arXiv subjects

Valiollah Khalili

Publications and source records attributed to Valiollah Khalili.

6 recordsLinked to original sources

arXiv2023

On the structure of graded $3$-Lie-Rinehart algebras

We study the structure of a graded $3$-Lie-Rinehart algebra $\mathcal{L}$ over an associative and commutative graded algebra $A.$ For $G$ an abelian group, we show that if $(L, A)$ is a tight $G$-graded 3-Lie-Rinehart algebra, then $\mathcal{L}$ and $A$ decompose as $\mathcal{L} =\bigoplus_{i\in I}\mathcal{L}_i$ and $A =\bigoplus_{j\in J}A_j,$ where any $\mathcal{L}_i$ is a non-zero graded ideal of $\mathcal{L}$ satisfying $[\mathcal{L}_{i_1}, \mathcal{L}_{i_2}, \mathcal{L}_{i_3}]=0$ for any $i_1, i_2, i_3\in I$ different from each other, and any $A_j$ is a non-zero graded ideal of $A$ satisfying $A_j A_l=0$ for any $l, j\in J$ such that $j\neq l,$ and both decompositions satisfy that for any $i\in I$ there exists a unique $j\in J$ such that $A_j \mathcal{L}_i\neq 0.$ Furthermore, any $(\mathcal{L}_i, A_j)$ is a graded 3-Lie-Rinehart algebra. Also, under certain conditions, it is shown that the above decompositions of $\mathcal{L}$ and $A$ are by means of the family of their, respective, graded simple ideals.

math.RA↗

arXiv2023

Double derivations of $n-$Hom-Lie color algebras

We study the double derivation algebra $\mathcal{D}(\mathcal{L})$ of $n-$Hom Lie color algebra $\mathcal{L}$ and describe the relation between $\mathcal{D}(\mathcal{L})$ and the usual derivation Hom-Lie color algebra $Der(\mathcal{L}).$ We prove that the inner derivation algebra $Inn(\mathcal{L})$ is an ideal of the double derivation algebra $\mathcal{D}(\mathcal{L}).$ We also show that if $\mathcal{L}$ is a perfect $n-$Hom Lie color algebra with certain constraints on the base field, then the centralizer of $Inn(\mathcal{L})$ in $\mathcal{D}(\mathcal{L})$ is trivial. In addition, we obtain that for every centerless perfect $n-$Hom Lie color algebra $\mathcal{L}$, the triple derivations of the derivation algebra $Der(\mathcal{L})$ are exactly the derivations of $Der(\mathcal{L}).$

math.RA↗

arXiv2023

On the structure of graded Poisson color algebras

In this paper we introduce the class of graded Poisson color algebras as the natural generalization of graded Poisson algebras and graded Poisson superalgebras. For $Λ$ an arbitrary abelian group, we show that any of such $Λ$-graed Poisson color algebra $\mathcal{P}$, with a symmetric $Λ$-support is of the form $\mathcal{P} = \mathcal{U}\oplus\sum_j I_j$, with $\mathcal{U}$ a subspace of $\mathcal{P}_1$ and any $I_j$ a well described graded ideal of $\mathcal{P},$ satisfying $\{I_j, I_k\}+I_j I_k=0$ if $j\neq i.$ Furthermore, under certain conditions, the gr-simplicity of $\mathcal{P}$ is characterized and it is shown that $\mathcal{P}$ is the direct sum of the family of its graded simple ideals.

math-ph↗

arXiv2023

Non-commutative Poisson algebras with a set grading

In this paper we study of the structure of non-commutative Poisson algebras with an arbitrary set $ß.$ We show that any of such an algebra $\pp$ decomposes as $$\pp=\uu\oplus\sum_{[λ]\in(Λ_ß\setminus\{0\})/\sim}\pp_{[λ]},$$ where $\uu$ is a linear subspace complement of $\span_{\bbbf}\{ [\pp_μ, \pp_η]+\pp_μ\pp_η : μ, η\in[\lam]\}\cap\pp_0$ in $\pp_0$ and any $\pp_{[λ]}$ a well-described graded ideals of $\pp,$ satisfying $[\pp_{[λ]}, \pp_{[μ]}]+\pp_{[λ]} \pp_{[μ]}=0$ if $[λ]\neq[μ].$ Under certain conditions, the simplicity of $\pp$ is characterized and it is shown that $\pp$ is the direct sum of the family of its graded simple ideals.

math.RA↗

arXiv2021

On the structure of graded $3-$Leibniz algebras

We study the structure of a $3-$Leibniz algebra $T$ graded by an arbitrary abelian group $G,$ which is considered of arbitrary dimension and over an arbitrary base field $\bbbf.$ We show that $T$ is of the form $T=\uu\oplus\sum_jI_j,$ with $\uu$ a linear subspace of $T_1,$ the homogeneous component associated to the unit element $1$ in $G,$ and any $I_j$ a well described graded ideal of $T,$ satisfying $$ [I_j, T, I_k] = [I_j, I_k, T] = [T, I_j, I_k] = 0, $$ if $j\neq k.$ In the case of $T$ being of maximal length, we characterize the gr-simplicity of the algebra in terms of connections in the support of the grading.

math.RA↗

arXiv2021

Split 3-Lie-Rinehart color algebras

In this paper we introduce a class of $3-$color algebras which are called split $3-$Lie-Rinehart color algebras as the natural generalization of the one of split LieRinehart algebras. We characterize their inner structures by developing techniques of connections of root systems and weight systems associated to a splitting Cartan subalgebra. We show that such a tight split $3-$Lie-Rinehart color algebras $(\LL, A)$ decompose as the orthogonal direct sums $\LL =\oplus_{i\in I}\LL_i$ and $A =\oplus_{j\in J}A_j,$ where any $\LL_i$ is a non-zero graded ideal of $\LL$ satisfying $[\LL{i_1}, \LL{i_2}, \LL{i_3}]=0$ if $i_1, i_2, i_3\in I$ be different from each other and any $A_j$ is a non-zero graded ideal of A satisfying $A_{j_1}A_{j_2}=0$ if $J_1\neq j_2.$ Both decompositions satisfy that for any $i\in I$ there exists a unique $j\in J$ such that $A_j\LL_i = 0$. Furthermore, any $(\LL_i , A_j )$ is a split $3-$LieRinehart color algebra. Also, under certain conditions, it is shown that the above decompositions of $\LL$ and $A$ are by means of the family of their, respective, simple ideals.

math.RA↗