Abstract
This paper is concerned with the algebraic and geometric classification of the commutative power-associative algebras of dimension over an algebraically closed field k with characteristic relative prime with 30. We prove that an algebra of this variety is Jordan if its dimension as vector space over k is not greater than 3. Albeit in dimension 4, non-Jordan commutative power-associative algebras exist, we can show that such algebras yet admits Wedderburn decomposition. We use the action of over the affine variety CPA4 of four-dimensional commutative power-associative algebras over to get 12 irreducible components that correspond to the Zariski closure of the orbit of rigid algebras, two of which are non-Jordan algebras.
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