Abstract
Для произвольного поля ${\mathbb F}$ мы рассматриваем коммутативную неассоциативную четырёхмерную алгебру ${\mathfrak M}$ камня, ножниц и бумаги с единичным элементом над полем ${\mathbb F}$ и доказываем, что образ произвольного неассоциативного мультилинейного полинома над ${\mathfrak M}$ является линейным пространством. Тот же вопрос мы рассматриваем и для двух подалгебр: алгебры камня, ножниц и бумаги без единицы, а также, алгебры элементов нулевого следа и нулевой скалярной части.Кроме того, в работе поставлены задачи и рассмотрены вопросы о возможных образах однородных полиномов на этих алгебрах.
Highlights
The study of images of polynomials evaluated on algebras is one of the most important branches of modern algebra
Similar questions for matrix rings were investigated by Bresar ([6])
One of the central conjectures regarding possible evaluations of multilinear polynomials on matrix algebras was attributed to Kaplansky and formulated by L’vov in [7]: Conjecture 1 (L’vov-Kaplansky)
Summary
The study of images of polynomials evaluated on algebras is one of the most important branches of modern algebra. In [10] the same question was considered for the algebra of quaternions with the Hamilton multiplication and it was shown that any evaluation of a multilinear polynomial is a vector space. In the same paper it was said that this question is interesting only for simple algebras, since for non-simple algebras it may be answered negatively. This conjecture fails for the Grassmann algebra. There is an interest in the investigation of this question for non-simple algebras: for some of them the Kaplansky question can be answered positively. The question of possible multilinear non-associative evaluations was considered in [4] where the Kaplansky conjecture was considered for Lie polynomials.
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