication rules that generalize the Peirce decomposition for idempotents in Jordan algebras. A universal $3$-generated algebra of Jordan type $\frac{1}{2}$ as an algebra with $4$ parameters was constructed by I. Gorshkov and A. Staroletov. Depending on the value of the parameter, the universal algebra may contain a non-trivial form radical. In this paper, we describe all semisimple $3$-generated algebras of Jordan type $\frac{1}{2}$ over a quadratically closed field.<br />Comment: 12 pages
- Subjects :
- Mathematics - Rings and Algebras
Mathematics - Group Theory
Subjects
Details
- Database :
- arXiv
- Journal :
- Communications in Mathematics, Volume 33 (2025), Issue 3 (Special issue: European Non-Associative Algebra Seminar) (October 7, 2024) cm:13307
- Publication Type :
- Report
- Accession number :
- edsarx.2309.10680
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.46298/cm.13307