Transposed Poisson structures111 The authors thank Pasha Zusmanovich, Viktor
Lopatkin, Zerui Zhang, Vladimir Dotsenko and Baurzhan Sartayev for useful
discussions. The first part of this work is supported by FCT UIDB/00212/2020
and UIDP/00212/2020. The second part of this work is supported by the Russian
Science Foundation under grant 22-11-00081.

Patrícia Damas Beites222Departamento de Matemática and Centro de Matemática e
Aplicações, Universidade da Beira Interior, Covilhã, Portugal<EMAIL_ADDRESS>Bruno Leonardo Macedo Ferreira333Federal University of Technology, Guarapuava,
Brazil<EMAIL_ADDRESS>& Ivan Kaygorodov444CMA-UBI, Universidade da
Beira Interior, Covilhã, Portugal; Saint Petersburg University, Russia; Moscow
Center for Fundamental and Applied Mathematics, Moscow, Russia;
<EMAIL_ADDRESS>

Abstract. Complex transposed Poisson structures on some Lie and associative
algebras are studied. A deep connection between unital transposed Poisson
algebras and generalized Poisson algebras is established. An analogue of a
theorem due to Farkas for unital transposed Poisson algebras is presented. All
transposed Poisson structures on simple graded Lie algebras of finite growth
are described. It is proved that the oscillator Lie algebras have non-trivial
transposed Poisson structures. Finally, a list of open questions for future
research is given.

Keywords: Lie algebra, transposed Poisson algebra, Hom-Lie algebra,
$\delta$-derivation.

MSC2020: 17A30, 17B40, 17B61, 17B63.

###### Contents

  1. 1 Preliminaries and some trivial results
    1. 1.1 Isomorphism problem for transposed Poisson algebras on a certain Lie algebra
    2. 1.2 Transposed Poisson structures on unital associative algebras
    3. 1.3 Relations to other Poisson type algebras
      1. 1.3.1 Quasi-Poisson algebras
      2. 1.3.2 Generalized Poisson algebras and algebras of Jordan brackets
      3. 1.3.3 Algebras of Jordan brackets
      4. 1.3.4 Gelfand-Dorfman algebras (GD-algebras) and $F$-manifold algebras
    4. 1.4 PI Transposed Poisson algebras
    5. 1.5 Transposed Poisson fields
    6. 1.6 Transposed Poisson structures on nilpotent $n$-Lie algebras
    7. 1.7 Transposed Poisson $n$-Lie algebras and Poisson $n$-Lie algebras
    8. 1.8 Transposed Poisson structures and quasi-automorphisms
    9. 1.9 Transposed Poisson structures and ${\rm Hom}$-Lie algebra structures
    10. 1.10 Transposed Poisson structures on simple graded Lie algebras of finite growth
  2. 2 $\frac{1}{2}$-derivations of oscillator Lie algebras and transposed Poisson algebras
    1. 2.1 Isomorphism problem for transposed Poisson algebras on generic oscillator Lie algebras
  3. 3 Open questions

## Introduction

Poisson algebras arose from the study of Poisson geometry in the 1970s and
have appeared in an extremely wide range of areas in mathematics and physics,
such as Poisson manifolds, algebraic geometry, operads, quantization theory,
quantum groups, and classical and quantum mechanics. The study of Poisson
algebras also led to other algebraic structures, such as noncommutative
Poisson algebras [80], generic Poisson algebras [41, 43], Poisson bialgebras
[66, 50], algebras of Jordan brackets and generalized Poisson algebras [12,
13, 31, 34, 90, 61], Gerstenhaber algebras [48], $F$-manifold algebras [17],
Novikov-Poisson algebras [81], quasi-Poisson algebras [9], double Poisson
algebras [75], Poisson $n$-Lie algebras [14], etc. The study of all possible
Poisson structures with a certain Lie or associative part is an important
problem in the theory of Poisson algebras [35, 1, 83]. Recently, a dual notion
of the Poisson algebra (transposed Poisson algebra), by exchanging the roles
of the two binary operations in the Leibniz rule defining the Poisson algebra,
has been introduced in the paper [4] of Bai, Bai, Guo, and Wu. They have shown
that the transposed Poisson algebra defined in this way not only shares common
properties with the Poisson algebra, including the closure undertaking tensor
products and the Koszul self-duality as an operad but also admits a rich class
of identities. More significantly, a transposed Poisson algebra naturally
arises from a Novikov-Poisson algebra by taking the commutator Lie algebra of
the Novikov algebra. Consequently, the classic construction of a Poisson
algebra from a commutative associative algebra with a pair of commuting
derivations has a similar construction to a transposed Poisson algebra when
there is one derivation. More broadly, the transposed Poisson algebra also
captures the algebraic structures when the commutator is taken in pre-Lie
Poisson algebras and two other Poisson-type algebras. The ${\rm Hom}$\- and
${\rm BiHom}$-versions of transposed Poisson algebras are considered in [49,
53]. Some new examples of transposed Poisson algebras are constructed by
applying the Kantor product of multiplications on the same vector space [21].
More recently, in a paper by Ferreira, Kaygorodov and Lopatkin, a relation
between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson
algebras has been established [24]. These ideas were used for describing all
transposed Poisson structures on the Witt algebra [24], the Virasoro algebra
[24], the algebra $\mathcal{W}(a,b)$[24], the thin Lie algebra [24], the
twisted Heisenberg-Virasoro algebra [84], the Schrodinger-Virasoro algebra
[84], the extended Schrodinger-Virasoro algebra [84], the 3-dimensional
Heisenberg Lie algebra [84], Block Lie algebras and superalgebras [36], Witt
type algebras [37], generalized Witt Lie algebras [38], Lie algebra of upper
triangular matrices [39], Schrodinger algebra in $(n+1)$-dimensional space-
time [82], and so on. The complete algebraic and geometric classification of
complex $3$-dimensional transposed Poisson algebras is given in [7].

The class of oscillator Lie algebras, which are the only noncommutative
solvable Lie algebras that carry a bi-invariant Lorentzian metric [63], is a
class of quadratic Lie algebras. The former algebras, which are the Lie
algebras of oscillator Lie groups, have aroused some interest [8, 1, 11, 16,
10, 5]. Ballesteros and Herranz obtained all possible coboundary Lie
bialgebras for the four-dimensional oscillator algebra [5]. Biggs and Remsing
examined the algebraic structure of the four-dimensional oscillator Lie
algebra[8]. Firstly, the adjoint orbits are determined (and graphed); these
orbits turn out to be linearly isomorphic to the coadjoint orbits. This
prompts the identification of a family of invariant scalar products (from
which the Casimir functions can be recovered). They identified the group of
automorphisms (resp. inner automorphisms); subsequently, they classified all
linear subspaces. As a by-product, they arrived at classifications of the
full-rank subspaces, the subalgebras, and the ideals. Based on the four-
dimensional oscillator Lie algebra, Crampé, van de Vijver, and Vinet
introduced an associative algebra which they call the oscillator Racah
algebra, and explain a relation between the representation theory of and
multivariate Krawtchouk polynomials [16]. In [11], Calvaruso and Zaeim
obtained a classification of all Ricci, curvature, Weyl and matter
collineations for the four-dimensional oscillator group equipped with a
1-parameter family of left-invariant Lorentzian metrics. Boucetta and Medina
determined the Lie bialgebra structures and the solutions of the classical
Yang–Baxter equation on a generic class of oscillator Lie algebras [10].
Albuquerque, Barreiro, Benayadi, Boucetta, and Sánchez described all Poisson
algebra structures and symmetric Leibniz bialgebra structures on generic
oscillator Lie algebras [1].

The structure of the present article is as follows. In section 1, we discuss
some definitions and previous results. This section establishes many relations
between transposed Poisson algebras and other well-known structures, such as
generalized Poisson algebras, fields of fractions, ${\rm Hom}$-Lie structures,
quasi-automorphisms, etc. In subsection 1.2, we describe the Lie bracket in a
unital transposed Poisson algebra. Namely, we prove that each bracket $[x,y]$
is given in the following way, $[x,y]=\mathfrak{D}(x)\cdot
y-x\cdot\mathfrak{D}(y)$ for a derivation $\mathfrak{D}$ of commutative
associative part of the transposed Poisson algebra. Hence, each unital
transposed Poisson algebra is a generalized Poisson algebra (and also it is an
algebra of Jordan brackets and a quasi-Poisson algebra) and on the other side,
each transposed Poisson algebra is a commutative GD-algebra and a $F$-manifold
algebra (subsection 1.3). In subsection 1.4, we find customary identities for
unital transposed Poisson algebras (namely, we prove that each transposed
Poisson PI algebra satisfies some identities of a special type). In subsection
1.5, we extend the bracket of a unital transposed Poisson algebra on their
field of fractions, hence we have the notion of transposed Poisson fields. In
subsection 1.6, we prove that each nilpotent finite-dimensional $n$-Lie
algebra admits a non-trivial transposed Poisson structure. In subsection 1.7,
we describe the intersection of transposed Poisson $n$-Lie algebras and
Poisson $n$-Lie algebras. In subsection 1.8, we discuss a relation between
quasi-automorphisms and transposed Poisson structures. As a corollary, we have
that Borel or parabolic subalgebras of a complex finite-dimensional simple Lie
algebra of rank $l>1$ do not admit non-trivial transposed Poisson structures.
In subsection 1.9, we discuss a relation between ${\rm Hom}$-Lie structures
and transposed Poisson structures. As a corollary, we have that complex
infinite-dimensional Lie superalgebras of vectors fields and affine Kac–Moody
algebras do not admit non-trivial transposed Poisson structures. In subsection
1.10, we give a complete description of transposed Poisson structures on
simple graded Lie algebras of finite growth. Namely, we prove that only the
Witt algebra ${\rm W}$ and the Cartan algebra ${\rm W}(1)$ admit non-trivial
transposed Poisson structures, and we describe them.

In section 2, we describe $\frac{1}{2}$-derivations of oscillator Lie
algebras. As a consequence, we have new examples of non-trivial ${\rm
Hom}$-Lie structures and transposed Poisson structures on oscillator Lie
algebras. We give the full description of all non-isomorphic transposed
Poisson structures on generic oscillator Lie algebras.

In the ending section 3, we give a list of some open questions for future
research.

##  1\. Preliminaries and some trivial results

Although all algebras and vector spaces are considered over the complex field,
many results can be proven over other fields without modifications of proofs.
The definition of transposed Poisson algebra was given in a paper by Bai, Bai,
Guo, and Wu [4]. The concept of $\frac{1}{2}$-derivations as a particular case
of $\delta$-derivations was presented in an article of Filippov [25] (see also
[89, 33] and references therein).

######  Definition 1.

Let ${\mathfrak{L}}$ be a vector space equipped with two nonzero bilinear
operations $\cdot$ and $[\cdot,\cdot].$ The triple
$({\mathfrak{L}},\cdot,[\cdot,\cdot])$ is called a transposed Poisson algebra
if $({\mathfrak{L}},\cdot)$ is a commutative associative algebra and
$({\mathfrak{L}},[\cdot,\cdot])$ is a Lie algebra that satisfies the following
compatibility condition

$2z\cdot[x,y]=[z\cdot x,y]+[x,z\cdot y].$

######  Definition 2.

Let $({\mathfrak{L}},[\cdot,\cdot])$ be an algebra with multiplication
$[\cdot,\cdot],$ $\varphi$ be a linear map and $\phi$ be a bilinear map. Then
$\varphi$ is a $\frac{1}{2}$-derivation if it satisfies

$\varphi[x,y]=\frac{1}{2}\big{(}[\varphi(x),y]+[x,\varphi(y)]\big{)};$

$\phi$ is a $\frac{1}{2}$-biderivation if it satisfies

$\phi([x,y],z)$ | $=$ | $\frac{1}{2}\big{(}[\phi(x,z),y]+[x,\phi(y,z)]\big{)},$
---|---|---
$\phi(x,[y,z])$ | $=$ | $\frac{1}{2}\big{(}[\phi(x,y),z]+[y,\phi(x,z)]\big{)}.$

Summarizing Definitions 1 and 2, we have the following key Lemma.

######  Lemma 3.

Let $({\mathfrak{L}},\cdot,[\cdot,\cdot])$ be a transposed Poisson algebra and
$z$ an arbitrary element from ${\mathfrak{L}}.$ Then the right multiplication
$R_{z}$ in the commutative associative algebra $({\mathfrak{L}},\cdot)$ gives
a $\frac{1}{2}$-derivation of the Lie algebra
$({\mathfrak{L}},[\cdot,\cdot]).$

The main example of $\frac{1}{2}$-derivations is the multiplication by an
element from the ground field. Let us call such $\frac{1}{2}$-derivations as
trivial $\frac{1}{2}$-derivations. As a consequence of the following theorem,
we are not interested in trivial $\frac{1}{2}$-derivations.

######  Theorem 4.

Let ${\mathfrak{L}}$ be a Lie algebra without non-trivial
$\frac{1}{2}$-derivations. Then every transposed Poisson structure defined on
${\mathfrak{L}}$ is trivial.

###  1.1. Isomorphism problem for transposed Poisson algebras on a certain Lie
algebra

######  Definition 5.

Let $({\mathfrak{L}}_{1},\cdot_{1},[\cdot,\cdot]_{1})$ and
$({\mathfrak{L}}_{2},\cdot_{2},[\cdot,\cdot]_{2})$ be two transposed Poisson
algebras. Then $({\mathfrak{L}}_{1},\cdot_{1},[\cdot,\cdot]_{1})$ and
$({\mathfrak{L}}_{2},\cdot_{2},[\cdot,\cdot]_{2})$ are isomorphic if and only
if there exists a liner map $\varphi$ such that

$\varphi([x,y]_{1})=[x,y]_{2},\ \varphi(x\cdot_{1}y)=x\cdot_{2}y.$

Our main strategy for classifying all non-isomorphic transposed Poisson
structures on a certain Lie algebra $({\mathfrak{L}},[\cdot,\cdot])$ is as
follows.

  1. (1)

Find all automorphisms ${\rm Aut}({\mathfrak{L}},[\cdot,\cdot])$.

  2. (2)

Consider the multiplication table of $({\mathfrak{L}},\cdot)$ under the action
of elements from ${\rm Aut}({\mathfrak{L}},[\cdot,\cdot])$ and separate all
non-isomorphic cases.

###  1.2. Transposed Poisson structures on unital associative algebras

As we will see in the present subsection, unital transposed Poisson algebras
are closely related to generalized Poisson algebras. Some theorems proved for
generalized Poisson algebras can give some interesting results in the theory
of transposed Poisson algebras.

######  Theorem 6.

Let $(\mathbf{A},\cdot,[\cdot,\cdot])$ be a complex transposed Poisson
algebra. Let $(\mathbf{A},\cdot)$ be a unital associative (not necessarily
commutative) algebra. Then there is a derivation $\mathfrak{D}$ on
$(\mathbf{A},\cdot)$ such that

$[x,y]=\mathfrak{D}(x)\cdot y-x\cdot\mathfrak{D}(y).$ (1)

In particular, all transposed Poisson structures on a finite-dimensional
semisimple commutative associative algebra $(\mathbf{A},\cdot)$ are trivial.

###### Proof.

It is easy to see that

$[x\cdot y,1]$ | $=$ | $2[x,1]\cdot y-[x,y],$
---|---|---
$[x\cdot y,1]$ | $=$ | $2x\cdot[y,1]-[y,x],$

then, summarizing and subtracting it, we have

$[x\cdot y,1]$ | $=$ | $x\cdot[y,1]+[x,1]\cdot y,$
---|---|---
$[x,y]$ | $=$ | $[x,1]\cdot y-x\cdot[y,1].$

Hence, the linear mapping $\mathfrak{D}$ given by $\mathfrak{D}(x)=[x,1]$ is a
derivation and the first statement of the theorem is proved.

Each finite-dimensional semisimple commutative associative algebra is a direct
sum of one-dimensional ideals. On the other hand, it is known that all
derivations of this algebra are trivial and it is unital. Hence, we have our
statement. ∎

###  1.3. Relations to other Poisson type algebras

####  1.3.1. Quasi-Poisson algebras

Recently, Billig gave a definition of quasi-Poisson algebras [9]. Namely,

######  Definition 7.

Let $(\mathbf{A},\cdot)$ be a unital commutative associative algebra and
$(\mathbf{A},\\{\cdot,\cdot\\})$ be a Lie algebra.
$(\mathbf{A},\cdot,\\{\cdot,\cdot\\})$ with a derivation $\mathfrak{D}$ of
$(\mathbf{A},\cdot)$ is called a quasi-Poisson algebra if

$a\cdot\big{(}\mathfrak{D}(\\{b,c\\})+\\{b,c\\}\big{)}=\\{a\cdot(\mathfrak{D}(b)+b),c\\}+\\{b,a\cdot(\mathfrak{D}(c)+c)\\}+\\{a,b\\}\cdot(\mathfrak{D}(c)+c)-(\mathfrak{D}(b)+b)\cdot\\{a,c\\}.$

The main example of quasi-Poisson algebras is given by relation (1) [9, Lemma
6.3].

######  Corollary 8.

Let $(\mathbf{A},\cdot,[\cdot,\cdot])$ be a unital transposed Poisson algebra.
Then $(\mathbf{A},\cdot,[\cdot,\cdot])$ is a quasi-Poisson algebra.

####  1.3.2. Generalized Poisson algebras and algebras of Jordan brackets

Let us remember now the definition of generalized Poisson bracket and Jordan
brackets [12, 34].

######  Definition 9.

Let $(\mathbf{A},\cdot)$ be a unital commutative associative algebra.

  1. $\bullet$

An anticommutative product
$\\{\cdot,\cdot\\}:\mathbf{A}\otimes\mathbf{A}\to\mathbf{A}$ is called a
generalized Poisson bracket (it is also known as the contact bracket [90]) if
$(\mathbf{A},\\{\cdot,\cdot\\})$ is a Lie algebra and

$\\{x,y\cdot z\\}=\\{x,y\\}\cdot z+y\cdot\\{x,z\\}-\\{x,1\\}\cdot y\cdot z.$

  2. $\bullet$

An anticommutative product
$\\{\cdot,\cdot\\}:\mathbf{A}\otimes\mathbf{A}\to\mathbf{A}$ is called a
Jordan bracket (see, [34]) if

$\\{x,\\{y,z\\}\\}=\\{\\{x,y\\},z\\}+\\{y,\\{x,z\\}\\}+\\{x,1\\}\cdot\\{y,z\\}+\\{y,1\\}\cdot\\{z,x\\}+\\{z,1\\}\cdot\\{x,y\\}.$

The main example of generalized Poisson brackets and Jordan brackets is given
by relation (1). Hence, we have the following corollary.

######  Corollary 10.

Let $(\mathbf{A},\cdot,[\cdot,\cdot])$ be a unital transposed Poisson algebra.
Then $(\mathbf{A},\cdot,[\cdot,\cdot])$ is a generalized Poisson algebra and
an algebra of Jordan brackets.

Martínez, Shestakov and Zelmanov have shown that all Jordan brackets are
embeddable into Poisson brackets [61]. In this sense, summarizing Corollary 11
and [61], we have the following statement.

######  Corollary 11.

Unital transposed Poisson brackets are embeddable into Poisson brackets.

####  1.3.3. Algebras of Jordan brackets

Let $(\mathbf{A},\cdot)$ be a commutative associative algebra and
$(\mathbf{A},\\{\cdot,\cdot\\})$ be an anticommutative algebra.
$(\mathbf{A},\cdot,\\{\cdot,\cdot\\})$ is called an algebra of Jordan brackets
if the Kantor double of $(\mathbf{A},\cdot,\\{\cdot,\cdot\\})$ is a Jordan
superalgebra [31]. If $(\mathbf{A},\cdot)$ is not a unital algebra, then the
conditions of Jordan brackets are equivalent to (see [32]):

$\\{\\{x,y\\}\cdot z,t\\}+\\{\\{y,t\\}\cdot z,x\\}+\\{\\{t,x\\}\cdot z,y\\}$ | $=$ | $\\{x,y\\}\cdot\\{z,t\\}+\\{y,t\\}\cdot\\{z,x\\}+\\{t,x\\}\cdot\\{z,y\\},$
---|---|---
$\\{y\cdot t,z\\}\cdot x+\\{x,z\\}\cdot y\cdot t$ | $=$ | $\\{t\cdot x,z\\}\cdot y+\\{y,z\\}\cdot t\cdot x,$
$\\{t\cdot x,y\cdot z\\}+\\{t\cdot y,x\cdot z\\}+\\{x\cdot y\cdot z,t\\}$ | $=$ | $\\{t\cdot y,z\\}\cdot x+\\{t\cdot x,z\\}\cdot y+x\cdot y\cdot\\{z,t\\}.$

It is easy to see (maybe using the identities from [4, Theorem 2.5.]) that
each transposed Poisson algebra is an algebra of Jordan brackets (see, also
[22]).

######  Corollary 12.

The Kantor double of a transposed Poisson algebra is a Jordan superalgebra.

####  1.3.4. Gelfand-Dorfman algebras (GD-algebras) and $F$-manifold algebras

######  Definition 13.

Let $(\mathbf{A},\cdot)$ be a Novikov algebra and $(\mathbf{A},[\cdot,\cdot])$
be a Lie algebra. $(\mathbf{A},\cdot,[\cdot,\cdot])$ is called a Gelfand-
Dorfmann algebra if

$[x,y\cdot z]-[z,y\cdot x]+[y,x]\cdot z-[y,z]\cdot x-y\cdot[x,z]$ | $=$ | $0.$
---|---|---

######  Definition 14.

Let $(\mathbf{A},\cdot)$ be a commutative associative algebra and
$(\mathbf{A},[\cdot,\cdot])$ be a Lie algebra.
$(\mathbf{A},\cdot,[\cdot,\cdot])$ is called an $F$-manifold algebra if

$[x\cdot y,z\cdot t]$ | $=$ | $[x\cdot y,z]\cdot t+[x\cdot y,t]\cdot z+x\cdot[y,z\cdot t]+y\cdot[x,z\cdot t]-$
---|---|---
|  | $\big{(}x\cdot z\cdot[y,t]+y\cdot z\cdot[x,t]+y\cdot t\cdot[x,z]+x\cdot t\cdot[y,z]\big{)}.$

Recently, Sartayev proved that the variety of transposed Poisson algebras
coincides to the variety of commutative GD-algebras and each commutative GD-
algebra is an $F$-manifold algebra [68]. Let us note that each Poisson algebra
is also an $F$-manifold algebra. Hence, the variety of $F$-manifold algebras
is the first interesting variety of algebras which includes all Poisson and
all transposed Poisson algebras.

###  1.4. PI Transposed Poisson algebras

The celebrated Amitsur–Levitsky theorem states that the algebra $M_{k}(R)$ of
$k\times k$ matrices over a commutative ring $R$ satisfies the identity
$s_{2k}=0$. Furthermore, every associative PI algebra satisfies
$(s_{k})^{l}=0$ by Amitsur’s theorem. Farkas defined customary polynomials

$g=\sum_{\sigma\in{\mathbb{S}}_{m}}c_{\sigma}\\{x_{\sigma(1)},x_{\sigma(2)}\\}\cdot\ldots\cdot\\{x_{\sigma(2i-1)},x_{\sigma(2i)}\\}$

and proved that every Poisson PI algebra satisfies some customary identity
[20]. Farkas’ theorem was established for generic Poisson algebras in [43].

Kaygorodov found an analog of customary identities for generalized Poisson
algebras and algebras of Jordan brackets [34]:

$\displaystyle{\mathfrak{g}}=\sum\limits_{i=0}^{[m/2]}\sum_{\sigma\in{\mathbb{S}}_{m}}c_{\sigma,i}\langle
x_{\sigma(1)},x_{\sigma(2)}\rangle\cdot\ldots\cdot\langle
x_{\sigma(2i-1)},x_{\sigma(2i)}\rangle\cdot{\mathfrak{D}}(x_{\sigma(2i+1)})\cdot\ldots\cdot{\mathfrak{D}}(x_{\sigma(m)}),$
(2)

where $\langle x,y\rangle:=\\{x,y\\}-\big{(}{\mathfrak{D}}(x)\cdot
y-x\cdot{\mathfrak{D}}(y)\big{)}\mbox{ and }{\mathfrak{D}}\mbox{ is the
related derivation, i.e., }{\mathfrak{D}}(x)=\\{x,1\\}.$

######  Theorem 15 (Theorem 13, [34]).

If a unital generalized Poisson algebra $\mathbf{A}$ satisfies a polynomial
identity, then $\mathbf{A}$ satisfies a polynomial identity ${\mathfrak{g}}$
of type (2).

Consequently, we obtain the following statement.

######  Theorem 16.

555An analogue of the present theorem was proven in [18]. Namely, the authors
proved that if a unital transposed Poisson algebra satisfies a
$[\cdot,\cdot]$-free polynomial identity (depending only on multiplication
$\cdot$), then the algebra satisfies an identity of the type
${\mathfrak{D}}(x_{1})\cdot\ldots\cdot{\mathfrak{D}}(x_{m})=0.$ The present
results motivate the following interesting question. Open question. Let $\bf
A$ be a commutative associative algebra with a derivation $\mathfrak{D}$
satisfying a derivation polynomial identity (i.e., a functional identity with
the derivation $\mathfrak{D}$). Is there a special type of identities for $\bf
A$?

If a unital transposed Poisson algebra $\mathbf{A}$ satisfies a polynomial
identity, then $\mathbf{A}$ satisfies polynomial identities ${\mathfrak{g}}$
and ${\mathfrak{f}}$ of the following types

${\mathfrak{g}}(x_{1},\ldots,x_{m})=\sum_{\sigma\in{\mathbb{S}}_{m}}c_{\sigma}{\mathfrak{D}}(x_{\sigma(1)})\cdot\ldots\cdot{\mathfrak{D}}(x_{\sigma(m)}),$

where ${\mathfrak{D}}$ is a derivation; or

${\mathfrak{f}}(x_{1},\ldots,x_{3m})=\sum_{\sigma\in\mathbb{S}_{m}}c_{\sigma}\prod\limits_{k=1}^{m}\Big{(}[x_{\sigma(i)}\cdot
x_{m+2k-1},x_{m+2k}]+[x_{\sigma(i)}\cdot x_{m+2k},x_{m+2k-1}]\Big{)}.$

###### Proof.

Note that, in our case, $\langle x,y\rangle=0$ and the identity (2) gives the
identity ${\mathfrak{g}}$ from our statement. Now, multiplying the identity
(2) by $\prod\limits_{k=m+1}^{3m}x_{i}$ and using the following relation,
which is a consequence of the transposed Poisson identity and the generalized
Poisson identity,

${\mathfrak{D}}(x)\cdot y\cdot z=-\frac{1}{2}\big{(}[x\cdot y,z]+[x\cdot
z,y]\big{)}.$

Hence, the identity ${\mathfrak{g}}$ gives the identity ${\mathfrak{f}}.$ ∎

###  1.5. Transposed Poisson fields

Denote by $\mathcal{Q}(\mathfrak{P})$ the field of fractions of the
commutative associative algebra $\mathfrak{P}.$ The Poisson bracket
$\\{\cdot,\cdot\\}$ on $\mathfrak{P}$ can be extended (see, for example, [30,
58, 54]) to a Poisson bracket on $\mathcal{Q}(\mathfrak{P})$ and

$\left\\{\frac{a}{b},\frac{c}{d}\right\\}=\frac{\\{a,c\\}\cdot b\cdot
d-\\{a,d\\}\cdot b\cdot c-\\{b,c\\}\cdot a\cdot d+\\{b,d\\}\cdot a\cdot
c}{b^{2}\cdot d^{2}}.$

Our aim in the present subsection is to define a similar notion to that of the
transposed Poisson case.

######  Theorem 17.

Let $(\mathfrak{L},\cdot,[\cdot,\cdot])$ be a unital transposed Poisson
algebra. Then
$(\mathcal{Q}(\mathfrak{L}),\bullet,\llbracket\cdot,\cdot\rrbracket),$ where

$\frac{a}{b}\bullet\frac{c}{d}=\frac{a\cdot c}{b\cdot d}\ \mbox{\ \ and \ \ }\
\left\llbracket\frac{a}{b},\frac{c}{d}\right\rrbracket=\frac{[a,b]\cdot c\cdot
d-a\cdot b\cdot[c,d]}{b^{2}\cdot d^{2}},$

is a unital transposed Poisson field.

###### Proof.

It is easy to see that $\bullet$ is a commutative associative multiplication
and $\llbracket\cdot,\cdot\rrbracket$ is anticommutative. Hence, we should
check only two identities to complete the proof (for simplification of our
proof the multiplication $\cdot$ will be omitted). Thanks to theorem 6, our
bracket $[x,y]$ can be written as $[x,y]=\mathfrak{D}(x)y-x\mathfrak{D}(y)$
for a derivation $\mathfrak{D}.$ Hence,

$\frac{a}{b}\bullet\left\llbracket\frac{c}{d},\frac{e}{f}\right\rrbracket-\frac{1}{2}\left(\left\llbracket\frac{a}{b}\bullet\frac{c}{d},\frac{e}{f}\right\rrbracket+\left\llbracket\frac{c}{d},\frac{a}{b}\bullet\frac{e}{f}\right\rrbracket\right)=$
$\frac{[c,d]aef-acd[e,f]}{bd^{2}f^{2}}-\frac{[ac,bd]ef-acbd[e,f]+[c,d]aebf-
cd[ac,bf]}{2b^{2}d^{2}f^{2}}=$

$(2b^{2}d^{2}f^{2})^{-1}$ | $\Big{(}$ | $2ab\mathfrak{D}(c)def-2abc\mathfrak{D}(d)ef-2abcd\mathfrak{D}(e)f+2abcde\mathfrak{D}(f)-$ |  |
---|---|---|---|---
|  | $(\mathfrak{D}(a)bcdef+ab\mathfrak{D}(c)def-a\mathfrak{D}(b)cdef-abc\mathfrak{D}(d)ef)+$ |  |
|  | $(abcd\mathfrak{D}(e)f-abcde\mathfrak{D}(f))-(ab\mathfrak{D}(c)def-abc\mathfrak{D}(d)ef)+$ |  |
|  | $(\mathfrak{D}(a)bcdef+ab\mathfrak{D}(c)def-a\mathfrak{D}(b)cdef-abcde\mathfrak{D}(f))$ | $\Big{)}$ | $=0.$

Now we should check the Jacobi identity. For a function
$f(x_{1},x_{2},y_{1},y_{2},z_{1},z_{2})$, we denote

$f(x_{1},x_{2},y_{1},y_{2},z_{1},z_{2})+f(y_{1},y_{2},z_{1},z_{2},x_{1},x_{2})+f(z_{1},z_{2},x_{1},x_{2},y_{1},y_{2})$

as $\circlearrowright f(x_{1},x_{2},y_{1},y_{2},z_{1},z_{2}).$

Hence,

$\circlearrowright\left\llbracket\frac{x_{1}}{x_{2}},\left\llbracket\frac{y_{1}}{y_{2}},\frac{z_{1}}{z_{2}}\right\rrbracket\right\rrbracket=\circlearrowright\left\llbracket\frac{x_{1}}{x_{2}},\frac{[y_{1},y_{2}]z_{1}z_{2}-y_{1}y_{2}[z_{1},z_{2}]}{y_{2}^{2}z_{2}^{2}}\right\rrbracket=$
$\frac{\circlearrowright\big{(}[x_{1},x_{2}]([y_{1},y_{2}]z_{1}z_{2}-y_{1}y_{2}[z_{1},z_{2}])\big{)}}{x_{2}^{2}y_{2}^{2}z_{2}^{2}}-\frac{\circlearrowright\big{(}x_{1}x_{2}^{3}[[y_{1},y_{2}]z_{1}z_{2}-y_{1}y_{2}[z_{1},z_{2}],y_{2}^{2}z_{2}^{2}]\big{)}}{x_{2}^{4}y_{2}^{4}z_{2}^{4}}.$

We will prove that each part of the last sum is equal to zero. For the first
summand of our sum, we have the following.

$\circlearrowright$ | $\big{(}[x_{1},x_{2}]([y_{1},y_{2}]z_{1}z_{2}-y_{1}y_{2}[z_{1},z_{2}])\big{)}=$ |
---|---|---
$\circlearrowright$ | $\big{(}\mathfrak{D}(x_{1})x_{2}\mathfrak{D}(y_{1})y_{2}z_{1}z_{2}-x_{1}\mathfrak{D}(x_{2})\mathfrak{D}(y_{1})y_{2}z_{1}z_{2}+\mathfrak{D}(x_{1})x_{2}y_{1}\mathfrak{D}(y_{2})z_{1}z_{2}-$ |
| $x_{1}\mathfrak{D}(x_{2})y_{1}\mathfrak{D}(y_{2})z_{1}z_{2}-\mathfrak{D}(x_{1})x_{2}y_{1}y_{2}\mathfrak{D}(z_{1})z_{2}+\mathfrak{D}(x_{1})x_{2}y_{1}y_{2}z_{1}\mathfrak{D}(z_{2})-$ |
| $x_{1}\mathfrak{D}(x_{2})y_{1}y_{2}\mathfrak{D}(z_{1})z_{2}+x_{1}\mathfrak{D}(x_{2})y_{1}y_{2}z_{1}\mathfrak{D}(z_{2})\big{)}$ | $=0.$

We will rewrite the second summand in a sum of two other sums in the following
way.

$\circlearrowright\big{(}x_{1}x_{2}^{3}[[y_{1},y_{2}]z_{1}z_{2}-y_{1}y_{2}[z_{1},z_{2}],y_{2}^{2}z_{2}^{2}]\big{)}=$

$\circlearrowright\big{(}x_{1}x_{2}^{3}[\mathfrak{D}(y_{1})y_{2}z_{1}z_{2}-y_{1}\mathfrak{D}(y_{2})z_{1}z_{2}-y_{1}y_{2}\mathfrak{D}(z_{1})z_{2}+y_{1}y_{2}z_{1}\mathfrak{D}(z_{2}),y_{2}^{2}z_{2}^{2}]\big{)}=$

$\circlearrowright x_{1}x_{2}^{3}\big{(}$ | $\par\mathfrak{D}(\mathfrak{D}(y_{1})y_{2}z_{1}z_{2})y_{2}^{2}z_{2}^{2}-\mathfrak{D}(y_{1})y_{2}z_{1}z_{2}\mathfrak{D}(y^{2}_{2}z^{2}_{2})-$ |
---|---|---
| $\mathfrak{D}(y_{1}\mathfrak{D}(y_{2})z_{1}z_{2})y_{2}^{2}z_{2}^{2}+y_{1}\mathfrak{D}(y_{2})z_{1}z_{2}\mathfrak{D}(y_{2}^{2}z_{2}^{2})-$ |
| $\mathfrak{D}(y_{1}y_{2}\mathfrak{D}(z_{1})z_{2})y_{2}^{2}z_{2}^{2}+y_{1}y_{2}\mathfrak{D}(z_{1})z_{2}\mathfrak{D}(y_{2}^{2}z_{2}^{2})+$ |
| $\mathfrak{D}(y_{1}y_{2}z_{1}\mathfrak{D}(z_{2}))y_{2}^{2}z_{2}^{2}-y_{1}y_{2}z_{1}\mathfrak{D}(z_{2})\mathfrak{D}(y_{2}^{2}z_{2}^{2})$ | $\big{)}.$

Hence, the first subsum is given by the following terms.

$\circlearrowright x_{1}x_{2}^{3}y_{2}^{2}z_{2}^{2}\big{(}$ | $\mathfrak{D}(\mathfrak{D}(y_{1})y_{2}z_{1}z_{2})-\mathfrak{D}(y_{1}\mathfrak{D}(y_{2})z_{1}z_{2})-\mathfrak{D}(y_{1}y_{2}\mathfrak{D}(z_{1})z_{2})+\mathfrak{D}(y_{1}y_{2}z_{1}\mathfrak{D}(z_{2}))$ | $\big{)}=$
---|---|---
$\circlearrowright x_{1}x_{2}^{3}y_{2}^{2}z_{2}^{2}\big{(}$ | $\mathfrak{D}^{2}(y_{1})y_{2}z_{1}z_{2}+\mathfrak{D}(y_{1})\mathfrak{D}(y_{2})z_{1}z_{2}+\mathfrak{D}(y_{1})y_{2}\mathfrak{D}(z_{1})z_{2}+\mathfrak{D}(y_{1})y_{2}z_{1}\mathfrak{D}(z_{2})-$ |
| $\mathfrak{D}(y_{1})\mathfrak{D}(y_{2})z_{1}z_{2}-y_{1}\mathfrak{D}^{2}(y_{2})z_{1}z_{2}-y_{1}\mathfrak{D}(y_{2})\mathfrak{D}(z_{1})z_{2}-y_{1}\mathfrak{D}(y_{2})z_{1}\mathfrak{D}(z_{2})-$ |
| $\mathfrak{D}(y_{1})y_{2}\mathfrak{D}(z_{1})z_{2}-y_{1}\mathfrak{D}(y_{2})\mathfrak{D}(z_{1})z_{2}-y_{1}y_{2}\mathfrak{D}^{2}(z_{1})z_{2}-y_{1}y_{2}\mathfrak{D}(z_{1})\mathfrak{D}(z_{2})+$ |
| $\mathfrak{D}(y_{1})y_{2}z_{1}\mathfrak{D}(z_{2})+y_{1}\mathfrak{D}(y_{2})z_{1}\mathfrak{D}(z_{2})+y_{1}y_{2}\mathfrak{D}(z_{1})\mathfrak{D}(z_{2})+y_{1}y_{2}z_{1}\mathfrak{D}^{2}(z_{2})$ | $\big{)}=$
| $\circlearrowright\big{(}x_{1}\mathfrak{D}^{2}(y_{1})z_{1}-x_{1}y_{1}\mathfrak{D}^{2}(z_{1})\big{)}(x_{2}y_{2}z_{2})^{3}+$ |
| $\circlearrowright\big{(}x_{2}\mathfrak{D}^{2}(y_{2})z_{2}-x_{2}y_{2}\mathfrak{D}^{2}(z_{2})\big{)}x_{1}y_{1}z_{1}(x_{2}y_{2}z_{2})^{2}+$ |
| $2\circlearrowright\big{(}x_{1}x_{2}\mathfrak{D}(y_{1})y_{2}z_{1}\mathfrak{D}(z_{2})-x_{1}x_{2}y_{1}\mathfrak{D}(y_{2})\mathfrak{D}(z_{1})z_{2}\big{)}(x_{2}y_{2}z_{2})^{3}=$ |
|
$2\circlearrowright\big{(}x_{1}x_{2}\mathfrak{D}(y_{1})y_{2}z_{1}\mathfrak{D}(z_{2})-x_{1}x_{2}y_{1}\mathfrak{D}(y_{2})\mathfrak{D}(z_{1})z_{2}\big{)}(x_{2}y_{2}z_{2})^{2}.$

And the second subsum is given by the following terms.

$\circlearrowright x_{1}x_{2}^{3}\mathfrak{D}(y_{2}^{2}z_{2}^{2})\big{(}$ | $\par-\mathfrak{D}(y_{1})y_{2}z_{1}z_{2}+y_{1}\mathfrak{D}(y_{2})z_{1}z_{2}+y_{1}y_{2}\mathfrak{D}(z_{1})z_{2}-y_{1}y_{2}z_{1}\mathfrak{D}(z_{2})$ | $\big{)}=$
---|---|---
$2\circlearrowright x_{1}x_{2}^{3}\big{(}$ | $-\mathfrak{D}(y_{1})y_{2}^{2}\mathfrak{D}(y_{2})z_{1}z_{2}^{3}-\mathfrak{D}(y_{1})y_{2}^{3}z_{1}z_{2}^{2}\mathfrak{D}(z_{2})+y_{1}y_{2}(\mathfrak{D}(y_{2}))^{2}z_{1}z_{2}^{3}+$ |
| $y_{1}y_{2}^{2}\mathfrak{D}(y_{2})z_{1}z_{2}^{2}\mathfrak{D}(z_{2})+y_{1}y_{2}^{2}\mathfrak{D}(y_{2})\mathfrak{D}(z_{1})z_{2}^{3}+y_{1}y_{2}^{3}\mathfrak{D}(z_{1})z_{2}^{2}\mathfrak{D}(z_{2})-$ |
| $y_{1}y_{2}^{2}\mathfrak{D}(y_{2})z_{1}z_{2}^{2}\mathfrak{D}(z_{2})-y_{1}y_{2}^{3}z_{1}z_{2}^{2}(\mathfrak{D}(z_{2}))^{2}$ | $\big{)}=$
| $2\circlearrowright\big{(}x^{2}_{2}(\mathfrak{D}(y_{2}))^{2}z_{2}^{2}-x_{2}^{2}y_{2}^{2}(\mathfrak{D}(z_{2}))^{2}\big{)}x_{1}x_{2}y_{1}y_{2}z_{1}z_{2}-$ |
| $2\circlearrowright\big{(}x_{1}x_{2}\mathfrak{D}(y_{1})\mathfrak{D}(y_{2})z_{1}z_{2}-x_{1}x_{2}y_{1}y_{2}\mathfrak{D}(z_{1})\mathfrak{D}(z_{2})\big{)}(x_{2}y_{2}z_{2})^{2}-$ |
| $2\circlearrowright\big{(}x_{1}x_{2}\mathfrak{D}(y_{1})y_{2}z_{1}\mathfrak{D}(z_{2})-x_{1}x_{2}y_{1}\mathfrak{D}(y_{2})\mathfrak{D}(z_{1})z_{2}\big{)}(x_{2}y_{2}z_{2})^{2}=$ |
|
$-2\circlearrowright\big{(}x_{1}x_{2}\mathfrak{D}(y_{1})y_{2}z_{1}\mathfrak{D}(z_{2})-x_{1}x_{2}y_{1}\mathfrak{D}(y_{2})\mathfrak{D}(z_{1})z_{2}\big{)}(x_{2}y_{2}z_{2})^{2}.$

To summarize, the sum of these two subsums gives zero and we have that the
Jacobi identity holds and we finish the proof. ∎

###  1.6. Transposed Poisson structures on nilpotent $n$-Lie algebras

Let us recall the definition of transposed Poisson structures on $n$-Lie
algebras.

######  Definition 18.

Let ${\mathfrak{L}}$ be a vector space equipped with two nonzero operations –
a bilinear operation $\cdot$ and an $n$-linear operation
$[\cdot,\ldots,\cdot].$ The triple
$({\mathfrak{L}},\cdot,[\cdot,\ldots,\cdot])$ is called a transposed Poisson
$n$-Lie algebra if $({\mathfrak{L}},\cdot)$ is a commutative associative
algebra and $({\mathfrak{L}},[\cdot,\ldots,\cdot])$ is an $n$-Lie algebra that
satisfies the following compatibility condition

$z\cdot[x_{1},\ldots,x_{n}]=\frac{1}{n}\sum\limits_{i=1}^{n}[x_{1},\ldots,z\cdot
x_{i},\ldots,x_{n}].$ (3)

######  Theorem 19.

Let $\mathfrak{L}$ be a nilpotent $k$-dimensional $n$-Lie algebra. Then
$\mathfrak{L}$ admits a non-trivial transposed Poisson $n$-Lie structure.

###### Proof.

It is easy to see that, in the case, of $n\geq k,$ $\mathfrak{L}$ has zero
multiplication and, obviously, it admits a non-trivial transposed Poisson
$n$-Lie structure. Let us now consider only the case $n<k.$ Choose the basis
$\\{e_{1},e_{2},\ldots,e_{k}\\}$ of $\mathfrak{L}$ in the following way:
$e_{1},e_{2},\ldots,e_{n}\in\mathfrak{L}\setminus\mathfrak{L}^{2}$ and
$e_{k}\in{\rm Ann}(\mathfrak{L}).$ It is easy to see that the linear mapping
$\varphi$ given by $\varphi(e_{1})=\varphi(e_{2})=\ldots=\varphi(e_{n})=e_{k}$
is a $\frac{1}{n}$-derivation of $\mathfrak{L}.$ Then the commutative
associative multiplication $\cdot$ given by $e_{i}\cdot e_{j}=e_{k}$ $(1\leq
i,j\leq n)$ gives a non-trivial structure of transposed Poisson $n$-Lie
algebra. ∎

###  1.7. Transposed Poisson $n$-Lie algebras and Poisson $n$-Lie algebras

The following result is a direct generalization of a result from [4] for
$3$-ary case.

######  Definition 20.

Let ${\mathfrak{L}}$ be a vector space equipped with two nonzero operations –
a bilinear operation $\cdot$ and an $n$-linear operation
$[\cdot,\ldots,\cdot].$ The triple
$({\mathfrak{L}},\cdot,[\cdot,\ldots,\cdot])$ is called a Poisson $n$-Lie
algebra if $({\mathfrak{L}},\cdot)$ is a commutative associative algebra and
$({\mathfrak{L}},[\cdot,\ldots,\cdot])$ is an $n$-Lie algebra that satisfies
the following compatibility condition

$[x\cdot
y,z_{2}\ldots,z_{n}]=x\cdot[y,z_{2},\ldots,z_{n}]+[x,z_{2},\ldots,z_{n}]\cdot
y.$ (4)

######  Proposition 21.

Let $(\mathfrak{L},\cdot)$ be a commutative algebra and
$(\mathfrak{L},[\cdot,\ldots,\cdot])$ be an $n$-Lie algebra. Then
$(\mathfrak{L},\cdot,[\cdot,\ldots,\cdot])$ is both a Poisson $n$-Lie and a
transposed Poisson $n$-Lie algebra if and only if

$x\cdot[y_{1},\ldots,y_{n}]=[x\cdot y_{1},\ldots,y_{n}]=0.$

###### Proof.

Let $u,x_{1},\ldots,x_{n}\in\mathfrak{L}$. Substituting Eq. (3) into Eq. (4),
we have

$nx\cdot[y_{1},\ldots,y_{n}]$ | $=$ | $\sum\limits_{i=1}^{n}[y_{1},\ldots,x\cdot y_{i},\ldots,y_{n}]$
---|---|---
| $=$ | $\sum\limits_{i=1}^{n}x\cdot[y_{1},\ldots,y_{n}]+\sum\limits_{i=1}^{n}y_{i}\cdot[y_{1},\ldots,y_{i-1},x,y_{i+1},\ldots,y_{n}].$

Substituting Eq. (4) again into the above equation, we have

$0$ | $=$ | $\sum\limits_{i=1}^{n}y_{i}\cdot[y_{1},\ldots,y_{i-1},x,y_{i+1},\ldots,y_{n}]$ |  |
---|---|---|---|---
| $=$ | $\sum\limits_{i=1}^{n}\big{(}\sum\limits_{t=1}^{i-1}[y_{1},\ldots y_{i}\cdot y_{t},\ldots,y_{i-1},x,y_{i+1},\ldots,y_{n}]\big{)}$ | $+$ |
|  | $\sum\limits_{i=1}^{n}[y_{1},\ldots y_{i-1},y_{i}\cdot x,y_{i+1},\ldots,y_{n}]$ | $+$ |
|  | $\sum\limits_{i=1}^{n}\big{(}\sum\limits_{t=i+1}^{n}[y_{1},\ldots y_{i-1},x,y_{i+1},\ldots,y_{i}\cdot y_{t},\ldots,y_{n}]\big{)}$ |  |
| $=$ | $\sum\limits_{i=1}^{n}[y_{1},\ldots y_{i-1},y_{i}\cdot x,y_{i+1},\ldots,y_{n}]$ | $=$ | $nx\cdot[y_{1},\ldots,y_{n}].$

Hence $x\cdot[y_{1},\ldots,y_{n}]=0$ and $[x\cdot y_{1},\ldots,y_{n}]=0.$

∎

###  1.8. Transposed Poisson structures and quasi-automorphisms

Let us recall the definition of quasi-automorphisms on Lie algebras.

######  Definition 22.

Let $({\mathfrak{L}},[\cdot,\cdot])$ be a Lie algebra and $\varphi$ be a
linear map. Then $\varphi$ is a quasi-automorphism on
$({\mathfrak{L}},[\cdot,\cdot])$ if there exist another linear map ${\phi},$
such that

$[\varphi(x),\varphi(y)]=\phi[x,y].$

Thanks to [4, Proposition 2.9], each transposed Poisson structure
$({\mathfrak{L}},\cdot,[\cdot,\cdot])$ introduces a quasi-automorphism
$\varphi_{h}$ on $({\mathfrak{L}},[\cdot,\cdot]),$ such that

$\varphi_{h}^{2}[x,y]=[\varphi_{h}(x),\varphi_{h}(y)],$ where
$\varphi_{h}(x)=h\cdot x$ for a fixed element $h.$

Automorphisms and scalar multiplication maps are trivial quasi-automorphisms.
Hence, we have the following statement.

######  Proposition 23.

If a Lie algebra $({\mathfrak{L}},[\cdot,\cdot])$ does not admit non-trivial
quasi-automorphisms, then $({\mathfrak{L}},[\cdot,\cdot])$ does not admit non-
trivial transposed Poisson structures.

Zhou, Feng, Kong, Wang and Chen gave the classification of quasi-automorphisms
of Borel and parabolic subalgebras of a complex finite-dimensional simple Lie
algebra of rank $l.$ Namely, they proved that in the case $l>1$, each quasi-
automorphism is trivial [77, 88]. Summarizing with Proposition 23, we have the
following statement.

######  Lemma 24.

Let ${\mathfrak{L}}$ be a complex finite-dimensional simple Lie algebra of
rank $l>1$ and $\mathfrak{p}$ be an arbitrary Borel or parabolic subalgebra of
${\mathfrak{L}}$. Then $\mathfrak{p}$ does not admit non-trivial transposed
Poisson structures.

###  1.9. Transposed Poisson structures and ${\rm Hom}$-Lie algebra structures

Let us recall the definition of ${\rm Hom}$-structures on Lie algebras.

######  Definition 25.

Let $({\mathfrak{L}},[\cdot,\cdot])$ be a Lie algebra and $\varphi$ be a
linear map. Then $({\mathfrak{L}},[\cdot,\cdot],\varphi)$ is a ${\rm Hom}$-Lie
structure on $({\mathfrak{L}},[\cdot,\cdot])$ if

$[\varphi(x),[y,z]]+[\varphi(y),[z,y]]+[\varphi(z),[x,y]]=0.$

Let us give some special cases of ${\rm Hom}$-Lie structures.

  1. $\bullet$

$({\mathfrak{L}},[\cdot,\cdot],\varphi)$ is a trivial ${\rm Hom}$-Lie
structure on $({\mathfrak{L}},[\cdot,\cdot])$ if $\varphi=\mathbb{C}\cdot{\rm
id}.$

  2. $\bullet$

$({\mathfrak{L}},[\cdot,\cdot],\varphi)$ is a central ${\rm Hom}$-Lie
structure on $({\mathfrak{L}},[\cdot,\cdot])$ if
$\varphi({\mathfrak{L}})\subseteq Z({\mathfrak{L}}).$

  3. $\bullet$

$({\mathfrak{L}},[\cdot,\cdot],\varphi)$ is a $\mathfrak{c}$-trivial ${\rm
Hom}$-Lie structure on $({\mathfrak{L}},[\cdot,\cdot])$ if $\varphi$ is a
linear combination of trivial and central ${\rm Hom}$-Lie structures.

Thanks to [4, Proposition 2.9], each transposed Poisson structure
$({\mathfrak{L}},\cdot,[\cdot,\cdot])$ introduces a ${\rm Hom}$-Lie structure
$\varphi_{h}$ on $({\mathfrak{L}},[\cdot,\cdot]),$ such that
$\varphi_{h}(x)=h\cdot x$ for a fixed element $h.$ Hence, we have the
following statement.

######  Proposition 26.

If a Lie algebra $({\mathfrak{L}},[\cdot,\cdot])$ does not admit non-trivial
${\rm Hom}$-Lie structures, then $({\mathfrak{L}},[\cdot,\cdot])$ does not
admit non-trivial transposed Poisson structures.

The subsequent list of algebras gives algebras without non-trivial ${\rm
Hom}$-Lie structures (known until now):

  1. (1)

complex finite-dimensional simple Lie algebras (dim $\neq 3$) and
superalgebras [78, 29, 85];

  2. (2)

infinite-dimensional Lie superalgebras of vectors fields [86, 70];

  3. (3)

infinite-dimensional Cartan algebras $W(n>1),$ $S(n),$ $H(2r),$ $K(2r+1)$
[79].

Hence, using Proposition 26, we have the following statement.

######  Lemma 27.

Complex finite-dimensional simple Lie algebras (dim $\neq 3$) and
superalgebras, infinite-dimensional Lie superalgebras of vectors fields, and
infinite-dimensional Cartan algebras $W(n>1),$ $S(n),$ $H(2r),$ $K(2r+1)$ do
not admit non-trivial transposed Poisson structures.

Let us give one more useful trivial observation.

######  Proposition 28.

If a Lie algebra $({\mathfrak{L}},[\cdot,\cdot])$ with one-dimensional center
$Z({\mathfrak{L}})$ admits only $\mathfrak{c}$-trivial ${\rm Hom}$-Lie
structures, then $({\mathfrak{L}},[\cdot,\cdot])$ does not admit non-trivial
transposed Poisson structures.

Thanks to [59], all ${\rm Hom}$-Lie structures on affine Kac–Moody algebras
are $\mathfrak{c}$-trivial. Hence, summarizing Proposition 28 and results from
[59], we have the following statement.

######  Lemma 29.

Complex affine Kac–Moody algebras do not admit non-trivial Transposed Poisson
structures.

###  1.10. Transposed Poisson structures on simple graded Lie algebras of
finite growth

Let us recall the definition of a complex simple graded Lie algebra of finite
growth. A complex graded Lie algebra is a Lie algebra $\mathfrak{L}$ endowed
with a decomposition $\mathfrak{L}=\bigoplus\mathfrak{L}_{i},$ such that
$[\mathfrak{L}_{i},\mathfrak{L}_{j}]\subseteq\mathfrak{L}_{i+j}$ for every
$i,j\in\mathbb{Z}.$ We always assume that every homogeneous component
$\mathfrak{L}_{i}$ is of finite dimension. A graded Lie algebra $\mathfrak{L}$
is called simple graded if $\mathfrak{L}$ is not abelian and does not contain
any non-trivial graded ideal. We say that $\mathfrak{L}$ is of finite growth
if the function $n\to{\rm dim}\ \mathfrak{L}_{n}$ is bounded by some
polynomial. In [62], Mathieu proved the following theorem, which was
conjectured by Kac.

######  Theorem 30.

Let $\mathfrak{L}$ be a graded simple Lie algebra of finite growth. Then
$\mathfrak{L}$ is isomorphic to one of the following Lie algebras:

  1. (1)

A finite-dimensional simple Lie algebra;

  2. (2)

A loop algebra;

  3. (3)

A Cartan algebra;

  4. (4)

The Witt algebra ${\rm W}$.

Xie and Liu described all ${\rm Hom}$-Lie structures on simple graded Lie
algebras of finite growth [79]. Summarizing results [62] and [79], it is
possible to prove the following theorem.

######  Theorem 31.

Let $\mathfrak{L}$ be a complex graded simple Lie algebra of finite growth. If
$\mathfrak{L}$ admits a non-trivial Transposed Poisson structure, then it is
isomorphic to one of the following Lie algebras:

  1. (1)

The Witt algebra ${\rm W}$, or

  2. (2)

The Cartan algebra ${\rm W}(1)$.

###### Proof.

Thanks to [79], we know that only a loop algebra, the Witt algebra ${\rm W}$
and the Cartan algebra ${\rm W}(1)$ have non-trivial ${\rm Hom}$-Lie
structures.

  1. $\bullet$

It is easy to see (from the description of ${\rm Hom}$-Lie structures), that a
loop algebra does not admit non-trivial Transposed Poisson structures.

  2. $\bullet$

All non-trivial transposed Poisson structures on the Witt algebra are
described in [24, Theorem 20].

  3. $\bullet$

The Cartan algebra ${\rm W}(1)$ admits the following multiplication table

$[e_{i},e_{j}]=(i-j)e_{i+j},\ \ i,j\geq-1.$

Using the same idea that was used for the description of transposed Poisson
structures on the Witt algebra [24, Theorem 20], we have that each transposed
Poisson structure on ${\rm W}(1)$ is given by the following additional
multiplication

$e_{i}\cdot e_{j}=\sum\limits_{t>0}\alpha_{t}e_{i+j+t},$

for a finite set $\\{\alpha_{t}\\}$. On the other hand, each finite set
$\\{\alpha_{t}\\}$ provides a structure of transposed Poisson algebra by the
given way.

∎

######  Remark 32.

Let us note that the Cartan algebra ${\rm W}(1)$ gives an example of a simple
Lie algebra that admits only non-unital transposed Poisson structures.

##  2\. $\frac{1}{2}$-derivations of oscillator Lie algebras and transposed
Poisson algebras

######  Definition 33.

The oscillator Lie algebra $\mathfrak{L_{\lambda}}$ is a $(2n+2)$-dimensional
Lie algebra with its canonical basis
$\mathbb{B}=\\{e_{-1},e_{0},e_{j},\check{e}_{j}\\}_{j=1,\ldots,n}$ and the Lie
brackets are given by

$[e_{-1},e_{j}]=\lambda_{j}\check{e}_{j},\ \ \
[e_{-1},\check{e}_{j}]=-\lambda_{j}e_{j},\ \ \ [e_{j},\check{e}_{j}]=e_{0},$

for $j=1,\ldots,n$ and
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{R}^{n}$ with
$0<\lambda_{1}\leq\ldots\leq\lambda_{n}$.

######  Proposition 34.

A linear map $\varphi:\mathfrak{L}_{\lambda}\rightarrow\mathfrak{L}_{\lambda}$
is a $\frac{1}{2}$-derivation of the algebra
$(\mathfrak{L}_{\lambda},[\cdot,\cdot])$ if and only if

$\varphi(e_{-1})$ | $=$ |  $\gamma e_{-1}+\mu e_{0}-\sum_{j=1}^{n}2\lambda_{j}\alpha_{j}e_{j}-\sum_{j=1}^{n}2\lambda_{j}\beta_{j}\check{e}_{j}$,
---|---|---
$\varphi(e_{0})$ | $=$ |  $\gamma e_{0}$,
$\varphi(e_{j})$ | $=$ |  $\alpha_{j}e_{0}+\gamma e_{j},\ \ \ (j=1,\ldots,n)$,
$\varphi(\check{e}_{j})$ | $=$ | $\beta_{j}e_{0}+\gamma\check{e}_{j},\ \ \ (j=1,\ldots,n).$

###### Proof.

For each $\ell\in\mathbb{B},$ we consider

$\varphi(\ell)=\varphi(\ell)_{-1}e_{-1}+\varphi(\ell)_{0}e_{0}+\sum\limits_{k=1}^{n}(\varphi(\ell)_{k}e_{k}+\varphi(\ell)_{\check{k}}\check{e}_{k})$.

Since $e_{0}=[e_{j},\check{e}_{j}]$ for each $j\in\\{1,\ldots,n\\},$ we get

$\varphi(e_{0})$ | $=$ | $\varphi([e_{j},\check{e}_{j}])=\frac{1}{2}([\varphi(e_{j}),\check{e}_{j}]+[e_{j},\varphi(\check{e}_{j})])$
---|---|---
| $=$ | $\frac{1}{2}(-\lambda_{j}\varphi(e_{j})_{-1}e_{j}+\varphi(e_{j})_{j}e_{0}-\lambda_{j}\varphi(\check{e}_{j})_{-1}\check{e}_{j}+\varphi(\check{e}_{j})_{\check{j}}e_{0}).$

Thus, $\varphi(e_{0})=\gamma e_{0},$ where
$\gamma=\frac{1}{2}(\varphi(e_{j})_{j}+\varphi(\check{e}_{j})_{\check{j}}).$

It is easy to see that

$\lambda_{j}^{2}\varphi(e_{j})$ | $=$ | $\varphi([e_{-1},[e_{j},e_{-1}]])=-\frac{1}{2}\lambda_{j}[\varphi(e_{-1}),\check{e}_{j}]-\frac{1}{4}[e_{-1},[e_{-1},\varphi(e_{j})]]+\frac{1}{4}[e_{-1},[e_{j},\varphi(e_{-1})]]$
---|---|---
| $=$ | $\frac{1}{2}\lambda_{j}^{2}\varphi(e_{-1})_{-1}e_{j}-\frac{1}{2}\lambda_{j}\varphi(e_{-1})_{j}e_{0}+\frac{1}{4}\sum\limits_{t=1}^{n}\lambda_{t}^{2}(\varphi(e_{j})_{t}+\varphi(e_{j})_{\check{t}}\check{e}_{t})+\frac{1}{4}\lambda_{j}^{2}\varphi(e_{-1})_{-1}e_{j},$

hence,

$\varphi(e_{-1})_{-1}=\varphi(e_{j})_{j},$
$\varphi(e_{-1})_{j}=-2\lambda_{j}\varphi(e_{j})_{0}$ and
$\varphi(e_{j})_{t}=0,$ for
$t\in\\{-1,1,\ldots,n,\check{1},\ldots,\check{n}\\}\setminus\\{j\\}.$

In a similar way, we have

$\varphi(e_{-1})_{-1}=\varphi(e_{\check{j}})_{\check{j}},$
$\varphi(e_{-1})_{\check{j}}=-2\lambda_{j}\varphi(e_{\check{j}})_{0}$ and
$\varphi(e_{\check{j}})_{t}=0,$ for
$t\in\\{-1,1,\ldots,n,\check{1},\ldots,\check{n}\\}\setminus\\{\check{j}\\}.$

Hence, our map $\varphi$ has the following type:

$\varphi(e_{-1})$ | $=$ |  $\gamma e_{-1}+\mu e_{0}-\sum_{j=1}^{n}2\lambda_{j}\alpha_{j}e_{j}-\sum_{j=1}^{n}2\lambda_{j}\beta_{j}\check{e}_{j}$,
---|---|---
$\varphi(e_{0})$ | $=$ |  $\gamma e_{0}$,
$\varphi(e_{j})$ | $=$ |  $\alpha_{j}e_{0}+\gamma e_{j},\ \ \ (j=1,\ldots,n)$,
$\varphi(\check{e}_{j})$ | $=$ | $\beta_{j}e_{0}+\gamma\check{e}_{j},\ \ \ (j=1,\ldots,n).$

It is easy to see that all linear maps given by the present way are
$\frac{1}{2}$-derivations. It finishes the proof.

∎

Filippov proved that each $\delta$-derivation ($\delta\neq 0,1$) gives a non-
trivial ${\rm Hom}$-Lie algebra structure [25, Theorem 1]. Hence, by
Proposition 34, we have the following corollary.

######  Corollary 35.

The oscillator Lie algebra $\mathfrak{L_{\lambda}}$ admits non-trivial ${\rm
Hom}$-Lie algebra structures.

######  Theorem 36.

Let $(\mathfrak{L}_{\lambda},[\cdot,\cdot])$ be the oscillator Lie algebra.
Then every symmetric $\frac{1}{2}$-biderivation gives a transposed Poisson
structure. If $(\mathfrak{L}_{\lambda},\cdot,[\cdot,\cdot])$ is a transposed
Poisson algebra, then $(\mathfrak{L}_{\lambda},\cdot)$ has the following
multiplication:

$e_{-1}\cdot e_{-1}$ | $=$ | $\gamma e_{-1}+\mu e_{0}-2\sum_{k=1}^{n}\lambda_{k}(\alpha_{k}e_{k}+\beta_{k}\check{e}_{k}),$
---|---|---
$e_{-1}\cdot e_{0}$ | $=$ | $\gamma e_{0},$
$e_{-1}\cdot e_{j}$ | $=$ | $\alpha_{j}e_{0}+\gamma e_{j},$
$e_{-1}\cdot\check{e}_{j}$ | $=$ | $\beta_{j}e_{0}+\gamma\check{e}_{j},$
$e_{j}\cdot e_{j}$ | $=$ | $\check{e}_{j}\cdot\check{e}_{j}=-\frac{\gamma}{2\lambda_{j}}e_{0}.$

$(\mathfrak{L}_{\lambda},\cdot,[\cdot,\cdot])$ is a Poisson algebra if and
only if $(\gamma,\alpha_{1},\ldots\alpha_{n},\beta_{1},\ldots,\beta_{n})=0.$

###### Proof.

We aim to describe the multiplication $\cdot.$ By Lemma 3, for each
$\ell\in\mathbb{B}$, there is a related $\frac{1}{2}$-derivation
$\varphi_{\ell}$ of $(\mathfrak{L}_{\lambda},[\cdot,\cdot])$ such that
$\varphi_{\ell_{1}}(\ell_{2})=\ell_{1}\cdot\ell_{2}=\varphi_{\ell_{2}}(\ell_{1}),$
with $\ell_{1},\ell_{2}\in\mathbb{B}.$ By Proposition 34, for each
$\ell\in\mathbb{B}$, we have

$\varphi_{\ell}(e_{-1})$ | $=$ |  $\gamma^{\ell}e_{-1}+\mu^{\ell}e_{0}-\sum_{k=1}^{n}2\lambda_{k}\alpha^{\ell}_{k}e_{k}-\sum_{k=1}^{n}2\lambda_{k}\beta^{\ell}_{k}\check{e}_{k}$,
---|---|---
$\varphi_{\ell}(e_{0})$ | $=$ |  $\gamma^{\ell}e_{0}$,
$\varphi_{\ell}(e_{j})$ | $=$ |  $\alpha^{\ell}_{j}e_{0}+\gamma^{\ell}e_{j},\ \ \ (j=1,\ldots,n)$,
$\varphi_{\ell}(\check{e}_{j})$ | $=$ |  $\beta^{\ell}_{j}e_{0}+\gamma^{\ell}\check{e}_{j},\ \ \ (j=1,\ldots,n)$.

In the first, for all $i,j\in\\{1,\ldots,n\\},$ we have

$\alpha^{e_{i}}_{j}e_{0}+\gamma^{e_{i}}e_{j}$ | $=$ | $\varphi_{e_{i}}(e_{j})$ | $=$ | $\varphi_{e_{j}}(e_{i})$ | $=$ |  $\alpha^{e_{j}}_{i}e_{0}+\gamma^{e_{j}}e_{i}$,
---|---|---|---|---|---|---
$\alpha^{\check{e}_{i}}_{j}e_{0}+\gamma^{\check{e}_{i}}e_{j}$ | $=$ | $\varphi_{\check{e}_{i}}(e_{j})$ | $=$ | $\varphi_{e_{j}}(\check{e}_{i})$ | $=$ |  $\beta^{e_{j}}_{i}e_{0}+\gamma^{e_{j}}\check{e}_{i}$,
$\beta^{\check{e}_{i}}_{j}e_{0}+\gamma^{\check{e}_{i}}\check{e}_{j}$ | $=$ | $\varphi_{\check{e}_{i}}(\check{e}_{j})$ | $=$ | $\varphi_{\check{e}_{j}}(\check{e}_{i})$ | $=$ | $\beta^{\check{e}_{j}}_{i}e_{0}+\gamma^{\check{e}_{j}}\check{e}_{i},$

hence $\beta^{\check{e}_{j}}_{i}=\beta^{\check{e}_{i}}_{j}$,
$\alpha^{e_{i}}_{j}=\alpha^{e_{j}}_{i}$,
$\alpha^{\check{e}_{i}}_{j}=\beta^{e_{j}}_{i}$ and
$\gamma^{e_{j}}=\gamma^{\check{e}_{j}}=0$.

Second, for all $j\in\\{1,\ldots,n\\},$ we have

$\gamma^{e_{-1}}e_{0}$ | $=$ | $\varphi_{e_{-1}}(e_{0})$ | $=$ | $\varphi_{e_{0}}(e_{-1})$ | $=$ |  $\gamma^{e_{0}}e_{-1}+\mu^{e_{0}}e_{0}-2\sum_{k=1}^{n}\lambda_{k}(\alpha_{k}^{e_{0}}e_{k}+\beta_{k}^{e_{0}}\check{e}_{k})$,
---|---|---|---|---|---|---
$0$ | $=$ | $\varphi_{e_{j}}(e_{0})$ | $=$ | $\varphi_{e_{0}}(e_{j})$ | $=$ |  $\alpha^{e_{0}}_{j}e_{0}+\gamma^{e_{0}}e_{j}$,
$0$ | $=$ | $\varphi_{\check{e}_{j}}(e_{0})$ | $=$ | $\varphi_{e_{0}}(\check{e}_{j})$ | $=$ | $\beta^{e_{0}}_{j}e_{0}+\gamma^{e_{0}}\check{e}_{j},$

hence $\alpha^{e_{0}}_{j}=\beta^{e_{0}}_{j}=\gamma^{e_{0}}=0$ and
$\gamma^{e_{-1}}=\mu^{e_{0}}$.

Also, for all $j\in\\{1,\ldots,n\\},$ we have

$\alpha^{e_{-1}}_{j}e_{0}+\gamma^{e_{-1}}e_{j}$ | $=$ | $\varphi_{e_{-1}}(e_{j})$ | $=$ | $\varphi_{e_{j}}(e_{-1})$ | $=$ | $\mu^{e_{j}}e_{0}-2\sum_{k=1}^{n}\lambda_{k}(\alpha^{e_{j}}_{k}e_{k}+\beta^{e_{j}}_{k}\check{e}_{k}),$
---|---|---|---|---|---|---
$\beta^{e_{-1}}_{j}e_{0}+\gamma^{e_{-1}}\check{e}_{j}$ | $=$ | $\varphi_{e_{-1}}(\check{e}_{j})$ | $=$ | $\varphi_{\check{e}_{j}}(e_{-1})$ | $=$ | $\mu^{\check{e}_{j}}e_{0}-2\sum_{k=1}^{n}\lambda_{k}(\alpha^{\check{e}_{j}}_{k}e_{k}+\beta^{\check{e}_{j}}_{k}\check{e}_{k}),$

hence

$\gamma^{e_{-1}}=-2\lambda_{j}\alpha^{e_{j}}_{j},$ | $\alpha^{e_{j}}_{j}=\beta^{\check{e}_{j}}_{j},$ | $\alpha^{\check{e}_{j}}_{k}=0,$ | $\beta^{e_{j}}_{k}=0,$
---|---|---|---
$\alpha^{e_{j}}_{k}=0\ (j\neq k),$ | $\beta^{\check{e}_{j}}_{k}=0\ (j\neq k),$ | $\alpha^{e_{-1}}_{j}=\mu^{e_{j}},$ |  $\beta^{e_{-1}}_{j}=\mu^{\check{e}_{j}}$.

It follows that, for all $j\in\\{1,\ldots,n\\}$, the multiplication $\cdot$
can be written as

$e_{-1}\cdot e_{-1}$ | $=$ | $\gamma e_{-1}+\mu e_{0}-2\sum_{k=1}^{n}\lambda_{k}(\alpha_{k}e_{k}+\beta_{k}\check{e}_{k}),$
---|---|---
$e_{-1}\cdot e_{0}$ | $=$ | $\gamma e_{0},$
$e_{-1}\cdot e_{j}$ | $=$ | $\alpha_{j}e_{0}+\gamma e_{j},$
$e_{-1}\cdot\check{e}_{j}$ | $=$ | $\beta_{j}e_{0}+\gamma\check{e}_{j},$
$e_{j}\cdot e_{j}$ | $=$ | $\check{e}_{j}\cdot\check{e}_{j}=-\frac{\gamma}{2\lambda_{j}}e_{0}.$

It is easy to see that $\cdot$ is a commutative associative multiplication.

∎

###  2.1. Isomorphism problem for transposed Poisson algebras on generic
oscillator Lie algebras

An oscillator Lie algebra is called generic if
$0<\lambda_{1}<\ldots<\lambda_{n}$ and
$\lambda_{i}+\lambda_{j}\neq\lambda_{k}$ for all $1\leq i<j<k\leq n.$ It is
easy to see that
$(\mathfrak{L}_{\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\\}},[\cdot,\cdot])$
is isomorphic to
$(\mathfrak{L}_{\left\\{1,\frac{\lambda_{2}}{\lambda_{1}},\ldots,\frac{\lambda_{n}}{\lambda_{1}}\right\\}},[\cdot,\cdot]).$
Hence, we will consider only the case $\lambda_{1}=1.$

For proving the main result of this part of the paper, we should give the
following useful Lemma.

######  Lemma 37.

Let $\phi$ be an automorphism of $(\mathfrak{L}_{\lambda},[\cdot,\cdot])$.
Then $\phi$ has the following types:

$\phi(e_{-1})$ | $=$ | $\pm e_{-1}+\nu e_{0}+\sum\limits_{i=1}^{n}\nu_{i}e_{i}+\sum\limits_{i=1}^{n}\check{\nu}_{i}\check{e}_{i},$
---|---|---
$\phi(e_{0})$ | $=$ | $\pm\xi e_{0},$
$\phi(e_{i})$ | $=$ | $\frac{\check{\nu}_{i}\check{\mu}_{i}\mp\nu_{i}\mu_{i}}{\lambda_{i}}e_{0}+\mu_{i}e_{i}\mp\check{\mu}_{i}\check{e}_{i},$
$\phi(\check{e}_{i})$ | $=$ | $\frac{-\check{\nu}_{i}\check{\mu}_{i}\mp\nu_{i}\mu_{i}}{\lambda_{i}}e_{0}+\check{\mu}_{i}e_{i}\pm\mu_{i}\check{e}_{i},$

where $\xi=\mu_{i}^{2}+\check{\mu}_{i}^{2}.$

###### Proof.

Let us say that

$\phi(e_{m})=\alpha^{m}_{-1}e_{-1}+\alpha^{m}_{0}e_{0}+\sum\limits_{i=1}^{n}(\alpha_{i}^{m}e_{m}+\beta_{i}^{m}\check{e}_{m}),$
where $m\in\\{-1,0,1,\ldots,n\\}.$

Then, for each $j\in\\{1,\ldots,n\\},$ we have

$\lambda_{j}^{2}\phi(e_{j})=-[\phi(e_{-1}),[\phi(e_{-1}),\phi(e_{j})]]=(\alpha^{-1}_{-1})^{2}\sum\limits_{i=1}^{n}\lambda_{i}^{2}(\alpha_{i}^{j}e_{i}+\beta_{i}^{j}\check{e}_{j})+\alpha^{-1}_{-1}\sum\limits_{i=1}^{n}(\alpha_{i}^{-1}\alpha_{i}^{j}-\beta_{i}^{-1}\beta_{i}^{j})e_{0}.$

Hence $\lambda_{i}\neq\lambda_{j},$ we have that
$\phi(e_{j})=\alpha_{0}^{j}e_{0}+\alpha_{j}^{j}e_{j}+\beta^{j}_{j}\check{e}_{j}$
and $\alpha_{-1}^{-1}=\pm 1.$ In a similar way, it is possible to find that
$\phi(\check{e}_{j})$ has the same type. Obviously, $\phi$ has the form from
our statement. ∎

For our main theorem from this part, we need to define the following set

$\mathbb{C}_{>0}=\\{a+b{\rm i}\\}_{a>0}\cup\\{b{\rm i}\\}_{b\geq 0}.$

######  Theorem 38.

Let $(\mathfrak{L}_{\lambda},[\cdot,\cdot])$ be the generic oscillator Lie
algebra. If $(\mathfrak{L}_{\lambda},\cdot,[\cdot,\cdot])$ is a transposed
Poisson algebra, then $(\mathfrak{L}_{\lambda},\cdot)$ has the following
multiplication.

  1. (A)

If $(\mathfrak{L}_{\lambda},\cdot)$ is non-nilpotent, then, for a $\gamma\neq
0,$ we have:

$e_{-1}\cdot e_{-1}=\gamma e_{-1},\ e_{-1}\cdot e_{0}=\gamma e_{0},\
e_{-1}\cdot e_{j}=\gamma e_{j},\
e_{-1}\cdot\check{e}_{j}=\gamma\check{e}_{j},\ e_{j}\cdot
e_{j}=\check{e}_{j}\cdot\check{e}_{j}=-\frac{\gamma}{2\lambda_{j}}e_{0},$

corresponding transposed Poisson algebra
$(\mathfrak{L}_{\\{\lambda,\gamma\\}},\cdot,[\cdot,\cdot])$ is isomorphic only
to $(\mathfrak{L}_{\\{\lambda,-\gamma\\}},\cdot,[\cdot,\cdot])$; all these
algebras are non-Poisson.

  2. (B)

If $(\mathfrak{L}_{\lambda},\cdot)$ is nilpotent, then there is a sequence
$\\{\beta_{q}\in\mathbb{C}_{>0}\\}_{1\leq q\leq n},$ where the first nonzero
element is equal to $1,$ and $(\mathfrak{L}_{\lambda},\cdot)$ has the
following multiplication table:

    1. (a)

$e_{-1}\cdot
e_{-1}=e_{0}-2\sum\limits_{j=1}^{n}\lambda_{j}\beta_{j}\check{e}_{j},\
e_{-1}\cdot\check{e}_{j}=\beta_{j}\check{e}_{j},$

    2. (b)

$e_{-1}\cdot e_{-1}=-2\sum\limits_{j=1}^{n}\lambda_{j}\beta_{j}\check{e}_{j},\
e_{-1}\cdot\check{e}_{j}=\beta_{j}\check{e}_{j}.$

The corresponding transposed Poisson algebras
$(\mathfrak{L}_{\lambda,\beta},\cdot,[\cdot,\cdot])$ are not isomorphic; and
they are non-Poisson if there exists $\beta_{j}\neq 0$.

###### Proof.

Let us consider the transposed Poisson structure
$(\mathfrak{L}_{\lambda},\cdot,[\cdot,\cdot])$ on the generic oscillator Lie
$(\mathfrak{L}_{\lambda},[\cdot,\cdot])$ and $\cdot$ is given as in theorem
36. Under the action of a positive automorphism of the Lie algebra
$(\mathfrak{L}_{\lambda},[\cdot,\cdot]),$ given in Lemma 37, we rewrite the
multiplication table of $(\mathfrak{L}_{\lambda},\cdot)$ by the following way:

$e_{-1}\cdot e_{-1}$ | $=$ | $\gamma e_{-1}+\left(\frac{\mu+\gamma\nu}{\xi}+\sum\limits_{i=1}^{n}\frac{\gamma(\nu_{i}^{2}+\check{\nu}_{i}^{2})}{2\lambda_{i}\xi}\right)e_{0}+$
---|---|---
|  | $\sum\limits_{k=1}^{n}\left(\frac{\gamma(\mu_{k}\nu_{k}-\check{\mu}_{k}\check{\nu}_{k})-2\lambda_{k}(\alpha_{k}\mu_{k}-\beta_{k}\check{\mu}_{k})}{\xi}\right)e_{k}+\sum\limits_{k=1}^{n}\left(\frac{\gamma(\check{\mu}_{k}\nu_{k}+\mu_{k}\check{\nu}_{k})-2\lambda_{k}(\alpha_{k}\check{\mu}_{k}+\beta_{k}\mu_{k})}{\xi}\right)\check{e}_{k}$
$e_{-1}\cdot e_{0}$ | $=$ | $\gamma e_{0}$
$e_{-1}\cdot e_{k}$ | $=$ | $\left(\frac{\gamma(\mu_{k}\nu_{k}-\check{\mu}_{k}\check{\nu}_{k})-2\lambda_{k}(\alpha_{k}\mu_{k}-\beta_{k}\check{\mu}_{k})}{2\lambda_{k}\xi}\right)e_{0}+\gamma e_{k}$
$e_{-1}\cdot\check{e}_{k}$ | $=$ | $\left(\frac{\gamma(\check{\mu}_{k}\nu_{k}+\mu_{k}\check{\nu}_{k})-2\lambda_{k}(\alpha_{k}\check{\mu}_{k}+\beta_{k}\mu_{k})}{2\lambda_{k}\xi}\right)e_{0}+\gamma\check{e}_{k}$
$e_{k}\cdot e_{k}$ | $=$ | $\check{e}_{k}\cdot\check{e}_{k}=-\frac{\gamma}{2\lambda_{k}}e_{0}.$

Hence, we will consider two different situations.

  1. $\gamma\neq 0.$

By taking

$\nu_{k}=\frac{2\alpha_{k}\lambda_{k}}{\gamma},$
$\check{\nu}_{k}=\frac{2\beta_{k}\lambda_{k}}{\gamma}$ and
$\nu=-\sum\limits_{i=1}^{n}\frac{\gamma(\nu_{i}^{2}+\check{\nu}_{i}^{2})}{2\gamma\lambda_{i}}-\frac{\mu}{\gamma},$

we have a multiplication table from part $(A)$ in our statement. It is easy to
see that, under the action of a positive automorphism of the Lie algebra
$(\mathfrak{L}_{\lambda},[\cdot,\cdot]),$ given in Lemma 37, all
$\cdot$-multiplications give non-isomorphic transposed Poisson structures.
Under the action of a negative automorphism given in Lemma 37, we have the
isomorphism between structures corresponding to $\gamma$ and $-\gamma.$
Insofar as $\gamma\neq 0,$ $(\mathfrak{L}_{\lambda},\cdot)$ is non-nilpotent.

  2. $\gamma=0.$

In this case, $(\mathfrak{L}_{\lambda},\cdot)$ is a nilpotent algebra. Note
that,

    1. $\bullet$

If $\\{\alpha_{k}\\}_{1\leq k\leq n}=0,$ then transposed Poisson structures
corresponding to $\\{\beta_{1},\ldots,\beta_{j},\ldots,\beta_{n}\\}$ and
$\\{\beta_{1},\ldots,-\beta_{j},\ldots,\beta_{n}\\}$ are isomorphic.

    2. $\bullet$

If $(\alpha_{k},\beta_{k})\neq 0,$ then after an action of a suitable
automorphism, we can suppose that $\alpha_{k}\neq 0$ and $\beta_{k}\neq 0.$ In
the case $(\alpha_{k},\beta_{k})=0$ after each automorphism action, we will
take the same situation $(\alpha_{k},\beta_{k})=0.$ Let $t$ be the maximal
number such that $(\alpha_{1},\beta_{1})=\ldots=(\alpha_{t-1},\beta_{t-1})=0.$

We should consider two subcases.

    1. $\mu\neq 0.$

By taking

$\mu=\xi,$ $\check{\mu}_{t}=\alpha_{t}$ and
$\mu_{k}=\frac{\beta_{k}\check{\mu}_{k}}{\alpha_{k}}$ for all $k$ such that
$(\alpha_{k},\beta_{k})\neq 0.$

Hence, $(\mathfrak{L}_{\lambda},\cdot)$ has a multiplication table from the
part $(B.a)$ of our statement.

    2. $\mu=0.$

By taking

$\check{\mu}_{t}=\alpha_{t}$ and
$\mu_{k}=\frac{\beta_{k}\check{\mu}_{k}}{\alpha_{k}}$ for all $k$ such that
$(\alpha_{k},\beta_{k})\neq 0.$

Hence, $(\mathfrak{L}_{\lambda},\cdot)$ has a multiplication table from the
part $(B.b)$ of our statement.

∎

##  3\. Open questions

######  Question 1 (Pasha Zusmanovich).

Formulate the notion of ”transposed” for any variety of algebras/operads with
two $n$-ary operations. Could this be done in either operadic or in
categorical theoretical language? This general notion should explain the
appearance of ”scaling by 2” in the definition of transposed Poisson algebras.
What would be the transposed analogues of dendriform, post-Lie, Novikov-
Poisson, etc.? What would be self-transposed algebras/operads?

Thanks to [24, Theorem 8], we know that if a Lie algebra does not admit non-
trivial $\frac{1}{2}$-derivations, then it does not admit non-trivial
$\frac{1}{2}$-biderivations and transposed Poisson structures. All algebras
which admit non-trivial $\frac{1}{2}$-derivations (from the present paper and
[24, 84]) also admit non-trivial $\frac{1}{2}$-biderivations and non-trivial
transposed Poisson structures. The last results motivate the following
question.

######  Question 2.

Is there a Lie algebra that admits non-trivial $\frac{1}{2}$-derivations and
does not admit non-trivial $\frac{1}{2}$-biderivations? Is there a Lie algebra
that admits non-trivial $\frac{1}{2}$-derivations and
$\frac{1}{2}$-biderivations, and does not admit non-trivial transposed Poisson
structures?

One of the first examples of non-trivial transposed Poisson algebras was given
for the Witt algebra [24, Theorem 20]. The central extension of the Witt
algebra is the Virasoro algebra and it has no non-trivial
$\frac{1}{2}$-derivations, and $\frac{1}{2}$-biderivations and transposed
Poisson structures [24, Theorem 27]. Thanks to [79] we know that the Cartan
(infinite-dimensional) algebras $S_{n}$ and $H_{n}$ do not have non-trivial
transposed Poisson structures (and $\frac{1}{2}$-derivations,
$\frac{1}{2}$-biderivations). It is easy to see that if the dimension of the
annihilator of an algebra is greater than $1,$ then it admits non-trivial
transposed Poisson structures (and $\frac{1}{2}$-derivations,
$\frac{1}{2}$-biderivations). Dzhumadildaev proved that the dimensions of
annihilators of central extensions of $S_{n}$ and $H_{n}$ is greater than $1$
in [19]. It follows that there are infinite-dimensional Lie algebras without
non-trivial transposed Poisson structures (and $\frac{1}{2}$-derivations,
$\frac{1}{2}$-biderivations), but their central extensions admit non-trivial.
The last results motivate the following question.

######  Question 3.

Is there a Lie algebra that does not admit non-trivial transposed Poisson
structures ($\frac{1}{2}$-derivations, $\frac{1}{2}$-biderivations), but its
(indecomposable) one-dimensional central extension admits them?

There are no non-trivial transposed Poisson structures defined on a complex
semisimple finite-dimensional Lie algebra [24, Corollary 9]. On the other
hand, the Schrödinger algebra, which is the semidirect product of a simple and
a nilpotent algebra, also does not have non-trivial transposed Poisson
structures (other perfect non-simple Lie algebras without nontrivial
$\frac{1}{2}$-derivations (also known as Galilean algebras) can be found in
[40]). Theorem 19 states that each finite-dimensional nilpotent Lie algebra
has a non-trivial transposed Poisson structure ($\frac{1}{2}$-derivations,
$\frac{1}{2}$-biderivations). Results from [40] state that each finite-
dimensional solvable non-nilpotent Lie algebra has a non-trivial transposed
Poisson structure ($\frac{1}{2}$-derivations, $\frac{1}{2}$-biderivations).
The last results motivate the following question.

######  Question 4.

Is there a non-perfect (non-solvable) Lie algebra that does not admit non-
trivial transposed Poisson structures ($\frac{1}{2}$-derivations,
$\frac{1}{2}$-biderivations)? Is there a perfect (non-simple) Lie algebra that
admits non-trivial transposed Poisson structures ($\frac{1}{2}$-derivations,
$\frac{1}{2}$-biderivations)?

The description of $\frac{1}{2}$-derivations gives a good criterion for the
detection of Lie algebras that do not admit non-trivial transposed Poisson
structures. Another criterion can be given by using ${\rm Hom}$-Lie structures
on Lie algebras (Proposition 26). However, the last criterion is not
sufficient: the simple $3$-dimensional algebra $\mathfrak{sl}_{2}$ has non-
trivial ${\rm Hom}$-Lie structures [78], but it does not admit non-trivial
transposed Poisson structures [24, Corollary 9]. Filippov proved that each
$\delta$-derivation ($\delta\neq 0,1$) gives a non-trivial ${\rm Hom}$-Lie
algebra structure [25, Theorem 1]. The last results motivate the following
questions.

######  Question 5.

Describe Lie algebras such that each ${\rm Hom}$-Lie algebra structure gives a
$\frac{1}{2}$-derivation. Which types of ${\rm Hom}$-Lie structures can
guarantee the existence of non-trivial transposed Poisson structures on a Lie
algebra?

Let us denote the associative algebra generated by $\frac{1}{2}$-derivations
of $\mathfrak{L}$ as $\Delta_{\frac{1}{2}}(\mathfrak{L})$. The Witt algebra
${\rm W}$ has the following interesting property:
$\mathfrak{Der}(\Delta_{\frac{1}{2}}({\rm W}))\cong{\rm W}$ [24, Theorem 19].
The last results motivate the following question.

######  Question 6.

Is there a Lie algebra $\mathfrak{L}$ ($\not\cong\rm W$) such that
$\mathfrak{Der}(\Delta_{\frac{1}{2}}(\mathfrak{L}))\cong\mathfrak{L}$?

Thanks to [24, Corollary 9], there are no non-trivial transposed Poisson
structures defined on a complex semisimple finite-dimensional Lie algebra;
Theorem 6 establishes that there are no non-trivial transposed Poisson
structures defined on a complex semisimple commutative associative algebra. On
the other hand, there are simple Lie algebras (the Witt algebra ${\rm W}$ and
the Cartan algebra ${\rm W}(1)$) that admit non-trivial transposed Poisson
structures, but in these cases, the associative part is not simple (Theorem
31). The last results motivate the following question.

######  Question 7.

Is there a (not necessarily commutative) transposed Poisson algebra
$(\mathfrak{L},\cdot,[\cdot,\cdot])$ such that its Lie and associative parts
are simple (semisimple, prime, semiprime) algebras?

Combing results from Theorem 6 and [21, Proposition 28], we have that the
Kantor product of a unital transposed Poisson algebra gives a new transposed
Poisson algebra. Furthermore, the Kantor product of transposed Poisson
algebras constructed on the Witt algebra gives new transposed Poisson algebras
[21, Proposition 27]. The last results motivate the following question.

######  Question 8.

Is there a transposed Poisson algebra such that the algebra constructed by the
Kantor product of multiplications (see [21]) is not a transposed Poisson
algebra?

The Kantor double gives way for the construction of Jordan superalgebras from
Poisson algebras and from algebras of Jordan brackets [31]. As we can see from
Corollary 11, each unital transposed Poisson algebra, by the Kantor double
process, gives a Jordan superalgebra. Moreover, the Kantor double of the
commutator of a Novikov-Poisson algebra is a Jordan superalgebra [87]. Thanks
to [22], each transposed Poisson superalgebra under the Kantor double process
gives a Jordan superalgebra. The last results motivate the following question.

######  Question 9.

Describe and study the class of superalgebras (=subvariety of Jordan
superalgebras) obtained by the Kantor double process from transposed Poisson
algebras.

The celebrated Amitsur–Levitsky theorem states that the algebra ${\rm
Mat}_{k}$ of $k\times k$ matrices over a commutative ring ${\rm R}$ satisfies
the identity $s_{2k}=0.$ Furthermore, every associative PI algebra satisfies
$(s_{k})^{l}=0$ by Amitsur’s theorem. Farkas defined customary Poisson
polynomials and proved that every Poisson PI algebra satisfies some customary
identity [20]. Recently, the Farkas’ theorem was established for generic
Poisson algebras in [43], for generalized Poisson algebras in [34] and unital
transposed Poisson algebras (Theorem 16). The last results motivate the
following question.

######  Question 10.

Is there a special type of polynomial identities for (not necessarily unital)
transposed Poisson algebras satisfying a polynomial identity?

In 1972, Regev proved that the tensor product of two associative PI algebras
is an associative PI algebra. Recently, a similar result was obtained for
Poisson algebras [64]. The last results motivate the following question.

######  Question 11.

Is a tensor product of two transposed Poisson PI algebras a transposed Poisson
PI algebra?

It is known that the tensor product of two Poisson algebras gives a Poisson
algebra [64] and also the tensor product of two transposed Poisson algebras
gives a transposed Poisson algebra [4]. In the case of the standard
multiplication of tensor Poisson algebras, it is not true for $F$-manifold
algebras [51]. On the other hand, in the case of generalized Poisson algebras,
their tensor product does not necessarily give a new generalized Poisson
algebra [90]. The last results motivate the following question.

######  Question 12 (Pasha Zusmanovich).

Find identities for algebras formed by the tensor product of a transposed
Poisson algebra and a Poisson algebra.

The notion of a depolarized Poisson algebra (i.e., a binary algebra such that
the under commutator and the anticommutator products give a Poisson algebra)
was introduced and studied in [60, 28]. The last results motivate the
following question.

######  Question 13.

666Askar Dzhumadildaev proved that weak Leibniz algebras (defined by the
following two identities $(ab-ba)c=2a(bc)-2b(ac)$ and $a(bc-
cb)=2(ab)c-2(ac)b$) give the depolarization of transposed Poisson algebras
(see, https://cmup.fc.up.pt/nonassociative2020/wp-
content/uploads/2022/07/booklet-principal-1.pdf, page 37).

Define and study the variety of depolarized transposed Poisson algebras.

In 1950, Hall constructed a basis for free Lie algebras. Later on: Shestakov
constructed a basis for free Poisson algebras; Shestakov and Zhukavets
constructed a basis for free Poisson–Malcev superalgebra with one generator;
Kaygorodov, Shestakov and Umirbaev constructed a basis for free generic
Poisson algebras in [41]; and Kaygorodov constructed a basis for free unital
generalized Poisson algebras in [34]. The last results motivate the following
question.

######  Question 14.

777Roughly speaking, a basis of a free Poisson algebra is a basis of a
polynomial ring on a free Lie algebra. Our first impression (Chengming Bai
mentioned in one of his talks, that it was also the first impression of his
team) was that a basis of a free transposed Poisson algebra can be found as a
free Lie algebra on a polynomial ring, but it is not true.

Construct a basis for free transposed Poisson algebras.

In 1930, Magnus proved one of the most important theorems of the combinatorial
group theory. Let $G=\langle x_{1},x_{2},\ldots,x_{n}\ |\ r=1\rangle$ be a
group defined by a single cyclically reduced relator $r.$ If $x_{n}$ appears
in $r,$ then the subgroup of $G$ generated by $x_{1},\ldots,x_{n-1}$ is a free
group, freely generated by $x_{1},\ldots,x_{n-1}$. He called it the
Freiheitssatz and used it to give several applications, the decidability of
the word problem for groups with a single defining relation among them. The
Freiheitssatz was studied in various varieties of groups and algebras. Thus,
Shirshov established it for Lie algebras; Romanovskii researched it for
solvable and nilpotent groups; Makar-Limanov proved it for associative
algebras over a field of characteristic zero. Recently it was confirmed for
Novikov and right-symmetric algebras, as well as for Poisson algebras [57] and
generic Poisson algebras [43]. The last results motivate the following
question.

######  Question 15.

Is the Freiheitssatz true for transposed Poisson algebras?

In 1942, Jung proved that all automorphisms of the polynomial algebra in two
variables are tame. Later, in 1970, Makar-Limanov established the same result
for the free associative algebra in two variables. Cohn proved that the
automorphisms of a free Lie algebra with a finite set of generators are tame.
Shestakov and Umirbaev proved that polynomial algebras and free associative
algebras in three variables in the case of characteristic zero have wild
automorphisms [69, 73]. At the same time, Abdykhalykov, Mikhalev and Umirbaev
found wild automorphisms in two generated free Leibniz algebras. It was
recently proved that free Poisson and generic Poisson algebras in two
variables do not have wild automorphisms [41, 55]. The last results motivate
the following question.

######  Question 16.

Are the automorphisms of a two-generated free transposed Poisson algebra tame?

Makar-Limanov and Umirbaev [56] proved an analog of the Bergman Centralizer
Theorem: the centralizer of every non-constant element in a free Poisson
algebra is a polynomial algebra in one variable. The last results motivate the
following question.

######  Question 17 (Zerui Zhang).

Describe the centralizer of a non-constant element in a free transposed
Poisson algebra.

Thanks to Corollary 11, we know that each unital transposed Poisson algebra is
an algebra of Jordan brackets. Martínez, Shestakov and Zelmanov have shown
that all Jordan brackets are embeddable into Poisson brackets [61]. Hence, all
unital transposed Poisson brackets are embeddable into Poisson brackets. The
last results motivate the following question.

######  Question 18.

Are transposed Poisson brackets embeddable into Poisson brackets?

Thanks to a famous relationship between Poisson algebras and Jordan
superalgebras, also known as Kantor double, the description of simple Jordan
superalgebras describes simple Poisson algebras. Also, recently, simple
generalized Poisson algebras and simple generalized Gerstenhaber algebras (odd
generalized Poisson superalgebras) have been described [12, 13]. The last
results motivate the following question.

######  Question 19.

888Thanks to [22], there are no non-trivial complex simple finite-dimensional
transposed Poisson algebras.

Classify simple transposed Poisson algebras.

The study of some generalizations of Poisson algebras (noncommutative Poisson
algebras; Leibniz-Poisson algebras [15]; generic Poisson algebras[41], etc)
motivate the consideration of some generalizations of transposed Poisson
algebras.

######  Question 20.

Define and study the following generalizations of transposed Poisson algebras:

  1. (a)

noncommutative transposed Poisson algebras;

  2. (b)

transposed Poisson-Leibniz (in particular, symmetric Leibniz) algebras;

  3. (c)

transposed generic Poisson algebras.

One of the important questions in the study of Poisson and transposed Poisson
algebras is a description of (transposed) Poisson algebra structures on a
certain Lie algebra or on a certain associative algebra. The most useful way
for a description of Poisson structures on a certain associative algebra is
given by the study of derivations and biderivations of this algebra. On the
other hand, a description of transposed Poisson structures on a certain Lie
algebra may be given by the study of $\frac{1}{2}$-derivations and
$\frac{1}{2}$-biderivations of this algebra. Recently, a constructive method
for describing Poisson algebra structures on a certain Lie algebra has been
given in a study of Leibniz bialgebras and symmetric Leibniz algebras [1]. The
last results and Theorem 6 motivate the following question.

######  Question 21.

Give a constructive method for describing transposed Poisson algebras on a
certain non-unital associative (not necessarily commutative) algebra.

It is known that the Poisson bracket admits an extension to the field of
fractions [30, 58, 54, 41]. As we proved in Theorem 17, a unital transposed
Poisson bracket admits an extension to the field of fractions. The last
results motivate the following question.

######  Question 22.

Does the bracket in Theorem 17 give an extension of a transposed Poisson
algebra to the field of fractions in the non-unital case?

The universal multiplicative enveloping algebra of free Poisson and free
generic Poisson fields are studied in [58, 41]. In particular, in these
papers, it was proved that the universal multiplicative enveloping algebra is
a free ideal ring. The last results motivate the following question.

######  Question 23.

Is the universal multiplicative enveloping algebra of a free transposed
Poisson field a free ideal ring?

Multiplicative enveloping algebras are often much simpler than the algebras
themselves, and in the case where the variety is not obvious, they can help a
lot. So, a basis of the universal enveloping algebra $P^{e}$ of a free Poisson
algebra $P$ is constructed and it is proved that the left dependency of a
finite number of elements of $P^{e}$ over $P^{e}$ is algorithmically
recognizable [74]. So it’s useful to label the following question.

######  Question 24 (Vladimir Dotsenko).

Construct a basis of the universal enveloping algebra $TP^{e}$ of a free
transposed Poisson algebra $TP$. Is left dependency of a finite number of
elements of $TP^{e}$ over $TP^{e}$ algorithmically recognizable?

It is known that each Novikov-Poisson algebra under commutator product on non-
associative multiplication gives a transposed Poisson algebra [4]. Let us say
that a transposed Poisson algebra is special if it can be embedded into a
Novikov-Poisson algebra relative to the commutator bracket. Similarly, let us
say that a transposed Poisson algebra is $D$-special (from “differentially”)
if it embeds into a commutative algebra with a derivation relative to the
bracket $[x,y]=\mathfrak{D}(x)y-x\mathfrak{D}(y).$ Obviously, every
$D$-special transposed Poisson algebra is a special one. It is known, that the
class of special Jordan algebras (i.e. embedded into associative algebras
relative to the multiplication $x\circ y=xy+yx$) is a quasivariety, but it is
not a variety of algebras [72]. On the other hand, the class of all special
Gelfand-Dorfman algebras (i.e. embedded into Poisson algebras with derivation
relative to the multiplication $x\circ y=xd(y)$) is closed with respect to
homomorphisms and thus forms a variety [45]. The last results motivate the
following question.

######  Question 25 (Pavel Kolesnikov).

Whether the class of special (resp., $D$-special) transposed Poisson algebras
forms a variety or a quasivariety?

Let us call by special identities all identities that hold on all special
transposed Poisson algebras but do not hold on the class of all transposed
Poisson algebras. The study of special and non-special identities is a popular
topic in non-associative algebras (see, for example, Jordan algebra case in
[27], Gelfand-Dorfman algebra case in [46], dialgebras [47], Jordan trialgebra
and post-Jordan algebra cases in [3], etc.). The last results motivate the
following question.

######  Question 26 (Pavel Kolesnikov).

Find the identities separating the variety generated by special (resp.,
$D$-special) transposed Poisson algebras in the variety of all transposed
Poisson algebras. Find the identities separating the variety generated by
$D$-special transposed Poisson algebras in the variety generated by special
transposed Poisson algebras.

Also known as each two-generated Jordan algebra is special [71]; each one-
generated Jordan dialgebra is special [76]; each $2$-dimensional Gelfand-
Dorfman algebra is special [45]. The last results motivate the following
question.

######  Question 27.

Are one-generated transposed Poisson algebras special?

Among Gelfand–Dorfman algebras, a special place is occupied by special ones;
i.e., those that can be embedded into differential Poisson algebras (see
[46]). Namely, a Gelfand–Dorfman algebra $V$ with operations $[\cdot,\cdot]$
and $*$ is special if there exists a Poisson algebra
$(P,\cdot,\\{\cdot,\cdot\\})$ and derivation $d$ such that $V\subseteq P$ and
$[u,v]=\\{u,v\\},$ with $u*v=d(u)v$ for all $u,v\in V.$ Recently. Kolesnikov
and Nesterenko proved that each transposed Poisson algebra obtained from a
Novikov-Poisson algebra is a special Gelfand–Dorfman algebra [44]. Sartayev
proved that transposed Poisson algebras satisfy special identities of
Gelfand–Dorfman algebras [68]. The last results motivate the following
question.

######  Question 28 (Pavel Kolesnikov, Bauyrzhan Sartayev).

Are transposed Poisson algebras special Gelfand–Dorfman algebras?

On the geometric point of view, the variety of $n$-dimensional algebras
defined by a family of polynomial identities gives a subvariety in
$\mathbb{C}^{n^{3}}.$ The geometric classification of a variety of algebras is
a description of his geometric variety: defining irreducible components, their
dimensions, and rigid algebras (see, [2, 23, 42] and references in therein).
The variety of transposed Poisson algebras gives a subvariety in
$\mathbb{C}^{2n^{3}}.$ The geometric classification of complex $3$-dimensional
transposed Poisson algebras is given in [7]. Neretin in his paper [65]
considered $n$-dimensional associative, commutative and Lie algebras and gave
the upper bound for dimensions of their geometric varieties. The last results
motivate the following question.

######  Question 29.

Give the upper bound for dimensions of geometric varieties of $n$-dimensional
transposed Poisson algebras.

The “Koszul” property of an (quadratric) operad is quite important. The
explicit definition needs more notions on operads [26, 52]. If an operad ${\rm
P}$ is Koszul, one can gives the $\infty$-structure of such an operad, ${\rm
P}_{\infty}$-algebra. For example, Lie, associative, pre-Lie, and Poisson are
Koszul and hence we have ${\rm L}_{\infty}$-, ${\rm A}_{\infty}$-, ${\rm
PL}_{\infty}$-, ${\rm P}_{\infty}$-algebras.

######  Question 30 (Li Guo).

999Here is a short proof of the fact that the operad $\mathrm{TPA}$ of
transposed Poisson algebras is not Koszul (communicated by Vladimir Dotsenko).
Using Gröbner bases for Operads
(http://irma.math.unistra.fr/~dotsenko/Operads.html), one finds that the
dimensions of components of the operad $\mathrm{TPA}$ up to parity five are
given by 1,2,6,20,74. Moreover, this operad is known to be self-dual under the
Koszul duality for operads. Therefore, if it were Koszul, the exponential
generating function $f(x)$ of dimensions of its components would satisfy the
functional equation $f(-f(-x))=x$. However, using the dimensions of components
listed above, we find $f(-f(-x))=x+7/30x^{5}+O(x^{6})$, and therefore the
operad $\mathrm{TPA}$ is not Koszul.101010Another proof is given by Askar
Dzhumadildaev: he defined the weak Leibniz algebras, as algebras satisfying
$(ab-ba)c=2a(bc)-2b(ac)$ and $a(bc-cb)=2(ab)c-2(ac)b.$ By his theorem, weak
Leibniz operad is not Koszul; operads of weak Leibniz algebras and transposed
Poisson algebras are isomorphic.

Whether the operad of the transposed Poisson is Koszul? If so, there might an
${\rm TPA}_{\infty}$-algebra structure.

Some recent works about the cohomology theory of Poisson algebras [67, 6] are
motivating the following question.

######  Question 31.

Describe the cohomology theory of transposed Poisson algebras.

Poisson algebras are the semi-limit of quantization deformation of commutative
associative algebras into associative algebras. $F$-manifold algebras are the
semi-limit of quantization deformation of commutative associative algebras (or
commutative pre-Lie algebras) into pre-Lie algebras. The last results motivate
the following question.

######  Question 32 (Chengming Bai).

Into which algebras, the transposed Poisson algebras are the semi-limit of
quantization deformation of commutative algebras?

The study of commutative and noncommutative Poisson bialgebras [50, 66] are
motivating the following question.

######  Question 33 (Chengming Bai).

Define and study transposed Poisson bialgebras.

The study of double Poisson algebras [75] is motivating the following
question.

######  Question 34.

Define and study double transposed Poisson algebras.

Bai, Bai, Guo and Wu proved that each transposed Poisson algebra
$({\mathfrak{L}},\cdot,[\cdot,\cdot])$ gives a transposed Poisson $3$-Lie
algebra by defining the following multiplication:

$[x,y,z]=\mathfrak{D}(x)\cdot[y,z]-\mathfrak{D}(y)\cdot[x,z]+\mathfrak{D}(z)\cdot[x,y].$

The last results motivate the following question.

######  Question 35 (Chengming Bai, Ruipu Bai, Li Guo & Yong Wu).

Let $n\geq 2$ be an integer. Let $({\mathfrak{L}},\cdot,[\cdot,\ldots,\cdot])$
be a transposed Poisson $n$-Lie algebra and let $\mathfrak{D}$ be a derivation
of $({\mathfrak{L}},\cdot)$ and $({\mathfrak{L}},[\cdot,\ldots,\cdot])$.
Define an $(n+1)$-ary operation

$\llbracket
x_{1},\ldots,x_{n+1}\rrbracket:=\sum\limits_{i=1}^{n+1}(-1)^{i+1}{\mathfrak{D}}(x_{i})\cdot[x_{1},\ldots,\hat{x_{i}},\ldots,x_{n+1}],$

where $\hat{x_{i}}$ means that the $i$-th entry is omitted. Is
$({\mathfrak{L}},\cdot,\llbracket\cdot,\ldots,\cdot\rrbracket)$ a transposed
Poisson $(n+1)$-Lie algebra?

Cantarini and Kac classified all complex linearly compact (generalized)
Poisson $n$-Lie algebras in [14], which motivates the following question.

######  Question 36.

Give examples and classify simple transposed Poisson $n$-Lie algebras.

It is known, that the variety of Gerstenhaber algebras coincides with odd
Poisson superalgebras (see, for example, [13]). It motivates the following
question.

######  Question 37.

Define and study transposed Gerstenhaber algebras.

It is known [4] that the tensor product of transposed Poisson algebras is
equipped with a canonical transposed Poisson algebra structure. On the level
of operads, this means that the operad $\mathrm{TPA}$ of transposed Poisson
algebras is a Hopf operad. Many natural examples of Hopf operads arise in
algebraic topology when one computes the homology of a topological operad.
This suggests the following question. One may define, by analogy with
transposed Poisson algebras, and transposed Gerstenhaber algebras, which are
given by similar axioms, but with the shifted Lie bracket of homological
degree one (and appropriate signs arising from the “Koszul sign rule”). The
corresponding operad seems to be a Hopf operad as well.

######  Question 38 (Vladimir Dotsenko).

Is it the homology operad of some operad made of topological spaces? In the
case of the Gerstenhaber operad, the corresponding topological operad is the
operad of little 2-disks, one of the protagonists of the operad theory at its
early days.

## References

  * [1] Albuquerque H., Barreiro E., Benayadi S., Boucetta M., Sánchez J.M., Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras, Journal of Geometry and Physics, 160 (2021), 103939.
  * [2] Alvarez M., Kaygorodov I., The algebraic and geometric classification of nilpotent weakly associative and symmetric Leibniz algebras, Journal of Algebra, 588 (2021), 278–314.
  * [3] Bagherzadeh F., Bremner M., Madariaga S., Jordan trialgebras and post-Jordan algebras, Journal of Algebra, 486 (2017), 360–395.
  * [4] Bai C., Bai R., Guo L., Wu Y., Transposed Poisson algebras, Novikov-Poisson algebras, and 3-Lie algebras, arXiv:2005.01110.
  * [5] Ballesteros A., Herranz F., Lie bialgebra quantizations of the oscillator algebra and their universal $R$-matrices, Journal of Physics A, 29 (1996), 15, 4307–4320.
  * [6] Bao Y.-H., Ye Yu, Cohomology structure for a Poisson algebra: II, Science China Mathematics, 64 (2021), 5, 903–920.
  * [7] Beites P., Fernández Ouaridi A., Kaygorodov I., The algebraic and geometric classification of transposed Poisson algebras, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 117 (2023), 2, 55.
  * [8] Biggs R., Remsing C., Some remarks on the oscillator group, Differential Geometry and its Applications, 35 (2014), suppl., 199–209.
  * [9] Billig Yu., Towards Kac-van de Leur conjecture: locality of superconformal algebras, Advances in Mathematics, 400 (2022), 108295.
  * [10] Boucetta M., Medina A., Solutions of the Yang-Baxter equations on quadratic Lie groups: the case of oscillator groups, Journal of Geometry and Physics, 61 (2011), 12, 2309–2320.
  * [11] Calvaruso G., Zaeim A., On the symmetries of the Lorentzian oscillator group, Collectanea Mathematica, 68 (2017), 1, 51–67.
  * [12] Cantarini N., Kac V., Classification of linearly compact simple Jordan and generalized Poisson superalgebras, Journal of Algebra, 313 (2007), 1, 100–124.
  * [13] Cantarini N., Kac V., Classification of linearly compact simple rigid superalgebras, International Mathematics Research Notices, 17 (2010), 3341–3393.
  * [14] Cantarini N., Kac V., Classification of linearly compact simple Nambu-Poisson algebras, Journal of Mathematical Physics, 57 (2016), 5, 051701.
  * [15] Casas J., Datuashvili T., Noncommutative Leibniz-Poisson algebras, Communications in Algebra, 34 (2006), 7, 2507–2530.
  * [16] Crampé N., van de Vijver W., Vinet L., Racah problems for the oscillator algebra, the Lie algebra $\mathfrak{sl}_{n}$, and multivariate Krawtchouk polynomials, Annales Henri Poincaré, 21 (2020), 12, 3939–3971.
  * [17] Dotsenko V., Algebraic structures of $F$-manifolds via pre-Lie algebras, Annali di Matematica Pura ed Applicata, 198 (2019), 2, 517–527.
  * [18] Dotsenko V., Ismailov N., Umirbaev U., Polynomial identities in Novikov algebras, Mathematische Zeitschrift, 303 (2023), 3, 60.
  * [19] Dzhumadildaev A., Central extensions of infinite-dimensional Lie algebras, Functional Analysis and Its Applications, 26 (1992), 4, 247–253.
  * [20] Farkas D., Poisson polynomial identities, Communications in Algebra, 26 (1998), 2, 401–416.
  * [21] Fehlberg Júnior R., Kaygorodov I., On the Kantor product, II, Carpathian Mathematical Publications, 14 (2022), 2, 543–563.
  * [22] Fernández Ouaridi A., On the simple transposed Poisson algebras and Jordan superalgebras, arXiv:2305.13848.
  * [23] Fernández Ouaridi A., Kaygorodov I., Khrypchenko M., Volkov Yu., Degenerations of nilpotent algebras, Journal of Pure Applied Algebra, 226 (2022), 3, 106850.
  * [24] Ferreira B. L. M., Kaygorodov I., Lopatkin V., $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 115 (2021), 3, 142.
  * [25] Filippov V., $\delta$-Derivations of Lie algebras, Siberian Mathematical Journal, 39 (1998), 6, 1218–1230.
  * [26] Ginzburg V., Kapranov M., Koszul duality for operads, Duke Mathematical Journal, 76 (1994), 1, 203–272.
  * [27] Glennie C., Some identities valid in special Jordan algebras but not valid in all Jordan algebras, Pacific Journal of Mathematics, 16 (1966), 47–59.
  * [28] Goze M., Remm E., Poisson algebras in terms of non-associative algebras, Journal of Algebra, 320 (2008), 1, 294–317.
  * [29] Jin Q., Li X., ${\rm Hom}$-Lie algebra structures on semi-simple Lie algebras, Journal of Algebra, 319 (2008), 4, 1398–1408.
  * [30] Kaledin D., Normalization of a Poisson algebra is Poisson, Proceedings of the Steklov Institute of Mathematics, 264 (2009), 1, 70–73.
  * [31] Kantor I., Jordan and Lie superalgebras determined by a Poisson algebra, Algebra and analysis (Tomsk, 1989), 55–80, Amer. Math. Soc. Transl. Ser. 2, 151, Amer. Math. Soc., Providence, RI, 1992.
  * [32] Kaygorodov I., Generalized Kantor Double, Vestnik Samarskogo Gosudarstvennogo Universiteta, 78 (2010), 4, 42–50.
  * [33] Kaygorodov I., $\delta$-superderivations of semisimple finite-dimensional Jordan superalgebras, Mathematical Notes, 91 (2012), 1-2, 187–197.
  * [34] Kaygorodov I., Algebras of Jordan brackets and generalized Poisson algebras, Linear and Multilinear Algebra, 65 (2017), 6, 1142–1157.
  * [35] Kaygorodov I., Khrypchenko M., Poisson structures on finitary incidence algebras, Journal of Algebra, 578 (2021), 402–420.
  * [36] Kaygorodov I., Khrypchenko M., Transposed Poisson structures on Block Lie algebras and superalgebras, Linear Algebra and Its Applications, 656 (2023), 167–197.
  * [37] Kaygorodov I., Khrypchenko M., Transposed Poisson structures on Witt type algebras, Linear Algebra and its Applications, 665 (2023), 196–210.
  * [38] Kaygorodov I., Khrypchenko M., Transposed Poisson structures on generalized Witt algebras and Block Lie algebras, arXiv:2302.00403.
  * [39] Kaygorodov I., Khrypchenko M., Transposed Poisson structures on the Lie algebra of upper triangular matrices, arXiv:2305.00727.
  * [40] Kaygorodov I., Lopatkin V., Zhang Z., Transposed Poisson structures on Galilean and solvable Lie algebras, Journal of Geometry and Physics, 187 (2023), 104781.
  * [41] Kaygorodov I., Shestakov I., Umirbaev U., Free generic Poisson fields and algebras, Communications in Algebra, 46 (2018), 4, 1799–1812.
  * [42] Kaygorodov I., Volkov Yu., Degenerations of Filippov algebras, Journal of Mathematical Physics, 61 (2020), 2, 021701.
  * [43] Kolesnikov P., Makar-Limanov L., Shestakov I., The Freiheitssatz for generic Poisson algebras, SIGMA, 10 (2014), 115.
  * [44] Kolesnikov P., Nesterenko A., Conformal Envelopes of Novikov–Poisson Algebras, Siberian Mathematical Journal, 64 (2023), 598–610.
  * [45] Kolesnikov P., Sartayev B., On the special identities of Gelfand-Dorfman algebras, Experimental Mathematics, 2022, DOI: 10.1080/10586458.2022.2041134.
  * [46] Kolesnikov P., Sartayev B., Orazgaliev A., Gelfand-Dorfman algebras, derived identities, and the Manin product of operads, Journal of Algebra, 539 (2019), 260–284.
  * [47] Kolesnikov P., Voronin V., On special identities for dialgebras, Linear and Multilinear Algebra, 61 (2013), 3, 377–391.
  * [48] Kosmann-Schwarzbach Y., From Poisson algebras to Gerstenhaber algebras, Annales de l’Institut Fourier (Grenoble), 46 (1996), 5, 1243–1274.
  * [49] Laraiedh I., Silvestrov S., Transposed ${\rm Hom}$-Poisson and ${\rm Hom}$-pre-Lie Poisson algebras and bialgebras, arXiv:2106.03277
  * [50] Liu J., Bai C., Sheng Y., Noncommutative Poisson bialgebras, Journal of Algebra, 556 (2020), 35–66.
  * [51] Liu J., Sheng Y., Bai C., Corrigendum to ”F-manifold algebras and deformation quantization via pre-Lie algebras” [J. Algebra 559 (2020) 467–495], Journal of Algebra, 574 (2021), 571–572.
  * [52] Loday J.-L., Vallette B., Algebraic operads. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 346. Springer, Heidelberg, 2012. xxiv+634 pp.
  * [53] Ma T., Li B., Transposed ${\rm BiHom}$-Poisson algebras, Communications in Algebra, 51 (2023), 2, 528–551.
  * [54] Makar-Limanov L., Shestakov I., Polynomial and Poisson dependence in free Poisson algebras and free Poisson fields, Journal of Algebra, 349 (2012), 372–379.
  * [55] Makar-Limanov L., Turusbekova U., Umirbaev U., Automorphisms and derivations of free Poisson algebras in two variables, Journal of Algebra, 322 (2009), 9, 3318–3330.
  * [56] Makar-Limanov L., Umirbaev U., Centralizers in free Poisson algebras, Proceedings of the American Mathematical Society, 135 (2007), 7, 1969–1975.
  * [57] Makar-Limanov L., Umirbaev U., The Freiheitssatz for Poisson algebras, Journal of Algebra, 328 (2011), 495–503.
  * [58] Makar-Limanov L., Umirbaev U., Free Poisson fields and their automorphisms, Journal of Algebra and its Applications, 15 (2016), 10, 1650196.
  * [59] Makhlouf A., Zusmanovich P., ${\rm Hom}$-Lie structures on Kac-Moody algebras, Journal of Algebra, 515 (2018), 278–297.
  * [60] Markl M., Remm E., Algebras with one operation including Poisson and other Lie-admissible algebras, Journal of Algebra, 299 (2006), 1, 171–189.
  * [61] Martínez C., Shestakov I., Zelmanov E., Jordan superalgebras defined by brackets, Journal of the London Mathematical Society (2), 64 (2001), 2, 357–368.
  * [62] Mathieu O., Classification of simple graded Lie algebras of finite growth, Inventiones Mathematicae, 108 (1992), 3, 455–519.
  * [63] Medina A., Groupes de Lie munis de métriques bi-invariantes, Tohoku Mathematical Journal (2), 37 (1985), 4, 405–421.
  * [64] Mishchenko S., Petrogradsky V., Regev A., Poisson PI algebras, Transactions of the American Mathematical Society, 359 (2007), 10, 4669–4694.
  * [65] Neretin Yu., An estimate for the number of parameters defining an $n$-dimensional algebra, Mathematics of the USSR-Izvestiya, 30 (1988), 2, 283–294
  * [66] Ni X., Bai C., Poisson bialgebras, Journal of Mathematical Physics, 54 (2013), 2, 023515.
  * [67] Penkava M., Pichereau A., $\mathbb{Z}_{2}$-graded Poisson algebras, their deformations and cohomology in low dimensions, Transformation Groups, 23 (2018), 4, 1091–1127.
  * [68] Sartayev B., Some generalizations of the variety of transposed Poisson algebras, arXiv:2305.12869.
  * [69] Shestakov I., Umirbaev U., Tame and wild automorphisms of rings of polynomials in three variables, Journal of the American Mathematical Society, 17 (2004), 197–227.
  * [70] Sun L., Liu W., ${\rm Hom}$-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields, Open Mathematics, 15 (2017), 1, 1332–1343.
  * [71] Shirshov A., Selected works of A. I. Shirshov, Contemporary Mathematicians. Birkhäuser Verlag, Basel, 2009. viii+242 pp.
  * [72] Sverchkov S., A quasivariety of special Jordan algebras, Algebra and Logic, 22 (1983), 5, 563–573.
  * [73] Umirbaev U., The Anick automorphism of free associative algebras, Journal für die reine und angewandte Mathematik, 605 (2007), 165–178.
  * [74] Umirbaev U., Universal enveloping algebras and universal derivations of Poisson algebras, Journal of Algebra, 354 (2012), 77–94.
  * [75] Van den Bergh M., Double Poisson algebras, Transactions of the American Mathematical Society, 360 (2008), 11, 5711–5769.
  * [76] Voronin V., Special and exceptional Jordan dialgebras, Journal of Algebra and its Applications, 11 (2012), 2, 1250029.
  * [77] Wang D., Chen Zh., Quasi-automorphisms of Lie algebras, Communications in Algebra, 39 (2011), 7, 2388–2395.
  * [78] Xie W., Jin Q., Liu W., ${\rm Hom}$-structures on semi-simple Lie algebras, Open Mathematics, 13 (2015), 1, 617–630.
  * [79] Xie W., Liu W., ${\rm Hom}$-structures on simple graded Lie algebras of finite growth, Journal of Algebra and Its Applications, 16 (2017), 8, 1750154.
  * [80] Xu P., Noncommutative Poisson algebras, American Journal of Mathematics, 116 (1994), 1, 101–125.
  * [81] Xu X., Novikov-Poisson algebras, Journal of Algebra, 190 (1997), 2, 253–279.
  * [82] Yang Ya., Tang X., Khudoyberdiyev A., Transposed Poisson structures on Schrodinger algebra in $(n+1)$-dimensional space-time, arXiv:2303.08180.
  * [83] Yao Y., Ye Y., Zhang P., Quiver Poisson algebras, Journal of Algebra, 312 (2007), 2, 570–589.
  * [84] Yuan L., Hua Q., $\frac{1}{2}$-(bi)derivations and transposed Poisson algebra structures on Lie algebras, Linear and Multilinear Algebra, 70 (2022), 22, 7672–7701
  * [85] Yuan J., Liu W., ${\rm Hom}$-structures on finite-dimensional simple Lie superalgebras, Journal of Mathematical Physics, 56 (2015), 6, 061702.
  * [86] Yuan J. X., Sun L. P., Liu W. D., ${\rm Hom}$-Lie superalgebra structures on infinite-dimensional simple Lie superalgebras of vector fields, Journal of Geometry and Physics, 84 (2014), 1–7.
  * [87] Zakharov A., Novikov–Poisson algebras and superalgebras of Jordan Brackets, Communications in Algebra, 42 (2014), 5, 2285–2298.
  * [88] Zhou L., Feng C., Kong X., Quasi-automorphisms of Borel subalgebras (Chinese), Nanjing Daxue Xuebao Shuxue Bannian Kan, 28 (2011), 1, 106–117.
  * [89] Zusmanovich P., On $\delta$-derivations of Lie algebras and superalgebras, Journal of Algebra, 324 (2010), 12, 3470–3486.
  * [90] Zusmanovich P., On contact brackets on the tensor product, Linear and Multilinear Algebra, 70 (2022), 19, 4695–4706.