Abstract
It is known the classification of commutative power-associative nilalgebras of dimension ? 4 (see, [4]). In [2], we give a description of commutative power-associative nilalgebras of dimension 5. In this work we describe Jordan nilalgebras of dimension 6.
Highlights
It is known the classification of commutative power-associative nilalgebras of dimension ≤ 4
A is called a nilalgebra of nilindex n ≥ 2, if yn = 0 for all y ∈ A and there is x ∈ A such that xn−1 = 0
It is known that any Jordan algebra is power-associative, and that any finite-dimensional Jordan nilalgebra is nilpotent
Summary
Let A be a commutative algebra over a field K. An element x ∈ A is called nilpotent, if there is an integer r ≥ 1 such that xr = 0. If any element in A is nilpotent, A is called a nilalgebra. A is called a nilalgebra of nilindex n ≥ 2, if yn = 0 for all y ∈ A and there is x ∈ A such that xn−1 = 0. It is known that any Jordan algebra is power-associative, and that any finite-dimensional Jordan nilalgebra (of characteristic = 2) is nilpotent (see, [5]). Throughout, A will denote a commutative nilalgebra of nilindex n ≥ 3 over a field K of characteristic = 2, 3. We will denote by < x1, ..., xj >K the subspace generated over K by the elements x1, ..., xj in A. It is clear that x, x2, ..., xn−1 are linearly independent and so dimK(A2) ≥ n − 2 and dimK (A3) ≥ n − 3
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