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Commutative finitely generated algebras satisfying \(((yx)x)x=0\) are solvable. (English) Zbl 1323.17001

From the text: The authors study commutative, nonassociative algebras satisfying the identity (1) \(((yx)x)x=0\). They show that finitely generated algebras over a field \(K\) of characteristic \(\neq 2\) satisfying (1) are solvable. For \(x\) in an algebra \(A\), define the multiplication operator \(R_x\) by \(yR_x=yx\), for all \(y\in A\). Their identity is then that \(R^3_x=0\).
The author’s interest in this problem arose from attempts to prove the Albert-Gerstenhaber conjecture. The conjecture asks if every commutative, power-associative, finite-dimensional, nil algebra is solvable. In such algebras the multiplication operator \(R_x\) is nilpotent for each \(x\). The authors’ result forms part of the solution for the Albert-Gerstenhaber conjecture in the particular case of nilindex four.

MSC:

17A30 Nonassociative algebras satisfying other identities

Software:

Albert
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References:

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