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arXiv · 2508.16427

Axial identities

Louis Halle Rowen
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Abstract

Axial algebras are algebras generated by semisimple idempotents whose eigenspace decompositions satisfy prescribed fusion rules, such as Jordan fusion rules, which hold in Jordan algebras and Matsuo algebras. Axial algebras are studied here in terms of identities involving idempotents and/or axes, and at times involving Frobenius forms. After laying out the general theory, organized as a PI-type theory of marked varieties, we turn to Jordan type, providing insight into a major theorem of I.~Gorshkov, S.~Shpectorov, and A.~Staroletov about solid subalgebras. This approach also leads to generic constructions of primitive $N$-generated axial algebras of Jordan type $\gl$, for $N\le 4$, and for arbitrary $N$ we obtain corresponding ``$M$-primitive'' constructions. We discuss the relevance to the problem of finding a primitive nonsingular axial algebra of Jordan type~$\half$, which is neither Jordan nor a homomorphic image of a Matsuo algebra, although we cannot provide an answer at this stage.

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Louis Halle Rowen. 2025-08-22. Axial identities. https://arxiv.org/abs/2508.16427

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