3-Hom-Lie Algebras Based on $\sigma$-Derivation and Involution

Abstract

We show that, having a Hom-Lie algebra and an element of its dual vector space that satisfies certain conditions, one can construct a ternary totally skew-symmetric bracket and prove that this ternary bracket satisfies the Hom-Filippov-Jacobi identity, i.e. this ternary bracket determines the structure of 3-Hom-Lie algebra on the vector space of a Hom-Lie algebra. Then we apply this construction to two Hom-Lie algebras constructed on an associative, commutative algebra using $\sigma$-derivation and involution, and we obtain two 3-Hom-Lie algebras.

1 Introduction

An important generalization of the concept of Lie algebra is a notion of n-Lie algebra, where n is any integer greater than 2. An integer n indicates the number of elements of algebra necessary to compose a product, which is called in this case n-ary bracket. Thus, we can say that a notion of n-Lie algebra is an extension of the concept of Lie algebra to algebraic structures with n-ary multiplication laws. In particular, when n is 2 we obtain the usual binary Lie algebra. The definition of n-Lie algebra is very similar to the definition of usual binary Lie algebra. The definition of a binary Lie algebra is based on the skew-symmetry of a bracket and the Jacobi identity, which can be written in a form that shows that a double bracket behaves like differentiation, that is, according to the Leibniz rule. Similarly, in a definition of n-Lie algebra it is required for an n-ary bracket to be skew-symmetric and for a double n-ary bracket to satisfy the Leibniz rule. A concept of n-Lie algebra was proposed and developed by Filippov [13]. The most known example of n-Lie algebra can be constructed by means of analog of cross product of n vectors in $(n+1)$-dimensional vector space.

Later it was shown that n-ary Lie algebras can be used in theoretical physics and this gave a strong impetus to their study and development. Even before Filippov introduced the concept of n-Lie algebra, Nambu constructed an extension of Hamiltonian mechanics to odd-dimensional space, particularly to 3-dimensional space [18]. The basic element of the Nambu approach in 3-dimensional space is a ternary bracket of functions, which is an analog of the Poisson bracket and is now called the Nambu–Poisson bracket. Later it was shown [19] that Nambu–Poisson bracket satisfied an analog of Jacobi identity of n-Lie algebra, i.e. Nambu–Poisson bracket determines a structure of 3-Lie (more generally, n-Lie) algebra on a space of functions.

The second circumstance, which greatly stimulated research on n-Lie algebras, is their appearance and use in M-theory. In M-theory, there is the Nahm equation [11], which describes the effective dynamics of stacks of D1-branes ending on a D3-brane in type IIB superstring theory. In [9] A. Basu and Harvey proposed an equation (now called Basu–Harvey equation) that describes dynamics of stacks of M2-branes ending on M5-branes and this equation is a generalization of Nahm’s equation. The transverse scalar fields $T^i$ of Basu–Harvey equation form a 3-Lie algebra and that is how 3-Lie algebras appear in M-theory.

A quantization of generalization of Hamiltonian mechanics proposed by Nambu is a hard problem. In search for a solution to this problem, in [7] the authors constructed a matrix 3-Lie algebra, whose ternary bracket is constructed by means the usual commutator of two matrices and the trace of a matrix. This construction was generalized and its various aspects were investigated in a number of papers [4,5,6, 16] (and references therein). The method of constructing 3-Lie superalgebras, similar to the method proposed in [7], but based on the supertrace of a supermatrix, was proposed in [1] and developed in [2, 3]. Later, this method of constructing 3-Lie superalgebras was extended to 3-Hom-Lie superalgebras [14] and 3-BiHom-Lie superalgebras [10].

The purpose of this paper is to construct 3-Hom-Lie algebras if we are given an associative commutative algebra with a $\sigma$-derivation and involution. To this end, we will use the particular case of the theorem, proved in [4] (Theorem 3.3), which generalizes the method, proposed in [7], to ternary Hom-Lie algebras. In our case, the theorem states (Theorem 3.2) that, given a Hom-Lie algebra $(\mathfrak {g},[\;,\;],\alpha )$ and an element $\omega$ of its dual vector space $\mathfrak {g}^*$, one can define the totally skew-symmetric ternary bracket

$$\begin{aligned}{}[u,v,w]=\omega (u)\,[v,w]+\omega (v)\,[w,u]+\omega (w)\,[u,v],\;\;u,v,w\in \mathfrak {g}, \end{aligned}$$

which satisfies the Hom-Filippov–Jacobi identity

$$\begin{aligned} \big [\alpha (x),\alpha (y),[u,v,w]\big ]= & {} \big [[x,y,u],\alpha (v),\alpha (w)\big ]\\&+\big [\alpha (u),[x,y,v],\alpha (w)\big ]+\big [\alpha (u),\alpha (v),[x,y,w]\big ], \end{aligned}$$

if $\omega$ satisfies the conditions

$$\begin{aligned} \omega ([u,v,w])= & {} 0, \end{aligned}$$

(1.1)

$$\begin{aligned} \omega (u)\,\omega (\alpha (v))-\omega (v)\,\omega (\alpha (u))= & {} 0. \end{aligned}$$

(1.2)

To prove this theorem, we expand all double ternary brackets in the Hom-Filippov–Jacobi identity and show that the conditions (1.1) and (1.2) together with the Hom-Jacobi identity make it possible to split all the terms into groups, such that the sum of all terms in each group is zero. Then we take an associative commutative algebra over $\mathbb C$ and use it to construct two Hom-Lie algebras, where the Lie bracket of the first is constructed with the help of $\sigma$-derivation D of $\mathcal A$ and the Lie bracket of second Hom-Lie algebra is constructed with the help of involution $\varrho$ of $\mathcal A$. Next we apply Theorem (3.2) to these two Hom-Lie algebras and construct two 3-Hom-Lie algebras.

2 Quasi-Hom-Lie Algebras Constructed by Means of $\sigma$-Derivation and Involution

Let $\mathfrak {g}$ be a vector space over the complex numbers. The vector space of linear transformations of $\mathfrak {g}$ will be denoted by $\text{ Lin }(\mathfrak {g})$. A quasi-hom-Lie algebra is a generalization of the concept of Lie algebra and this generalization is based on an analog of Jacobi identity, which contains six terms [17]. The definition of Quasi-Hom-Lie algebra (QHL-algebra) can be given as follows:

Definition 2.1

A tuple $(\mathfrak {g},[\;,\;],\alpha ,\beta ,\varsigma )$, where $[\;,\;]:\mathfrak {g}\times \mathfrak {g}\rightarrow \mathfrak {g}$ is a bilinear mapping called a bracket in $\mathfrak {g}$, $\alpha ,\beta \in \text{ Lin }(\mathfrak {g})$ and $\varsigma :(u,v)\in \mathfrak {g}\times \mathfrak {g}\rightarrow \varsigma _{u,v}\in \text{ Lin }(\mathfrak {g})$, is said to be a QHL-algebra if for any $u,v,w\in \mathfrak {g}$ it holds

• 1. $[u,v]=\varsigma _{u,v}(\,[v,u])$,

• 2. $[\alpha (u),\alpha (v)]=\beta \circ \alpha ([u,v])$,

• 3. $\varsigma _{w,u}([\alpha (u),[v,w]])+\varsigma _{u,v}([\alpha (v),[w,u]])+\varsigma _{v,w}([\alpha (w),[u,v]])+\varsigma _{w,u}\circ \beta ([u,[v,w]])+\varsigma _{u,v}\circ \beta ([v,[w,u]])+\varsigma _{v,w}\circ \beta ([w,[u,v]])=0.$

The last relation will be called a QHL-Jacobi identity.

A Hom-Lie algebra is a particular case of QHL-algebra such that the Jacobi identity is twisted by means of a linear transformation of the vector space of Lie algebra. A linear transformation, which twists the Jacobi identity, is usually called a twist. If twist is the identity transformation, then a definition of Hom-Lie algebra reduces to that of Lie algebra. For the convenience of reader, we remind a definition of Hom-Lie algebra [12, 15, 17].

Definition 2.2

A Hom-Lie algebra is a triple $(\mathfrak {g},[\;,\;],\alpha )$, where $[\;,\;]:\mathfrak {g}\times \mathfrak {g}\rightarrow \mathfrak {g}$ is a bilinear mapping, called bracket, and $\alpha :\mathfrak {g}\rightarrow \mathfrak {g}$ is a linear mapping, called twist, which for all $u,v,w\in \mathfrak {g}$ satisfy the following conditions:

• 1. $[u,v]=-[v,u]$, i.e. a bracket is skew-symmetric,

• 2. $\big [\alpha (u),[v,w]\big ]+\big [\alpha (v),[w,u]\big ]+\big [\alpha (w),[u,v]\big ]=0$.

The second condition is called a Hom-Jacobi identity.

It is easy to show that Hom-Lie algebra is a particular case of QHL-algebra. Indeed, if in the definition of QHL-algebra for any pair $(u,v)\in \mathfrak {g}\times \mathfrak {g}$ we take $\varsigma _{u,v}=-\text{ id }_{\mathfrak {g}}$, then, according to condition 1 of Definition 2.1, a bracket becomes skew-symmetric. Next, if in Definition 2.1 we take $\beta =\text{ id }_{\mathfrak {g}}$, then the QHL-Jacobi identity can be written in the form of Hom-Jacobi identity, where $\alpha +\text{ id }_{\mathfrak {g}}$ is a twist.

Let $\mathcal A$ be a unital associative commutative algebra over the field of complex numbers. Each element $u\in \mathcal A$ determines the linear transformation $\mu _u:\mathcal A\rightarrow \mathcal A$ defined by $\mu _u(v)=uv$. Let $\sigma ,\tau$ be two endomorphisms of algebra $\mathcal A$. Let D be a $(\sigma ,\tau )$-derivation of an algebra $\mathcal A$, that is, D is a linear transformation of $\mathcal A$, which satisfies the $(\sigma ,\tau )$-twisted Leibniz rule

$$\begin{aligned} D(uv)=D(u)\tau (v)+\sigma (u)\,D(v). \end{aligned}$$

(2.1)

If $\tau =\text{ id }_{\mathfrak {g}}$, then $(\sigma ,\tau )$-derivation D is called a $\sigma$-derivation of an algebra $\mathcal A$. If $\tau =\sigma$, then D will be called a $(\sigma ,\sigma )$-derivation.

Lemma 2.3

Let D is a $\sigma$-derivation of an algebra $\mathcal A$. Define the bracket

$$\begin{aligned}{}[u,v]_{\mathtt {D}}=\sigma (u)\,D(v)-\sigma (v)\,D(u). \end{aligned}$$

(2.2)

If for any $u\in \mathcal A$ an endomorphism $\sigma$ and $\sigma$-derivation D satisfy the relation $D(\sigma (u))=\delta \,\sigma (D(u))$, where $\delta$ is an element of algebra $\mathcal A$, then $(\mathcal A, [\;,\;]_{\mathtt D},\sigma ,\mu _\delta ,\varsigma )$, where $\varsigma _{u,v}=-\text{ id }_{\mathcal A}$, is a QHL-algebra.

Proof

Obviously, (2.2) is skew-symmetric, hence the first condition of Definition 2.1 is satisfied, because, according to the assumption of lemma, $\varsigma _{u,v}=-\text{ id }_{\mathcal A}$. Next, we show that the second condition of Definition (2.1) is also satisfied. Indeed, we have

$$\begin{aligned}{}[\sigma (u),\sigma (v)]_{\mathtt D}= & {} \sigma ^2(u)\,D\big (\sigma (v)\big )-\sigma ^2(v)\,D\big (\sigma (u)\big )\\= & {} \delta \,\big (\sigma ^2(u)\sigma (D(v))-\sigma ^2(v)\sigma (D(u))\big )\\= & {} \delta \,\sigma \big (\sigma (u)\,D (v)-\sigma (v)\,D (u)\big )\\= & {} \delta \,\sigma ([u,v]_{\mathtt D})=\mu _\delta \circ \sigma ([u,v]_{\mathtt D}). \end{aligned}$$

The last condition, which we need to prove is the QHL-Jacobi identity. We can split the QHL-Jacobi identity into the sum of two parts, where the terms of the first part contain $\sigma$ and the terms of the second part contain multiplication by $\delta$, i.e.

$$\begin{aligned}&\big [\sigma (u),[v,w]_{\mathtt D}\big ]_{\mathtt D}+\big [\sigma (v),[w,u]_{\mathtt D}\big ]_{\mathtt D}+\big [\sigma (w),[u,v]_{\mathtt D}\big ]_{\mathtt D}\\&\quad +\delta \,\big [u,[v,w]_{\mathtt D}\big ]_{\mathtt D}+\delta \,\big [v,[w,u]_{\mathtt D}\big ]_{\mathtt D}+\delta \,\big [w,[u,v]_{\mathtt D}\big ]_{\mathtt D}=0. \end{aligned}$$

Applying twice the formula (2.2) to the first term in the first part of the above Hom-Jacobi identity, we obtain

$$\begin{aligned} \big [\sigma (u),[v,w]_{\mathtt {D}}\big ]_{\mathtt {D}}= & {} \big [\sigma (u),\sigma (v)\,D(w)-\sigma (w)\,D(v)\big ]_{\mathtt D}\\= & {} \sigma ^2(u)\,D(\sigma (v))\,D(w)+\underbrace{\sigma ^2(u)\sigma ^2(v)\,D^2(w)}_{I}\\&- \sigma ^2(u)\,D(\sigma (w))\,D(v)\\&-\underbrace{\sigma ^2(u)\sigma ^2(w)D^2(v)}_{II}-\underbrace{\sigma ^2(v)\sigma (D(w))D(\sigma (u))}_{1}\\&+\underbrace{\sigma ^2(w)\sigma (D(v))D(\sigma (u))}_{2}. \end{aligned}$$

Analogously, the second and third term in the first part of the Hom-Jacobi identity give respectively

$$\begin{aligned} \big [\sigma (v),[w,u]_{\mathtt {D}}\big ]_{\mathtt {D}}= & {} \big [\sigma (v),\sigma (w)\,D(u)-\sigma (u)\,D(w)\big ]_{\mathtt D}\\= & {} \sigma ^2(v)\,D(\sigma (w))\,D(u)+\underbrace{\sigma ^2(v)\sigma ^2(w)\,D^2(u)}_{III}\\&- \sigma ^2(v)\,D(\sigma (u))\,D(w)\\&-\underbrace{\sigma ^2(v)\sigma ^2(u)D^2(w)}_{I}-\underbrace{\sigma ^2(w)\sigma (D(u))D(\sigma (v))}_{2}\\&+\underbrace{\sigma ^2(u)\sigma (D(w))D(\sigma (v))}_{3}, \end{aligned}$$

and

$$\begin{aligned}&\big [\sigma (w),[u,v]_{\mathtt {D}}\big ]_{\mathtt {D}} = \big [\sigma (w),\sigma (u)\,D(v)-\sigma (v)\,D(u)\big ]_{\mathtt D}\\&\quad =\sigma ^2(w)\,D(\sigma (u))\,D(v)+\underbrace{\sigma ^2(w)\sigma ^2(u)\,D^2(v)}_{II}- \sigma ^2(w)\,D(\sigma (v))\,D(u)\\&\qquad -\underbrace{\sigma ^2(w)\sigma ^2(v)D^2(u)}_{III}-\underbrace{\sigma ^2(u)\sigma (D(v))D(\sigma (w))}_{3}+\underbrace{\sigma ^2(v)\sigma (D(u))D(\sigma (w))}_{1}. \end{aligned}$$

Taking the sum of the expressions, standing at the right-hand sides of the above relations and making use of the commutativity of algebra $\mathcal A$ as well as of the relation $D\big (\sigma (u)\big )=\delta \,\sigma \big (D(u)\big )$, we see that the terms labeled with the same numbers 1, 2, 3, I, II, III cancel each other. Finally, after all cancellations, the first part of QHL-Jacobi identity reduces to the following expression

$$\begin{aligned}&\big [\sigma (u),[v,w]_{\mathtt {D}}\big ]_{\mathtt {D}}+ \big [\sigma (v),[w,u]_{\mathtt {D}}\big ]_{\mathtt {D}}+ \big [\sigma (w),[u,v]_{\mathtt {D}}\big ]_{\mathtt {D}}\\&\quad = \underbrace{\sigma ^2(u)\,D(\sigma (v))\,D(w)}_{a}-\underbrace{\sigma ^2(u)\,D(\sigma (w))\,D(v)}_{b}\\&\qquad +\underbrace{\sigma ^2(v)\,D(\sigma (w))\,D(u)}_{d}-\underbrace{\sigma ^2(v)\,D(\sigma (u))\,D(w)}_{e}\\&\qquad +\underbrace{\sigma ^2(w)\,D(\sigma (u))\,D(v)}_{c}-\underbrace{\sigma ^2(w)\,D(\sigma (v))\,D(u)}_{f} \end{aligned}$$

Now we expand the double brackets of the second part of the QHL-Jacobi identity. We obtain

$$\begin{aligned}&\big [u,[v,w]_{\mathtt D}\big ]_{\mathtt D}=\big [u,D(w)\sigma (v)-D(v)\,\sigma (w)\big ]_{\mathtt D}\\&\quad =\underbrace{\sigma (u)D^2(w)\sigma (v)}_{I}+\underbrace{\sigma (u)\sigma (D(w))D(\sigma (v))}_{1}-\underbrace{\sigma (u)D^2(v)\sigma (w)}_{II}\\&\qquad -\underbrace{\sigma (u)\sigma (D(v))D(\sigma (w))}_{1}-\sigma (D(w))\sigma ^2(v)D(u)+\sigma (D(v))\sigma ^2(w)D(u) \end{aligned}$$

and

$$\begin{aligned}&\big [v,[w,u]_{\mathtt D}\big ]_{\mathtt D}=\big [v,D(u)\,\sigma (w)-D(w)\,\sigma (u)\big ]_{\mathtt D}\\&\quad =\underbrace{\sigma (v)D^2(u)\sigma (w)}_{III}+\underbrace{\sigma (v)\sigma (D(u))D(\sigma (w))}_{2}-\underbrace{\sigma (v)D^2(w)\sigma (u)}_{I}\\&\qquad -\underbrace{\sigma (v)\sigma (D(w))D(\sigma (u))}_{2}-\sigma (D(u))\sigma ^2(w)D(v)+\sigma (D(w))\sigma ^2(u)D(v)\\&\qquad \big [w,[u,v]_{\mathtt D}\big ]_{\mathtt D}=\big [w,D(v)\,\sigma (u)-D(u)\,\sigma (v)\big ]_{\mathtt D}\\&\quad =\underbrace{\sigma (w)D^2(v)\sigma (u)}_{II}+\underbrace{\sigma (w)\sigma (D(v))D(\sigma (u))}_{3}-\underbrace{\sigma (w)D^2(u)\sigma (v)}_{III}\\&\qquad -\underbrace{\sigma (w)\sigma (D(u))D(\sigma (v))}_{3}-\sigma (D(v))\sigma ^2(u)D(w)+\sigma (D(u))\sigma ^2(v)D(w). \end{aligned}$$

Note that applying the right hand side of (2.2) to the internal bracket in the first double bracket, we swap the factors $\sigma (v),D(w)$ and $\sigma (w),D(v)$ and do the similar transformation in the second and third double bracket. Again, due to the commutativity of an algebra $\mathcal A$ and relation $D\big (\sigma (u)\big )=\delta \,\sigma \big (D(u)\big )$, the terms labelled with 1, 2, 3, I, II, III cancel each other. The second part of QHL-Jacobi identity reduces to the expression

$$\begin{aligned}&\delta \,\big [u,[v,w]_{\mathtt {D}}\big ]_{\mathtt {D}}+ \delta \,\big [v,[w,u]_{\mathtt {D}}\big ]_{\mathtt {D}}+ \delta \,\big [w,[u,v]_{\mathtt {D}}\big ]_{\mathtt {D}}\nonumber \\&\quad =-\underbrace{\delta \,\sigma (D(w))\sigma ^2(v)D(u)}_{d}+\underbrace{\delta \,\sigma (D(v))\sigma ^2(w)D(u)}_{f}\nonumber \\&\qquad -\underbrace{\delta \,\sigma (D(u))\sigma ^2(w)D(v)}_{c}+\underbrace{\delta \,\sigma (D(w))\sigma ^2(u)D(v)}_{b}\nonumber \\&\qquad -\underbrace{\delta \,\sigma (D(v))\sigma ^2(u)D(w)}_{a}+\underbrace{\delta \,\sigma (D(u))\sigma ^2(v)D(w)}_{e}. \end{aligned}$$

(2.3)

Finally, taking the sum of both parts of QHL-Jacobi identity and cancelling terms labelled with a, b, c, d, f, we obtain the QHL-Jacobi identity

$$\begin{aligned}&\big [\sigma (u),[v,w]_{\mathtt {D}}\big ]_{\mathtt {D}}+ \big [\sigma (v),[w,u]_{\mathtt {D}}\big ]_{\mathtt {D}}+ \big [\sigma (w),[u,v]_{\mathtt {D}}\big ]_{\mathtt {D}}\\&\quad +\big [u,[v,w]_{\mathtt {D}}\big ]_{\mathtt {D}}+ \big [v,[w,u]_{\mathtt {D}}\big ]_{\mathtt {D}}+ \big [w,[u,v]_{\mathtt {D}}\big ]_{\mathtt {D}}=0. \end{aligned}$$

$\square$

Corollary 2.4

Let $\mathcal A$ be a unital associative commutative algebra, $\sigma$ be an endomorphism of $\mathcal A$ and D be a $\sigma$-derivative of $\mathcal A$. If $\sigma$ and D satisfy the relation $D\circ \sigma =\lambda \,\sigma \circ D$, where $\lambda$ is a non-zero complex number, then the bracket (2.2) determines a Hom-Lie algebra structure on an algebra $\mathcal A$, whose twist is $\sigma +\lambda \,\text{ id }_{\mathcal A}$.

Indeed, in this case, that is, when $\lambda$ is a non-zero complex number, we can bring it inside the double brackets of the second part of QHL-Jacobi identity, then combine it with an endomorphism $\sigma$ into the sum of two linear mappings $\sigma +\lambda \,\text{ id }_{\mathcal A}$. After this, the QHL-Jacobi identity takes the form of Hom-Jacobi identity and we obtain a structure of Hom-Lie algebra with bracket defined in (2.2) and twist $\sigma +\lambda \,\text{ id }_{\mathcal A}$.

In the paper [8] the authors constructed the Lie bracket on a commutative associative algebra with involution. Our aim is to show that their construction can be extended to Hom-Lie algebras. Hence, we assume that $\mathcal A$ is a commutative associative algebra endowed with a mapping $\varrho :\mathcal A\rightarrow \mathcal A$, which for all $u,v\in \mathcal A$ satisfies

• (a) $\varrho$ is a linear transformation of $\mathcal A$,

• (b) $\varrho (u\,v)=\varrho (u)\,\varrho (v)$, i.e. $\varrho$ is an endomorphism of algebra $\mathcal A$,

• (c) $\varrho ^2(u)=\varrho \circ \varrho (u)=\text{ id }(u)$.

An endomorphism $\varrho$ of an algebra $\mathcal A$ satisfying the properties (a), (b), (c) will be referred to as an involution of a commutative algebra $\mathcal A$. First, note that we call an involution a linear transformation of $\mathcal A$, and not an anti-linear one. Second, the condition (b) in the case of involution usually has the form $\varrho (u\,v)=\varrho (v)\,\varrho (u)$, but in the case of commutative algebra, which we assume, it can be written due to commutativity as $\varrho (u\,v)=\varrho (u)\,\varrho (v)$.

It is useful to define two linear mappings $\varrho _{-1},\varrho _1:\mathcal A\rightarrow \mathcal A$ as follows

$$\begin{aligned} \varrho _{-1}=\text{ id }_{\mathcal A}-\varrho ,\quad \varrho _{1}=\text{ id }_{\mathcal A}+\varrho . \end{aligned}$$

For any element $u\in \mathcal A$ we denote $u_{-1}=\varrho _{-1}(u),\,u_{1}=\varrho _{1}(u)$. Then

$$\begin{aligned} \varrho (u_{-1})=-u_{-1},\quad \varrho (u_{1})=u_{1},\quad u_{1}+u_{-1}=2\,u,\quad u_{1}-u_{-1}=2\,\varrho (u). \end{aligned}$$

(2.4)

Lemma 2.5

Define the bracket

$$\begin{aligned}{}[u,v]_\varrho =\sigma (u)\,\varrho (v)-\sigma (v)\,\varrho (u). \end{aligned}$$

(2.5)

If an involution $\varrho$ commutes with an endomorphism $\sigma$, i.e. $\varrho \circ \sigma =\sigma \circ \varrho$, then $(\mathcal A, [\;,\;]_\varrho ,\gamma )$ is a Hom-Lie algebra, where the twist has the form $\gamma =\sigma +\varrho$.

Proof

Clearly, the bracket (2.5) is skew-symmetric. As the twist $\gamma$ has the form of sum of two endomorphism, we can split the whole Hom-Jacobi identity into sum of two parts, where the terms of the first part contain $\sigma$ and the terms of the second part contain involution $\varrho$. Thus,

$$\begin{aligned}&\big [\sigma (u),[v,w]\big ]+\big [\sigma (v),[w,u]\big ]+\big [\sigma (w),[u,v]\big ]\\&\qquad \big [\varrho (u),[v,w]\big ]+\big [\varrho (v),[w,u]\big ]+\big [\varrho (w),[u,v]\big ]=0. \end{aligned}$$

Applying twice the definition of bracket (2.5), we obtain

$$\begin{aligned} \big [\sigma (u),[v,w]\big ]= & {} \underbrace{\sigma ^2(u)\sigma (\varrho (v))\,w}_{V}-\underbrace{\sigma ^2(u)\,v\,\sigma (\varrho (w))}_{IV}\\&\quad -\underbrace{\varrho (\sigma (u)\sigma ^2(v))\,\sigma (\varrho (w))}_{1}+\underbrace{\varrho (\sigma (u))\sigma (\varrho (v))\sigma ^2(w)}_{2},\\ \big [\sigma (v),[w,u]\big ]= & {} \underbrace{\sigma ^2(v)\sigma (\varrho (w))\,u}_{I}-\underbrace{\sigma ^2(v)\,w\,\sigma (\varrho (u))}_{VI}\\&\quad -\underbrace{\varrho (\sigma (v)\sigma ^2(w))\,\sigma (\varrho (u))}_{2}+\underbrace{\varrho (\sigma (v))\sigma (\varrho (w))\sigma ^2(u)}_{3},\\ \big [\sigma (w),[u,v]\big ]= & {} \underbrace{\sigma ^2(w)\sigma (\varrho (u))\,v}_{III}-\underbrace{\sigma ^2(w)\,u\,\sigma (\varrho (v))}_{II}\\&\quad -\underbrace{\varrho (\sigma (w)\sigma ^2(u))\,\sigma (\varrho (v))}_{3}+\underbrace{\varrho (\sigma (w))\sigma (\varrho (u))\sigma ^2(v)}_{1}, \end{aligned}$$

and we see that due to the commutativity of $\sigma$ and $\rho$ the terms, labelled with one and the same integer from the set 1, 2, 3, cancel each other.

The second part of the Hom-Jacobi identity gives

$$\begin{aligned} \big [\varrho (u),[v,w]\big ]= & {} \underbrace{\sigma (\varrho (u))\varrho (\sigma (v))\,w}_{a}-\underbrace{\sigma (\varrho (u))\,v\,\varrho (\sigma (w))}_{b}\\&\quad -\underbrace{u\,\sigma ^2(v))\,\sigma (\varrho (w))}_{I}+\underbrace{u\,\sigma (\varrho (v))\sigma ^2(w)}_{II},\\ \big [\varrho (v),[w,u]\big ]= & {} \underbrace{\sigma (\varrho (v))\varrho (\sigma (w))\,u}_{c}-\underbrace{\sigma (\varrho (v))\,w\,\varrho (\sigma (u))}_{a}\\&\quad -\underbrace{v\,\sigma ^2(w))\,\sigma (\varrho (u))}_{III}+\underbrace{v\,\sigma (\varrho (w))\sigma ^2(u)}_{IV},\\ \big [\varrho (w),[u,v]\big ]= & {} \underbrace{\sigma (\varrho (w))\varrho (\sigma (u))\,v}_{b}-\underbrace{\sigma (\varrho (w))\,u\,\varrho (\sigma (v))}_{c}\\&\quad -\underbrace{w\,\sigma ^2(u))\,\sigma (\varrho (v))}_{V}+\underbrace{w\,\sigma (\varrho (u))\sigma ^2(v)}_{VI}. \end{aligned}$$

By canceling the terms labeled with the same symbol, we obtain the Hom-Jacobi identity. $\square$

3 Induced 3-Hom-Lie Algebras

Definition 3.1

A 3-Hom-Lie algebra is a quadruple $(\mathfrak {g},[\;,\;,\;],\alpha ,\beta )$, where $\mathfrak {g}$ is a vector space, $[\;,\;,\;]:\mathfrak {g}\times \mathfrak {g}\times \mathfrak {g}\rightarrow \mathfrak {g}$ is a ternary bracket and $\alpha ,\beta :\mathfrak {g}\rightarrow \mathfrak {g}$ are linear mappings, called twists of 3-Hom-Lie algebra. A ternary bracket is trilinear, totally skew-symmetric mapping and it satisfies the Hom-Filippov-Jacobi identity

$$\begin{aligned}{}[\alpha (x),\beta (y),[u,v,w]]= & {} [[x,y,u],\alpha (v),\beta (w)]+[\alpha (u),[x,y,v],\beta (w)]\\&+[\alpha (u),\beta (v),[x,y,w]]. \end{aligned}$$

Particularly, if $\alpha =\beta$, then a 3-Hom-Lie algebra will be denoted as the triple $(\mathfrak {g},[\;,\;,\;],\alpha )$.

Theorem 3.2

Let $(\mathfrak {g},[\;,\;],\alpha )$ be a Hom-Lie algebra and $\omega :\mathfrak {g}\rightarrow \mathbb C$ be a linear function, i.e. $\omega \in \mathfrak {g}^*$, where $\mathfrak {g}^*$ is the dual vector space of $\mathfrak {g}$. Define the ternary bracket by the formula

$$\begin{aligned}{}[u,v,w]=\omega (u)\,[v,w]+\omega (v)\,[w,u]+\omega (w)\,[u,v]. \end{aligned}$$

(3.1)

If for all $u,v,w\in \mathfrak {g}$ a linear function $\omega$ satisfies the conditions

$$\begin{aligned} \omega ([u,v,w])= & {} 0, \end{aligned}$$

(3.2)

$$\begin{aligned} \omega (u)\,\omega (\alpha (v))-\omega (v)\,\omega (\alpha (u))= & {} 0, \end{aligned}$$

(3.3)

then the ternary bracket (3.1) determines the structure of 3-Hom-Lie algebra on the vector space $\mathfrak {g}$, i.e. the triple $(\mathfrak {g},[\;,\;,\;]_\omega ,\alpha )$ is a 3-Hom-Lie algebra.

Proof

To prove this theorem, we need to show that the ternary bracket (3.1) defines the structure of 3-Hom-Lie algebra described in Definition (3.1). First of all, it is easy to see that the ternary bracket (3.1) is trilinear, totally skew-symmetric. Thus, all that is required is to prove the Hom-Filippov-Jacoby identity

$$\begin{aligned} \big [\alpha (x),\alpha (y),[u,v,w]\big ]= & {} \big [[x,y,u],\alpha (v),\alpha (w)\big ]\\&+\big [\alpha (u),[x,y,v],\alpha (w)\big ]+\big [\alpha (u),\alpha (v),[x,y,w]\big ]. \end{aligned}$$

For this purpose, it is convenient to write condition (3.2) as follows. Substituting the expression for the ternary commutator (3.1) into the condition (3.2) we can write it in the form

$$\begin{aligned} \omega (u)\,\omega ([v,w])+\omega (w)\,\omega ([u,v])+\omega (v)\,\omega ([w,u])=0. \end{aligned}$$

(3.4)

Now expanding the double ternary bracket at the left hand side of Hom-Filippov–Jacobi identity, we obtain the expression

$$\begin{aligned}&\omega (u)\omega (\alpha (x))[\alpha (y),[v,w]]+\omega (u)\omega (\alpha (y))[[v,w],\alpha (x)]\\&\quad +\omega (v)\omega (\alpha (x))[\alpha (y),[w,u]]+\omega (v)\omega (\alpha (y))[[w,u],\alpha (x)]\\&\quad +\omega (w)\omega (\alpha (x))[\alpha (y),[u,v]]+\omega (w)\omega (\alpha (y))[[u,v],\alpha (x)]\\&\quad +\big (\omega (u)\omega ([v,w])+\omega (v)\omega ([w,u])+\omega (w)\omega ([u,v])\big )[\alpha (x),\alpha (y)] \end{aligned}$$

The coefficient of the last term in the above expression (last line) vanishes because of the condition (3.4). Thus, the left hand side of the Hom-Filippov–Jacobi identity can be written in the form

$$\begin{aligned}&\underbrace{\omega (u)\omega (\alpha (x))[\alpha (y),[v,w]]}_1+\underbrace{\omega (u)\omega (\alpha (y))[[v,w],\alpha (x)]}_2\nonumber \\&\quad +\underbrace{\omega (v)\omega (\alpha (x))[\alpha (y),[w,u]]}_3+\underbrace{\omega (v)\omega (\alpha (y))[[w,u],\alpha (x)]}_4\nonumber \\&\quad +\underbrace{\omega (w)\omega (\alpha (x))[\alpha (y),[u,v]]}_5+\underbrace{\omega (w)\omega (\alpha (y))[[u,v],\alpha (x)]}_6 \end{aligned}$$

(3.5)

Expanding the first double ternary bracket at the right hand side of the Hom-Filippov–Jacobi identity, we obtain the expression

$$\begin{aligned}&\underbrace{\omega (x)\omega (\alpha (v))\big [\alpha (w),[y,u]\big ]}_3+\underbrace{\omega (x)\omega (\alpha (w))\big [[y,u],\alpha (v)\big ]}_5\nonumber \\&\quad +\underbrace{\omega (y)\omega (\alpha (v))\big [\alpha (w),[u,x]\big ]}_4+\underbrace{\omega (y)\omega (\alpha (w))\big [[u,x],\alpha (v)\big ]}_6\nonumber \\&\quad +\underbrace{\omega (u)\omega (\alpha (v))\big [\alpha (w),[x,y]\big ]}_a+\underbrace{\omega (u)\omega (\alpha (w))\big [[x,y],\alpha (v)\big ]}_b. \end{aligned}$$

(3.6)

In this expression, the three terms ($\omega (x)\omega ([y,u])[\alpha (v),\alpha (w)]$ and expressions derived from this by cyclic permutations of x, y, u) are not shown, because they all together give zero by virtue of condition (3.4). Expanding the second and third double ternary brackets in the Hom-Filippov–Jacobi identity, we obtain two more expressions

$$\begin{aligned}&\underbrace{\omega (x)\omega (\alpha (w))\big [\alpha (u),[y,v]\big ]}_5+\underbrace{\omega (x)\omega (\alpha (u))\big [[y,v],\alpha (w)\big ]}_1\nonumber \\&\quad +\underbrace{\omega (y)\omega (\alpha (w))\big [\alpha (u),[v,x]\big ]}_6+\underbrace{\omega (y)\omega (\alpha (u))\big [[v,x],\alpha (w)\big ]}_2\nonumber \\&\quad +\underbrace{\omega (v)\omega (\alpha (w))\big [\alpha (u),[x,y]\big ]}_c+\underbrace{\omega (v)\omega (\alpha (u))\big [[x,y],\alpha (w)\big ]}_a. \end{aligned}$$

(3.7)

and

$$\begin{aligned}&\underbrace{\omega (x)\omega (\alpha (u))\big [\alpha (v),[y,w]\big ]}_1+\underbrace{\omega (x)\omega (\alpha (v))\big [[y,w],\alpha (u)\big ]}_3\nonumber \\&\quad +\underbrace{\omega (y)\omega (\alpha (u))\big [\alpha (v),[w,x]\big ]}_2+\underbrace{\omega (y)\omega (\alpha (v))\big [[w,x],\alpha (u)\big ]}_4\nonumber \\&\quad +\underbrace{\omega (w)\omega (\alpha (u))\big [\alpha (v),[x,y]\big ]}_b+\underbrace{\omega (w)\omega (\alpha (v))\big [[x,y],\alpha (u)\big ]}_c. \end{aligned}$$

(3.8)

In the case of the second double ternary bracket at the right hand side of the Hom-Filippov–Jacobi identity, we omitted the summands $\omega (x)\omega ([y,u][\alpha (v),\alpha (w)])$ and expressions derived from this by cyclic permutations of x, y, u, which together give zero by virtue of condition (3.4). In the case of the third double bracket at the right hand side of the Hom-Filippov–Jacobi identity, we omitted the summands $\omega (x)\omega ([y,e])[\alpha (u),\alpha (v)]$ and the expressions derived from this by cyclic permutations of x, y, w. In order to show that the terms labeled with number 1 give zero, we will use the condition (3.3) and then the binary Hom-Jacobi identity. By collecting all the terms labeled with number 1 on the left hand side of Hom-Filippov–Jacobi identity, we obtain

$$\begin{aligned}&\omega (u)\omega (\alpha (x))[\alpha (y),[v,w]]-\omega (x)\omega (\alpha (u))\big ([[y,v],\alpha (w)]+[\alpha (v),[y,w]]\big )\\&\quad =\omega (u)\omega (\alpha (x))[\alpha (y),[v,w]]-\omega (u)\omega (\alpha (x))\big ([[y,v],\alpha (w)]+[\alpha (v),[y,w]]\big )\\&\quad =\omega (u)\omega (\alpha (x))\big ([\alpha (y),[v,w]]+[\alpha (w),[y,v]]+[\alpha (v),[w,y]] \big )=0. \end{aligned}$$

Similarly we prove that the terms labeled by the integers 2, 3, 4, 5, 6 give zero. Now we consider the terms labeled with the letter a. In this case in order to show that the sum of these terms is zero, we only need to apply the condition (3.3). Indeed, we have

$$\begin{aligned}&\omega (u)\omega (\alpha (v))\,[\alpha (w),[x,y]]+\omega (v)\omega (\alpha (u))[[x,y],\alpha (w)]\nonumber \\&\quad =[\alpha (w),[x,y]]\big (\omega (u)\omega (\alpha (v))-\omega (v)\omega (\alpha (u))\big )=0. \end{aligned}$$

(3.9)

Analogously we prove that the terms labeled with b and c give zero and this ends the proof. $\square$

Note that the condition (3.4) will be satisfied if we, in particular, assume

$$\begin{aligned} \omega ([u,v])=0,\;\;\forall u,v\in \mathfrak {g}. \end{aligned}$$

(3.10)

This particular case of the condition (3.4) gives a key to practical construction of 3-Hom-Lie algebras. Indeed, the condition (3.10) shows that a linear function $\omega$ is an analogue of the trace, since it vanishes on any binary skew-symmetric bracket.

We can apply Theorem (3.2) to Hom-Lie algebras constructed in the previous section to construct 3-Hom-Lie algebras.

Proposition 3.3

Let $\mathcal A$ be a commutative algebra, $\sigma :\mathcal A\rightarrow \mathcal A$ be an endomorphism of $\mathcal A$, D be a $\sigma$-derivation of $\mathcal A$ and $\psi \in \mathcal A^*$. Define the ternary bracket

$$\begin{aligned}{}[u,v,w]_{\mathtt D}=\psi (u)[v,w]_{\mathtt D}+\psi (v)[w,u]_{\mathtt D}+\psi (w)[u,v]_{\mathtt D}, \end{aligned}$$

(3.11)

where $[\;,\;]_{\mathtt D}$ is defined in (2.2). If $\sigma$-derivation D and endomorphism $\sigma$ satisfy the relation $D(\sigma (u))=\lambda \,\sigma (D(u))$, where $\lambda$ is a non-zero complex number, and for all $u,v\in \mathcal A$ a linear function $\psi :\mathcal A\rightarrow \mathbb C$ satisfies the conditions

$$\begin{aligned} \psi (u)\,\psi \big (\sigma (u)\big )= & {} \psi (v)\,\psi \big (\sigma (u)\big ),\\ \psi \big (\sigma (v)\,D(u)\big )= & {} \psi \big (\sigma (u)\,D(v)\big ), \end{aligned}$$

then $(\mathcal A,[\;,\;,\;]_{\mathtt D}, \sigma +\lambda \,\text{ id }_{\mathcal A})$ is a 3-Hom-Lie algebra.

This proposition follows from Lemma (2.3), Theorem (3.2) and Corollary 2.4.

Proposition 3.4

Let $\mathcal A$ be a commutative algebra, $\varrho$ be an involution of $\mathcal A$, $\sigma$ be an endomorphism of $\mathcal A$ and $\varphi \in \mathcal A^*$. Define the ternary bracket

$$\begin{aligned}{}[u,v,w]_{\varrho }=\varphi (u)\,[v,w]_{\varrho }+\varphi (v)\,[w,u]_{\varrho }+\varphi (w)\,[u,v]_{\varrho }, \end{aligned}$$

(3.12)

where the binary bracket $[\;,\;]_{\varrho }$ is defined in (2.5). If involution $\varrho$ commutes with an endomorphism $\sigma$ and for all $u,v\in \mathcal A$ a linear function $\varphi :\mathcal A\rightarrow \mathbb C$ satisfies the conditions

$$\begin{aligned} \varphi (u)\,\varphi (\sigma (v))+\varphi (u)\,\varphi \circ \varrho (v)= & {} \varphi (v)\,\varphi (\sigma (u))+ \varphi (v)\,\varphi \circ \varrho (u),\\ \varphi \big (\sigma (u)\varrho (v)\big )= & {} \varphi \big (\sigma (v)\varrho (u)\big ), \end{aligned}$$

then $(\mathcal A,[\;,\;,\;]_{\varrho }, \sigma +\varrho )$ is a 3-Hom-Lie algebra.

This proposition follows immediately from Lemma (2.5) and Theorem (3.2).