arXiv subjects

Sehmus Findik

Publications and source records attributed to Sehmus Findik.

15 recordsLinked to original sources

arXiv2025

On the Nowicki Conjecture for the free Lie algebra of rank 2

Let K[X_n]=K[x_1,\ldots,x_n] be the polynomial algebra in n variables over a field K of characteristic zero. A locally nilpotent linear derivation \delta of K[X_n] is called Weitzenb\"ock due to his well known result from 1932 stating that the algebra \text{\rm ker}(\delta)=K[X_n]^{\delta} of constants of $\delta$ is finitely generated. The explicit form of a generating set of $K[X_n,Y_n]^{\delta}$ was conjectured by Nowicki in 1994 in the case \delta was such that \delta(y_{i})=x_{i}$, $\delta(x_i)=0, i=1,\ldots,n. Nowicki's conjecture turned out to be true and, recently, has been applied to several relatively free associative algebras. In this paper, we consider the free Lie algebra \mathcal{L}(x,y) of rank 2 generated by x and y over K and we assume the Weitzenb\"ock derivation \delta sending y to x, and x to zero. We introduce the idea of pseudodeterminants and we present a characterization of Hall monomials that are constants showing they are not so far from being pseudodeterminants. We also give a complete list of generators of the constants of degree less than 7 which are, of course, pseudodeterminants.

math.RA↗

arXiv2024

Generalizations of almost prime and right $S$-prime ideals in noncommutative rings

Let $R$ be a noncommutative ring, and let $S$ be an $m$-system of $R$. In this paper, we give more results on the concept of almost prime (right) ideals, that were introduced by the first two authors, especially in (right) $S$-unital rings, local rings, and decomposable rings. In addition, we introduce the concept of almost right $S$-prime ideals, and we show how some findings regarding almost prime ideals can be derived as consequences of almost right $S$-prime ideals. Besides, we show how almost right $S$-prime ideals behave in related rings such as homomorphic images, quotient rings, and decomposable rings. Finally, we construct almost right $S$-prime ideals using the Nagata method of idealization.

math.RA↗

arXiv2022

Almost prime ideals in noncommutative rings

A proper ideal $P$ of a commutative ring with identity is an almost prime ideal if $ab \in P{\setminus}P^2$ implies $a \in P$ or $b \in P$. In this paper we define almost prime ideals of a noncommutative ring, and provide some equivalent definitions. We also examine some cases such that all right ideals of a noncommutative ring are almost prime right ideals.

math.RA↗

arXiv2022

On Nowicki's conjecture: a survey and a new result

The goal of the paper is twofold: it aims to give an extensive set of tools and bibliography towards Nowicki's conjecture both in an associative setting; it establishes a new result about Nowicki's conjecture for the free metabelian Poisson algebra.

math.RA↗

arXiv2021

Symmetric polynomials in the variety generated by Grassmann algebras

Let $\mathcal{G}$ denote the variety generated by infinite dimensional Grassmann algebras; i.e., the collection of all unitary associative algebras satisfying the identity $[[z_1,z_2],z_3]=0$, where $[z_i,z_j]=z_iz_j-z_jz_i$. Consider the free algebra $F_3$ in $\mathcal{G}$ generated by $X_3=\{x_1,x_2,x_3\}$. The commutator ideal $F_3'$ of the algebra $F_3$ has a natural $K[X_3]$-module structure. We call an element $p\in F_3$ symmetric if $p(x_1,x_2,x_3)=p(x_{\xi1},x_{\xi2},x_{\xi3})$ for each permutation $ξ\in S_3$. Symmetric elements form the subalgebra $F_3^{S_3}$ of invariants of the symmetric group $S_3$ in $F_3$. We give a free generating set for the $K[X_3]^{S_3}$-module $(F_3')^{S_3}$.

math.RA↗

arXiv2020

Symmetric polynomials in Leibniz algebras and their inner automorphisms

Let $L_n$ be the free metabelian Leibniz algebra generated by the set $X_n=\{x_1,\ldots,x_n\}$ over a field $K$ of characteristic zero. This is the free algebra of rank $n$ in the variety of solvable of class $2$ Leibniz algebras. We call an element $s(X_n)\in L_n$ symmetric if $s(x_{σ(1)},\ldots,x_{σ(n)})=s(x_1,\ldots,x_n)$ for each permutation $σ$ of $\{1,\ldots,n\}$. The set $L_n^{S_n}$ of symmetric polynomials of $L_n$ is the algebra of invariants of the symmetric group $S_n$. Let $K[X_n]$ be the usual polynomial algebra with indeterminates from $X_n$. The description of the algebra $K[X_n]^{S_n}$ is well known, and the algebra $(L_n')^{S_n}$ in the commutator ideal $L_n'$ is a right $K[X_n]^{S_n}$-module. We give explicit forms of elements of the $K[X_n]^{S_n}$-module $(L_n')^{S_n}$. Additionally, we determine the description of the group ${\rm Inn}(L_{n}^{S_n})$ of inner automorphisms of the algebra $L_n^{S_n}$. The findings can be considered as a generalization of the recent results obtained for the free metabelian Lie algebra of rank $n$.

math.RA↗

arXiv2020

Inner automorphisms of Lie algebras of symmetric polynomials

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

math.RA↗

arXiv2019

Symmetric polynomials in the free metabelian Lie algebras

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 $n\geq 2$ 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 $F_n^{S_n}$.

math.RA↗

arXiv2019

The Nowicki Conjecture for relatively free algebras

A linear locally nilpotent derivation of the polynomial algebra $K[X_m]$ in $m$ variables over a field $K$ of characteristic 0 is called a Weitzenböck derivation. It is well known from the classical theorem of Weitzenböck that the algebra of constants $K[X_{m}]^δ$ of a Weitzenböck derivation $δ$ is finitely generated. Assume that $δ$ acts on the polynomial algebra $K[X_{2d}]$ in $2d$ variables as follows: $δ(x_{2i})=x_{2i-1}$, $δ(x_{2i-1})=0$, $i=1,\ldots,d$. The Nowicki conjecture states that the algebra $K[X_{2d}]^δ$ is generated by $x_1,x_3.\ldots,x_{2d-1}$, and $x_{2i-1}x_{2j}-x_{2i}x_{2j-1}$, $1\leq i<j\leq d$. The conjecture was proved by several authors based on different techniques. We apply the same idea to two relatively free algebras of rank $2d$. We give the infinite set of generators of the algebra of constants in the the free metabelian associative algebras $F_{2d}(\mathfrak A)$, and finite set of generators in the free algebra $F_{2d}(\mathcal G)$ in the variety determined by the identities of the infinite dimensional Grassmann algebra.

math.RA↗

arXiv2019

Classical invariant theory for free metabelian Lie algebras

Let $KX_d$ be a vector space with basis $X_d=\{x_1,\ldots,x_d\}$ over a field $K$ of characteristic 0. One of the main topics of classical invariant theory is the study of the algebra of invariants $K[X_d]^{SL_2(K)}$, where $KX_d$ is a module of the special linear group $SL_2(K)$ isomorphic to a direct sum $V_{k_1}\oplus\cdots\oplus V_{k_r}$ and $V_k$ is the $SL_2(K)$-module of binary forms of degree $k$. Noncommutative invariant theory deals with the algebra of invariants $F_d({\mathfrak V})^G$ of the group $G<GL_d(K)$ acting on the relatively free algebra $F_d({\mathfrak V})$ of a variety of $K$-algebras $\mathfrak V$. In this paper we consider the free metabelian Lie algebra $F_d({\mathfrak A}^2)$ which is the relatively free algebra in the variety ${\mathfrak A}^2$ of metabelian (solvable of class 2) Lie algebras. We study the algebra $F_d({\mathfrak A}^2)^{SL_2(K)}$ of $SL_2(K)$-invariants of $F_d({\mathfrak A}^2)$. We describe the cases when this algebra is finitely generated. This happens if and only if $KX_d\cong V_1\oplus V_0\oplus\cdots\oplus V_0$ or $KX_d\cong V_2$ as an $SL_2(K)$-module (and in the trivial case $KX_d\cong V_0\oplus\cdots\oplus V_0$). For small $d$ we give a list of generators even when $F_d({\mathfrak A}^2)^{SL_2(K)}$ is not finitely generated. The methods for establishing that the algebra $F_d({\mathfrak A}^2)^{SL_2(K)}$ is not finitely generated work also for other relatively free algebras $F_d({\mathfrak V})$ and for other groups $G$.

math.RA↗

arXiv2015

Weitzenboeck derivations of free metabelian associative algebras

By the classical theorem of Weitzenboeck the algebra of constants (i.e., the kernel) of a nonzero locally nilpotent linear derivation of the polynomial algebra K[X] in d variables over a field K of characteristic 0 is finitely generated. As a noncommutative generalization one considers the algebra of constants of a locally nilpotent linear derivation of a d-generated relatively free algebra F(V) in a variety V of unitary associative algebras over K. It is known that the algebra of constants of F(V) is finitely generated if and only if V satisfies a polynomial identity which does not hold for the algebra of 2 x 2 upper triangular matrices. Hence the free metabelian associative algebra F(M) is a crucial object to study. We show that the vector space of the constants in the commutator ideal F'(M) is a finitely generated module of the algebra of constants of the polynomial algebra K[U,V] in 2d variables, where the derivation acts on U and V in the same way as on X. For small d, we calculate the Hilbert series of the constants in F'(M) and find the generators of the related module. This gives also an (infinite) set of generators of the algebra of constants in F(M).

math.RA↗

arXiv2013

Weitzenboeck derivations of free metabelian Lie algebras

A nonzero locally nilpotent linear derivation of the polynomial algebra K[X] in d variables over a field K of characteristic 0 is called a Weitzenboeck derivation. The classical theorem of Weitzenboeck states that the algebra of constants (which coincides with the algebra of invariants of a single unipotent transformation) is finitely generated. Similarly one may consider the algebra of constants of a locally nilpotent linear derivation of a finitely generated (not necessarily commutative or associative) algebra which is relatively free in a variety of algebras over K. Now the algebra of constants is usually not finitely generated. Except for some trivial cases this holds for the algebra of constants of the free metabelian Lie algebra L/L" with d generators. We show that the vector space of the constants in the commutator ideal L'/L" is a finitely generated module over the algebra of constants in K[X]. For small d, we calculate the Hilbert series of the algebra of constants in L/L" and find the generators of the module of the constants in L'/L". This gives also an (infinite) set of generators of the Lie algebra of constants in L/L".

math.RA↗

arXiv2012

Inner Automorphisms of Lie Algebras Related with Generic 2 x 2 Matrices

Let F_m=F_m(var(sl(2,K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl(2,K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion of F_m with respect to the formal power series topology. Our results are more precise for m=2 when F_2 is isomorphic to the Lie algebra L generated by two generic traceless 2 x 2 matrices. We give a complete description of the group of inner automorphisms of the completion of L. As a consequence we obtain similar results for the automorphisms of the relatively free algebra F_m/F_m^{c+1} in the subvariety of var(sl(2,K)) consisting of all nilpotent algebras of class at most c in var(sl(2,K)).

math.RA↗

arXiv2010

Normal and normally outer automorphisms of free metabelian nilpotent Lie algebras

Let L be the free m-generated metabelian nilpotent of class c Lie algebra over a field of characteristic 0. An automorphism f of L is called normal if f(I)=I for every ideal I of the algebra L. Such automorphisms form a normal subgroup N(L) of Aut(L) containing the group of inner automorphisms. We describe the group of normal automorphisms of L and the quotient group of Aut(L) modulo N(L).

math.RA↗

arXiv2010

Inner and Outer Automorphisms of Free Metabelian Nilpotent Lie algebras

We describe the groups of inner and outer automorphisms of the free metabelian nilpotent Lie algebra of finite rank over a field of characteristic 0. To obtain this result we first describe the groups of inner and continuous outer automorphisms of the completion with respect to the formal power series topology of the free metabelian Lie algebra finite rank.

math.RA↗