Equality of local and ordinary derivations for genetic Volterra algebras

Let AA be an nn-dimensional genetic Volterra algebra. A local derivation is a linear map on AA such that, for every xAx\in A, there exists a derivation Dx:AAD_x:A\to A with Δ(x)=Dx(x)\Delta(x)=D_x(x). The derivations of AA form the usual class of linear maps satisfying the derivation identity.

Local-derivation conjecture. The category of local derivations of AA coincides with the category of derivations of AA.

The preceding theorem establishes this equality for three-dimensional genetic Volterra algebras. The assertion asks whether the same conclusion holds in every dimension.

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Primary source

Rasul Ganikhodzhaev, Farrukh Mukhamedov, Abror Pirnapasov and Izzat Qaralleh, “Genetic Volterra algebras and their derivations”, arXiv:1706.01667 (2017).

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