Abstract
It is proven that in free Lie nilpotent n-class algebra \(F_{2}^{{(n)}}\) of rank 2 over the field of characteristics \(p \geqslant n \geqslant 4\) there exists a finite decreasing series of T-ideals \({{T}_{0}} \supseteq {{T}_{1}} \supseteq \ldots {{T}_{k}} \supseteq {{T}_{{k + 1}}} = 0\), such as \({{T}_{0}} = {{T}^{{(3)}}}\), where the T-ideal, generated by commutator \([{{x}_{1}},{{x}_{2}},{{x}_{3}}]\) and factors \({{T}_{i}}{\text{/}}{{T}_{{i + 1}}}\) do not contain the proper T-spaces. This implies that every T-space of algebra \(F_{2}^{{(n)}}\), which is contained in T-ideal \({{T}^{{(3)}}}\), has a finite system of generators. This result is an answer to the question of Grishin, formulated in the study “On T-spaces in a relatively free two-generated Lie nilpotent associative algebra of index 4,” J. Math. Sci. 191, 686–690 (2013).
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Glizburg, V.I., Pchelintsev, S.V. Finitely Based T-Spaces of Free Lie Nilpotent Algebras of Rank 2. Russ Math. 66, 1–7 (2022). https://doi.org/10.3103/S1066369X2210005X
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DOI: https://doi.org/10.3103/S1066369X2210005X