Abstract
Let \(K[X_n]\) be the commutative polynomial algebra in the variables \(X_n=\{x_1,\ldots ,x_n\}\) over a field K of characteristic zero. A theorem from undergraduate course of algebra states that the algebra \(K[X_n]^{S_n}\) of symmetric polynomials is generated by the elementary symmetric polynomials which are algebraically independent over K. In the present paper, we study a noncommutative and nonassociative analogue of the algebra \(K[X_n]^{S_n}\) replacing \(K[X_n]\) with the free metabelian Lie algebra \(F_n\) of rank \(n\ge 2\) over K. It is known that the algebra \(F_n^{S_n}\) is not finitely generated, but its ideal \((F_n')^{S_n}\) consisting of the elements of \(F_n^{S_n}\) in the commutator ideal \(F_n'\) of \(F_n\) is a finitely generated \(K[X_n]^{S_n}\)-module. In our main result, we describe the generators of the \(K[X_n]^{S_n}\)-module \((F_n')^{S_n}\) which gives the complete description of the algebra \(F_n^{S_n}\).
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Acknowledgements
The authors are very grateful to the anonymous referee for the careful reading of the manuscript and the valuable suggestions for the improvement of the exposition. The research of the first named author was partially supported by Grant KP-06-N-32/1 “Groups and Rings—Theory and Applications” of the Bulgarian National Science Fund.
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Drensky, V., Fındık, Ş. & Öüşlü, N.Ş. Symmetric Polynomials in the Free Metabelian Lie Algebras. Mediterr. J. Math. 17, 151 (2020). https://doi.org/10.1007/s00009-020-01582-8
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DOI: https://doi.org/10.1007/s00009-020-01582-8