Let \({\mathbb{C}}\) be the field of complex numbers and let \({\mathbb{C}}\left[ {W_{n} } \right]\) be the polynomial algebra over \({\mathbb{C}}\) in the set \(W_{n} = \left\{ {w_{1} , \ldots ,w_{n} } \right\}\) of \(n\) commuting variables. This is the free associative algebra of rank \(n\) in the variety of commutative associative algebras over the field \({\mathbb{C}}\) . A polynomial \(p \in {\mathbb{C}}\left[ {W_{n} } \right]\) is called symmetric if \(p\left( {w_{1} , \ldots ,w_{n} } \right) = p\left( {w_{\pi \left( 1 \right)} , \ldots ,w_{\pi \left( n \right)} } \right) \) for each permutation \(\in S_{n}\) , where \(S_{n}\) denotes the symmetric group \(n\) elements. It is folklorely known that such polynomials form a subalgebra of \({\mathbb{C}}\left[ {W_{n} } \right]\) , which is equal to the algebra \({\mathbb{C}}\left[ {W_{n} } \right]^{{S_{n} }}\) of invariants of the symmetric group \(S_{n}\) in \({\mathbb{C}}\left[ {W_{n} } \right]\) . The algebra \({\mathbb{C}}\left[ {W_{n} } \right]^{{S_{n} }}\) is finitely generated by algebraically independent elementary symmetric polynomials. The concept of symmetric polynomials can be generalized to other finitely generated free or relatively free objects in non-commutative and non-associative varieties such as Lie, Poisson, or Leibniz algebras. In this study, the symmetric polynomials of free metabelian Lie algebras, Poisson algebras, and Leibniz algebras were surveyed.

Advances in Symmetric Polynomials

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2025年

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Let \({\mathbb{C}}\) be the field of complex numbers and let \({\mathbb{C}}\left[ {W_{n} } \right]\) be the polynomial algebra over \({\mathbb{C}}\) in the set \(W_{n} = \left\{ {w_{1} , \ldots ,w_{n} } \right\}\) of \(n\) commuting variables. This is the free associative algebra of rank \(n\) in the variety of commutative associative algebras over the field \({\mathbb{C}}\) . A polynomial \(p \in {\mathbb{C}}\left[ {W_{n} } \right]\) is called symmetric if \(p\left( {w_{1} , \ldots ,w_{n} } \right) = p\left( {w_{\pi \left( 1 \right)} , \ldots ,w_{\pi \left( n \right)} } \right) \) for each permutation \(\in S_{n}\) , where \(S_{n}\) denotes the symmetric group \(n\) elements. It is folklorely known that such polynomials form a subalgebra of \({\mathbb{C}}\left[ {W_{n} } \right]\) , which is equal to the algebra \({\mathbb{C}}\left[ {W_{n} } \right]^{{S_{n} }}\) of invariants of the symmetric group \(S_{n}\) in \({\mathbb{C}}\left[ {W_{n} } \right]\) . The algebra \({\mathbb{C}}\left[ {W_{n} } \right]^{{S_{n} }}\) is finitely generated by algebraically independent elementary symmetric polynomials. The concept of symmetric polynomials can be generalized to other finitely generated free or relatively free objects in non-commutative and non-associative varieties such as Lie, Poisson, or Leibniz algebras. In this study, the symmetric polynomials of free metabelian Lie algebras, Poisson algebras, and Leibniz algebras were surveyed.