Ravil Bildanov ; Ilya Gorshkov - On $3$-generated axial algebras of Jordan type $\frac{1}{2}$

cm:13307 - Communications in Mathematics, October 7, 2024, Volume 33 (2025), Issue 3 (Special issue: European Non-Associative Algebra Seminar) - https://doi.org/10.46298/cm.13307
On $3$-generated axial algebras of Jordan type $\frac{1}{2}$Article

Authors: Ravil Bildanov ; Ilya Gorshkov

    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.


    Volume: Volume 33 (2025), Issue 3 (Special issue: European Non-Associative Algebra Seminar)
    Published on: October 7, 2024
    Accepted on: June 7, 2024
    Submitted on: March 28, 2024
    Keywords: Mathematics - Rings and Algebras,Mathematics - Group Theory

    Consultation statistics

    This page has been seen 59 times.
    This article's PDF has been downloaded 27 times.