Hom-Lie algebras are a new class of algebraic structures, which can be regarded as the deformation of Lie algebras. The study of Hom-Lie algebras originated from applications in Lie algebras and physics, especially in quantum field theory and string theory, where they provide new mathematical tools for the study of certain symmetries in these theories. Hom-Lie algebras are a class of nonassociative algebras satisfying antisymmetry and the Hom-Jacobi identity. In 2006, Hartwing, Larsson and Silvestrov introduced the structures of Hom-Lie algebras in order to study the deformable Witt algebras and Virasoro algebras [1]. In recent years, Hom-Lie algebras have received wide attention, for example, literature [2] describes cohomology and deformations on Hom-Lie algebras. The literature [3] provides the adjoint representation, trivial representation, derivations, deformations, central extensions of Hom-Lie algebras, etc. The literature [4] provides a complete classification of Hom-Lie algebras on the Lie algebra $ gl(2, C) $. The literature [5] extends the concept of Hom-Lie algebra to Hom-Lie superalgebra, and thus the study of Hom-Lie superalgebra is developed. Hom-structures have been widely studied by researchers, since Hom-algebras were proposed. For example, the literature [6] proves that the Hom-Lie algebra structures on finite-dimensional simple Lie algebras are trivial, and the literature [7] proves that the Hom-Lie superalgebra structures on finite-dimensional simple Lie superalgebras are also trivial.

By Levi's theorem, we know that any finite-dimensional Lie algebra is isomorphic to a direct sum of a semi-simple Lie algebra and a maximal solvable ideal [8]. In this paper, we will focus on studying a class of solvable Lie algebras. Since any solvable Lie algebra has a uniquely determined nilradical, the main work of studying solvable Lie algebras comes down to studying their nilradicals. The literature [9] describes $ \frac{1}{2} $-derivations of finite-dimensional solvable Lie algebras with a filiform nilradical and the literature [10] describes local and 2-local $ \frac{1}{2} $-derivation of these solvable Lie algebras. The literature [11] finds one-dimensional extensions of solvable Lie algebras with nilradical $ {{n}_{n, 1}} $. In this paper, we mainly study the Hom-structures on solvable Lie algebras with filiform nilradical $ {{n}_{n, 1}} $ over an algebraically closed field $ \mathbb{F} $ of zero characteristic.

In this section, we mainly introduce the definitions of the Lie algebra, Hom-Lie algebra and Hom-structure, then we recall the definitions of the solvable Lie algebra and nilpotent Lie algebra.

Definition 2.1   ^[12] A Lie algebra over the field $ \mathbb{F} $ is a vector space $ \mathcal{G} $ together with a bilinear map $ [\cdot, \cdot]:\mathcal{G}\times \mathcal{G}\to \mathcal{G} $, called the Lie bracket of $ \mathcal{G} $, which is skew symmetric, i.e.

$ [a, b]=-[b, a], \forall a, b\in \mathcal{G}, $

and satisfies the Jacobi identity, i.e.

$ [a, [b, c]]+[b, [c, a]]+[c, [a, b]]=0, \forall a, b, c\in \mathcal{G}. $

The Hom-Lie algebra can be regarded as a deformation of the Lie algebra. Next, we will introduce the definition of the Hom-Lie algebra.

Definition 2.2   ^[13] A Hom-Lie algebra ($ \mathcal{G} $, $ \varphi $) is a non-associative algebra $ \mathcal{G} $ together with an algebra homomorphism $ \varphi:\mathcal{G} \to \mathcal{G} $, such that

$ \begin{aligned} &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[a, b]=-[b, a], \forall a, b \in \mathcal{G},\\ \\ &[\varphi(a),[b, c]]+[\varphi(b),[c, a]]+[\varphi(c),[a, b]]=0, \forall a, b, c \in \mathcal{G}, \end{aligned} $

(2.1)

where $ [\cdot, \cdot] $ denotes the product in $ \mathcal{G} $. We call the equation (2.1) the Hom-Jacobi identity. Let ($ \mathcal{G} $, $ [\cdot, \cdot] $) be a Lie algebra and $ \varphi $ be a linear map satisfying the Hom-Jacobi identity, then $ \varphi $ is a Hom-structure on ($ \mathcal{G} $, $ [\cdot, \cdot] $).

Obviously, all the Hom-structures of Lie algebra $ \mathcal{G} $ constitute a vector space, denoted $ \rm HS\left( \mathcal{G} \right) $.

For a Lie algebra $ \mathcal{G} $, consider the following lower central and derived sequences:

$ ~~~~~~~~~~ {{\mathcal{G}}^{1}}=\mathcal{G}, {{\mathcal{G}}^{k+1}}=\left[ \mathcal{G}, {{\mathcal{G}}^{k}} \right], k\ge 1,\\ {{\mathcal{G}}^{\left( 0 \right)}}=\mathcal{G}, {{\mathcal{G}}^{\left( 1 \right)}}=\left[ {\mathcal{G}}, {\mathcal{G}} \right], {{\mathcal{G}}^{\left( s+1 \right)}}=\left[ {{\mathcal{G}}^{\left( s \right)}}, {{\mathcal{G}}^{\left( s \right)}} \right], s\ge 1. $

Definition 2.3   ^[12] A Lie algebra $ \mathcal{G} $ is called nilpotent (respectively, solvable), if there exists $ p\in \mathbb{N} $ ($ q\in \mathbb{N} $) such that $ {{\mathcal{G}}^{p}}=0 $ (respectively, $ {{\mathcal{G}}^{\left( q \right)}}=0 $).

Any solvable Lie algebra $ \mathcal{G} $ contains a unique maximal nilpotent ideal, called the nilradical of this Lie algebra $ \mathcal{G} $. Thus, we can consider a given nilpotent Lie algebra as a nilradical, and then find all of its extensions to solvable Lie algebras.

It is well known that there are two types of naturally graded filiform Lie algebras, the second type only occurs if the dimension of the algebra is even, and any naturally graded filiform Lie algebra is isomorphic to one of the following non-isomorphic algebras [14].

$ {{n}_{n, 1}}:\left[ {{e}_{i}}, {{e}_{1}} \right]=-\left[ {{e}_{i}}, {{e}_{1}} \right]={{e}_{i+1}}, \quad 2\le i\le n-1. $

$ {{Q}_{2n}}:\left\{ \begin{aligned} &\left[ {{e}_{i}}, {{e}_{1}} \right]=-\left[ {{e}_{1}}, {{e}_{i}} \right]={{e}_{i+1}}, 2\le i\le 2n-2, \\ &\left[ {{e}_{i}}, {{e}_{2n+1-i}} \right]=-\left[ {{e}_{2n+1-i}}, {{e}_{i}} \right]={{(-1)}^{i}}{{e}_{2n}}, 2\le i\le n. \\ \end{aligned} \right. $

All solvable Lie algebras whose nilradical are naturally graded filiform Lie algebras $ {{n}_{n, 1}} $ are classified in [15]. Here we introduce the solvable Lie algebras with nilradical $ {{n}_{n, 1}} $:

$ S_{n+1, 1}(\beta ):\left\{\begin{aligned} & \left[ {{e}_{i}}, {{e}_{1}} \right]=- \left[ {{e}_{1}}, {{e}_{i}} \right]={{e}_{i+1}}, 2\le i\le n-1, \\ & \left[ {{e}_{i}}, x \right]=-\left[ x, {{e}_{i}} \right]=(i-2+\beta ){{e}_{i}}, 2\le i\le n, \\ & \left[ {{e}_{1}}, x \right]=-\left[ {x, {e}_{1}} \right]={{e}_{1}}; \\ \end{aligned}\right. $

$ S_{n+1, 2}({{\alpha }_{3}}, \cdots , {{\alpha }_{n-1}}):\left\{ \begin{aligned} &\left[ {{e}_{i}}, {{e}_{1}} \right]=-\left[ {{e}_{1}}, {{e}_{i}} \right]={{e}_{i+1}}, 2\le i\le n-1, \\ &\left[ {{e}_{i}}, x \right]=-\left[ x, {{e}_{i}} \right]={{e}_{i}}+\sum\limits_{l=i+2}^{n}{{{\alpha }_{l+1-i}}{{e}_{l}}, 2\le i\le n}, \\ \end{aligned} \right. $

if the first non-vanishing parameter $ \left\{ {{\alpha }_{3}}, \cdots , {{\alpha }_{n-1}} \right\} $ exists, it can be assumed to be equal to 1.

$ S_{n+1, 3}:\left\{ \begin{aligned} &\left[ {{e}_{i}}, {{e}_{1}} \right]=-\left[ {{e}_{1}}, {{e}_{i}} \right]={{e}_{i+1}}, 2\le i\le n-1, \\ &\left[ {{e}_{i}}, x \right]=-\left[ x, {{e}_{i}} \right]=(i-1){{e}_{i}}, 2\le i\le n, \\ &\left[ {{e}_{1}}, x \right]=-\left[ x, {{e}_{1}} \right]={{e}_{1}}+{{e}_{2}}; \\ \end{aligned} \right. $

$ {{S}_{n+2}}:\left\{ \begin{aligned} &\left[ {{e}_{i}}, {{e}_{1}} \right]=-\left[ {{e}_{1}}, {{e}_{i}} \right]={{e}_{i+1}}, 2\le i\le n-1, \\ &\left[ {{e}_{i}}, {{x}_{1}} \right]=-\left[ {{x}_{1}}, {{e}_{i}} \right]=(i-2){{e}_{i}}, 3\le i\le n, \\ &\left[ {{e}_{i}}, {{x}_{2}} \right]=-\left[ {{x}_{2}}, {{e}_{i}} \right]={{e}_{i}}, 2\le i\le n, \\ &\left[ {{e}_{1}}, {{x}_{1}} \right]=-\left[ {{x}_{1}} , {{e}_{1}} \right]={{e}_{1}}, \\ \end{aligned} \right. $

where $ \left\{ {{e}_{1}}, {{e}_{2}}, \cdots , {{e}_{n}}, x \right\} $ is a standard basis of $ S_{n+1, 1}(\beta ) $, $ S_{n+1, 2}({{\alpha }_{3}}, \cdots , {{\alpha }_{n-1}}) $, $ S_{n+1, 3} $, and $ \left\{ {{e}_{1}}, {{e}_{2}}, \cdots , {{e}_{n}}, {{x}_{1}}, {{x}_{2}} \right\} $ is a standard basis of $ {{S}_{n+2}} $.

In this section, we will calculate the Hom-structures on solvable Lie algebras with nilradical $ {{n}_{n, 1}} $.

Theorem 3.1   The bases of $ {\rm HS}\left({S_{n+1, 1}(\beta )} \right) $, $ {\rm HS}\left( {S_{n+1, 2}({{\alpha }_{3}}, \cdots , {{\alpha }_{n-1}})}\right) $, $ {\rm HS}\left( {S_{n+1, 3}}\right) $ and $ {\rm HS}\left( {{S}_{n+2}}\right) $ are as follows:

In this paper, we provide the proof only for $ {\rm HS}\left({S_{n+1, 1}(\beta )} \right) $.

Proof   Suppose

$ \begin{aligned} \Gamma=&\left\{ \sum\limits_{i=2}^{n-1}{(i-1+\beta ){{E}_{ii}}}+{{E}_{n+1, n+1}} \right\}\cup\left\{{{E}_{in}}, {{E}_{n+1, j}}, {{E}_{1k}}\left| 2\le i, j, k\le n \right. \right\} \\ &\cup\left\{\sum\limits_{i=2}^{n-2}{{{E}_{i, i+1}}}+{{E}_{n+1, 1}}, {{E}_{11}}-\sum\limits_{i=2}^{n-1}{(i-2+\beta ){{E}_{ii}}}\right\}. \end{aligned} $

Obviously, $ \Gamma $ is a linearly independent set and $ \Gamma \subseteq {\rm HS}\left({S_{n+1, 1}(\beta )} \right) $. For all $ {\varphi} \in{\rm HS}\left({S_{n+1, 1}(\beta )} \right) $, suppose that the matrix of $ {\varphi} $ in the standard basis is $ {\rm A}={{\left( {{a}_{ij}} \right)}_{\left( n+1 \right)\times \left( n+1 \right)}} $, where $ {{a}_{ij}}\in \mathbb{F} $, that is

$ {{\left( \begin{matrix} {{\varphi }}({{e}_{1}}) & {{\varphi }}({{e}_{2}}) & \cdots & {{\varphi }}({{e}_{n}}) & {{\varphi }}(x) \\ \end{matrix} \right)}^{\rm T}}={\rm A}{{\left( \begin{matrix} {{e}_{1}} & {{e}_{2}} & \cdots & {{e}_{n}} & x \\ \end{matrix} \right)}^{\rm T}}. $

Putting $ a=e_{k}, b=e_{l}, c=x $ in equation (2.1), where $ k\ne l, k, l=1, 2, \cdots , n. $

Case 1   If $ 2\le k, l\le n-1, $ we can obtain

$ \begin{aligned} &\left(k-2+\beta\right)^{2}a_{l, n+1}e_{k}+\left(k-2+\beta\right)a_{l, 1}e_{k+1}-\left(l-2+\beta\right)a_{k, 1}e_{l+1} \\ &-\left(l-2+\beta\right)^{2}a_{k, n+1}e_{l}=0, \end{aligned} $

then

$ \begin{eqnarray} {{a}_{l, n+1}}={{a}_{l, 1}}=0, \quad 2\le l\le n-1. \end{eqnarray} $

(3.1)

Case 2   If $ k=1, 2\le l\le n-2, $ we can obtain

$ \begin{aligned} &a_{l, n+1}e_{1}-\left(l-2+\beta\right)^{2}a_{1, n+1}e_{l}+\left[\left(l-1+\beta\right)a_{n+1, n+1}-\left(l-2+\beta\right)a_{11}\right]e_{l+1} \\ &+a_{n+1, 1}e_{l+2}-\sum\limits_{i=3}^{n}a_{l, i-1}e_{i}=0, \end{aligned} $

then

$ \begin{eqnarray} {{a}_{l, 2}}={{a}_{l, 3}}=\cdots ={{a}_{l, l-2}}=0, \quad 4\le l\le n-2; \end{eqnarray} $

(3.2)

$ \begin{gathered} a_{l, l-1}=-(l-2+\beta)^2 a_{1, n+1}, \quad 3 \leq l \leq n-2 ; \\ \\ a_{l, l}=(l-1+\beta) a_{n+1, n+1}-(l-2+\beta) a_{11}, \quad 2 \leq l \leq n-2 ; \end{gathered}$

(3.3)

$ \begin{eqnarray} {{a}_{l, l+1}}={{a}_{n+1, 1}}, 2\le l\le n-2; {{a}_{l, l+2}}={{a}_{l, l+3}}=\cdots ={{a}_{l, n-1}}=0, 2\le l\le n-3. \end{eqnarray} $

(3.4)

In particular, when $ l=2 $, we can obtain $ {{a}_{1, n+1}}=0 $, then

$ \begin{eqnarray} {{a}_{l, l-1}}=0, \quad 3\le l\le n-2. \end{eqnarray} $

(3.5)

Case 3   If $ k=1, l=n-1, $ we can obtain

$ \begin{aligned} &a_{n-1, n+1}e_{1}-\sum\limits_{i=3}^{n-2}a_{n-1, i-1}e_{i}-\left[a_{n-1, n-2}+\left(n-3+\beta\right)^{2}a_{1, n+1}\right]e_{n-1} \\ &+\left[\left(n-2+\beta\right)a_{n+1, n+1}-a_{n-1, n-1}-\left(n-3+\beta\right)a_{11}\right]e_{n}=0, \end{aligned} $

then

$ \begin{eqnarray} {{a}_{n-1, 2}}={{a}_{n-1, 3}}=\cdots={{a}_{n-1, n-3}}=0; \end{eqnarray} $

(3.6)

$ \begin{eqnarray} {{a}_{n-1, n-2}}=-{{\left( n-3+\beta \right)}^{2}}{{a}_{1, n+1}}=0; \end{eqnarray} $

(3.7)

$ \begin{eqnarray} {{a}_{n-1, n-1}}=\left( n-2+\beta \right){{a}_{n+1, n+1}}-\left( n-3+\beta \right){{a}_{11}}. \end{eqnarray} $

(3.8)

Case 4   If $ k=n, 2\le l\le n-1, $ we can obtain

$ \begin{aligned} -\left(l-2+\beta\right)^2a_{n, n+1}e_{l}-\left(l-2+\beta\right)a_{n1}e_{l+1}+\left(n-2+\beta\right)^2a_{l, n+1}e_{n}=0, \end{aligned} $

then

$ \begin{eqnarray} {{a}_{n1}}={{a}_{n, n+1}}=0. \end{eqnarray} $

(3.9)

Case 5   If $ k=n, l=1, $ we can obtain

$ \begin{aligned} -a_{n, n+1}e_{1}+\sum\limits_{i=3}^{n-1}a_{n, i-1}e_{i}+\left[a_{n, n-1}+\left(n-2+\beta\right)^{2}a_{1, n+1}\right]e_{n}=0, \end{aligned} $

then

$ \begin{eqnarray} {{a}_{n, 2}}={{a}_{n, 3}}=\cdots ={{a}_{n, n-2}}=0; \end{eqnarray} $

(3.10)

$ \begin{eqnarray} {{a}_{n, n-1}}=-{{\left( n-2+\beta \right)}^{2}}{{a}_{1, n+1}}=0. \end{eqnarray} $

(3.11)

By (3.1)–(3.11) and $ {{a}_{1, n+1}}=0 $, we have

$ {\rm A}=\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} & \cdots & {{a}_{1, n-1}} & {{a}_{1n}} & 0 \\ 0 & {{a}_{22}} & {{a}_{n+1, 1}} & \cdots & 0 & {{a}_{2n}} & 0 \\ 0 & 0 & {{a}_{33}} & \cdots & 0 & {{a}_{3n}} & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & {{a}_{n-1, n-1}} & {{a}_{n-1, n}} & 0 \\ 0 & 0 & 0 & \cdots & 0 & {{a}_{nn}} & 0 \\ {{a}_{n+1, 1}} & {{a}_{n+1, 2}} & {{a}_{n+1, 3}} & \cdots & {{a}_{n+1, n-1}} & {{a}_{n+1, n}} & {{a}_{n+1, n+1}} \\ \end{matrix} \right), $

where $ {{a}_{ii}}=(i-1+\beta ){{a}_{n+1, n+1}}-(i-2+\beta ){{a}_{11}}, 2\le i\le n-1 $. Then the matrix $ \rm A $ can be expressed linearly by $ \Gamma $.