Abstract In the class of left nilpotent algebras of index two it was proved that there are no varieties of fractional polynomial growth ≈ n α with 1 α 2 and 2 α 3 instead it was established the existence of a variety of fractional polynomial growth with α = 7 2 . In this paper we investigate similar problems for varieties of commutative or anticommutative metabelian algebras. We construct a correspondence between left nilpotent algebras of index two and commutative metabelian algebras or anticommutative metabelian algebras and we prove that the codimensions sequences of the corresponding algebras coincide up to a constant. This allows us to transfer the above results concerning varieties of left nilpotent algebras of index two to varieties of commutative or anticommutative metabelian algebras.

中文翻译：

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一些metabelian变体与左幂零变体的对应关系

摘要 在指数二的左幂零代数类中，证明了分数多项式增长 ≈ n α 不存在 1 α 2 和 2 α 3 的变体，而是建立了分数多项式增长的变体，其中 α = 7 2 . 在本文中，我们研究了各种交换或反交换元代数的类似问题。我们构建了索引为 2 的左幂零代数和交换元代数或反交换元代数之间的对应关系，并且我们证明了相应代数的余维数列重合到一个常数。这允许我们将上述关于指数为 2 的左幂零代数的结果转移到交换或反交换元代数的变体。