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<p>The main goal of this article is to investigate the relationship between action accessibility and weak action representability in the context of varieties of non-associative algebras over a field. Specifically, using an argument of J.&#xa0;R.&#xa0;A.&#xa0;Gray in the setting of groups, we prove that the varieties of <i>k</i>-nilpotent Lie algebras (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k \geqslant 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>⩾</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>) and the varieties of <i>n</i>-solvable Lie algebras (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n \geqslant 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) do not form weakly action representable categories. These are the first known examples of action accessible varieties of non-associative algebras that fail to be weakly action representable, establishing that a subvariety of a (weakly) action representable variety of non-associative algebras needs not be weakly action representable. Eventually, we refine J.&#xa0;R.&#xa0;A.&#xa0;Gray’s result by proving that the varieties of <i>k</i>-nilpotent groups (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(k \geqslant 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>⩾</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>) and that of 2-solvable groups are not weakly action representable.</p>

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Action accessible and weakly action representable varieties of algebras

  • 期刊论文

  • 发表日期：

    2025年7月21日

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  • Xabier García-Martínez,
  • Manuel Mancini

摘要

The main goal of this article is to investigate the relationship between action accessibility and weak action representability in the context of varieties of non-associative algebras over a field. Specifically, using an argument of J. R. A. Gray in the setting of groups, we prove that the varieties of k-nilpotent Lie algebras ( \(k \geqslant 3\) k ⩾ 3 ) and the varieties of n-solvable Lie algebras ( \(n \geqslant 2\) n ⩾ 2 ) do not form weakly action representable categories. These are the first known examples of action accessible varieties of non-associative algebras that fail to be weakly action representable, establishing that a subvariety of a (weakly) action representable variety of non-associative algebras needs not be weakly action representable. Eventually, we refine J. R. A. Gray’s result by proving that the varieties of k-nilpotent groups ( \(k \geqslant 3\) k ⩾ 3 ) and that of 2-solvable groups are not weakly action representable.