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arXiv · 2607.15154

On the Grothendieck Ring of Finite-Dimensional Non-Degenerate Evolution Algebras

Idrees Alshatnawi·Cristina Costoya·Antonio Viruel
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Abstract

We study the Grothendieck ring of finite-dimensional non-degenerate evolution algebras over a field $\mathbb{K}$, with addition induced by direct sum and multiplication induced by tensor product. Although its underlying abelian group is freely generated by indecomposable isomorphism classes, the ring structure is much smaller: tensor products create systematic non-cancellation phenomena and many zero-divisors. The key invariant is the balance of the directed graph associated with a natural basis. We prove that, for non-degenerate evolution algebras, this balance is independent of the chosen natural basis. We then show that balance controls cancellation after tensoring: indecomposable factors of balance $1$ cancel, whereas factors of larger balance produce zero-divisor relations under mild hypotheses. Finally, we analyze the subring generated by cyclic evolution algebras and prove the predicted zero-divisor criterion for all finite direct sums of cyclic algebras.

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Idrees Alshatnawi, Cristina Costoya, Antonio Viruel. 2026-07-16. On the Grothendieck Ring of Finite-Dimensional Non-Degenerate Evolution Algebras. https://arxiv.org/abs/2607.15154

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