Palindromes in the free metabelian Lie algebras

Findik, Sehmus; Oguslu, Nazar Sahin

doi:10.1142/s0218196719500334

Cited by 9 publications

(5 citation statements)

References 5 publications

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“…of symmetric polynomials coincide with the algebras of invariants of the group S n . See the work [5] for F S2 2 , and its generalization [4] for the full description of the algebra F Sn n . The decription of the algebra L Sn n,c is a direct consequence of the known results on the algebra…”

Section: Preliminariesmentioning

confidence: 99%

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On Automorphisms of Lie Algebra of Symmetric Polynomials

Findik

Oguslu

2023

Journal of Universal Mathematics

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Let $L_{n}$ be the free Lie algebra of rank $n$ over a field $K$ of characteristic zero, $L_{n,c}=L_{n}/(L_{n}''+\gamma_{c+1}(L_{n}))$ be the free metabelian nilpotent of class $c$ Lie algebra, and $F_{n}=L_{n}/L_{n}''$ be the free metabelian Lie algebra generated by $x_1,\ldots,x_n$ over a field $K$ of characteristic zero. We call a polynomial $p(X_n)$ in these Lie algebras {\it symmetric} if $p(x_1,\ldots,x_n)=p(x_{\pi(1)},\ldots,x_{\pi(n)})$ for each element of the symmetric group $S_n$. The sets $L_n^{S_n}$, $F_n^{S_n}$, and $L_{n,c}^{S_n}$ of symmetric polynomials coincides with the algebras of invariants of the group $S_n$ in $L_{n}$, $F_{n}$, and $L_{n,c}$, respectively. We determine the groups $\text{\rm Inn}(L_{n,c}^{S_n})\cap \text{\rm Inn}(L_{n,c})$ and $\text{\rm Inn}(F_{n}^{S_n})\cap \text{\rm Inn}(F_{n})$ of inner automorphisms of the algebras $L_{n,c}^{S_n}$ and $F_{n}^{S_n}$ in the groups $\text{\rm Inn}(L_{n,c})$ and $\text{\rm Inn}(F_{n})$, respectively. In particular, we obtain the descriptions of the groups $\text{\rm Aut}(L_{2}^{S_2})\cap \text{\rm Aut}(L_{2})$ and $\text{\rm Aut}(F_{2}^{S_2})\cap \text{\rm Aut}(F_{2})$ of automorphisms of the algebras $L_{2}^{S_2}$ and $F_{2}^{S_2}$ in the groups $\text{\rm Aut}(L_{2})$ and $\text{\rm Aut}(F_{2})$, respectively.

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Section: Preliminariesmentioning

confidence: 99%

“…[3], that the algebra F Sn n of symmetric polynomials is not finitely generated. Recently the authors [5] have provided an infinite set of generators for F S2 2 , later the result was generalized in [4].…”

Section: Introductionmentioning

confidence: 99%

On Automorphisms of Lie Algebra of Symmetric Polynomials

Findik

Oguslu

2023

Journal of Universal Mathematics

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“…The polynomials 𝑝 1 , … , 𝑝 𝑛 ∈ 𝐾[𝑥 1 , … , 𝑥 𝑛 ] form another generating set for the symmetric polynomials, where Cox et al, 2015;Strumfels, 2008;van der Waerden, 1949. ) We refer the readers to the work by Fındık and Öğüşlü (2019) respectively. These sets are subalgebras of invariants of the symmetric group 𝑆 2 .…”

Section: Preliminariesmentioning

confidence: 99%

On the symmetric polynomials in the variety of Grassmann algebras

Akdoğan

2021

Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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“…In the present paper, we describe the algebra F Sn n of the symmetric polynomials in the free metabelian Lie algebra F n , n ≥ 2, and give a generating set of the K[X n ] Sn -module (F n ) Sn . The case n = 2 was handled by Fındık andÖǧüşlü [9]. They gave the description of the symmetric polynomials and a nonfinite generating set for the algebra F S2…”

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

confidence: 99%

Symmetric Polynomials in the Free Metabelian Lie Algebras

2020

Self Cite

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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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Palindromes in the free metabelian Lie algebras

Cited by 9 publications

References 5 publications

On Automorphisms of Lie Algebra of Symmetric Polynomials

On Automorphisms of Lie Algebra of Symmetric Polynomials

On the symmetric polynomials in the variety of Grassmann algebras

Symmetric Polynomials in the Free Metabelian Lie Algebras

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