Equality of local and ordinary derivations for genetic Volterra algebras

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Let AA be an nn-dimensional genetic Volterra algebra. A local derivation is a linear map on AA such that, for every x∈Ax\in A, there exists a derivation Dx:A→AD_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.

References

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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