# On Ternary $F$-manifold Algebras and their Representations

A. Ben Hassine  Department of Mathematics, Faculty of Science and Arts at
Belqarn, University of Bisha, Kingdom of Saudi Arabia & Faculty of Sciences,
University of Sfax, Tunisia<EMAIL_ADDRESS>,  T. Chtioui  Faculty of
Sciences, University of Sfax, Tunisia<EMAIL_ADDRESS>,  M.
Elhamdadi  Department of Mathematics, University of South Florida, Tampa, FL
33620, U.S.A<EMAIL_ADDRESS>and  S. Mabrouk  Faculty of Sciences,
University of Gafsa, BP 2100, Gafsa, Tunisia<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We introduce a notion of ternary $F$-manifold algebras which is a
generalization of $F$-manifold algebras. We study representation theory of
ternary $F$-manifold algebras. In particular, we introduce a notion of dual
representation which requires additional conditions similar to the binary
case. Then, we establish a notion of a coherence ternary $F$-manifold algebra.
Moreover, we investigate the construction of ternary $F$-manifold algebras
using $F$-manifold algebras. Furthermore, we introduce and investigate a
notion of a relative Rota-Baxter operator with respect to a representation and
use it to construct ternary pre-$F$-manifold algebras.

Keywords: Ternary $F$-manifold algebra, representation, relative Rota-Baxter
operators, ternary pre-$F$-manifold algebras.

MSC(2020):17B10, 17B56, 17A42

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Ternary $F$-manifold algebras
  4. 4 Representations of ternary $F$-manifold algebras
  5. 5 Ternary pre-$F$-manifold algebras
    1. 5.1 Definition and elementary results
    2. 5.2 Relative Rota-Baxter operators on ternary-$F$-manifold algebras

##  1\. Introduction

Ternary Lie algebras and more generally $n$-ary Lie algebras are natural
generalizations of Lie algebras. They were introduced and studied by Filippov
in [21]. This type of algebras appeared also in the algebraic formulation of
Nambu Mechanics [41], generalizing Hamiltonian mechanics by considering two
hamiltonians (see [22, 45]). Moreover, $3$-Lie algebras appeared in string
theory and $M$-theory. Basu and Harvey suggested to replace the Lie algebra
appearing in the Nahm equation by a $3$-Lie algebra for the lifted Nahm
equations ([11]. Furthermore, Bagger-Lambert managed to construct, using a
ternary bracket, an $N=2$ supersymmetric version of the world volume theory of
the $M$-theory membrane, in the context of Bagger-Lambert-Gustavsson model of
multiple $M2$-branes (for more details see [7, 8, 23, 26, 42]. The notion of
$n$-Lie algebra was studied in recent years in several algebraic contexts. In
[6, 9], the authors gave construction, realization and classifications of
$3$-Lie algebras and $n$-Lie algebras. Representation theory of $n$-Lie
algebras was first introduced by Kasymov in [35] and cohomology theory of
Fillippov algebras was studied in [30]. The adjoint representation is defined
by the ternary bracket in which two elements are fixed. The notion of Rota-
Baxter operators on associative algebras was introduced in 1960 by G.Baxter
[3] in his study of fluctuation theory in probability. Recently, there has
been many applications, including in Connes-Kreimer’s algebraic approach to
renormalization in perturbative quantum field theory [13]. In the Lie algebra
context, a Rota-Baxter operator of weight zero was introduced independently in
the 1980s as the operator form of the classical Yang Baxter equation, whereas
the classical Yang-Baxter equation plays important roles in many areas of
mathematics and mathematical physics such as quantum groups and integrable
systems [12, 43]. Rota-Baxter operators on super-type algebras were studied in
[1], which build relationships between associative superalgebras, Lie
superalgebras, $L$-dendriform superalgebras and pre-Lie superalgebras.
Recently relative Rota-Baxter operators on Leibniz algebras were studied in
[44]. The notion of relative Rota-Baxter operators on $3$-Lie algebras with
respect to a representation thanks to the solutions of $3$-Lie classical Yang-
Baxter equation [4]. In particular, Rota-Baxter operators on $3$-Lie algebras,
introduced in [5], are relative Rota-Baxter operators on a $3$-Lie algebra
with respect to the adjoint representation.
On the other hand, in order to give a geometrical expression of the WDV
equations, Dubrov introduced the notion of Frobenius manifolds [18]. The
concept of $F$-manifolds was introduced by Hertling and Manin, as a relaxation
of the conditions of Frobenius manifolds [28]. $F$-manifolds appear in many
areas of mathematics such as singularity theory [27], quantum $K$-theory [36],
integrable systems [15, 18, 37], operads [38] and so on. Inspired by the
investigation of algebraic structures of $F$-manifolds, the notion of
$F$-manifold algebras was given by Dotsenko in [19]. Recently, the notion of
representations of $F$-manifold algebras which gives rise to pre-$F$-manifold
algebras was introduced in [31]. Later, the notion of $F$-manifold color
algebras and Hom-$F$-manifolds algebras and some constructions were given in
([10, 40]).

A natural question of introducing a notion of ternary $F$-manifold algebras
then arises. This is the goal of this article. We introduce a notion of
ternary $F$-manifold algebras by first, defining a structure which can be
induced by an $F$-manifold algebra exactly as for $3$-Lie algebras. Second,
the structure must generalize ternary Nambu-Poisson algebras.
The paper is organized as follows. In Section 2, we review some basics of
$3$-Lie algebras, $3$-pre-Lie algebras, ternary Nambu-Poisson algebras and
$F$-manifold algebras. In Section 3, we introduce a notion of _ternary
$F$-manifold_ algebras. We also give a construction of ternary $F$-manifold
algebras with the help of a given $F$-manifold algebra equipped with a trace
function. Section 4 is dedicated to representations and dual representations
of ternary $F$-manifold algebras. In Section 5 we introduce a notion of
ternary pre-$F$-manifold algebras and relative Rota-Baxter operators on a
ternary $F$-manifold algebra. We show that on one hand, a relative Rota-Baxter
operator on a ternary $F$-manifold algebra gives rise to a ternary
pre-$F$-manifold algebra. On the other hand, a ternary pre-$F$-manifold
algebra is naturally built via symplectic structures.

In this paper, all the vector spaces are over an algebraically closed field
$\mathbb{K}$ of characteristic $0$.

##  2\. Preliminaries

An associative algebra is a pair $(A,\cdot)$, where $A$ is a vector space and
$\cdot:A\otimes A\longrightarrow A$ is a linear map satisfying that for all
$x,y,z\in A$, the associator $\mathfrak{as}(x,y,z)=(x\cdot y)\cdot
z-x\cdot(y\cdot z)=0$, i.e.

$(x\cdot y)\cdot z=x\cdot(y\cdot z).$

Furthermore, if $x\cdot y=y\cdot x$ for all $x,y\in A$, then $(A,\cdot)$ is
called a commutative associative algebra.

A representation of commutative associative algebra $(A,\cdot)$ on a vector
space $V$ is a linear map $\mu:A\longrightarrow\mathfrak{gl}(V)$, such that
for any $x,y\in A$, the following equalities are satisfied:

(2.1) $\displaystyle\mu(x\cdot y)$ $\displaystyle=$
$\displaystyle\mu(x)\mu(y),\forall x,y\in A.$

If there exists an element $1_{A}\in A$ satisfying $x\cdot 1_{A}=1_{A}\cdot
x=x$, then $A$ is called unital with unit $1_{A}$.

Let $(V;\mu)$ be a representation of a commutative associative algebra
$(A,\cdot)$. In the sequel, define
$\mu^{*}:A\longrightarrow\mathfrak{gl}(V^{*})$ by

$\langle\mu^{*}(x)\xi,v\rangle=-\langle\xi,\mu(x)v\rangle,\quad\forall~{}x\in
A,\xi\in V^{*},v\in V.$

######  Lemma 2.1.

Under the above notation, $(V^{*},\mu^{*})$ is a representation of $(A,\cdot)$
which is called the dual representation of $(V;\mu)$.

A Zinbiel algebra is a pair $(A,\diamond)$, where $A$ is a vector space,
$\diamond:A\otimes A\longrightarrow A$ is a binary multiplication such that
for all $x,y,z\in A$,

(2.2) $x\diamond(y\diamond z)=(y\diamond x)\diamond z+(x\diamond y)\diamond
z.$

######  Lemma 2.2.

Let $(A,\diamond)$ be a Zinbiel algebra. Then $(A,\cdot)$ is a commutative
associative algebra, where $x\cdot y=x\diamond y+y\diamond x$. Moreover, for
$x\in A$, define $L_{\diamond}(x):A\longrightarrow\mathfrak{gl}(A)$ by

(2.3) $L_{\diamond}(x)(y)=x\diamond y,\quad\forall~{}y\in A.$

Then $(A;L_{\diamond})$ is a representation of the commutative associative
algebra $(A,\cdot)$.

######  Definition 2.3.

A $3$-Lie algebra consists of a vector space $A$ equipped with a skew-
symmetric linear map called $3$-Lie bracket
$[\cdot,\cdot,\cdot]:\otimes^{3}A\rightarrow A$ such that the following
Fundamental Identity holds (for $x_{i}\in A,1\leq i\leq 5$)

(2.4)
$[x_{1},x_{2},[x_{3},x_{4},x_{5}]]=[[x_{1},x_{2},x_{3}],x_{4},x_{5}]+[x_{3},[x_{1},x_{2},x_{4}],x_{5}]+[x_{3},x_{4},[x_{1},x_{2},x_{5}]]$

In other words, for $x_{1},x_{2}\in A$, the operator

(2.5) $\displaystyle ad_{x_{1},x_{2}}:A\to A,\quad
ad_{x_{1},x_{2}}x:=[x_{1},x_{2},x],\quad\forall x\in A,$

is a derivation in the sense that

$ad_{x_{1},x_{2}}[x_{3},x_{4},x_{5}]=[ad_{x_{1},x_{2}}x_{3},x_{4},x_{5}]+[x_{3},ad_{x_{1},x_{2}}x_{4},x_{5}]+[x_{3},x_{4},ad_{x_{1},x_{2}}x_{5}],\forall
x_{3},x_{4},x_{5}\in A.$

A morphism between 3-Lie algebras is a linear map that preserves the 3-Lie
brackets.

######  Example 2.4.

[21] Consider $4$-dimensional $3$-Lie algebra $A$ generated by
$(e_{1},e_{2},e_{3},e_{4})$ with the following multiplication

$\displaystyle[e_{1},e_{2},e_{3}]=e_{4},\ [e_{1},e_{2},e_{4}]=e_{3},\
[e_{1},e_{3},e_{4}]=e_{2},\ [e_{2},e_{3},e_{4}]=e_{1}.$

The notion of a representation of an $n$-Lie algebra was introduced in [35].
See also [20].

######  Definition 2.5.

A representation of a 3-Lie algebra $A$ on a vector space $V$ is a skew-
symmetric linear map $\rho:\otimes^{2}A\rightarrow gl(V)$ satisfying

  1. (i)

$\rho(x_{1},x_{2})\rho(x_{3},x_{4})-\rho(x_{3},x_{4})\rho(x_{1},x_{2})=\rho([x_{1},x_{2},x_{3}],x_{4})-\rho([x_{1},x_{2},x_{4}],x_{3})$,

  2. (ii)

$\rho([x_{1},x_{2},x_{3}],x_{4})=\rho(x_{1},x_{2})\rho(x_{3},x_{4})+\rho(x_{2},x_{3})\rho(x_{1},x_{4})+\rho(x_{3},x_{1})\rho(x_{2},x_{4})$,

for $x_{i}\in A,1\leq i\leq 4$.

Let $(V,\rho)$ be a representation of a $3$-Lie algebra $A$. Define
$\rho^{\ast}:\otimes^{2}A\to\mathfrak{gl}(V^{\ast})$ by

(2.6)
$\langle\rho^{\ast}(x_{1},x_{2})\xi,v\rangle=-\langle\xi,\rho(x_{1},x_{2})v\rangle,\forall\xi\in
V^{\ast},x_{1},x_{2}\in A,v\in V.$

######  Proposition 2.6.

With the above notations, $(V^{\ast},\rho^{\ast})$ is a representation of $A$,
called the dual representation.

######  Example 2.7.

Let $A$ be a $3$-Lie algebra. The linear map
$ad:\otimes^{2}A\to\mathfrak{gl}(A)$ with $x_{1},x_{2}\to ad_{x_{1},x_{2}}$
for any $x_{1},x_{2}\in A$ defines a representation $(A,ad)$ which is called
the adjoint representation of $A$, where $ad_{x_{1},x_{2}}$ is given by Eq.
(2.5). The dual representation $(A^{\ast},ad^{\ast})$ of the adjoint
representation $(A,ad)$ of a $3$-Lie algebra $A$ is called the coadjoint
representation.

######  Definition 2.8.

A $3$-pre-Lie algebra is a pair $(A,\\{\cdot,\cdot,\cdot\\})$ consisting of a
a vector space $A$ and a linear map $\\{\cdot,\cdot,\cdot\\}:A\otimes A\otimes
A\rightarrow A$ such that the following identities hold:

(2.7) $\displaystyle\\{x,y,z\\}$ $\displaystyle=$ $\displaystyle-\\{y,x,z\\},$
(2.8) $\displaystyle\\{x_{1},x_{2},\\{x_{3},x_{4},x_{5}\\}\\}$
$\displaystyle=$
$\displaystyle\\{[x_{1},x_{2},x_{3}]^{C},x_{4},x_{5}\\}+\\{x_{3},[x_{1},x_{2},x_{4}]^{C},x_{5}\\}$
$\displaystyle+\\{x_{3},x_{4},\\{x_{1},x_{2},x_{5}\\}\\},$ (2.9)
$\displaystyle\\{[x_{1},x_{2},x_{3}]^{C},x_{4},x_{5}\\}$ $\displaystyle=$
$\displaystyle\\{x_{1},x_{2},\\{x_{3},x_{4},x_{5}\\}\\}+\\{x_{2},x_{3},\\{x_{1},x_{4},x_{5}\\}\\}$
$\displaystyle+\\{x_{3},x_{1},\\{x_{2},x_{4},x_{5}\\}\\},$

where $x,y,z,x_{i}\in A,1\leq i\leq 5$ and $[\cdot,\cdot,\cdot]^{C}$ is
defined by

(2.10) $[x,y,z]^{C}=\\{x,y,z\\}+\\{y,z,x\\}+\\{z,x,y\\},\quad\forall x,y,z\in
A.$

######  Proposition 2.9.

Let $(A,\\{\cdot,\cdot,\cdot\\})$ be a $3$-pre-Lie algebra. Then the induced
$3$-commutator given by Eq. (2.10) defines a $3$-Lie algebra.

Let $(A,\\{\cdot,\cdot,\cdot\\})$ be a $3$-pre-Lie algebra. Define a skew-
symmetric linear map $\mathbb{L}:\otimes^{2}A\to gl(A)$ by

(2.11) $\displaystyle\mathbb{L}(x,y)z=\\{x,y,z\\},\quad\forall x,y,z\in A.$

By the definitions of a $3$-pre-Lie algebra and a representation of a $3$-Lie
algebra, we have

######  Lemma 2.10.

With the above notations, $(A,\mathbb{L})$ is a representation of the $3$-Lie
algebra $(A,[\cdot,\cdot,\cdot]^{C})$.

######  Definition 2.11.

[39] A ternary Nambu-Poisson algebra is a triple
$(A,\cdot,[\cdot,\cdot,\cdot])$ consisting of a $\mathbb{K}$-vector space $A$,
two linear maps $\cdot:A\otimes A\rightarrow A$ and
$[\cdot,\cdot,\cdot]:A\otimes A\otimes A\rightarrow A$ such that

  1. (1)

$(A,\cdot)$ is a commutative associative algebra,

  2. (2)

$(A,[\cdot,\cdot,\cdot])$ is a $3$-Lie algebra,

  3. (3)

the following Leibniz rule

$\displaystyle[x_{1},x_{2},x_{3}\cdot
x_{4}]=x_{3}\cdot[x_{1},x_{2},x_{4}]+[x_{1},x_{2},x_{3}]\cdot x_{4}$

holds for all $x_{1},x_{2},x_{3}\in A$.

Now, we recall the notion of $F$-manifold algebra given in [31].

######  Definition 2.12.

A $F$-manifold algebra is a tuple $(A,\cdot,[\cdot,\cdot])$, where $(A,\cdot)$
is a commutative associative algebra and $(A,[\cdot,\cdot])$ is a Lie algebra,
such that for all $x,y,z,w\in A$, the Hertling-Manin relation holds :

(2.12) $\mathcal{L}(x\cdot
y,z,w)=x\cdot\mathcal{L}(y,z,w)+y\cdot\mathcal{L}(x,z,w),$

where $\mathcal{L}(x,y,z)$ is the Leibnizator define by

(2.13) $\mathcal{L}(x,y,z)=[x,y\cdot z]-[x,y]\cdot z-y\cdot[x,z].$

Let $(A,\cdot,[\cdot,\cdot])$ be a $F$-manifold algebra, $(V;\rho)$ be a
representation of the Lie algebra $(A,[\cdot,\cdot])$ and $(V;\mu)$ be a
representation of the commutative associative algebra $(A,\cdot)$. Define the
three linear maps $\mathcal{L}_{1},\mathcal{L}_{2}:A\otimes A\otimes V\to V$
given by

(2.14) $\displaystyle\mathcal{L}_{1}(x,y,u)$ $\displaystyle=$
$\displaystyle\rho(x)\mu(y)(u)-\mu(y)\rho(x)(u)-\mu([x,y])(u),$ (2.15)
$\displaystyle\mathcal{L}_{2}(x,y,u)$ $\displaystyle=$
$\displaystyle\mu(x)\rho(y)(u)+\mu(y)\rho(x)(u)-\rho(x\cdot y)(u),$

for all $x,y\in A$ and $u\in V$.

######  Definition 2.13.

With the above notations, the tuple $(V;\rho,\mu)$ is a representation of $A$
if the following conditions hold:

(2.16) $\displaystyle\mathcal{L}_{1}(x\cdot y,z,u)$
$\displaystyle=\mu(x)\mathcal{L}_{1}(y,z,u)+\mu(y)\mathcal{L}_{1}(x,z,u),$
(2.17) $\displaystyle\mu(\mathcal{L}(x,y,z))(u)$
$\displaystyle=\mathcal{L}_{2}(y,z,\mu(x)(u))-\mu(x)\mathcal{L}_{2}(y,z,u)$

for all $x,y,z\in A$ and $u\in V$.

##  3\. Ternary $F$-manifold algebras

In this section, we introduce the notion of ternary $F$-manifold algebras as a
generalization of $F$-manifold algebras given in [19] see also [31]. They are
the generalization of $F$-manifold algebra and ternary Nambu-Poisson algebra.

######  Definition 3.1.

A tuple $(A,\cdot,[\cdot,\cdot,\cdot])$ is called a ternary $F$-manifold
algebra if $(A,\cdot)$ is a commutative associative algebra and
$(A,[\cdot,\cdot,\cdot])$ is a $3$-Lie algebra, such that for all
$x_{1},x_{2},x_{3},x_{4},x_{5}\in A$, holds :

(3.18) $\mathcal{L}(x_{1}\cdot
x_{2},x_{3},x_{4},x_{5})=x_{1}\cdot\mathcal{L}(x_{2},x_{3},x_{4},x_{5})+x_{2}\cdot\mathcal{L}(x_{1},x_{3},x_{4},x_{5}),$

where $\mathcal{L}(x_{1},x_{2},x_{3},x_{4})$ is the $3$-Leibnizator define by

(3.19) $\mathcal{L}(x_{1},x_{2},x_{3},x_{4})=[x_{1},x_{2},x_{3}\cdot
x_{4}]-x_{3}\cdot[x_{1},x_{2},x_{4}]-[x_{1},x_{2},x_{3}]\cdot x_{4}.$

######  Example 3.2.

Every ternary Nambu-Poisson algebras is a ternary $F$-manifold.

######  Definition 3.3.

Let $(A,\cdot_{A},[\cdot,\cdot,\cdot]_{A})$ and
$(B,\cdot_{B},[\cdot,\cdot,\cdot]_{B})$ be two ternary $F$-manifold algebras.
A homomorphism between $A$ and $B$ is a linear map $f:A\rightarrow B$ such
that

(3.20) $\displaystyle f(x_{1}\cdot_{A}x_{2})$ $\displaystyle=$ $\displaystyle
f(x_{1})\cdot_{B}f(x_{2}),$ (3.21) $\displaystyle f[x_{1},x_{2},x_{3}]_{A}$
$\displaystyle=$
$\displaystyle[f(x_{1}),f(x_{2}),f(x_{3})]_{B},\quad\forall~{}x_{1},x_{2},x_{3}\in
A.$

######  Proposition 3.4.

Let $(A,\cdot_{A},[\cdot,\cdot,\cdot]_{A})$ and
$(B,\cdot_{B},[\cdot,\cdot,\cdot]_{B})$ be two ternary $F$-manifold algebras.
Then $(A\oplus B,\cdot_{A\oplus B},[\cdot,\cdot,\cdot]_{A\oplus B})$ is an
ternary $F$-manifold algebra, where the product $\cdot_{A\oplus B}$ and
bracket $[\cdot,\cdot,\cdot]_{A\oplus B}$ are given by

$\displaystyle(x_{1}+y_{1})\cdot_{A\oplus B}(x_{2}+y_{2})$ $\displaystyle=$
$\displaystyle x_{1}\cdot_{A}x_{2}+y_{1}\cdot_{B}y_{2},$
$\displaystyle{[(x_{1}+y_{1}),(x_{2}+y_{2}),(x_{3}+y_{3})]_{A\oplus B}}$
$\displaystyle=$
$\displaystyle[x_{1},x_{2},x_{3}]_{A}+[y_{1},y_{2},y_{3}]_{B}$

for all $x_{i}\in A,y_{i}\in B,\forall i=1,2,3.$

###### Proof.

It easy to check that $(A\oplus B,\cdot_{A\oplus B})$ is a commutative
associative algebra and $(A\oplus B,[\cdot,\cdot,\cdot]_{A\oplus B})$ is a
$3$-Lie algebra and for all $x_{i}\in A,y_{i}\in B,~{}~{}i=1,\cdots,4$

$\displaystyle\mathcal{L}_{A\oplus
B}(x_{1}+y_{1},x_{2}+y_{2},x_{3}+y_{3},x_{4}+y_{4})=$
$\displaystyle\mathcal{L}_{A}(x_{1},x_{2},x_{3},x_{4})+\mathcal{L}_{B}(y_{1},y_{2},y_{3},x_{4}).$

Using the above identity, we have

$\displaystyle\mathcal{L}_{A\oplus B}((x_{1}+y_{1})\cdot_{A\oplus
B}(x_{2}+y_{2}),x_{3}+y_{3},x_{4}+y_{4},x_{5}+y_{5})$ $\displaystyle=$
$\displaystyle\mathcal{L}_{A\oplus
B}(x_{1}\cdot_{A}x_{2}+y_{1}\cdot_{B}y_{2},x_{3}+y_{3},x_{4}+y_{4},x_{5}+y_{5})$
$\displaystyle=$
$\displaystyle\mathcal{L}_{A}(x_{1}\cdot_{A}x_{2},x_{3},x_{4},x_{5})+\mathcal{L}_{B}(y_{1}\cdot_{B}y_{2},y_{3},y_{4},y_{5})$
$\displaystyle=$ $\displaystyle
x_{1}\cdot_{A}\mathcal{L}_{A}(x_{2},x_{3},x_{4},x_{5})+x_{2}\cdot_{A}\mathcal{L}_{A}(x_{1},x_{3},x_{4},x_{5})$
$\displaystyle+y_{1}\cdot_{B}\mathcal{L}_{B}(y_{2},y_{3},y_{4},y_{5})+y_{2}\cdot_{B}\mathcal{L}_{B}(y_{1},y_{3},y_{4},y_{5})$
$\displaystyle=$ $\displaystyle(x_{1}+y_{1})\cdot_{A\oplus
B}\mathcal{L}_{A\oplus B}(x_{2}+y_{2},x_{3}+y_{3},x_{4}+y_{4},x_{5}+y_{5})$
$\displaystyle+(x_{2}+y_{2})\cdot_{A\oplus B}\mathcal{L}_{A\oplus
B}(x_{1}+y_{1},x_{3}+y_{3},x_{4}+y_{4},x_{5}+y_{5})$

Then, Eq. (3.18) holds. ∎

######  Proposition 3.5.

Let $(A,\cdot_{A},[\cdot,\cdot,\cdot]_{A})$ be a ternary $F$-manifold algebras
and $(B,\cdot_{B})$ be commutative associative algebra. Then $(A\otimes
B,\cdot_{A\otimes B},[\cdot,\cdot,\cdot]_{A\otimes B})$ is an ternary
$F$-manifold algebra, where the product $\cdot_{A\otimes B}$ is defined by

$\displaystyle(x_{1}\otimes y_{1})\cdot_{A\otimes B}(x_{2}\otimes
y_{2})=(x_{1}\cdot_{A}y_{1})\otimes(x_{2}\cdot_{B}y_{2}),$
$\displaystyle[x_{1}\otimes y_{1},x_{2}\otimes y_{2},x_{3}\otimes
y_{3}]_{A\otimes B}=[x_{1},x_{2},x_{3}]_{A}\otimes(y_{1}\cdot_{B}y_{2}\cdot
y_{3}),$

for all $x_{i}\in A,y_{i}\in B,\forall i=1,2,3.$

###### Proof.

It is obvious to check that $(A\otimes B,\cdot_{A\otimes B})$ is a commutative
associative algebra and $(A\otimes B,[\cdot,\cdot,\cdot]_{A\otimes B})$ is a
$3$-Lie algebra. For any $x_{i},y_{i}\in A,i=1,\cdots,4$ we have

$\displaystyle\mathcal{L}_{A\otimes B}(x_{1}\otimes y_{1},x_{2}\otimes
y_{2},x_{3}\otimes y_{3},x_{4}\otimes y_{4})$ $\displaystyle=$
$\displaystyle[x_{1}\otimes y_{1},x_{2}\otimes y_{2},(x_{3}\otimes
y_{3})\cdot_{A\otimes B}(x_{4}\otimes y_{4})]_{A\otimes B}-(x_{3}\otimes
y_{3})\cdot_{A\otimes B}[x_{1}\otimes y_{1},x_{2}\otimes y_{2},x_{4}\otimes
y_{4}]$ $\displaystyle-[x_{1}\otimes y_{1},x_{2}\otimes y_{2},x_{3}\otimes
y_{3}]\cdot_{A\otimes B}(x_{4}\otimes y_{4})$ $\displaystyle=$
$\displaystyle[x_{1}\otimes y_{1},x_{2}\otimes
y_{2},(x_{3}\cdot_{A}x_{4})\otimes(y_{3}\otimes_{B}y_{4})]_{A\otimes
B}-(x_{3}\otimes y_{3})\cdot_{A\otimes
B}([x_{1},x_{2},x_{4}]_{A}\otimes(y_{1}\cdot_{B}y_{2}\cdot_{B}y_{4}))$
$\displaystyle-([x_{1},x_{2},x_{3}]_{A}\otimes(y_{1}\cdot_{B}y_{2}\cdot_{B}y_{3}))\cdot_{A\otimes
B}(x_{4}\otimes y_{4})$ $\displaystyle=$
$\displaystyle[x_{1},x_{2},(x_{3}\cdot_{A}x_{4})]\otimes(y_{1}\cdot_{B}y_{2}\cdot_{B}y_{3}\cdot_{B}y_{4}))-(x_{3}\cdot_{A}[x_{1},x_{2},x_{4}]_{A})\otimes(y_{1}\cdot_{B}y_{2}\cdot_{B}y_{3}\cdot
y_{4})$
$\displaystyle-([x_{1},x_{2},x_{3}]_{A}\cdot_{A}x_{4}\otimes(y_{1}\cdot_{B}y_{2}\cdot_{B}y_{3}\cdot_{B}y_{4})$
$\displaystyle=$
$\displaystyle\mathcal{L}_{A}(x_{1},x_{2},x_{3},x_{4})\otimes(y_{1}\cdot_{B}y_{2}\cdot_{B}y_{3}\cdot_{B}y_{4}).$

Using the above equality, we obtain,

$\displaystyle\mathcal{L}_{A\otimes B}((x_{1}\otimes y_{1})\cdot_{A\otimes
B}(x_{2}\otimes y_{2}),(x_{3}\otimes y_{3}),(x_{4}\otimes y_{4}),(x_{5}\otimes
y_{5}))$ $\displaystyle=$ $\displaystyle\mathcal{L}_{A\otimes
B}((x_{1}\cdot_{A}x_{2})\otimes(y_{1}\cdot_{B}y_{2}),(x_{3}\otimes
y_{3}),(x_{4}\otimes y_{4}),(x_{5}\otimes y_{5}))$ $\displaystyle=$
$\displaystyle\mathcal{L}_{A}(x_{1}\cdot_{A}x_{2},x_{3},x_{4},x_{5})\otimes(y_{1}\cdot_{B}y_{2}\cdot_{B}y_{3}\cdot_{B}y_{4}\cdot_{B}y_{5})$
$\displaystyle=$
$\displaystyle(x_{1}\cdot_{A}\mathcal{L}_{A}(x_{2},x_{3},x_{4},x_{5})\otimes(y_{1}\cdot_{B}y_{2}\cdot_{B}y_{3}\cdot_{B}y_{4}\cdot_{B}y_{5})$
$\displaystyle+x_{2}\cdot_{A}\mathcal{L}_{A}(x_{1},x_{3},x_{4},x_{5})\otimes(y_{1}\cdot_{B}y_{2}\cdot_{B}y_{3}\cdot_{B}y_{4}\cdot_{B}y_{5})$
$\displaystyle=$ $\displaystyle(x_{1}\otimes y_{1})\cdot_{A\otimes
B}\mathcal{L}_{A}(x_{2},x_{3},x_{4},x_{5})\otimes(y_{2}\cdot_{B}y_{3}\cdot_{B}y_{4}\cdot_{B}y_{5})$
$\displaystyle+(x_{2}\cdot_{B}y_{2})\cdot_{A\otimes
B}\mathcal{L}_{A}(x_{1},x_{3},x_{4},x_{5})\otimes(y_{1}\cdot_{B}y_{3}\cdot_{B}y_{4}\cdot_{B}y_{5})$
$\displaystyle=$ $\displaystyle(x_{1}\otimes y_{1})\cdot_{A\otimes
B}\mathcal{L}_{A\otimes B}(x_{2}\otimes y_{2},x_{3}\otimes y_{2},x_{4}\otimes
y_{4},x_{5}\otimes y_{5})$ $\displaystyle+(x_{2}\cdot_{B}y_{2})\cdot_{A\otimes
B}\mathcal{L}_{A\otimes B}(x_{1}\otimes y_{1},x_{3}\otimes y_{3},x_{4}\otimes
y_{4},x_{5}\otimes y_{5}).$

Then, Eq. (3.18) holds and $(A\otimes B,\cdot_{A\otimes
B},[\cdot,\cdot,\cdot]_{A\otimes B})$ is an ternary $F$-manifold algebra ∎

######  Proposition 3.6.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary $F$-manifold algebra. Fixed
$a\in A$ and define a bracket $[\cdot,\cdot]_{a}:A\otimes A\to A$ by
$[x,y]_{a}=[x,a,y]$ for all $x,y\in A$. Then $(A,\cdot,[\cdot,\cdot]_{a})$ is
an $F$-manifold algebra.

###### Proof.

Thanks to [33], we have $(A,[\cdot,\cdot]_{a})$ is a Lie algebra. Now, we will
show Eq.(2.12) for any $x,y,z,w\in A$, observe that

$\displaystyle\mathcal{L}_{a}(x,y,z)$ $\displaystyle=[x,y\cdot
z]_{a}-[x,y]_{a}\cdot z-y\cdot[x,z]_{a}$ $\displaystyle=[x,a,y\cdot
z]-[x,a,y]\cdot z-y\cdot[x,a,z]$ $\displaystyle=\mathcal{L}(x,a,y,z).$

Then, we have

$\displaystyle\mathcal{L}_{a}(x\cdot
y,z,w)-x\cdot\mathcal{L}_{a}(y,z,w)+y\cdot\mathcal{L}_{a}(x,z,w)$
$\displaystyle=$ $\displaystyle\mathcal{L}(x\cdot
y,a,z,w)-x\cdot\mathcal{L}(y,a,z,w)+y\cdot\mathcal{L}(x,a,z,w)$
$\displaystyle=$ $\displaystyle 0.$

Therefore, $(A,\cdot,[\cdot,\cdot]_{a})$ is an $F$-manifold algebra. ∎

Now we generalize the result given in [39] to the notion of $F$-manifold. Let
$(A,[\cdot,\cdot])$ be a Lie algebra and $\tau:A\rightarrow\mathbb{K}$ a
linear map. For any $x_{1},x_{2},x_{3}\in A$, we define the 3-ary bracket by

(3.22)
$[x_{1},x_{2},x_{3}]_{\tau}=\tau(x_{1})\cdot[x_{2},x_{3}]-\tau(x_{2})\cdot[x_{1},x_{3}]+\tau(x_{3})\cdot[x_{1},x_{2}].$

The map $\tau$ is called trace or $[\cdot,\cdot]$-trace if
$\tau[\cdot,\cdot]:=0$.

######  Theorem 3.7.

[2] With the above notation and $\tau$ is a trace,
$(A,[\cdot,\cdot,\cdot]_{\tau})$ is $3$-Lie algebra.

######  Theorem 3.8.

Let $(A,\cdot,[\cdot,\cdot])$ be an $F$-manifold algebra and $\tau$ be a
trace. Then $(A,\cdot,[\cdot,\cdot,\cdot]_{\tau})$ is a ternary $F$-manifold
algebra if and only if

(3.23) $\tau(x_{1}\cdot
x_{2})\mathcal{L}(x_{3},x_{4},x_{5})=\tau(x_{2})x_{1}\cdot\mathcal{L}(x_{3},x_{4},x_{5})+\tau(x_{1})x_{2}\cdot\mathcal{L}(x_{3},x_{4},x_{5}).$

###### Proof.

We have $(A,\cdot)$ commutative associative algebra and
$(A,[\cdot,\cdot,\cdot]_{\tau})$ is $3$-Lie algebra. For any
$x_{i},i=1,\cdots,5$ we have

$\displaystyle\mathcal{L}_{\tau}(x_{1},x_{2},x_{3},x_{4})=$
$\displaystyle[x_{1},x_{2},x_{3}\cdot
x_{4}]_{\tau}-x_{3}\cdot[x_{1},x_{2},x_{4}]_{\tau}-[x_{1},x_{2},x_{3}]_{\tau}\cdot
x_{4}$ $\displaystyle=$ $\displaystyle\tau(x_{1})\cdot[x_{2},x_{3}\cdot
x_{4}]-\tau(x_{2})\cdot[x_{1},x_{3}\cdot x_{4}]+\tau(x_{3}\cdot
x_{4})\cdot[x_{1},x_{2}]$ $\displaystyle-
x_{3}\cdot(\tau(x_{1})\cdot[x_{2},x_{4}]-\tau(x_{2})\cdot[x_{1},x_{4}]+\tau(x_{4})\cdot[x_{1},x_{2}])$
$\displaystyle-(\tau(x_{1})\cdot[x_{2},x_{3}]-\tau(x_{2})[x_{1},x_{3}]+\tau(x_{3})\cdot[x_{1},x_{2}])\cdot
x_{4}$ $\displaystyle=$ $\displaystyle\tau(x_{1})([x_{2},x_{3}\cdot
x_{4}]-x_{3}\cdot[x_{2},x_{4}]-[x_{2},x_{3}]\cdot x_{4})$
$\displaystyle-\tau(x_{2})([x_{1},x_{3}\cdot
x_{4}]-x_{3}\cdot[x_{1},x_{4}]-[x_{1},x_{3}]\cdot x_{4})$
$\displaystyle+\tau(x_{3}\cdot
x_{4})[x_{1},x_{2}]-\tau(x_{4})x_{3}\cdot[x_{1},x_{2}]-\tau(x_{3})[x_{1},x_{2}]\cdot
x_{4}$ $\displaystyle=$
$\displaystyle\tau(x_{1})\mathcal{L}(x_{2},x_{3},x_{4})-\tau(x_{2})\mathcal{L}(x_{1},x_{3},x_{4})$
$\displaystyle+\tau(x_{3}\cdot
x_{4})[x_{1},x_{2}]-\tau(x_{4})x_{3}\cdot[x_{1},x_{2}]-\tau(x_{3})[x_{1},x_{2}]\cdot
x_{4}.$

Using the above relation, we obtain

$\displaystyle\mathcal{L}_{\tau}(x_{1}\cdot
x_{2},x_{3},x_{4},x_{5})-x_{1}\cdot\mathcal{L}_{\tau}(x_{2},x_{3},x_{4},x_{5})-x_{2}\cdot\mathcal{L}_{\tau}(x_{1},x_{3},x_{4},x_{5})$
$\displaystyle=$ $\displaystyle\tau(x_{1}\cdot
x_{2})\mathcal{L}(x_{3},x_{4},x_{5})-\tau(x_{3})\mathcal{L}(x_{1}\cdot
x_{2},x_{4},x_{5})-\tau(x_{2})x_{1}\cdot\mathcal{L}(x_{3},x_{4},x_{5})$
$\displaystyle-\tau(x_{3})x_{1}\cdot\mathcal{L}(x_{2},x_{4},x_{5}))-\tau(x_{1})x_{2}\cdot\mathcal{L}(x_{3},x_{4},x_{5})-\tau(x_{3})x_{2}\cdot\mathcal{L}(x_{1},x_{4},x_{5})$
$\displaystyle+\tau(x_{4}\cdot x_{5})[x_{1}\cdot
x_{2},x_{3}]-\tau(x_{5})x_{4}\cdot[x_{1}\cdot
x_{2},x_{3}]-\tau(x_{4})[x_{1}\cdot x_{2},x_{3}]\cdot x_{5}$
$\displaystyle-\tau(x_{4}\cdot
x_{5})x_{1}\cdot[x_{2},x_{3}]+\tau(x_{5})x_{1}\cdot
x_{4}\cdot[x_{2},x_{3}]+\tau(x_{4})x_{1}\cdot[x_{2},x_{3}]\cdot x_{5}$
$\displaystyle-\tau(x_{4}\cdot
x_{5})x_{2}\cdot[x_{1},x_{3}]+\tau(x_{5})x_{2}\cdot
x_{4}\cdot[x_{1},x_{3}]+\tau(x_{4})x_{2}\cdot[x_{1},x_{3}]\cdot x_{5}$
$\displaystyle=$ $\displaystyle\tau(x_{1}\cdot
x_{2})\mathcal{L}(x_{3},x_{4},x_{5})-\tau(x_{3})\mathcal{L}(x_{1}\cdot
x_{2},x_{4},x_{5})-\tau(x_{2})x_{1}\cdot\mathcal{L}(x_{3},x_{4},x_{5})$
$\displaystyle-\tau(x_{3})x_{1}\cdot\mathcal{L}(x_{2},x_{4},x_{5}))-\tau(x_{1})x_{2}\cdot\mathcal{L}(x_{3},x_{4},x_{5})-\tau(x_{3})x_{2}\cdot\mathcal{L}(x_{1},x_{4},x_{5}))$
$\displaystyle+\tau(x_{4}\cdot
x_{5})\mathcal{L}(x_{1},x_{2},x_{3})-\tau(x_{5})x_{4}\cdot\mathcal{L}(x_{1},x_{2},x_{3})-\tau(x_{4})\mathcal{L}(x_{1},x_{2},x_{3})\cdot
x_{5}$ $\displaystyle=$ $\displaystyle\tau(x_{1}\cdot
x_{2})\mathcal{L}(x_{3},x_{4},x_{5})-\tau(x_{1})x_{2}\cdot\mathcal{L}(x_{3},x_{4},x_{5})-\tau(x_{2})x_{1}\cdot\mathcal{L}(x_{3},x_{4},x_{5})$
$\displaystyle+\tau(x_{4}\cdot
x_{5})\mathcal{L}(x_{1},x_{2},x_{3})-\tau(x_{4})\mathcal{L}(x_{1},x_{2},x_{3})\cdot
x_{5}-\tau(x_{5})x_{4}\cdot\mathcal{L}(x_{1},x_{2},x_{3}).$

Then $(A,\cdot,[\cdot,\cdot,\cdot]_{\tau})$ is a ternary $F$-manifold algebra
if and only if Eq. (3.23) holds. ∎

######  Corollary 3.9.

Let $(A,\cdot,[\cdot,\cdot])$ be a $F$-manifold algebra and $\tau$ a trace. If
$(A,\cdot)$ is unital with unit $1_{A}$ and

$\tau(x\cdot y)1_{A}=\tau(x)y+\tau(y)x.$

Then $(A,\cdot,[\cdot,\cdot,\cdot]_{\tau})$ is ternary $F$-manifold algebra.

##  4\. Representations of ternary $F$-manifold algebras

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary $F$-manifold algebra,
$(V;\rho)$ be a representation of the $3$-Lie algebra
$(A,[\cdot,\cdot,\cdot])$ and $(V;\mu)$ be a representation of the commutative
associative algebra $(A,\cdot)$. Define the three linear maps
$\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3}:A\otimes A\otimes A\otimes
V\to V$ given by

(4.24) $\displaystyle\mathcal{L}_{1}(x,y,z,u)$ $\displaystyle=$
$\displaystyle\rho(x,y)\mu(z)(u)-\mu(z)\rho(x,y)(u)-\mu([x,y,z])u,$ (4.25)
$\displaystyle\mathcal{L}_{2}(x,y,z,u)$ $\displaystyle=$
$\displaystyle\mu(z)\rho(x,y)(u)+\mu(y)\rho(x,z)(u)-\rho(x,y\cdot z)u,$ (4.26)
$\displaystyle\mathcal{L}_{3}(x,y,z,u)$ $\displaystyle=$
$\displaystyle\rho(x,y)\mu(z)(u)+\rho(x,z)\mu(y)(u)-\rho(x,y\cdot z)u$

for all $x,y\in A$ and $u\in V$. Note that, if we consider $V=A$, then
$\mathcal{L}_{1}=\mathcal{L}$ and $\mathcal{L}_{2}=\mathcal{L}\sigma$, where
$\sigma(x,y,z)=(z,x,y)$.

######  Definition 4.1.

With the above notations, the tuple $(V;\rho,\mu)$ is a representation of $A$
if the following conditions hold:

(4.27) $\displaystyle\mathcal{L}_{1}(x_{1}\cdot
x_{2},x_{3},x_{4},u)=\mu(x_{1})\mathcal{L}_{1}(x_{2},x_{3},x_{4},u)+\mu(x_{2})\mathcal{L}_{1}(x_{1},x_{3},x_{4},u),$
(4.28) $\displaystyle\mathcal{L}_{2}(x_{1}\cdot
x_{2},x_{3},x_{4},u)=\mu(x_{1})\mathcal{L}_{2}(x_{2},x_{3},x_{4},u)+\mu(x_{2})\mathcal{L}_{2}(x_{1},x_{3},x_{4},u),$
(4.29)
$\displaystyle\mu(\mathcal{L}(x_{1},x_{2},x_{3},x_{4}))u=\mathcal{L}_{2}(x_{2},x_{3},x_{4},\mu(x_{1})(u))-\mu(x_{1})\mathcal{L}_{2}(x_{2},x_{3},x_{4},u),$

for all $x_{1},x_{2},x_{3},x_{4}\in A$ and $u\in V$.

######  Example 4.2.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary $F$-manifold algebra. Then
$(A,\mathrm{ad},L)$ is a representation of $A$ called the adjoint
representation.

######  Example 4.3.

Let $(V;\rho,\mu)$ be a representation of a ternary Nambu-Poisson algebra
$(P,\cdot,[\cdot,\cdot,\cdot])$, i.e. $(V;\rho,\phi)$ is a representation of
the $3$-Lie algebra $(P,[\cdot,\cdot,\cdot])$ and $(V;\mu)$ is a
representation of the commutative associative algebra $(P,\cdot)$ satisfying

$\displaystyle\mathcal{L}_{1}(x_{1},x_{2},x_{3},u)=\mathcal{L}_{2}(x_{1},x_{2},x_{3},u)=0,\quad\forall~{}x_{1},x_{2},x_{3}\in
P,u\in V.$

Then $(V;\rho,\mu)$ is also a representation of the ternary $F$-manifold
algebra given by this ternary Nambu-Poisson algebra $P$.

######  Example 4.4.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary $F$-manifold algebra and
$(V;\rho,\mu)$ be a representation of $A$. Fixed $a\in A$, we define a linear
map $\rho_{a}:A\to End(V)$ defined by $\rho_{a}(x)v=\rho(x,a)v$. Then,
$(V;\rho_{a},\mu)$ be a representation of the $F$-manifold algebra defined in
Proposition 3.6.

######  Proposition 4.5.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary $F$-manifold algebra. Then
$(V;\rho,\mu)$ is a representation of $A$ if and only if $(A\oplus
V,\cdot_{\mu},[\cdot,\cdot,\cdot]_{\rho})$ is a ternary $F$-manifold algebra,
where $(A\oplus V,\cdot_{\mu})$ is the semi-direct product commutative
associative algebra $A\ltimes_{\mu}V$, i.e.

$(x_{1}+v_{1})\cdot_{\mu}(x_{2}+v_{2})=x_{1}\cdot
x_{2}+\mu(x_{1})v_{2}+\mu(x_{2})v_{1},\quad$

and $(A\oplus V,[\cdot,\cdot,\cdot]_{\rho})$ is the semi-direct product
$3$-Lie algebra $A\ltimes_{\rho}V$, i.e.

$[x_{1}+v_{1},x_{2}+v_{2},x_{3}+v_{3}]_{\rho}=[x_{1},x_{2},x_{3}]+\rho(x_{1},x_{2})v_{3}-\rho(x_{1},x_{3})v_{2}+\rho(x_{2},x_{3})v_{1},$

$forall~{}x_{1},x_{2},x_{3}\in A,~{}v_{1},v_{2},v_{3}\in V.$

###### Proof.

It is obvious to check that $(A\oplus V,\cdot_{\mu})$ is commutative
associative algebra and $(A\oplus V,[\cdot,\cdot,\cdot]_{\rho})$ is a $3$-Lie
algebra. Observe that for any $x_{i}\in A$ and $u_{i}\in V$, $i=1,\cdots,4$

$\displaystyle\mathcal{L}_{A\oplus
V}(x_{1}+u_{1},x_{2}+u_{2},x_{3}+u_{3},x_{4}+u_{4})=$
$\displaystyle\mathcal{L}(x_{1},x_{2},x_{3},x_{4})+\mathcal{L}_{1}(x_{1},x_{2},x_{3},u_{4})+\mathcal{L}_{1}(x_{1},x_{2},x_{4},u_{3})$
$\displaystyle+\mathcal{L}_{2}(x_{1},x_{3},x_{4},u_{2})-\mathcal{L}_{2}(x_{2},x_{3},x_{4},u_{1}).$

So, we have

$\displaystyle\mathcal{L}_{A\oplus V}((x_{1}+u_{1})\cdot_{A\oplus
V}(x_{2}+u_{2}),x_{3}+u_{3},x_{4}+u_{4},x_{5}+u_{5})$
$\displaystyle-(x_{1}+u_{1})\cdot_{A\oplus V}\mathcal{L}_{A\oplus
V}(x_{2}+u_{2},x_{3}+u_{3},x_{4}+u_{4},x_{5}+u_{5})$
$\displaystyle-(x_{2}+u_{2})\cdot_{A\oplus V}\mathcal{L}_{A\oplus
V}(x_{1}+u_{1},x_{3}+u_{3},x_{4}+u_{4},x_{5}+u_{5})$ $\displaystyle=$
$\displaystyle\mathcal{L}_{A\oplus V}(x_{1}\cdot
x_{2}+\mu(x_{1})u_{2}+\mu(x_{2})u_{1},x_{3}+u_{3},x_{4}+u_{4},x_{5}+u_{5})$
$\displaystyle-(x_{1}+u_{1})\cdot_{A\oplus
V}\big{(}\mathcal{L}(x_{2},x_{3},x_{4},x_{5})+\mathcal{L}_{1}(x_{2},x_{3},x_{4},u_{5})$
$\displaystyle+\mathcal{L}_{1}(x_{2},x_{3},x_{5},u_{4})+\mathcal{L}_{2}(x_{2},x_{4},x_{5},u_{3})-\mathcal{L}_{2}(x_{3},x_{4},x_{5},u_{2})\big{)}$
$\displaystyle-(x_{2}+u_{2})\cdot_{A\oplus
V}\big{(}\mathcal{L}(x_{1},x_{3},x_{4},x_{5})+\mathcal{L}_{1}(x_{1},x_{3},x_{4},u_{5})$
$\displaystyle+\mathcal{L}_{1}(x_{1},x_{3},x_{5},u_{4})+\mathcal{L}_{2}(x_{1},x_{4},x_{5},u_{3})-\mathcal{L}_{2}(x_{3},x_{4},x_{5},u_{1})\big{)}$
$\displaystyle=$ $\displaystyle\mathcal{L}(x_{1}\cdot
x_{2},x_{3},x_{4},x_{5})+\mathcal{L}_{1}(x_{1}\cdot
x_{2},x_{3},x_{4},u_{5})+\mathcal{L}_{1}(x_{1}\cdot x_{2},x_{3},x_{5},u_{4})$
$\displaystyle+\mathcal{L}_{2}(x_{1}\cdot
x_{2},x_{4},x_{5},u_{3})-\mathcal{L}_{2}(x_{3},x_{4},x_{5},\mu(x_{1})u_{2})-\mathcal{L}_{2}(x_{3},x_{4},x_{5},\mu(x_{2})u_{1})$
$\displaystyle-
x_{1}\cdot\mathcal{L}(x_{2},x_{3},x_{4},x_{5})-\mu(\mathcal{L}(x_{2},x_{3},x_{4},x_{5}))u_{1}-\mu(x_{1})\mathcal{L}_{1}(x_{2},x_{3},x_{4},u_{5})$
$\displaystyle-\mu(x_{1})\mathcal{L}_{1}(x_{2},x_{3},x_{5},u_{4})-\mu(x_{1})\mathcal{L}_{2}(x_{2},x_{4},x_{5},u_{3})+\mu(x_{1})\mathcal{L}_{2}(x_{3},x_{4},x_{5},u_{2})$
$\displaystyle-
x_{2}\cdot\mathcal{L}(x_{1},x_{3},x_{4},x_{5})-\mu(\mathcal{L}(x_{1},x_{3},x_{4},x_{5}))u_{2}-\mu(x_{2})\mathcal{L}_{1}(x_{1},x_{3},x_{4},u_{5})$
$\displaystyle-\mu(x_{2})\mathcal{L}_{1}(x_{1},x_{3},x_{5},u_{4})-\mu(x_{2})\mathcal{L}_{2}(x_{1},x_{4},x_{5},u_{3})+\mu(x_{2})\mathcal{L}_{2}(x_{3},x_{4},x_{5},u_{1}).$

Then, $(V;\rho,\mu)$ is a representation of $A$ if and only if $(A\oplus
V,\cdot_{\mu},[\cdot,\cdot,\cdot]_{\rho})$ is a ternary $F$-manifold algebra.
∎

Let $(V;\rho,\mu)$ be a representation of a ternary Nambu-Poisson algebra
$(P,\cdot_{P},\\{\cdot,\cdot,\cdot\\}_{P})$. Then,
$(V^{*};\rho^{\ast},\mu^{\ast})$ is also a representation of $P$. But ternary
$F$-manifold algebras do not have this property. In fact, we have

######  Lemma 4.6.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary $F$-manifold algebra and
$(V;\rho,\mu)$ be a representation of $A$. Define the two linear maps
$\mathcal{L}_{1}^{\ast},\mathcal{L}_{2}^{\ast}:A\otimes A\otimes A\otimes
V^{*}\to V^{*}$ given by

$\displaystyle\mathcal{L}_{1}^{\ast}(x,y,z,\xi)$ $\displaystyle=$
$\displaystyle-\rho^{\ast}(x,y)\mu^{\ast}(z)(\xi)+\mu^{\ast}(z)\rho^{\ast}(x,y)(\xi)+\mu^{\ast}([x,y,z])(\xi),$
$\displaystyle\mathcal{L}_{2}^{\ast}(x,y,z,\xi)$ $\displaystyle=$
$\displaystyle-\mu^{\ast}(x)\rho^{\ast}(y,z)(\xi)-\mu^{\ast}(y)\rho^{\ast}(x,z)(\xi)+\rho^{\ast}(x,y\cdot
z)(\xi)$

for all $x,y,z\in A$ and $\xi\in V^{*}$. Then we have

$\langle\mathcal{L}_{1}^{\ast}(x,y,z,\xi),u\rangle=\langle\xi,\mathcal{L}_{1}(x,y,z,u)\rangle$

and

$\langle\mathcal{L}_{2}^{\ast}(x,y,z,\xi),u\rangle=-\langle\xi,\mathcal{L}_{3}(x,y,z,u)\rangle.$

for all $x,y,z\in A$ and $\xi\in V^{\ast}$.

###### Proof.

Let $x,y,z\in A$ and $\xi\in V^{\ast}$, we have

$\displaystyle\langle\mathcal{L}_{1}^{\ast}(x,y,z,\xi),u\rangle$
$\displaystyle=\langle-\rho^{\ast}(x,y)\mu^{\ast}(z)(\xi)+\mu^{\ast}(z)\rho^{\ast}(x,y)(\xi)+\mu^{\ast}([x,y,z])(\xi),u\rangle$
$\displaystyle=-\langle\rho^{\ast}(x,y)\mu^{\ast}(z)(\xi),u\rangle+\langle\mu^{\ast}(z)\rho^{\ast}(x,y)(\xi),u\rangle+\langle\mu^{\ast}([x,y,z])(\xi),u\rangle$
$\displaystyle=-\langle\xi,\mu(z)\rho(x,y)u\rangle+\langle\xi,\rho(x,y)\mu(z)(u)\rangle-\langle\xi,\mu([x,y,z])(u)\rangle$
$\displaystyle=\langle\xi,\mathcal{L}_{1}(x,y,z,u)\rangle.$

Similarly, we obtain

$\langle\mathcal{L}_{2}^{\ast}(x,y,z,\xi),u\rangle=-\langle\xi,\mathcal{L}_{3}(x,y,z,u)\rangle.$

∎

######  Proposition 4.7.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary-$F$-manifold algebra. If the
tuple $(V;\rho,\mu)$ is representation of $A$ satisfying the following axioms

(4.30) $\displaystyle\mathcal{L}_{1}(x\cdot
y,z,t,u)=\mathcal{L}_{1}(y,z,t,\mu(x)u)+\mathcal{L}_{1}(x,z,t,\mu(y)u),$
(4.31) $\displaystyle\mathcal{L}_{3}(x\cdot
y,z,t,u)=-\mathcal{L}_{3}(y,z,t,\mu(x)u)-\mathcal{L}_{3}(x,z,t,\mu(y)u),$
(4.32)
$\displaystyle\mu(\mathcal{L}(x,y,z,t))u=\mathcal{L}_{3}(y,z,t,\mu(x)u)-\mu(x)\mathcal{L}_{3}(y,z,t,u)$

for any $x,y,z,t\in A$ and $u\in V$. Then $(V^{*};\rho^{\ast},-\mu^{\ast})$ is
a representation of $A$ called the dual representation.

###### Proof.

For any $x,y,z,t\in A,v\in u,\xi\in V^{*}$, we have

$\displaystyle\langle\mathcal{L}_{1}^{\ast}(x\cdot
y,z,t,\xi)+\mu^{\ast}(x)\mathcal{L}_{1}^{\ast}(y,z,t,\xi)+\mu^{\ast}(y)\mathcal{L}_{1}^{\ast}(x,z,t,\xi),u\rangle$
$\displaystyle=$ $\displaystyle\langle\xi,\mathcal{L}_{1}(x\cdot
y,z,t,u)-\mathcal{L}_{1}(y,z,t,\mu(x)u)-\mathcal{L}_{1}(x,z,t,\mu(y)u)\rangle,$
$\displaystyle\langle\mathcal{L}_{2}^{\ast}(x\cdot
y,z,t,\xi)+\mu^{\ast}(x)\mathcal{L}_{2}^{\ast}(y,z,t,\xi)+\mu^{\ast}(y)\mathcal{L}_{2}^{\ast}(x,z,t,\xi),u\rangle$
$\displaystyle=$ $\displaystyle\langle\xi,-\mathcal{L}_{3}(x\cdot
y,z,t,u)+\mathcal{L}_{3}(y,z,t,\mu(x)u)+\mathcal{L}_{1}(x,z,t,\mu(y)u)\rangle$

and

$\displaystyle\langle-\mu^{\ast}(\mathcal{L}(x,y,z,t)\xi+\mathcal{L}_{2}^{\ast}(y,z,t,\mu^{\ast}(x)\xi-\mu^{\ast}(x)\mathcal{L}_{2}^{\ast}(y,z,t,\xi),u\rangle$
$\displaystyle=$
$\displaystyle\langle\xi,\mu(\mathcal{L}(x,y,z,t))u+\mu(x)\mathcal{L}_{3}(y,z,t,u)-\mathcal{L}_{3}(y,z,t,\mu(x)u)\rangle.$

According to Eqs. (4.30)– (4.32) and Definition 4.1, the conclusion follows
immediately. ∎

######  Corollary 4.8.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary-$F$-manifold algebra such
that the following relations hold:

(4.33) $\displaystyle\mathcal{L}(x\cdot y,z,t,u)=\mathcal{L}(y,z,t,x\cdot
u)+\mathcal{L}(x,z,t,y\cdot u),$ (4.34) $\displaystyle\mathcal{K}(x\cdot
y,z,t\cdot u)=-\mathcal{K}(x,z,t,y\cdot u)-\mathcal{K}(y,z,t,x\cdot u),$
(4.35) $\displaystyle\mathcal{L}(x,y,z,t)\cdot u=\mathcal{K}(y,z,t,x\cdot
u)-x\cdot\mathcal{K}(y,z,t,u)$

for all $x,y,z,t,u\in A$, where $\mathcal{K}:\otimes^{4}A\rightarrow A$ is
defined by

$\mathcal{K}(x,y,z,u)=\circlearrowleft_{y,z,u}[x,y,z\cdot u].$

Then $(A^{*};\mathrm{ad}^{\ast},-L^{\ast})$ is a representation of $A$ called
the coadjoint representation.

######  Definition 4.9.

A coherence ternary-$F$-manifold algebra is a ternary-$F$-manifold algebra
such that Eqs. (4.33)–(4.35) hold.

##  5\. Ternary pre-$F$-manifold algebras

In the following, we give a notion of ternary pre-$F$-manifold algebras which
is a generalization of pre-$F$-manifold algebra (see [31]) and give some
results.

###  5.1. Definition and elementary results

######  Definition 5.1.

A ternary pre-$F$-manifold algebra is a triple
$(A,\diamond,\\{\cdot,\cdot,\cdot\\})$, where $(A,\diamond)$ is a Zinbiel
algebra and $(A,\\{\cdot,\cdot,\cdot\\})$ is a $3$-pre-Lie algebra, such that:

(5.36) $\displaystyle F_{1}(x_{1}\cdot x_{2},x_{3},x_{4},x_{5})=x_{1}\diamond
F_{1}(x_{2},x_{3},x_{4},x_{5})+x_{2}\diamond F_{1}(x_{1},x_{3},x_{4},x_{5}),$
(5.37) $\displaystyle F_{2}(x_{1}\cdot x_{2},x_{3},x_{4},x_{5})=x_{1}\diamond
F_{1}(x_{2},x_{3},x_{4},x_{5})+x_{2}\diamond F_{1}(x_{1},x_{3},x_{4},x_{5}),$
(5.38) $\displaystyle\mathcal{L}(x_{1},x_{2},x_{3},x_{4})\diamond
x_{5}=F_{2}(x_{2},x_{3},x_{4},x_{1}\diamond x_{5})-x_{1}\diamond
F_{2}(x_{2},x_{3},x_{4},x_{5})$

where $F_{1},F_{2},\mathcal{L}:\otimes^{4}A\longrightarrow A$ are defined by

(5.39) $\displaystyle F_{1}(x_{1},x_{2},x_{3},x_{4})$ $\displaystyle=$
$\displaystyle\\{x_{1},x_{2},x_{3}\diamond
x_{4}\\}-x_{3}\diamond\\{x_{1},x_{2},x_{4}\\}-[x_{1},x_{2},x_{3}]\diamond
x_{4},$ (5.40) $\displaystyle F_{2}(x_{1},x_{2},x_{3},x_{4})$ $\displaystyle=$
$\displaystyle
x_{3}\diamond\\{x_{1},x_{2},x_{4}\\}+x_{2}\diamond\\{x_{1},x_{3},x_{4}\\}-\\{x_{1},x_{2}\cdot
x_{3},x_{4}\\},$ (5.41) $\displaystyle\mathcal{L}(x_{1},x_{2},x_{3},x_{4})$
$\displaystyle=$ $\displaystyle[x_{1},x_{2},x_{3}\cdot
x_{4}]-x_{3}\cdot[x_{1},x_{2},x_{4}]-[x_{1},x_{2},x_{3}]\cdot x_{4}$

and the operation $\cdot$ and the bracket $[\cdot,\cdot,\cdot]$ are defined by

(5.42) $x\cdot y=x\diamond y+y\diamond
x,\quad[x,y,z]=\circlearrowleft_{x,y,z}\\{x,y,z\\},$

for all $x,y,z,\in A$.

If $F_{1}=F_{2}=0$ in the above definition, then we obtain a particular case
of ternary pre-$F$-manifold algebras. It will be called ternary pre-Nambu-
Poisson algebra since it can be obtained by splitting of a ternary Nambu-
Poisson algebra introduced in [39].

######  Definition 5.2.

A ternary pre-Nambu-Poisson algebra is a tuple
$(A,\diamond,\\{\cdot,\cdot,\cdot\\})$, where $(A,\diamond)$ is a Zinbiel
algebra and $(A,\\{\cdot,\cdot,\cdot\\})$ is a $3$-pre-Lie algebra, such that:

$\displaystyle\\{x_{1},x_{2},x_{3}\diamond x_{4}\\}$ $\displaystyle=$
$\displaystyle
x_{3}\diamond\\{x_{1},x_{2},x_{4}\\}+[x_{1},x_{2},x_{3}]\diamond x_{4},$
$\displaystyle\\{x_{1},x_{2}\cdot x_{3},x_{4}\\}$ $\displaystyle=$
$\displaystyle
x_{3}\diamond\\{x_{1},x_{2},x_{4}\\}+x_{2}\diamond\\{x_{1},x_{3},x_{4}\\}$

for all $x_{1},x_{2},x_{3},x_{4}\in A.$

######  Theorem 5.3.

Let $(A,\diamond,\\{\cdot,\cdot,\cdot\\})$ be a ternary pre-$F$-manifold
algebra. Then $(A,\cdot,[\cdot,\cdot,\cdot])$ is a ternary $F$-manifold
algebra, where the operation $\cdot$ and bracket $[\cdot,\cdot,\cdot]$ are
given by Eq. (5.42), which is called the sub-adjacent ternary $F$-manifold
algebra of $(A,\diamond,\\{\cdot,\cdot,\cdot\\})$ and denoted by $A^{c}$.

###### Proof.

Thanks to Lemma 2.2 and Proposition 2.9, we have $(A,\cdot)$ is a commutative
associative algebra and $(A,[\cdot,\cdot,\cdot])$ is a $3$-Lie algebra. Using
Eqs. (5.39)-(5.41), we have

$\displaystyle\mathcal{L}(x_{1},x_{2},x_{3},x_{4})$ $\displaystyle=$
$\displaystyle[x_{1},x_{2},x_{3}\cdot
x_{4}]-x_{3}\cdot[x_{1},x_{2},x_{4}]-[x_{1},x_{2},x_{3}]\cdot x_{4}$
$\displaystyle=$ $\displaystyle\\{x_{1},x_{2},x_{3}\cdot
x_{4}\\}+\\{x_{2},x_{3}\cdot x_{4},x_{1}\\}+\\{x_{3}\cdot
x_{4},x_{1},x_{2},\\}$ $\displaystyle-
x_{3}\diamond[x_{1},x_{2},x_{4}]-[x_{1},x_{2},x_{4}]\diamond
x_{3}-[x_{1},x_{2},x_{3}]\diamond x_{4}-x_{4}\diamond[x_{1},x_{2},x_{3}]$
$\displaystyle-x_{3}\diamond[x_{1},x_{2},x_{4}]-[x_{1},x_{2},x_{4}]\diamond
x_{3}-[x_{1},x_{2},x_{3}]\diamond x_{4}-x_{4}\diamond[x_{1},x_{2},x_{3}]$
$\displaystyle=$ $\displaystyle\\{x_{1},x_{2},x_{3}\diamond
x_{4}\\}+\\{x_{1},x_{2},x_{4}\diamond x_{3}\\}+\\{x_{2},x_{3}\cdot
x_{4},x_{1}\\}-\\{x_{1},x_{3}\cdot x_{4},x_{2}\\}$ $\displaystyle-
x_{3}\diamond\\{x_{1},x_{2},x_{4}\\}-x_{3}\diamond\\{x_{2},x_{4},x_{1}\\}-x_{3}\diamond\\{x_{4},x_{1},x_{2}\\}$
$\displaystyle-
x_{4}\diamond\\{x_{1},x_{2},x_{3}\\}-x_{4}\diamond\\{x_{2},x_{3},x_{1}\\}-x_{4}\diamond\\{x_{3},x_{1},x_{2}\\}$
$\displaystyle-[x_{1},x_{2},x_{4}]\diamond x_{3}-[x_{1},x_{2},x_{3}]\diamond
x_{4}$ $\displaystyle=$ $\displaystyle
F_{1}(x_{1},x_{2},x_{3},x_{4})+F_{1}(x_{1},x_{2},x_{4},x_{3})+F_{2}(x_{1},x_{3},x_{4},x_{2})-F_{2}(x_{2},x_{3},x_{4},x_{1}).$

According to Eqs.(5.36), (5.37) and (5.38), we get

$\displaystyle\mathcal{L}(x_{1}\cdot
x_{2},x_{3},x_{4},x_{5})-x_{1}\cdot\mathcal{L}(x_{2},x_{3},x_{4},x_{5})-x_{2}\cdot\mathcal{L}(x_{1},x_{3},x_{4},x_{5})$
$\displaystyle=$ $\displaystyle F_{1}(x_{1}\cdot
x_{2},x_{3},x_{4},x_{5})+F_{1}(x_{1}\cdot
x_{2},x_{3},x_{5},x_{4})+F_{2}(x_{1}\cdot
x_{2},x_{4},x_{5},x_{3})-F_{2}(x_{3},x_{4},x_{5},x_{1}\cdot x_{2})$
$\displaystyle-
x_{1}\diamond\mathcal{L}(x_{2},x_{3},x_{4},x_{5})-\mathcal{L}(x_{2},x_{3},x_{4},x_{5})\diamond
x_{1}-x_{2}\diamond\mathcal{L}(x_{1},x_{3},x_{4},x_{5})-\mathcal{L}(x_{1},x_{3},x_{4},x_{5})\diamond
x_{2}$ $\displaystyle=$ $\displaystyle F_{1}(x_{1}\cdot
x_{2},x_{3},x_{4},x_{5})+F_{1}(x_{1}\cdot
x_{2},x_{3},x_{5},x_{4})+F_{2}(x_{1}\cdot x_{2},x_{4},x_{5},x_{3})$
$\displaystyle-F_{2}(x_{3},x_{4},x_{5},x_{1}\diamond
x_{2})-F_{2}(x_{3},x_{4},x_{5},x_{2}\diamond x_{1})-x_{1}\diamond
F_{1}(x_{2},x_{3},x_{4},x_{5})-x_{1}\diamond F_{1}(x_{2},x_{3},x_{5},x_{4})$
$\displaystyle-x_{1}\diamond F_{2}(x_{2},x_{4},x_{5},x_{3})+x_{1}\diamond
F_{2}(x_{3},x_{4},x_{5},x_{2})-\mathcal{L}(x_{2},x_{3},x_{4},x_{5})\diamond
x_{1}-x_{2}\diamond F_{1}(x_{1},x_{3},x_{4},x_{5})$ $\displaystyle-
x_{2}\diamond F_{1}(x_{1},x_{3},x_{5},x_{4})-x_{2}\diamond
F_{2}(x_{1},x_{4},x_{5},x_{3})+x_{2}\diamond
F_{2}(x_{3},x_{4},x_{5},x_{1})-\mathcal{L}(x_{1},x_{3},x_{4},x_{5})\diamond
x_{2}$ $\displaystyle=$ $\displaystyle 0.$

Therefore, $(A,\cdot,[\cdot,\cdot,\cdot])$ is a ternary $F$-manifold algebra.
∎

The following observation is a direct consequence of the above Theorem.

######  Corollary 5.4.

Let $(A,\diamond,\\{\cdot,\cdot,\cdot\\})$ be a ternary pre-Nambu-Poisson
algebra. Then $(A,\cdot,[\cdot,\cdot,\cdot])$ is a ternary Nambu-Poisson
algebra, where the operation $\cdot$ and bracket $[\cdot,\cdot,\cdot]$ are
given by Eq. (5.42).

######  Proposition 5.5.

The tuple $(A;\mathbb{L},L_{\diamond})$ is a representation of the sub-
adjacent ternary-$F$-manifold algebras $A^{c}$, where $\mathbb{L}$ and
$L_{\diamond}$ are given by Eqs. (2.11) and (2.3), respectively.

###### Proof.

Thanks to Lemma 2.2 and Lemma 2.10, $(A;L_{\diamond})$ is a representation of
the commutative associative algebra $(A,\cdot)$ as well as $(A;\mathbb{L})$ is
a representation of the sub-adjacent $3$-Lie algebra $A^{c}$.

According to Eqs. (5.36)-(5.38) we can easly check Eqs. (4.27)-(4.29).
Therefore, $(A;\mathbb{L},L_{\diamond})$ is a representation of the sub-
adjacent ternary-$F$-manifold algebra $A^{c}$. ∎

###  5.2. Relative Rota-Baxter operators on ternary-$F$-manifold algebras

The notion of relative Rota-Baxter operators (called also
$\mathcal{O}$-operators) was first given for Lie algebras by Kupershmidt in
[29] as a natural generalization of the classical Yang-Baxter equation and
then defined by analogy in other various (associative, alternative, Jordan,
…). In [4], authors extended this notion to $3$-Lie algebras.

A linear map $T:V\longrightarrow A$ is called a relative Rota-Baxter operator
on a commutative associative algebra $(A,\cdot)$ with respect to a
representation $(V;\mu)$ if $T$ satisfies

(5.43) $\displaystyle Tu\cdot Tv$
$\displaystyle=T(\mu(Tu)v+\mu(Tv)u),\quad\forall~{}u,v\in V.$

In particular, a Relative Rota-Baxter operator on a commutative associative
algebra $(A,\cdot)$ with respect to the adjoint representation is called a
Rota-Baxter operator of weight zero or briefly a Rota-Baxter operator on $A$.

It is obvious to obtain the following result.

######  Lemma 5.6.

Let $(A,\cdot)$ be a commutative associative algebra and $(V;\mu)$ a
representation. Let $T:V\rightarrow A$ be a relative Rota-Baxter operator on
$(A,\cdot)$ with respect to $(V;\mu)$. Then there exists a Zinbiel algebra
structure on $V$ given by

$u\diamond v=\mu(Tu)v,\quad\forall~{}u,v\in V.$

A linear map $T:V\longrightarrow A$ is called a relative Rota-Baxter operator
on a $3$-Lie algebra $(A,[\cdot,\cdot,\cdot])$ with respect to a
representation $(V;\rho)$ if $T$ satisfies

(5.44) $\displaystyle[Tu,Tv,Tw]$
$\displaystyle=T\Big{(}\rho(Tu,Tv)w+\rho(Tv,Tw)u+\rho(Tw,Tu)v\Big{)},\quad\forall~{}u,v,w\in
V.$

In particular, a relative Rota-Baxter operator on a $3$-Lie algebra
$(A,[\cdot,\cdot,\cdot])$ with respect to the adjoint representation $ad$ is
called a Rota-Baxter operator of weight zero or briefly a Rota-Baxter operator
on $A$.

######  Lemma 5.7.

([4]) Let $T:V\to A$ be a relative Rota-Baxter operator on a $3$-Lie algebra
$(A,[\cdot,\cdot,\cdot])$ with respect to a representation $(V;\rho)$. Define
the bracket $\\{\cdot,\cdot,\cdot\\}$ on $V$ by

(5.45) $\\{u,v,w\\}=\rho(Tu,Tv)w,\quad\forall~{}u,v,w\in V.$

Then $(V,\\{\cdot,\cdot,\cdot\\})$ is a $3$-pre-Lie algebra.

Let $(V;\rho,\mu)$ be a representation of a ternary-$F$-manifold algebra
$(A,\cdot,[\cdot,\cdot,\cdot])$.

######  Definition 5.8.

A linear operator $T:V\longrightarrow A$ is called a relative Rota-Baxter
operator on $(A,\cdot,[\cdot,\cdot,\cdot])$ if $T$ is both a relative Rota-
Baxter operator on the commutative associative algebra $(A,\cdot)$ and a
relative Rota-Baxter operator on the $3$-Lie algebra
$(A,[\cdot,\cdot,\cdot])$.

In particular, a linear operator $\mathcal{R}:A\longrightarrow A$ is called a
Rota-Baxter operator of weight zero or briefly a Rota-Baxter operator on $A$,
if $\mathcal{R}$ is both a Rota-Baxter operator on the commutative associative
algebra $(A,\cdot)$ and a Rota-Baxter operator on the $3$-Lie algebra
$(A,[\cdot,\cdot,\cdot])$.

In the following result, we give the construction of ternary pre-$F$-manifold
algebra using a relative Rota-Baxter operator on a ternary-$F$-manifold
algebra. In other words, we will give a dendrification of a
ternary-$F$-manifold algebra via relative Rota-Baxter operators.

######  Theorem 5.9.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary $F$-manifold algebra and
$T:V\longrightarrow A$ be a relative Rota-Baxter operator on $A$ with respect
to the representation $(V;\rho,\mu)$. Define two operations $\diamond$ and
$\\{\cdot,\cdot,\cdot\\}$ on $V$ as follow:

$u\diamond v=\mu(Tu)v,\quad\\{u,v,w\\}=\rho(Tu,Tv)w,\ \forall\ u,v,w\in V.$

Then $(V,\diamond,\\{\cdot,\cdot,\cdot\\})$ is a ternary pre-$F$-manifold
algebra and $T$ is a homomorphism from $V^{c}$ to
$(A,\cdot,[\cdot,\cdot,\cdot])$. Moreover,

###### Proof.

Since $T$ is a relative Rota-Baxter operator on the commutative associative
algebra $(A,\cdot)$ as well as a relative Rota-Baxter operator on the $3$-Lie
algebra $(A,[\cdot,\cdot,\cdot])$ with respect to the representations
$(V;\mu)$ and $(V;\rho)$ respectively. Using Lemma 5.6 and Lemma 5.7, we
deduce that $(V,\diamond)$ is a Zinbiel algebra and
$(V,\\{\cdot,\cdot,\cdot\\})$ is a $3$-pre-Lie algebra. Put, for any $u,v,w\in
V$,

$[u,v,w]_{T}:=\circlearrowleft_{u,v,w}\\{u,v,w\\}=\rho(Tu,Tv)w+\rho(Tv,Tw)u+\rho(Tw,Tu)v$

and

$u\cdot_{T}v:=u\diamond v+v\diamond u=\mu(Tu)v+\mu(Tv)u.$

By these facts and Eq.(4.27), for $v_{1},v_{2},v_{3},v_{4},v_{5}\in V$, one
has

$\displaystyle F_{1}(v_{1}\cdot_{T}v_{2},v_{3},v_{4},v_{5})-v_{1}\diamond
F_{1}(v_{2},v_{3},v_{4},v_{5})-v_{2}\diamond F_{1}(v_{1},v_{3},v_{4},v_{5})$
$\displaystyle=$ $\displaystyle\\{v_{1}\cdot_{T}v_{2},v_{3},v_{4}\diamond
v_{5}\\}-v_{4}\diamond\\{v_{1}\cdot v_{2},v_{3},v_{5}\\}-[v_{1}\cdot
v_{2},v_{3},v_{4}]_{T}\diamond v_{5}$ $\displaystyle-
v_{1}\diamond\\{v_{2},v_{3},v_{4}\diamond
v_{5}\\}-v_{1}\diamond(v_{4}\diamond\\{v_{2},v_{3},v_{5}\\})-v_{1}\diamond([v_{2},v_{3},v_{4}]_{T}\diamond
v_{5})$ $\displaystyle-v_{2}\diamond\\{v_{1},v_{3},v_{4}\diamond
v_{5}\\}-v_{2}\diamond(v_{4}\diamond\\{v_{1},v_{3},v_{5}\\})-v_{2}\diamond([v_{1},v_{3},v_{4}]_{T}\diamond
v_{5})$ $\displaystyle=$
$\displaystyle\rho(T(v_{1}\cdot_{T}v_{2}),Tv_{3})\mu(Tv_{4})v_{5}-\mu(Tv_{4})\rho(T(v_{1}\cdot
v_{2}),Tv_{3})v_{5}-\mu(T[v_{1}\cdot v_{2},v_{3},v_{4}]_{T})v_{5}$
$\displaystyle-\mu(Tv_{1})\rho(Tv_{2},Tv_{3})\mu(Tv_{4})v_{5}-\mu(Tv_{1})\mu(Tv_{4})\rho(Tv_{2},Tv_{3})v_{5}-\mu(Tv_{1})\mu(T[v_{2},v_{3},v_{4}]_{T})v_{5}$
$\displaystyle-\mu(Tv_{2})\rho(Tv_{1},Tv_{3})\mu(Tv_{4})v_{5}-\mu(Tv_{2})\mu(Tv_{4})\rho(Tv_{1},Tv_{3})v_{5}-\mu(Tv_{2})(\mu(T[v_{1},v_{3},v_{4}]_{T})v_{5}$
$\displaystyle=$
$\displaystyle\mathcal{L}_{1}(Tv_{1}\cdot_{T}Tv_{2},v_{3},v_{4},v_{5})-\mu(Tv_{1})\mathcal{L}_{1}(Tv_{2},Tv_{3},Tv_{4},v_{5})-\mu(Tv_{2}\mathcal{L}_{1}(Tv_{1},Tv_{3},Tv_{4},v_{5})$
$\displaystyle=$ $\displaystyle 0,$

Then we obtain Eq.(5.36). Similarly, using a similar computation we can obtain
Eqs. (5.37) and (5.38). Therefore, $(V,\diamond,\\{\cdot,\cdot,\cdot\\})$ is a
ternary pre-$F$-manifold algebra. On the other hand, for all $u,v,w\in V$ we
have

$[Tu,Tv,Tw)]=T\Big{(}\rho(Tu,Tv)w+\rho(Tv,Tw)u+\rho(Tw,Tu)v\Big{)}=T([u,v,w]_{T})$

and $T(u)\cdot T(v)=T(u\cdot_{T}v)$. Then, $T$ is a homomorphism from $V^{c}$
to $(A,\cdot,[\cdot,\cdot,\cdot])$. ∎

######  Corollary 5.10.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary $F$-manifold algebra and
$T:V\longrightarrow A$ an relative Rota Baxter on $A$ with respect to the
representation $(V;\rho,\mu)$. Then $T(V)=\\{T(v)\mid v\in V\\}\subset A$ is a
subalgebra of $A$ and there is an induced ternary pre-$F$-manifold algebra
structure on $T(V)$ given by

$Tu\diamond Tv=T(u\diamond
v),\quad\\{Tu,Tv,Tw\\}=T\\{u,v,w\\},~{}~{}~{}~{}\forall u,v,w\in V.$

######  Corollary 5.11.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary $F$-manifold algebra and
$\mathcal{R}:A\longrightarrow A$ a Rota-Baxter operator of weight $0$. Define
two operations on $A$ by

$x\diamond y=\mathcal{R}(x)\cdot
y,\quad\\{x,y,z\\}=[\mathcal{R}(x),\mathcal{R}(y),z].$

Then $(A,\diamond,\\{\cdot,\cdot,\cdot\\})$ is a ternary pre-$F$-manifold
algebra and $\mathcal{R}$ is a homomorphism from the sub-adjacent
ternary-$F$-manifold algebra
$(A,\cdot_{\mathcal{R}},[\cdot,\cdot,\cdot]_{\mathcal{R}})$ to
$(A,\cdot,[\cdot,\cdot,\cdot])$, where

$x\cdot_{\mathcal{R}}y=x\diamond y+y\diamond
x,~{}~{}\text{and}~{}~{}[x,y,z]_{\mathcal{R}}=\circlearrowleft_{x,y,z}\\{x,y,z\\}$

for all $x,y,z\in A.$

The following result is necessary and sufficient condition on an
ternary-$F$-manifold algebra to admit a ternary pre-$F$-manifold algebra
structure.

######  Proposition 5.12.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be an ternary $F$-manifold algebra. There
is a ternary pre-$F$-manifold algebra structure on $A$ such that its sub-
adjacent ternary $F$-manifold algebra is exactly
$(A,\cdot,[\cdot,\cdot,\cdot])$ if and only if there exists an invertible
relative Rota-Baxter operator on $(A,\cdot,[\cdot,\cdot,\cdot])$.

###### Proof.

Suppose $T:V\longrightarrow A$ is an invertible relative Rota-Baxter operator
on $A$ with respect to the representation $(V;\rho,\mu)$, then the compatible
ternary pre-$F$-manifold algebra structure on $A$ is given by

$x\diamond y=T(\mu(x)(T^{-1}(y))),\quad\quad\\{x,y,z\\}=T(\rho(x,y)(T^{-1}z))$

for all $x,y,y\in A$. Sine $T$ is a relative Rota-baxter operator on $A$, we
have

$\displaystyle x\diamond y+y\diamond x$
$\displaystyle=T(\mu(x)(T^{-1}(y)))+T(\mu(y)(T^{-1}(x)))$
$\displaystyle=T(\mu(TT^{-1}(x))(T^{-1}(y))+\mu(TT^{-1}(y))(T^{-1}(x)))$
$\displaystyle=TT^{-1}(x)\cdot TT^{-1}(y)=x\cdot y$

and

$\displaystyle\circlearrowleft_{x,y,z}\\{x,y,z\\}$
$\displaystyle=T\big{(}\rho(x,y)(T^{-1}z)+\rho(y,z)(T^{-1}x)+\rho(z,x)(T^{-1}y)\big{)}$
$\displaystyle=[TT^{-1}(x),TT^{-1}(y),TT^{-1}(z)]=[x,y,z].$

Conversely, let $(A,\diamond,\\{\cdot,\cdot,\cdot\\})$ be a ternary
pre-$F$-manifold algebra and $(A,\cdot,[\cdot,\cdot,\cdot])$ the sub-adjacent
ternry $F$-manifold algebra. Then the identity map $\rm{Id}$ is an relative
Rota-Baxter operator on $A$ with respect to the representation
$(A;\mathbb{L},L)$. ∎

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary $F$-manifold algebra and
$\mathcal{B}\in\wedge^{2}A^{*}$. We say that $\mathcal{B}$ is a cyclic
$2$-cocycle in the sense of Connes on the commutative associative algebra
$(A,\cdot)$ if

(5.46) $\displaystyle\mathcal{B}(x\cdot y,z)+\mathcal{B}(y\cdot
z,x)+\mathcal{B}(z\cdot x,y)=0$

and $\mathcal{B}$ is said to be a symplectic structure on the $3$-Lie algebra
$(A,[\cdot,\cdot,\cdot])$ if it satisfies

(5.47)
$\displaystyle\mathcal{B}([x,y,z],u)-\mathcal{B}([x,y,u],z)+\mathcal{B}([x,z,u],y)-\mathcal{B}([y,z,u],x)=0,$

for any $x,y,z,u\in A$.

######  Proposition 5.13.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a coherence ternary-$F$-manifold
algebra and $\mathcal{B}$ be a skew-symmetric bilinear form on $A$ satisfying
identities (5.46) and (5.47). Then there is a compatible ternary-
pre-$F$-manifold algebra structure on $A$ given by

$\mathcal{B}(x\diamond y,z)=\mathcal{B}(y,x\cdot
z),\quad\mathcal{B}(\\{x,y,z\\},u)=-\mathcal{B}(z,[x,y,u]).$

###### Proof.

Since $(A,\cdot,[\cdot,\cdot,\cdot])$ is a coherence ternary $F$-manifold
algebra, then $(A^{*};\mathrm{ad}^{\ast},-\mathcal{L}^{\ast})$ is a
representation of $A$. By the fact that $\mathcal{B}$ is a symplectic
structure, we deduce that the musical map
$\mathcal{B}^{\sharp}:A\longrightarrow A^{*}$ defined by
$\langle\mathcal{B}^{\sharp}(x),y\rangle=\mathcal{B}(x,y)$ is invertible and
$(\mathcal{B}^{\sharp})^{-1}$ is a relative Rota-Baxter operator on the
commutative associative algebra $(A,\cdot)$ with respect to the representation
$(A^{*};-L^{\ast})$.

In addition, $(\mathcal{B}^{\sharp})^{-1}$ is a relative Rota-Baxter operator
on the $3$-Lie algebra $(A,[\cdot,\cdot,\cdot])$ with respect to the
representation $(A^{*};\mathrm{ad}^{\ast})$. Then,
$(\mathcal{B}^{\sharp})^{-1}$ is a relative Rota-Baxter operator on the
coherence ternary $F$-manifold algebra $(A,\cdot,[\cdot,\cdot,\cdot])$ with
respect to the representation $(A^{*};\mathrm{ad}^{\ast},-L^{\ast})$. By
Proposition 5.12, there is a compatible ternary pre-$F$-manifold algebra
structure on $A$ given as above. ∎

Next, let $T$ be a realtive Rota-Baxter on a ternary $F$-manifold algebra
$(A,\cdot,[\cdot,\cdot,\cdot])$ with respect to a representation
$(V;\rho,\mu)$. According to Theorem 5.9,
$(V,\cdot_{T},[\cdot,\cdot,\cdot]_{T})$ is ternary $F$-manifold algebra. Is
there a representation of $V$ on the vector space $A$? The answer is
affirmative. In fact, define two linear maps $\rho_{T}:\wedge^{2}V\to End(A)$
and $\mu_{T}:V\to End(A)$ by

(5.48) $\displaystyle\rho_{T}(u,v)(x)$
$\displaystyle=[Tu,Tv,x]-T(\rho(Tv,x)u+\rho(x,Tu)v),$ (5.49)
$\displaystyle\mu_{T}(u)(x)$ $\displaystyle=Tu\cdot x-T(\mu(x)(u))$

for all $x\in A,u,v\in V.$

######  Theorem 5.14.

With the above notations, the triple $(A;\rho_{T},\mu_{T})$ is a
representation of $(V,\cdot_{T},[\cdot,\cdot,\cdot]_{T}).$

We may prove Theorem 5.14 by checking that $(A;\rho_{T},\mu_{T})$ satisfies
conditions of Definition 4.1, but here we will prove it by another way which
is more elegant. In order to do this, we should introduce and go back to some
notions. Recall that a Nijenhuis operator on an associative commutative
algebra $(A,\cdot)$ is a linear map $N:A\to A$ obeying to the following
integrability condition:

(5.50) $\displaystyle Nx\cdot Ny=N(Nx\cdot y+x\cdot Ny-N(x\cdot y)),\ \forall
x,y,\in A.$

On the other hand, a linear map $N$ on a $3$-Lie algebra
$(A,[\cdot,\cdot,\cdot])$ is called a Nijenhuis operator if it satisfies, for
any $x,y,z\in A$,

$\displaystyle[Nx,Ny,Nz]=N\big{(}[Nx,Ny,z][Nx,y,Nz]+[x,Ny,Nz]$ (5.51)
$\displaystyle-N[Nx,y,z]-N[x,Ny,z,]-N[x,y,Nz]+N^{2}[x,y,z]\big{)}.$

######  Definition 5.15.

Let $(A,\cdot,[\cdot,\cdot,\cdot])$ be a ternary $F$-manifold algebra. A
Nijenhuis operator on $A$ is a linear map $N:A\to A$ satisfying (5.50) and
(5.51).

Define two deformed products $\cdot_{N}$ and $[\cdot,\cdot,\cdot]_{N}$ on $A$
by, for all $x,y,z\in A,$

(5.52) $\displaystyle x\cdot_{N}y=Nx\cdot y+x\cdot Ny-N(x\cdot y),$
$\displaystyle[x,y,z]_{N}=[Nx,Ny,z][Nx,y,Nz]+[x,Ny,Nz]$ (5.53)
$\displaystyle-N[Nx,y,z]-N[x,Ny,z,]-N[x,y,Nz]+N^{2}[x,y,z].$

The following two lemmas are straightforward, so we omit the proofs.

######  Lemma 5.16.

With the above notations, the tuple $(A,\cdot_{N},[\cdot,\cdot,\cdot]_{N})$ is
a ternary $F$-manifold algebra.

######  Lemma 5.17.

Let $T:V\to A$ be a relative Rota-Baxter operator on a ternary $F$-manifold
algebra $(A,\cdot,[\cdot,\cdot,\cdot])$ with respect to a representation
$(V;\rho,\mu)$. Then the operator $N_{T}=\begin{pmatrix}0&T\\\ 0&0\\\
\end{pmatrix}$ is a Nijenhuis operator on the semi-direct product ternary
$F$-manifold algebra $A\oplus V$.

_Proof of Theorem_ 5.14: According to Lemma 5.17, $N_{T}$ is a Nijenhuis
operator on the semi-direct product ternary $F$-manifold algebra $A\oplus V$.
Then, we deduce that there is a ternary $F$-manifold structure on $V\oplus
A\cong A\oplus V$ given by, for all $x,y,z\in A,\ u,v,w\in V$,

$\displaystyle(x+u)\cdot_{N_{T}}(y+v)$ $\displaystyle=$ $\displaystyle
N_{T}(x+u)\cdot_{\mu}(y+v)+(x+u)\cdot_{\mu}N_{T}(y+v)-N_{T}((x+u)\cdot_{\mu}(y+v))$
$\displaystyle=$ $\displaystyle Tu\cdot_{\mu}(y+v)+(x+u)\cdot_{\mu}Tv-
N_{T}(x\cdot y+\mu(x)v+\mu(y)u)$ $\displaystyle=$ $\displaystyle Tu\cdot
y+\mu(Tu)v+x\cdot Tv+\mu(Tv)u-T(\mu(x)v+\mu(y)u)$ $\displaystyle=$
$\displaystyle u\cdot_{T}v+\mu_{T}(u)y+\mu_{T}(v)x$

and (since $N_{T}^{2}=0$), we have

$\displaystyle[x+u,y+v,z+w]_{N_{T}}$ $\displaystyle=$
$\displaystyle[N_{T}(x+u),N_{T}(y+v),z+w]_{\rho}+[N_{T}(x+u),y+v,N_{T}(z+w)]_{\rho}+[x+u,N_{T}(y+v),N_{T}(z+w)]_{\rho}$
$\displaystyle-$ $\displaystyle
N_{T}([N_{T}(x+u),y+v,z+w]+[x+u,N_{T}(y+v),z+w]+[x+u,y+v,N_{T}(z+w)]_{\rho})$
$\displaystyle=$
$\displaystyle[Tu,Tv,z+w]_{\rho}+[Tu,y+v,Tw]_{\rho}+[x+u,Tv,Tw]_{\rho}$
$\displaystyle-$ $\displaystyle
N_{T}([Tu,y+v,z+w]_{\rho}+[x+u,Tv,z+w]_{\rho}+[x+u,y+v,Tw]_{\rho})$
$\displaystyle=$
$\displaystyle[Tu,Tv,z]+\rho(Tu,Tv)w+[Tu,y,Tw]+\rho(Tw,Tu)v+[x,Tu,Tv]+\rho(Tv,Tw)u$
$\displaystyle-$ $\displaystyle
T\big{(}\rho(Tu,y)w+\rho(z,Tu)v+\rho(x,Tv)w+\rho(Tv,z)u+\rho(y,Tw)u+\rho(Tw,x)v\big{)}$
$\displaystyle=$
$\displaystyle[u,v,w]_{T}+\rho_{T}(u,v)z+\rho_{T}(v,w)x+\rho_{T}(w,u)y,$

which implies that $(A;\rho_{T},\mu_{T})$ is a representation of the ternary
$F$-manifold algebra $(V,\cdot_{T},[\cdot,\cdot,\cdot]_{T})$. This finishes
the proof. ∎

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Motivic stable cohomotopy and unimodular rows
Samuel Lerbet
Institut Fourier, UMR 5582, Laboratoire de mathématiques

Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France

We relate the group structure of van der Kallen on orbit sets of unimodular rows (<cit.>) with values in a smooth algebra $A$ over a field $k$ with the motivic cohomotopy groups of $X=\Spec A$ with coefficients in $\A^n\setminus 0$ in the sense of <cit.>. In the last section, we compare the motivic cohomotopy theory studied in this paper and defined by $\A^{n+1}\setminus 0$ or, equivalently, by an $\A^1$-weakly equivalent quadric $Q_{2n+1}$ to that considered in <cit.>, defined by a quadric $Q_{2n}$, by means of explicit morphisms $Q_{2n+1}\rightarrow Q_{2n}$, $Q_{2n}\times\mathbb{G}_m\rightarrow Q_{2n+1}$ of quadrics.

§ INTRODUCTION

§ UNIMODULAR ROWS

§ COHOMOTOPY GROUPS

§ A COHOMOTOPICAL INTERPRETATION OF VAN DER KALLEN'S LAW

We now come to the interplay between the two group structures introduced in the previous sections. Recall that the letter $k$ denotes a field. We let $\mathscr{X}$ be $\A^n\setminus 0$, pointed by $(0,\ldots,0,1)$. The goal of this section is to prove the following.

Let $X$ be a smooth affine $k$-scheme of Krull dimension $d$ and let $n$ be an integer such that $d\leqslant 2n-4$. Then the natural map $\Psi:\Um_n(k[X])/\Er_n(k[X])\rightarrow[X,\A^n\setminus 0]_{\A^1}$ is a group isomorphism, where the left-hand side is endowed with the structure of van der Kallen and the right-hand side is endowed with the group structure defined by means of Proposition <ref>.

The theorem contains two statements: the first is that $\Psi$ is a group homomorphism; the second is that $\Psi$ is a bijection. We study these two statements in two separate subsections to make the needed assumptions on $k$ clearer.

§.§ The map $\Psi$ is a bijection

§.§ The map $\Psi$ is a group homomorphism

§ COMPARING COHOMOTOPY THEORIES

The letter $k$ once again denotes an arbitrary field. In this section, we compare the cohomotopy theory $[\text{–},Q_{2n+1}]_{\A^1}$ represented by the quadric $Q_{2n+1}$ to the cohomotopy theory $[\text{–},Q_{2n}]_{\A^1}$ represented by the quadric $Q_{2n}$ and studied in <cit.>. We provide morphisms of schemes $Q_{2n+1}\rightarrow Q_{2n}$ and $Q_{2n}\times\mathbb{G}_m\rightarrow Q_{2n+1}$ to allow for these comparisons.

§.§ The morphism $\eta_n:Q_{2n+1}\rightarrow Q_{2n}$

§.§ The morphism $\mu_n:\mathbb{G}_m\times Q_{2n}\rightarrow Q_{2n+1}$

§.§ Euler class groups and motivic stable cohomotopy theories

Missing 'biblatex' package
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