Rationality of Hilbert series in noncommutative invariant theory

Domokos, M.; Drensky, Vesselin

doi:10.1142/s0218196717500394

Cited by 5 publications

(2 citation statements)

References 24 publications

(38 reference statements)

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“…Another popular topic in noncommutative invariant theory is the study of invariants of groups acting on relatively free algebras in varieties of associative algebras. See, e.g., the recent paper [6] and the references there for the common features and the differences with classical invariant theory.…”

Section: Theorem 1 the Algebra K[x N ]mentioning

confidence: 99%

Symmetric Polynomials in the Free Metabelian Lie Algebras

2020

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Let K[Xn] be the commutative polynomial algebra in the variables Xn = {x1,. .. , xn} over a field K of characteristic zero. A theorem from undergraduate course of algebra states that the algebra K[Xn] Sn 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[Xn] Sn replacing K[Xn] with the free metabelian Lie algebra Fn of rank n ≥ 2 over K. It is known that the algebra F Sn n is not finitely generated, but its ideal (F n) Sn consisting of the elements of F Sn n in the commutator ideal F n of Fn is a finitely generated K[Xn] Sn-module. In our main result, we describe the generators of the K[Xn] Sn-module (F n) Sn which gives the complete description of the algebra F Sn n .

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Section: Theorem 1 the Algebra K[x N ]mentioning

confidence: 99%

Symmetric Polynomials in the Free Metabelian Lie Algebras

2020

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“…The leading idea of the proof is to find an object which has nice properties from the point of view of classical invariant theory and then to transfer these properties to W G n . A similar idea was used in the proof of [DoDr,Proposition 2.2].…”

Section: Preliminariesmentioning

confidence: 99%

Symmetric polynomials in the free metabelian Lie algebras

Drensky¹,

Findik²,

Oguslu³

2019

Preprint

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Let K[Xn] be the commutative polynomial algebra in the variables Xn = {x 1 , . . . , xn} over a field K of characteristic zero. A theorem from undergraduate course of algebra states that the algebra K[Xn] Sn 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[Xn] Sn replacing K[Xn] with the free metabelian Lie algebra Fn of rank n ≥ 2 over K. It is known that the algebra F Sn n is not finitely generated but its ideal (F ′ n ) Sn consisting of the elements of F Sn n in the commutator ideal F ′ n of Fn is a finitely generated K[Xn] Sn -module. In our main result we describe the generators of the K[Xn] Sn -module (F ′ n ) Sn which gives the complete description of the algebra F Sn n .

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Graded Algebras, Algebraic Functions, Planar Trees, and Elliptic Integrals

Drensky

2020

Springer INdAM Series

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Rationality of Hilbert series in noncommutative invariant theory

Cited by 5 publications

References 24 publications

Symmetric Polynomials in the Free Metabelian Lie Algebras

Symmetric Polynomials in the Free Metabelian Lie Algebras

Symmetric polynomials in the free metabelian Lie algebras

Graded Algebras, Algebraic Functions, Planar Trees, and Elliptic Integrals

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