Preprint

Article

Generalized Reynolds Operators on Hom-Lie Triple Systems

Altmetrics

Downloads

Views

143

Comments

A peer-reviewed article of this preprint also exists.

Yunpeng Xiao,

Wen Teng^*,Fengshan Long

Yunpeng Xiao,

Wen Teng^*,Fengshan Long

This version is not peer-reviewed

Submitted:

31 December 2023

Posted:

02 January 2024

You are already at the latest version

Alerts

Abstract

Uchino first initiated the study of generalized Reynolds operators on associative algebras. Recently, related research has become a hot topic. In this paper, we first introduce the notion of generalized Reynolds operators on Hom-Lie triple systems associated to a representation and a 3-cocycle. Then, we develop cohomology of generalized Reynolds operators on Hom-Lie triple systems with coefficients in a suitable representation. As applications, we use the first cohomology group to classify linear deformations and we study the obstruction class of an extendable order $n$ deformation. Finally, we introduce and investigate Hom-NS-Lie triple system as the underlying structure of generalized Reynolds operators on Hom-Lie triple systems.

Keywords:

Subject: Computer Science and Mathematics - Algebra and Number Theory

1. Introduction

Lie triple system first appeared in Cartan’s work [4] on Riemannian geometry. Since then, Jacobson [14,15] studied Lie triple systems from Jordan theory and quantum mechanics. Lie triple systems extend the classical theory of Lie algebras and Lie groups by introducing a trilinear product, capturing the interplay between three elements. After that, Lie triple systems have found applications in diverse fields such as quantum mechanics, differential geometry and numerical analysis of differential equations. As a Hom-type algebra [11] generalization of Lie triple system, Hom-Lie triple system was introduced by Yau in [29]. Further, Ma et al. [20] established the cohomology, central extensions and deformations of Hom-Lie triple systems. More research on Hom-Lie triple systems have been developed, see [1,13,24,30] and references cited therein.

The notion of Rota-Baxter operators on associative algebras was introduced by Baxter [3] in his study of the fluctuation theory. Later, the notion of a relative Rota-Baxter operator (also called an

O

-operator) on a Lie algebra was independently introduced by Kupershmidt [17] to better understand the classical Yang-Baxter equation. Recently, relative Rota-Baxter operators were widely studied, see [2,5,6,18,21,26]. In addition, other operators related to (relative) Rota-Baxter operators are constantly emerging. Among them is Reynolds operator in fluid dynamics, which was originally proposed by Reynolds [23] in his renowned work on fluctuation theory. And then, Kamp

\overset{´}{e}

de F

\overset{´}{e}

riet [16] created the notion of the Reynolds operator as a mathematical subject in general. Inspired by the twisted Poisson structure, Uchino [22] introduced the generalized Reynolds operators on associative algebras, also known as twisted Rota-Baxter operators, and studied its relationship with NS-algebras.

In recent years, Das [8] introduced the cohomology of generalized Reynolds operators on associative algebras and considers NS-algebras as the underlying structure motivated by Uchino’s work. He also developed the notions of generalized Reynolds operators on Lie algebras and NS-Lie algebras in [9]. Generalized Reynolds operators on other algebraic structures have also been widely studied, including 3-Lie algebras [7,12], 3-Hom-Lie algebras [19], Lie-Yamaguti algebras [25], Lie triple systems [10,25] and Lie supertriple systems [27].

Inspired by these works, we propose generalized Reynolds operators on Hom-Lie triple systems, investigate the corresponding cohomology theory which will be used to describe deformations and establish Hom-NS-Lie triple system as the underlying structure in present paper.

The paper is organized as follows. In Section 2, we recall some basic notions and facts about Hom-Lie triple systems. In Section 3, we introduce the notion of generalized Reynolds operators on a Lie triple system and we give some constructions. In Section 4, we develop the cohomologies of generalized Reynolds operators on Hom-Lie triple systems with coefficients in a suitable representation.. In Section 5, we study linear deformations and higher order deformations of generalized Reynolds operators on Hom-Lie triple systems via the cohomology theory. In Section 6, we introduce the notion of Hom-NS-Lie triple systems, which is the underlying algebraic structure of generalized Reynolds operators on Hom-Lie triple systems.

Throughout this paper,

K

denotes a field of characteristic zero. All the vector spaces and (multi)linear maps are taken over

K

2. Preliminaires

In this section, we will briefly recall representations and cohomology of Hom-Lie triple systems from [29] and [20].

A Hom-Lie triple system (Hom-L.t.s.) is a triplet

(L, [-, -, -], α)

in which

L

is a vector space together with a trilinear operation

[-, -, -] : L \times L \times L \to L

and a linear map

α : L \to L

, called the twisted map, satisfying

α ([a, b, c]) = [α (a), α (b), α (c)]

such that

\begin{matrix} [a, b, c] + [b, a, c] = 0, \end{matrix}

(2.1)

\begin{matrix} [a, b, c] + [c, a, b] + [b, c, a] = 0, \end{matrix}

(2.2)

\begin{matrix} [α (a), α (b), [x, y, z]] = [[a, b, x], α (y), α (z)] + [α (x), [a, b, y], α (z)] + [α (x), α (y), [a, b, z]], \end{matrix}

(2.3)

where

x, y, z, a, b \in L

. In particular,

(L, [-, -, -], α)

is called regular Hom-Lie triple system if

α

is an algebraic automorphism of

L

A homomorphism between two Hom-Lie triple systems

(L_{1}, {[-, -, -]}_{1}, α_{1})

and

(L_{2}, [-, -,

{-]}_{2}, α_{2})

is a linear map

φ : L_{1} \to L_{2}

satisfying

φ (α_{1} (x)) = α_{2} (φ (x)), φ ({[x, y, z]}_{1}) = {[φ (x), φ (y), φ (z)]}_{2}, \forall x, y, z \in L_{1} .

Example 2.1.

Let

(L, [-, -], α)

be a Hom-Lie algebra, then

(L, [-, -, -], α)

is a Hom-Lie triple system, where

[x, y, z] = [[x, y], α (z)], \forall x, y, z \in L .

Note that Yamaguti [28] introduced the representation and cohomology theory of Lie triple system. Furthermore, based on Yamaguti’s work, the authors in [20] developed the representation and cohomology theory of Hom-Lie triple system, which can be described as follows.

Definition 2.2.

[20] A representation of a Hom-Lie triple system

(L, [-, -, -], α)

on a Hom-vector space

(V, β)

is a bilinear map

θ : L \times L \to End (V)

, such that for all

x, y, a, b \in L

\begin{matrix} θ (α (x), α (y)) \circ β = β \circ θ (x, y), \end{matrix}

(2.4)

\begin{matrix} θ (α (a), α (b)) θ (x, y) - θ (α (y), α (b)) θ (x, a) - θ (α (x), [y, a, b]) \circ β + D (α (y), α (a)) θ (x, b) = 0, \end{matrix}

(2.5)

\begin{matrix} θ (α (a), α (b)) D (x, y) - D (α (x), α (y)) θ (a, b) + θ ([x, y, a], α (b)) \circ β + θ (α (a), [x, y, b]) \circ β = 0, \end{matrix}

(2.6)

where

D (x, y) = θ (y, x) - θ (x, y)

. We also denote a representation of

L

(V, β)

(V, β; θ)

. In particular,

(V, β; θ)

is called regular representation of

L

if β is an automorphism of vector space V.

Example 2.3.

Let

(L, [-, -, -], α)

be a Hom-Lie triple system. Define bilinear map

R : L \times L \to End (L), (a_{1}, a_{2}) \mapsto (x \mapsto [x, a_{1}, a_{2}]),

with

L (a_{1}, a_{2}) (x) = R (a_{2}, a_{1}) x - R (a_{1}, a_{2}) x = [a_{1}, a_{2}, x]

. Then,

(L, α; R)

is a representation of the Hom-Lie triple system

L

, which is called the adjoint representation of

L

Let

(V, β; θ)

be a representation of a Hom-Lie triple system

(L, [-, -, -], α)

. Denote the

(2 n + 1)

-cochains of

L

with coefficients in representation

(V, β; θ)

\begin{matrix} C_{HLts}^{2 n + 1} (L, V) : = {f \in Hom (L^{\otimes 2 n + 1}, V) | β (f (a_{1}, \dots, a_{2 n + 1}) = f (α (a_{1}), \dots, α (a_{2 n + 1})), \\ f (a_{1}, \dots, a_{2 n - 2}, a, b, c) + f (a_{1}, \dots, a_{2 n - 2}, b, a, c) = 0, ⥁_{a, b, c} f (a_{1}, \dots, a_{2 n - 2}, a, b, c) = 0} . \end{matrix}

For

n \geq 1,

let

δ : C_{HLts}^{2 n - 1} (L, V) \to C_{HLts}^{2 n + 1} (L, V)

be the corresponding coboundary operator of the Hom-Lie triple system

(L, [-, -, - [, α)

with coefficients in the representation

(V, β; θ)

, More precisely, for

a_{1}, \dots, a_{2 n + 1} \in L

and

f \in C_{HLts}^{2 n - 1} (L, V)

, as

\begin{matrix} δ f (a_{1}, \dots, a_{2 n + 1}) \\ = & θ (α^{n - 1} (a_{2 n}), α^{n - 1} (a_{2 n + 1})) f (a_{1}, \dots, a_{2 n - 1}) - θ (α^{n - 1} (a_{2 n - 1}), α^{n - 1} (a_{2 n + 1})) f (a_{1}, \dots, a_{2 n - 2}, a_{2 n}) \\ + \sum_{i = 1}^{n} {(- 1)}^{i + n} D (α^{n - 1} (a_{2 i - 1}), α^{n - 1} (a_{2 i})) f (a_{1}, \dots, a_{2 i - 2}, a_{2 i + 1}, \dots, a_{2 n + 1}) \\ + \sum_{i = 1}^{n} \sum_{j = 2 i + 1}^{2 n + 1} {(- 1)}^{i + n + 1} f (α (a_{1}), \dots, α (a_{2 i - 2}), α (a_{2 i + 1}), \dots, [a_{2 i - 1}, a_{2 i}, a_{j}], \dots, α (a_{2 n + 1})) . \end{matrix}

δ \circ δ = 0 .

See [20] for more details.

In particular, for

f \in C_{HLts}^{1} (L, V)

, f is a 1-cocycle on

(L, [-, -, -], α)

with coefficients in

(V, β; θ)

δ f = 0,

i.e.,

θ (a_{2}, a_{3}) f (a_{1}) - θ (a_{1}, a_{3}) f (a_{2}) + D (a_{1}, a_{2}) f (a_{3}) - f ([a_{1}, a_{2}, a_{3}]) = 0 .

A 3-cochain

H \in C_{HLts}^{3} (L, V)

is a 3-cocycle on

(L, [-, -, -], α)

with coefficients in

(V, β; θ)

δ H = 0,

i.e.,

\begin{matrix} θ (α (a_{4}), α (a_{5})) H (a_{1}, a_{2}, a_{3}) - θ (α (a_{3}), α (a_{5})) H (a_{1}, a_{2}, a_{4}) - D (α (a_{1}), α (a_{2})) H (a_{3}, a_{4}, a_{5}) \\ + D (α (a_{3}), α (a_{4})) H (a_{1}, a_{2}, a_{5}) + H ([a_{1}, a_{2}, a_{3}], α (a_{4}), α (a_{5})) + H (α (a_{3}), [a_{1}, a_{2}, a_{4}], α (a_{5})) \\ + H (α (a_{3}), α (a_{4}), [a_{1}, a_{2}, a_{5}]) - H (α (a_{1}), α (a_{2}), [a_{3}, a_{4}, a_{5}]) = 0 . \end{matrix}

3. Generalized Reynolds operators on Hom-Lie triple systems

In this section, we introduce the notion of generalized Reynolds operators on Hom-Lie triple systems, which can be regarded as the generalization of relative Rota-Baxter operators on Hom-Lie triple systems [18,26] and generalized Reynolds operators on Lie triple systems [10,25]. We give its characterization by a graph and provide some examples.

Definition 3.1. (i) Let

(L, [-, -, -], α)

be a Hom-Lie triple system and

(V, β; θ)

be a representation of

L

. A linear operator

R : V \to L

is called a generalized Reynolds operator on Hom-Lie triple system associated to

(V, β; θ)

and 3-cocycle

H

if R satisfies:

\begin{matrix} α (R u) = R β (u), \end{matrix}

(3.1)

\begin{matrix} [R u, R v, R w] = R (θ (R v, R w) u + D (R u, R v) w - θ (R u, R w) v + H (R u, R v, R w)), \end{matrix}

(3.2)

where

u, v, w \in V

(ii) A morphism of generalized Reynolds operators from R to

R^{'}

consists of a pair

(η, ζ)

of a Hom-Lie triple system morphism

η : (L, [-, -, -], α) \to (L^{'}, {[-, -, -]}^{'}, α^{'})

and a linear map

ζ : V \to V^{'}

satisfying

\begin{matrix} α^{'} \circ η = η \circ α, β^{'} \circ ζ = ζ \circ β, \end{matrix}

(3.3)

\begin{matrix} η \circ R = R^{'} \circ ζ, \end{matrix}

(3.4)

\begin{matrix} ζ (θ (a, b) u) = θ^{'} (η (a), η (b)) ζ (u), \end{matrix}

(3.5)

\begin{matrix} ζ (H (a, b, c)) = H^{'} (η (a), η (b), η (c)), \end{matrix}

(3.6)

for

a, b, c \in L, u \in V .

Remark 3.2. (i) A generalized Reynolds operator R on Hom-Lie triple system

(L, [-, -, -], α)

with

α = id

is nothing but a generalized Reynolds operator R on Lie triple system

(L, [-, -, -])

. See [10,25] for more details about generalized Reynolds operators on Lie triple systems.

(ii) Any relative Rota-Baxter operator (in particular, Rota-Baxter operator of weight 0) on a Hom-Lie triple system is a generalized Reynolds operator with

H = 0

. See [18,26] for more details about relative Rota-Baxter operators on Hom-Lie triple systems.

Example 3.3.

Let

(V, β; θ)

be a representation of a Hom-Lie triple system

(L, [-, -, -], α)

. Suppose that

f : L \to V

is an invertible linear map and f satisfies

β \circ f = f \circ α

, take

H = - δ f

. Then

R = f^{- 1} : V \to L

is a generalized Reynolds operator.

Example 3.4.

In [13], Hou, Ma and Chen introduced the notion of Nijenhuis operator by the 2-order deformation of Hom-Lie triple system

(L, [-, -, -], α)

. More precisely, a linear map

N : L \to L

is called a Nijenhuis operator if for all

a, b, c \in L

, the following equations hold:

\begin{matrix} N \circ α = & α \circ N, \\ [N a, N b, N c] = & N ([a, N b, N c] + [N a, b, N c] + [N a, N b, c]) - N^{2} ([N a, b, c] + [a, N b, c] + [a, b, N c]) \\ + N^{3} [a, b, c] . \end{matrix}

In this case, the Hom-vector space

(L, α)

carries a new Hom-Lie triple system structure with bracket

\begin{matrix} {[a, b, c]}_{N} = & [a, N b, N c] + [N a, b, N c] + [N a, N b, c] - N ([N a, b, c] + [a, N b, c] + [a, b, N c]) \\ + N^{2} [a, b, c], \forall a, b, c \in L . \end{matrix}

This deformed Hom-Lie triple system

L_{N} = (L, {[-, -, -]}_{N}, α)

has a representation on

(L, α)

θ_{N} (a, b) c = [c, N a, N b]

, for

a, b \in L_{N}, c \in L

. The map

H : L_{N} \times L_{N} \times L_{N} \to L, H (a, b, c) = - N ([N a, b, c] + [a, N b, c] + [a, b, N c]) + N^{2} [a, b, c]

is a 3-cocycle with coefficients in

(L, α; θ_{N})

. Moreover, the identity map

i d : L \to L_{N}

is a generalized Reynolds operator.

Example 3.5.

Let

(L, [-, -, -], α)

be a Hom-Lie triple system and

(L, α; R)

the adjoint representation. Set the 3-cocycle

H (a, b, c) = - [a, b, c]

, for

a, b, c \in L

, then, a linear operator

T : L \to L

defined by Eqs. (3.1) and (3.2) is called a Reynolds operator on

(L, [-, -, -], α)

, more specifically, T satisfies:

\begin{matrix} α (T a) = T α (a), \end{matrix}

(3.7)

\begin{matrix} [T a, T b, T c] = T ([T a, T b, c] + [a, T b, T c] + [T a, b, T c] - [T a, T b, T c]), \end{matrix}

(3.8)

where

a, b, c \in L

Example 3.6.

Let

D : L \to L

be a derivation on a Hom-Lie triple system

(L, [-, -, -], α)

. If

D + \frac{1}{2} i d

is invertible, then

{(D + \frac{1}{2} i d)}^{- 1}

is a Reynolds operator on

(L, [-, -, -], α)

Given a 3-cocycle

H

in the cochain complex of

L

with coefficients in V, one can construct the twisted semidirect product Hom-Lie triple system. More precisely, the direct sum

L \oplus V

carries a Hom-Lie triple system structure with the bracket given by

\begin{matrix} {[a + u, b + v, c + w]}_{⋉_{H}} = [a, b, c] + D (a, b) w - θ (a, c) v + θ (b, c) u + H (a, b, c), \\ (α \oplus β) (a + u) = α (a) + β (u), \forall a, b, c \in L, u, v, w \in V . \end{matrix}

We denote this twisted semidirect product Hom-Lie triple system by

L ⋉_{H} V

Proposition 3.7.

A linear map

R : V \to L

is a generalized Reynolds operator on

L

if and only if the graph of R

G r (R) = {R u + u | u \in V}

is a subalgebra of the twisted semidirect product Hom-Lie triple system by

L ⋉_{H} V

Proof.

Let

R : V \to L

be a linear map, then for any

u, v, w \in V

, we have

\begin{matrix} α \oplus β (R u + u) = & α \circ R (u) + β (u), \\ {[R u + u, R v + v, R w + w]}_{⋉_{H}} = & [R u, R v, R w] + D (R u, R v) w - θ (R u, R w) v + θ (R v, R w) u \\ + H (R u, R v, R w), \end{matrix}

which implies that the graph

G r (R)

is a subalgebra of the twisted semidirect product Hom-Lie triple system

L ⋉_{H} V

if and only if R satisfies Eqs. (3.1) and (3.2), which means that R is a generalized Reynolds operator. □

Since

G r (R)

is isomorphic to V as a vector space. Define a trilinear operation on V by

\begin{matrix} {[u, v, w]}_{R} = D (R u, R v) w - θ (R u, R w) v + θ (R v, R w) u + H (R u, R v, R w), \forall u, v, w \in V . \end{matrix}

By Proposition 3.7, we get

(V, {[-, -, -]}_{R}, β)

is a Hom-Lie triple system.

4. Cohomologies of generalized Reynolds operators on Hom-Lie triple systems

In this section, first, we construct a representation of the Hom-Lie triple system

(V, {[-, -, -]}_{R}, β)

on the Hom-vector space

(L, α)

. Then we develop the cohomology of generalized Reynolds operator on the Hom-Lie triple system.

Lemma 4.1.

Let

R : V \to L

be a generalized Reynolds operator on a Hom-Lie triple system

(L, [-, -, -], α)

associated to

(V, β; θ)

and 3-cocycle

H

. For any

u, v \in V

a \in L

, define

θ_{R} : V \otimes V \to End (L)

\begin{matrix} θ_{R} (u, v) (a) = [a, R u, R v] + R (θ (a, R v) u - D (a, R u) v - H (a, R u, R v)), \end{matrix}

(4.1)

then

(L, α; θ_{R})

is a representation of the Hom-Lie triple system

(V, {[-, -, -]}_{R}, β)

Proof.

For any

u, v, s, t \in V, a \in L

, note that

\begin{matrix} D_{R} (u, v) (a) \\ = & θ_{R} (v, u) (a) - θ_{R} (u, v) (a) \\ = & [a, R v, R u] + R (θ (a, R u) v - D (a, R v) u - H (a, R v, R u)) - [a, R u, R v] \\ - R (θ (a, R v) u - D (a, R u) v - H (a, R u, R v)) \\ = & [R u, R v, a] + R (θ (R u, a) v - θ (R v, a) u - H (R u, R v, a)) . \end{matrix}

(4.2)

Further, we obtain that

\begin{matrix} θ_{R} (β (u), β (v)) α (a) \\ = & [α (a), R β (u), R β (v)] + R (θ (α (a), R β (v)) β (u) - D (α (a), R β (u)) β (v) - H (α (a), R β (u), R β (v))) \\ = & [α (a), α (R u), α (R v)] + R (θ (α (a), α (R v)) β (u) - D (α (a), α (R u)) β (v) - H (α (a), α (R u), α (R v))) \\ = & α ([a, R u, R v] + R (θ (a, R v) u - D (a, R u) v - H (a, R u, R v))) \\ = & α (θ_{R} (u, v) (a)), \\ θ_{R} (β (u), β (v)) θ_{R} (s, t) a - θ_{R} (β (t), β (v)) θ_{R} (s, u) a - θ_{R} (β (s), {[t, u, v]}_{R}) α (a) + D_{R} (β (t), β (u)) θ_{R} (s, v) a \\ = & [[a, R s, R t], R β (u), R β (v)] + R θ ([a, R s, R t], R β (v)) β (u) - R D ([a, R s, R t], R β (u)) β (v) \\ - R H ([a, R s, R t], R β (u), R β (v)) + [R θ (a, R t) s, R β (u), R β (v)] + R θ (R θ (a, R t) s, R β (v)) β (u) \\ - R D (R θ (a, R t) s, R β (u)) β (v) - R H (R θ (a, R t) s, R β (u), R β (v)) - [R D (a, R s) t, R β (u), R β (v)] \\ - R θ (R D (a, R s) t, R β (v)) β (u) + R D (R D (a, R s) t, R β (u)) β (v) + R H (R D (a, R s) t, R β (u), R β (v)) \\ - [R H (a, R s, R t), R β (u), R β (v)] - R θ (R H (a, R s, R t), R β (v)) β (u) + R D (R H (a, R s, R t), R β (u)) β (v) \\ + R H (R H (a, R s, R t), R β (u), R β (v)) - [[a, R s, R u], R β (t), R β (v)] - R θ ([a, R s, R u], R β (v)) β (t) \\ + R D ([a, R s, R u], R β (t)) β (v) + R H ([a, R s, R u], R β (t), R β (v)) - [R θ (a, R u) s, R β (t), R β (v)] \\ - R θ (R θ (a, R u) s, R β (v)) β (t) + R D (R θ (a, R u) s, R β (t)) β (v) + R H (R θ (a, R u) s, R β (t), R β (v)) \\ + [R D (a, R s) u, R β (t), R β (v)] + R θ (R D (a, R s) u, R β (v)) β (t) - R D (R D (a, R s) u, R β (t)) β (v) \\ - R H (R D (a, R s) u, R β (t), R β (v)) + [R H (a, R s, R u), R β (t), R β (v)] + R θ (R H (a, R s, R u), R β (v)) β (t) \\ - R D (R H (a, R s, R u), R β (t)) β (v) - R H (R H (a, R s, R u), R β (t), R β (v)) - [α (a), R β (s), R {[t, u, v]}_{R}] \\ - R θ (α (a), R {[t, u, v]}_{R}) β (s) + R D (α (a), R β (s)) {[t, u, v]}_{R} + R H (α (a), R β (s), R {[t, u, v]}_{R}) \\ + [R β (t), R β (u), [a, R s, R v]] + R θ (R β (t), [a, R s, R v]) β (u) - R θ (R β (u), [a, R s, R v]) β (t) \\ - R H (R β (t), R β (u), [a, R s, R v]) + [R β (t), R β (u), R θ (a, R v) s] + R θ (R β (t), R θ (a, R v) s) β (u) \\ - R θ (R β (u), R θ (a, R v) s) β (t) - R H (R β (t), R β (u), R θ (a, R v) s) - [R β (t), R β (u), R D (a, R s) v] \\ - R θ (R β (t), R D (a, R s) v) β (u) + R θ (R β (u), R D (a, R s) v) β (t) + R H (R β (t), R β (u), R D (a, R s) v) \\ - [R β (t), R β (u), R H (a, R s, R v)] - R θ (R β (t), R H (a, R s, R v)) β (u) + R θ (R β (u), R H (a, R s, R v)) β (t) \\ + R H (R β (t), R β (u), R H (a, R s, R v)) \\ = & 0 . \end{matrix}

Similarly, we also have

\begin{matrix} θ_{R} (β (u), β (v)) D_{R} (s, t) a - D_{R} (β (s), β (t)) θ_{R} (u, v) a + θ_{R} ({[s, t, u]}_{R}, β (v)) α (a) \\ + θ_{R} (β (u), {[s, t, v]}_{R}) α (a) = 0 . \end{matrix}

Therefore,

(L, α; θ_{R})

is a representation of

(V, {[-, -, -]}_{R}, β)

. □

Denote by

δ_{R} : C_{HLts}^{2 n - 1} (V, L) \to C_{HLts}^{2 n + 1} (V, L)

the coboundary operator of the Hom-Lie triple system

(V, {[-, -, -]}_{R}, β)

with coefficients in the representation

(L, α; θ_{R})

. More precisely, for

v_{1}, \dots, v_{2 n + 1} \in V

and

f \in C_{HLts}^{2 n - 1} (V, L)

, as

\begin{matrix} δ_{R} f (v_{1}, \dots, v_{2 n + 1}) \\ = & θ_{R} (β^{n - 1} (v_{2 n}), β^{n - 1} (v_{2 n + 1})) f (v_{1}, \dots, v_{2 n - 1}) - θ_{R} (β^{n - 1} (v_{2 n - 1}), β^{n - 1} (v_{2 n + 1})) f (v_{1}, \dots, v_{2 n - 2}, v_{2 n}) \\ + \sum_{i = 1}^{n} {(- 1)}^{i + n} D_{R} (β^{n - 1} (v_{2 i - 1}), β^{n - 1} (v_{2 i})) f (v_{1}, \dots, v_{2 i - 2}, v_{2 i + 1}, \dots, v_{2 n + 1}) \\ + \sum_{i = 1}^{n} \sum_{j = 2 i + 1}^{2 n + 1} {(- 1)}^{i + n + 1} f (β (v_{1}), \dots, β (v_{2 i - 2}), β (v_{2 i + 1}), \dots, {[v_{2 i - 1}, v_{2 i}, v_{j}]}_{R}, \dots, β (v_{2 n + 1})) . \end{matrix}

In particular, for

f \in C_{HLts}^{1} (V, L)

, f is a 1-cocycle on

(V, {[-, -, -]}_{R}, β)

with coefficients in

(L, α; θ_{R})

δ_{R} f (v_{1}, v_{2}, v_{3}) = 0,

i.e.,

θ_{R} (v_{2}, v_{3}) f (v_{1}) - θ_{R} (v_{1}, v_{3}) f (v_{2}) + D_{R} (v_{1}, v_{2}) f (v_{3}) - f ({[v_{1}, v_{2}, v_{3}]}_{R}) = 0

and

α (f (v_{1})) - f (β (v_{1})) = 0 .

Proposition 4.2.

Let

R : V \to L

be a generalized Reynolds operator on a Hom-Lie triple system

(L, [-, -, -], α)

associated to

(V, β; θ)

and 3-cocycle

H

. If there exist two elements

a, b \in L

such that

α (a) = a, α (b) = b

, we define

℘ (a, b) : V \to L

℘ (a, b) u = R D (a, b) β^{- 1} (u) - [a, b, R β^{- 1} (u)] + R H (a, b, R β^{- 1} (u)),

for any

u \in V .

Then

℘ (a, b)

is a 1-cocycle on

(V, {[-, -, -]}_{R}, β)

with coefficients in

(L, α; θ_{R})

Proof.

For any

v_{1}, v_{2}, v_{3} \in V

, first, obviously

α (℘ (a, b) (v_{1})) - ℘ (a, b) (β (v_{1})) = 0

. Next, by Eqs. (3.6), (4.1) and (4.2), we have

\begin{matrix} δ_{R} ℘ (a, b) (v_{1}, v_{2}, v_{3}) \\ = & θ_{R} (v_{2}, v_{3}) ℘ (a, b) (v_{1}) - θ_{R} (v_{1}, v_{3}) ℘ (a, b) (v_{2}) + D_{R} (v_{1}, v_{2}) ℘ (a, b) (v_{3}) - ℘ (a, b) ({[v_{1}, v_{2}, v_{3}]}_{R}) \\ = & θ_{R} (v_{2}, v_{3}) R D (a, b) β^{- 1} (v_{1}) - θ_{R} (v_{2}, v_{3}) [a, b, R β^{- 1} (v_{1})] + θ_{R} (v_{2}, v_{3}) R H (a, b, R β^{- 1} (v_{1})) \\ - θ_{R} (v_{1}, v_{3}) R D (a, b) β^{- 1} (v_{2}) + θ_{R} (v_{1}, v_{3}) [a, b, R β^{- 1} (v_{2})] - θ_{R} (v_{1}, v_{3}) R H (a, b, R β^{- 1} (v_{2})) \\ + D_{R} (v_{1}, v_{2}) R D (a, b) β^{- 1} (v_{3}) - D_{R} (v_{1}, v_{2}) [a, b, R β^{- 1} (v_{3})] + D_{R} (v_{1}, v_{2}) R H (a, b, R β^{- 1} (v_{3})) \\ - R D (a, b) β^{- 1} ({[v_{1}, v_{2}, v_{3}]}_{R}) - [a, b, R β^{- 1} ({[v_{1}, v_{2}, v_{3}]}_{R})] + R H (a, b, R β^{- 1} ({[v_{1}, v_{2}, v_{3}]}_{R})) \\ = & [R D (a, b) β^{- 1} (v_{1}), R v_{2}, R v_{3}] + R θ (R D (a, b) β^{- 1} (v_{1}), R v_{3}) v_{2} - R D (R D (a, b) β^{- 1} (v_{1}), R v_{2}) v_{3} \\ - R H (R D (a, b) β^{- 1} (v_{1}), R v_{2}, R v_{3}) - [[a, b, R β^{- 1} (v_{1})], R v_{2}, R v_{3}] - R θ ([a, b, R β^{- 1} (v_{1})], R v_{3}) v_{2} \\ + R D ([a, b, R β^{- 1} (v_{1})], R v_{2}) v_{3} + R H ([a, b, R β^{- 1} (v_{1})], R v_{2}, R v_{3}) + [R H (a, b, R β^{- 1} (v_{1})), R v_{2}, R v_{3}] \end{matrix}

\begin{matrix} + R θ (R H (a, b, R β^{- 1} (v_{1})), R v_{3}) v_{2} - R D (R H (a, b, R β^{- 1} (v_{1})), R v_{2}) v_{3} - R H (R H (a, b, R β^{- 1} (v_{1})), R v_{2}, R v_{3}) \\ - [R D (a, b) β^{- 1} (v_{2}), R v_{1}, R v_{3}] - R θ (R D (a, b) β^{- 1} (v_{2}), R v_{3}) v_{1} + R D (R D (a, b) β^{- 1} (v_{2}), R v_{1}) v_{3} \\ + R H (R D (a, b) β^{- 1} (v_{2}), R v_{1}, R v_{3}) + [[a, b, R β^{- 1} (v_{2})], R v_{1}, R v_{3}] + R θ ([a, b, R β^{- 1} (v_{2})], R v_{3}) v_{1} \\ - R D ([a, b, R β^{- 1} (v_{2})], R v_{1}) v_{3} - R H ([a, b, R β^{- 1} (v_{2})], R v_{1}, R v_{3}) - [R H (a, b, R β^{- 1} (v_{2})), R v_{1}, R v_{3}] \\ - R θ (R H (a, b, R β^{- 1} (v_{2})), R v_{3}) v_{1} + R D (R H (a, b, R β^{- 1} (v_{2})), R v_{1}) v_{3} + R H (R H (a, b, R β^{- 1} (v_{2})), R v_{1}, R v_{3}) \\ + [R v_{1}, R v_{2}, R D (a, b) β^{- 1} (v_{3})] + R θ (R v_{1}, R D (a, b) β^{- 1} (v_{3})) v_{2} - R θ (R v_{2}, R D (a, b) β^{- 1} (v_{3})) v_{1} \\ - R H (R v_{1}, R v_{2}, R D (a, b) β^{- 1} (v_{3})) - [R v_{1}, R v_{2}, [a, b, R β^{- 1} (v_{3})]] - R θ (R v_{1}, [a, b, R β^{- 1} (v_{3})]) v_{2} \\ + R θ (R v_{2}, [a, b, R β^{- 1} (v_{3})]) v_{1} + R H (R v_{1}, R v_{2}, [a, b, R β^{- 1} (v_{3})]) + [R v_{1}, R v_{2}, R H (a, b, R β^{- 1} (v_{3}))] \\ + R θ (R v_{1}, R H (a, b, R β^{- 1} (v_{3}))) v_{2} - R θ (R v_{2}, R H (a, b, R β^{- 1} (v_{3}))) v_{1} - R H (R v_{1}, R v_{2}, R H (a, b, R β^{- 1} (v_{3}))) \\ - R D (a, b) β^{- 1} (D (R v_{1}, R v_{2}) v_{3}) + R D (a, b) β^{- 1} (θ (R v_{1}, R v_{3}) v_{2}) - R D (a, b) β^{- 1} (θ (R v_{2}, R v_{3}) v_{1}) \\ - R D (a, b) β^{- 1} (H (R v_{1}, R v_{2}, R v_{3})) - [a, b, R β^{- 1} (D (R v_{1}, R v_{2}) v_{3})] + [a, b, R β^{- 1} (θ (R v_{1}, R v_{3}) v_{2})] \\ - [a, b, R β^{- 1} (θ (R v_{2}, R v_{3}) v_{1})] - [a, b, R β^{- 1} (H (R v_{1}, R v_{2}, R v_{3}))] + R H (a, b, R β^{- 1} (D (R v_{1}, R v_{2}) v_{3})) \\ - R H (a, b, R β^{- 1} (θ (R v_{1}, R v_{3}) v_{2})) + R H (a, b, R β^{- 1} (θ (R v_{2}, R v_{3}) v_{1})) + R H (a, b, R β^{- 1} (H (R v_{1}, R v_{2}, R v_{3}))) \\ = & 0 . \end{matrix}

Therefore,

δ_{R} ℘ (a, b) = 0 .

□

Definition 4.3.

Let

R : V \to L

be a generalized Reynolds operator on a Hom-Lie triple system

(L, [-, -, -], α)

associated to

(V, β; θ)

and 3-cocycle

H

. Define the set of n-cochains by

C_{R}^{n} (V, L) = \{\begin{matrix} C_{HLts}^{2 n - 1} (V, L), & n \geq 1; \\ {(a, b) \in \land^{2} L | α (a) = a, α (b) = b}, & n = 0 . \end{matrix}

Define

\partial_{R} : C_{R}^{n} (V, L) \to C_{R}^{n + 1} (V, L)

\partial_{R} = \{\begin{matrix} δ_{R}, & n \geq 1; \\ ℘, & n = 0 . \end{matrix}

δ \circ δ = 0

and Proposition 4.2, we know that

(\oplus_{n = 0}^{+ \infty} C_{R}^{n} (V, g), \partial)

is a cochain complex. Denote the set of n-cocycles by

Z_{R}^{n} (V, L)

, the set of n-coboundaries by

B_{R}^{n} (V, L)

, and n-th cohomology group by

H_{R}^{n} (V, L) = \frac{Z_{R}^{n} (V, L)}{B_{R}^{n} (V, L)}, n \geq 1,

which is taken to be the n-th cohomology group for the generalized Reynolds operator R.

Remark 4.4.

The cohomology theory for generalized Reynolds operators on Hom-Lie triple systems enjoys certain functorial properties. Let

R, R^{'} : V \to L

be two generalized Reynolds operators on a Hom-Lie triple system

(L, [-, -, -], α)

associated to

(V, β; θ)

and 3-cocycle

H

, and

(η, ζ)

be a homomorphism from R to

R^{'}

in which ζ is invertible. Define linear map

Φ : C_{R}^{n} (V, L) \to C_{R^{'}}^{n} (V, L)

Φ (f) (v_{1}, \dots, v_{2 n - 1}) = η (f (ζ^{- 1} (v_{1}), \dots, ζ^{- 1} (v_{2 n - 1}))),

for any

f \in C_{R}^{n} (V, L)

and

v_{1}, \dots, v_{2 n - 1} \in V .

Then it is straightforward to deduce that Φ is a cochain map from the cochain complex

(\oplus_{n = 1}^{+ \infty} C_{R}^{n} (V, L), \partial_{R})

to the cochain complex

(\oplus_{n = 1}^{+ \infty} C_{R^{'}}^{n} (V, L), \partial_{R^{'}})

. Consequently, it induces a homomorphism

Φ^{*}

from the cohomology group

H_{R}^{n} (V, L)

H_{R^{'}}^{n} (V, L)

5. Deformatons of generalized Reynolds operators on Hom-Lie triple systems

In this section, we study linear deformations and higher order deformations of generalized Reynolds operators on Hom-Lie triple systems via the cohomology theory established in the former section.

First, we use the cohomology constructed to characterize the linear deformations of generalized Reynolds operators on Hom-Lie triple systems.

Definition 5.1.

Let

R : V \to L

be a generalized Reynolds operator on a Hom-Lie triple system

(L, [-, -, -], α)

associated to

(V, β; θ)

and 3-cocycle

H

. A linear deformation of R is a generalized Reynolds operator of the form

R_{t} = R + t R_{1}

, where

R_{1} : V \to L

is a linear map and t is a parameter with

t^{2} = 0

Suppose

R + t R_{1}

is a linear deformation of R, direct deduction shows that

R_{1} \in C_{R}^{1} (V, L)

is a 1-cocycle on

(V, {[-, -, -]}_{R}, β)

with coefficients in

(L, α; θ_{R})

. So the cohomology class of

R_{1}

defines an element in

H_{R}^{1} (V, L)

. Furthermore, the 1-cocycle

R_{1}

is called the infinitesimal of the linear deformation

R_{t}

of R.

Definition 5.2.

Let

R : V \to L

be a generalized Reynolds operator on a regular Hom-Lie triple system

(L, [-, -, -], α)

associated to regular representation

(V, β; θ)

and 3-cocycle

H

. Two linear deformations

R_{t} = R + t R_{1}

and

R_{t}^{'} = R + t R_{1}^{'}

are called equivalent if there exist two elements

a, b \in L

such that

α (a) = a, α (b) = b

and the pair

(I d_{L} + t α^{- 1} (L (a, b) -), I d_{V} + t β^{- 1} (D (a, b) -) + t β^{- 1} (H (a, b, R -)))

is a homomorphism from

R_{t}

R_{t}^{'}

Suppose

R_{t}

and

R_{t}^{'}

are equivalent, then Eq. () yields that

(I d_{L} + t α^{- 1} (L (a, b) -)) R_{t} u = R_{t}^{'} (I d_{V} + t β^{- 1} (D (a, b) -) + t β^{- 1} (H (a, b, R -))) u, \forall u \in V .

which means that

\begin{matrix} R_{1} u - R_{1}^{'} u = & R β^{- 1} (D (a, b) u) - α^{- 1} ([a, b, R u]) + R β^{- 1} (H (a, b, R u)) \\ = & R D (a, b) β^{- 1} (u) - [a, b, R β^{- 1} (u)] + R H (a, b, R β^{- 1} (u)) . \end{matrix}

By Proposition 4.2, we have

R_{1} - R_{1}^{'} = ℘ (a, b) = \partial_{R} (a, b)

. So their cohomology classes are the same in

H_{R}^{1} (V, L)

Conversely, any 1-cocycle

R_{1}

gives rise to the linear deformation

R + t R_{1}

. To sum up, we have the following result.

Proposition 5.3.

Let

R : V \to L

be a generalized Reynolds operator on a regular Hom-Lie triple system

(L, [-, -, -], α)

associated to regular representation

(V, β; θ)

and 3-cocycle

H

. Then there is a bijection between the set of all equivalence classes of linear deformation of R and the first cohomology group

H_{R}^{1} (V, L)

Next, we introduce a special cohomology class associated to an order n deformation of a generalized Reynolds operator, and show that an order n deformation of a generalized Reynolds operator is extendable if and only if this cohomology class in the second cohomology group vanishes.

Definition 5.4.

Let

R : V \to L

be a generalized Reynolds operator on a Hom-Lie triple system

(L, [-, -, -], α)

associated to

(V, β; θ)

and 3-cocycle

H

. If

R_{t} = \sum_{i = 0}^{n} t^{i} R_{i}

with

R_{0} = R, R_{i} \in Hom (V, L), i = 1, \dots, n

, defines a

K [[t]] / (t^{n + 1})

-module map from

V [[t]] / (t^{n + 1})

to the Hom-Lie triple system

L [[t]] / (t^{n + 1})

satisfying

\begin{matrix} R_{t} \circ β = α \circ R_{t}, \\ [R_{t} u, R_{t} v, R_{t} w] = R_{t} (θ (R_{t} v, R_{t} w) u + D (R_{t} u, R_{t} v) w - θ (R_{t} u, R_{t} w) v + H (R_{t} u, R_{t} v, R_{t} w)), \end{matrix}

for any

u, v, w \in V

, we say that

R_{t}

is an order n deformation of R.

Definition 5.5.

Let

R : V \to L

be a generalized Reynolds operator on a Hom-Lie triple system

(L, [-, -, -], α)

associated to

(V, β; θ)

and 3-cocycle

H

. Let

R_{t} = \sum_{i = 0}^{n} t^{i} R_{i}

be an order n deformation of R. If there is a

R_{n + 1} \in C_{R}^{1} (V, L)

such that

R_{t}^{'} = R_{t} + t^{n + 1} R_{n + 1}

is an order

(n + 1)

deformation of R, then we say that

R_{t}

is extendable.

Proposition 5.6.

Let

R : V \to L

be a generalized Reynolds operator on a Hom-Lie triple system

(L, [-, -, -], α)

associated to

(V, β; θ)

and 3-cocycle

H

. Let

R_{t} = \sum_{i = 0}^{n} t^{i} R_{i}

be an order n deformation of R. Then

R_{t}

is extendable if and only if the cohomology class

[O b s^{n}] \in H_{R}^{2} (V, L)

vanishes, where

\begin{matrix} O b s^{n} (u_{1}, u_{2}, u_{3}) = \sum_{\begin{matrix} i + j + k = n + 1 \\ 0 \leq i, j, k \leq n \end{matrix}} ([R_{i} u_{1}, R_{j} u_{2}, R_{k} u_{3}] - R_{i} (D (R_{j} u_{1}, R_{k} u_{2}) u_{3} \\ - θ (R_{j} u_{1}, R_{k} u_{3}) u_{2} + θ (R_{j} u_{2}, R_{k} u_{3}) u_{1})) - \sum_{\begin{matrix} i + j + k + l = n + 1 \\ 0 \leq i, j, k, l \leq n \end{matrix}} R_{i} H (R_{j} u_{1}, R_{k} u_{2}, R_{l} u_{3}) . \end{matrix}

Proof.

Let

R_{t}^{'} = R_{t} + t^{n + 1} R_{n + 1}

be the extension of

R_{t}

, then for all

u_{1}, u_{2}, u_{3} \in V

\begin{matrix} [R_{t}^{'} u_{1}, R_{t}^{'} u_{2}, R_{t}^{'} u_{3}] = R_{t}^{'} (D (R_{t}^{'} u_{1}, R_{t}^{'} u_{2}) u_{3} - θ (R_{t}^{'} u_{1}, R_{t}^{'} u_{3}) u_{2} + θ (R_{t}^{'} u_{2}, R_{t}^{'} u_{3}) u_{1} + H (R_{t}^{'} u_{1}, R_{t}^{'} u_{2}, R_{t}^{'} u_{3})) . \end{matrix}

Expanding the equation and comparing the coefficients of

t^{n + 1}

yields that:

\begin{matrix} \sum_{\begin{matrix} i + j + k = n + 1 \\ 0 \leq i, j, k \leq n + 1 \end{matrix}} ([R_{i} u_{1}, R_{j} u_{2}, R_{k} u_{3}] - R_{i} (D (R_{j} u_{1}, R_{k} u_{2}) u_{3} - θ (R_{j} u_{1}, R_{k} u_{3}) u_{2} + θ (R_{j} u_{2}, R_{k} u_{3}) u_{1})) \\ - \sum_{\begin{matrix} i + j + k + l = n + 1 \\ 0 \leq i, j, k, l \leq n + 1 \end{matrix}} R_{i} H (R_{j} u_{1}, R_{k} u_{2}, R_{l} u_{3}) = 0, \end{matrix}

which is equivalent to

\begin{matrix} \sum_{\begin{matrix} i + j + k = n + 1 \\ 0 \leq i, j, k \leq n \end{matrix}} ([R_{i} u_{1}, R_{j} u_{2}, R_{k} u_{3}] - R_{i} (D (R_{j} u_{1}, R_{k} u_{2}) u_{3} - θ (R_{j} u_{1}, R_{k} u_{3}) u_{2} + θ (R_{j} u_{2}, R_{k} u_{3}) u_{1})) \\ - \sum_{\begin{matrix} i + j + k + l = n + 1 \\ 0 \leq i, j, k, l \leq n \end{matrix}} R_{i} H (R_{j} u_{1}, R_{k} u_{2}, R_{l} u_{3}) + [R_{n + 1} u_{1}, R u_{2}, R u_{3}] + [R u_{1}, R_{n + 1} u_{2}, R u_{3}] \\ + [R u_{1}, R u_{2}, R_{n + 1} u_{3}] - R_{n + 1} (D (R u_{1}, R u_{2}) u_{3} - θ (R u_{1}, R u_{3}) u_{2} + θ (R u_{2}, R u_{3}) u_{1} + H (R u_{1}, R u_{2}, R u_{3})) \\ - R (D (R_{n + 1} u_{1}, R u_{2}) u_{3} + D (R u_{1}, R_{n + 1} u_{2}) u_{3} - θ (R_{n + 1} u_{1}, R u_{3}) u_{2} - θ (R u_{1}, R_{n + 1} u_{3}) u_{2} \\ + θ (R_{n + 1} u_{2}, R u_{3}) u_{1} + θ (R u_{2}, R_{n + 1} u_{3}) u_{1} + H (R_{n + 1} u_{1}, R u_{2}, R u_{3}) \\ + H (R u_{1}, R_{n + 1} u_{2}, R u_{3}) + H (R u_{1}, R u_{2}, R_{n + 1} u_{3})) = 0, \end{matrix}

that is

O b s^{n} (u_{1}, u_{2}, u_{3}) + \partial_{R} R_{n + 1} (u_{1}, u_{2}, u_{3}) = 0

. Hence

O b s^{n} = - \partial_{R} R_{n + 1}

, further

\partial_{R} O b s^{n} = 0,

which implies that the cohomology class

[O b s^{n}] \in H_{R}^{2} (V, L)

vanishes.

Conversely, suppose that the cohomology class

[O b s^{n}]

vanishes, then there exists a 1-cochain

R_{n + 1} \in C_{R}^{1} (V, L)

such that

O b s^{n} = - \partial_{R} R_{n + 1}

. Set

R_{t}^{'} = R_{t} + t^{n + 1} R_{n + 1}

. Then

R_{t}^{'}

satisfies

\begin{matrix} \sum_{\begin{matrix} i + j + k = d \end{matrix}} ({[A_{i} u_{1}, A_{j} u_{2}, A_{k} u_{3}]}_{g} - A_{i} (D (A_{j} u_{1}, A_{k} u_{2}) u_{3} - θ (A_{j} u_{1}, A_{k} u_{3}) u_{2} + θ (A_{j} u_{2}, A_{k} u_{3}) u_{1})) \\ - \sum_{\begin{matrix} i + j + k + l = d \end{matrix}} R_{i} H (R_{j} u_{1}, R_{k} u_{2}, R_{l} u_{3}) = 0, 0 \leq d \leq n + 1, \end{matrix}

which implies that

R_{t}^{'}

is an order

(n + 1)

deformation of R. Hence it is an extension of

R_{t} .

□

6. Hom-NS-Lie triple systems

In this section, we introduce the notion of Hom-NS-Lie triple system, which is the underlying algebraic structure of generalized Reynolds operators. Moreover, we show that there exists a Hom-Lie triple system structure on a Hom-NS-Lie triple system.

Definition 6.1. (i) A Hom-NS-Lie triple system

(L, {-, -, -}, [-, -, -], α)

consists of a vector space

L

with trilinear products

{-, -, -}, [-, -, -] : L \otimes L \otimes L \to L

and an algebra morphism

α : L \to L

such that

\begin{matrix} [a_{1}, a_{2}, a_{3}] = & - [a_{2}, a_{1}, a_{3}], \end{matrix}

(6.1)

\begin{matrix} ⥀_{a_{1}, a_{2}, a_{3}} [a_{1}, a_{2}, a_{3}] = & 0, \\ {α (b_{1}), α (b_{2}), [[a_{1}, a_{2}, a_{3}]]} = & {{b_{1}, b_{2}, a_{1}}, α (a_{2}), α (a_{3})} - {{b_{1}, b_{2}, a_{2}}, α (a_{1}), α (a_{3})} \end{matrix}

(6.2)

\begin{matrix} + {α (a_{1}), α (a_{2}), {b_{1}, b_{2}, a_{3}}}^{*}, \\ {α (b_{1}), α (b_{2}), {a_{1}, a_{2}, a_{3}}}^{*} = & {{b_{1}, b_{2}, a_{1}}^{*}, α (a_{2}), α (a_{3})} + {α (a_{1}), [[b_{1}, b_{2}, a_{2}]], α (a_{3})} \end{matrix}

(6.3)

\begin{matrix} + {α (a_{1}), α (a_{2}), [[b_{1}, b_{2}, a_{3}]]}, \\ α (b_{1}), α (b_{2}), [[a_{1}, a_{2}, a_{3}]]] = & [[[b_{1}, b_{2}, a_{1}]], α (a_{2}), α (a_{3})] + [α (a_{1}), [[b_{1}, b_{2}, a_{2}]], α (a_{3})] \\ + [α (a_{1}), α (a_{2}), [[b_{1}, b_{2}, a_{3}]]] + {[b_{1}, b_{2}, a_{1}], α (a_{2}), α (a_{3})} \\ - {[b_{1}, b_{2}, a_{2}], α (a_{1}), α (a_{3})} + {α (a_{1}), α (a_{2}), [b_{1}, b_{2}, a_{3}]}^{*} \end{matrix}

(6.4)

\begin{matrix} - {α (b_{1}), α (b_{2}), [a_{1}, a_{2}, a_{3}]}^{*}, \end{matrix}

(6.5)

where

a_{1}, a_{2}, a_{3}, b_{1}, b_{2} \in L

{-, -, -}^{*}

and

[[-, -, -]]

are defined to be

\begin{matrix} {a_{1}, a_{2}, a_{3}}^{*} = & {a_{3}, a_{2}, a_{1}} - {a_{3}, a_{1}, a_{2}}, \end{matrix}

(6.6)

\begin{matrix} [[a_{1}, a_{2}, a_{3}]] = & {a_{1}, a_{2}, a_{3}}^{*} + {a_{1}, a_{2}, a_{3}} - {a_{2}, a_{1}, a_{3}} + [a_{1}, a_{2}, a_{3}] . \end{matrix}

(6.7)

(ii) A homomorphism between two Hom-NS-Lie triple systems

(L_{1}, {-, -, -}_{1}, {[-, -, -]}_{1}, α_{1})

and

(L_{2}, {-, -, -}_{2}, {[-, -, -]}_{2}, α_{2})

is a linear map

φ : L_{1} \to L_{2}

satisfying

φ (α_{1} (a_{1})) = α_{2} (φ (a_{1})), φ ({a_{1}, a_{2}, a_{3}}_{1}) = {φ (a_{1}), φ (a_{2}), φ (a_{3})}_{2}, φ ({[a_{1}, a_{2}, a_{3}]}_{1}) = {[φ (a_{1}), φ (a_{2}), φ (a_{3})]}_{2} .

Remark 6.2. (i) Let

(L, {-, -, -}, [-, -, -], α)

be a Hom-NS-Lie triple system. If the bracket

{-, -, -} = 0,

then we get

(L, [-, -, -], α)

is a Hom-Lie triple system.

(ii) A NS-Lie triple system is a Hom-NS-Lie triple system with

α = {id}_{L}

. See [10] for more details about NS-Lie triple systems.

Proposition 6.3.

Let

(L, {-, -, -}, [-, -, -], α)

be a Hom-NS-Lie triple system. Then,

(i) the triple

(L, [[-, -, -]], α)

is a Hom-Lie triple system, which is called the adjacent Hom-Lie triple system.

(ii) the triple

(L, α; ϑ)

is a representation of the adjacent Hom-Lie triple system

(L, [[-, -, -]], α)

, where

ϑ : L \otimes L \to End (L), (a_{1}, a_{2}) \mapsto (b \mapsto {b, a_{1}, a_{2}}), \forall a_{1}, a_{2}, b \in L .

Proof. (i) Obviously, for any

a_{1}, a_{2}, a_{3} \in L

, by Eqs. (6.1), (6.2), (6.6) and (6.7), we have

[[a_{1}, a_{2}, a_{3}]] = - [[a_{2}, a_{1}, a_{3}]]

and

⥀_{a_{1}, a_{2}, a_{3}} [[a_{1}, a_{2}, a_{3}]] = 0

. Further, for any

a_{1}, a_{2}, a_{3}, b_{1}, b_{2} \in L

, by Eqs. (6.3)-(6.7), we have

\begin{matrix} [[α (b_{1}), α (b_{2}), [[a_{1}, a_{2}, a_{3}]]]] - [[[[b_{1}, b_{2}, a_{1}]], α (a_{2}), α (a_{3})]] - [[α (a_{1}), [[b_{1}, b_{2}, a_{2}]], α (a_{3})]] \\ - [[α (a_{1}), α (a_{2}), [[b_{1}, b_{2}, a_{3}]]]] \\ = & {α (b_{1}), α (b_{2}), [[a_{1}, a_{2}, a_{3}]]}^{*} + {α (b_{1}), α (b_{2}), [[a_{1}, a_{2}, a_{3}]]} - {α (b_{2}), α (b_{1}), [[a_{1}, a_{2}, a_{3}]]} \\ + [α (b_{1}), α (b_{2}), [[a_{1}, a_{2}, a_{3}]]] - {[[b_{1}, b_{2}, a_{1}]], α (a_{2}), α (a_{3})}^{*} - {[[b_{1}, b_{2}, a_{1}]], α (a_{2}), α (a_{3})} \\ + {α (a_{2}), [[b_{1}, b_{2}, a_{1}]], α (a_{3})} - [[[b_{1}, b_{2}, a_{1}]], α (a_{2}), α (a_{3})] - {α (a_{1}), [[b_{1}, b_{2}, a_{2}]], α (a_{3})}^{*} \\ - {α (a_{1}), [[b_{1}, b_{2}, a_{2}]], α (a_{3})} + {[[b_{1}, b_{2}, a_{2}]], α (a_{1}), α (a_{3})} - [α (a_{1}), [[b_{1}, b_{2}, a_{2}]], α (a_{3})] \\ - {α (a_{1}), α (a_{2}), [[b_{1}, b_{2}, a_{3}]]}^{*} - {α (a_{1}), α (a_{2}), [[b_{1}, b_{2}, a_{3}]]} + {α (a_{2}), α (a_{1}), [[b_{1}, b_{2}, a_{3}]]} \\ - [α (a_{1}), α (a_{2}), [[b_{1}, b_{2}, a_{3}]]] \\ = & 0 . \end{matrix}

Hence,

(L, [[-, -, -]], α)

is a Hom-Lie triple system.

(ii) For all

a_{1}, a_{2}, a_{3} \in L

, we have

\begin{matrix} D (a_{1}, a_{2}) a_{3} = ϑ (a_{2}, a_{1}) a_{3} - ϑ (a_{1}, a_{2}) a_{3} = {a_{3}, a_{2}, a_{1}} - {a_{3}, a_{1}, a_{2}} = {a_{1}, a_{2}, a_{3}}^{*} . \end{matrix}

Obviously,

ϑ (α (a_{1}), α (a_{2})) α (a_{3}) = α (ϑ (a_{1}, a_{2}) a_{3}) .

Further, for any

a_{1}, a_{2}, a_{3}, b_{1}, b_{2} \in L

, by Eqs. (6.3) and (6.4), we get

\begin{matrix} ϑ (α (b_{1}), α (b_{2})) ϑ (a_{1}, a_{2}) a_{3} - ϑ (α (a_{2}), α (b_{2})) ϑ (a_{1}, b_{1}) a_{3} - ϑ (α (a_{1}), [[a_{2}, b_{1}, b_{2}]]) α (a_{3}) \\ + D (α (a_{2}), α (b_{1})) ϑ (a_{1}, b_{2}) a_{3} \\ = & {{a_{3}, a_{1}, a_{2}}, α (b_{1}), α (b_{2})} - {{a_{3}, a_{1}, b_{1}}, α (a_{2}), α (b_{2})} - {α (a_{3}), α (a_{1}), [[a_{2}, b_{1}, b_{2}]]} \\ + {α (a_{2}), α (b_{1}), {a_{3}, a_{1}, b_{2}}}^{*} \\ = & 0, \\ ϑ (α (b_{1}), α (b_{2})) D (a_{1}, a_{2}) a_{3} - D (α (a_{1}), α (a_{2})) ϑ (b_{1}, b_{2}) a_{3} + ϑ ([[a_{1}, a_{2}, b_{1}]], α (b_{2})) α (a_{3}) \\ + ϑ (α (b_{1}), [[a_{1}, a_{2}, b_{2}]]) α (a_{3}) \\ = & {{a_{1}, a_{2}, a_{3}}^{*}, α (b_{1}), α (b_{2})} - {α (a_{1}), α (a_{2}), {a_{3}, b_{1}, b_{2}}}^{*} + {α (a_{3}), [[a_{1}, a_{2}, b_{1}]], α (b_{2})} \\ + {α (a_{3}), α (b_{1}), [[a_{1}, a_{2}, b_{2}]]} \\ = & 0 . \end{matrix}

Therefore,

(L, α; ϑ)

is a representation of the adjacent Hom-Lie triple system

(L, [[-, -, -]], α)

. □

Corollary 6.4.

Let

φ : (L_{1}, {-, -, -}_{1}, {[-, -, -]}_{1}, α_{1}) \to (L_{2}, {-, -, -}_{2}, {[-, -, -]}_{2}, α_{2})

be a Hom-NS-Lie triple system homomorphism. Then, φ is also a Hom-Lie triple system homomorphism between the subadjacent Hom-Lie triple system from

(L_{1}, {[[-, -, -]]}_{1}, α_{1})

(L_{2}, {[[-, -, -]]}_{2}, α_{2})

The following proposition illustrate that Hom-NS-Lie triple systems can be viewed as the underlying algebraic structures of generalized Reynolds operators on Hom-Lie triple systems.

Proposition 6.5.

Let

R : V \to L

be a generalized Reynolds operator on a Hom-Lie triple system

(L, [-, -, -], α)

associated to

(V, β; θ)

and 3-cocycle

H

. Then, the 4-tuple

(V, {-, -, -}_{θ}, {[-, -, -]}_{H}, β)

is a Hom-NS-Lie triple system, where

{u, v, w}_{θ} = θ (R v, R w) u, {[u, v, w]}_{H} = H (R u, R v, R w), \forall u, v, w \in V .

Proof.

For any

u, v, w, s, t \in V

, first, obviously, we have

{[u, v, w]}_{H} = - {[v, u, w]}_{H}

and

⥀_{u, v, w} {[u, v, w]}_{H} = 0

. On the one hand

\begin{matrix} {u, v, w}_{θ}^{*} = & {w, v, u}_{θ} - {w, u, v}_{θ} = θ (R v, R u) w - θ (R u, R v) w = D (R u, R v) w, \\ {[[u, v, w]]}_{V} = & {u, v, w}_{θ}^{*} + {u, v, w}_{θ} - {v, u, w}_{θ} + {[u, v, w]}_{H} \\ = & D (R u, R v) w + θ (R v, R w) u - θ (R u, R w) v + H (R u, R v, R w) . \end{matrix}

On the other hand, by Eqs. (2.5),(2.6), (3.1)-(3.2), we get

\begin{matrix} {β (s), β (t), {[[u, v, w]]}_{V}}_{θ} - {{s, t, u}_{θ}, β (v), β (w)}_{θ} + {{s, t, v}_{θ}, β (u), β (w)}_{θ} \\ - {β (u), β (v), {s, t, w}_{θ}}_{θ}^{*} \\ = & θ (R β (t), R {[[u, v, w]]}_{V}) β (s) - θ (R β (v), R β (w)) θ (R t, R u) s + θ (R β (u), R β (w)) θ (R t, R v) s \\ - D (R β (u), R β (v)) θ (R t, R w) s \\ = & 0, \\ {{s, t, u}_{θ}^{*}, β (v), β (w)}_{θ} + {β (u), {[[s, t, v]]}_{V}, β (w)}_{θ} + {β (u), β (v), {[[s, t, w]]}_{V}}_{θ} \\ - {β (s), β (t), {u, v, w}_{θ}}_{θ}^{*} \\ = & θ (R β (v), R β (w)) D (R s, R t) u + θ (R {[[s, t, v]]}_{V}, R β (w)) β (u) + θ (R β (v), R {[[s, t, w]]}_{V}) β (u) \\ - D (R β (s), R β (t)) θ (R v, R w) u \\ = & 0, \\ {[{[[s, t, u]]}_{V}, β (v), β (w)]}_{H} + {[β (u), {[[s, t, v]]}_{V}, β (w)]}_{H} + {[β (u), β (v), {[[s, t, w]]}_{V}]}_{H} \\ + {{[s, t, u]}_{H}, α (v), α (w)}_{θ} - {{[s, t, v]}_{H}, β (u), β (w)}_{θ} + {β (u), β (v), {[s, t, w]}_{H}}_{θ}^{*} \\ - {β (s), β (t), {[u, v, w]}_{H}}_{θ}^{*} - {[β (s), β (t), {[[u, v, w]]}_{V}]}_{H} \\ = & H (R {[[s, t, u]]}_{V}, R β (v), R β (w)) + H (R β (u), R {[[s, t, v]]}_{V}, R β (w)) + H (R β (u), R β (v), R {[[s, t, w]]}_{V}) \\ + θ (R α (v), R α (w)) H (R s, R t, R u) - θ (R β (u), R β (w)) H (R s, R t, R v) + D (R β (u), R β (v)) H (R s, R t, R w) \\ - D (R β (s), R β (t)) H (R u, R v, R w) - H (R β (s), R β (t), R {[[u, v, w]]}_{V}) \\ = & 0 . \end{matrix}

Thus

(V, {-, -, -}_{θ}, {[-, -, -]}_{H}, β)

is a Hom-NS-Lie triple system. □

Example 6.6.

Let

(L, [-, -, -], α)

be a Hom-Lie triple system and

N : L \to L

be a Nijenhuis operator. Then

(L, {-, -, -}_{θ}, {[-, -, -]}_{H},

α)

is a Hom-NS-Lie triple system, where

\begin{matrix} {a, b, c}_{θ} = & [a, N b, N c], \\ {[a, b, c]}_{H} = & - N ([N a, b, c] + [a, N b, c] + [a, b, N c]) + N^{2} [a, b, c], \forall a, b, c \in L . \end{matrix}

Proposition 6.7.

Let

R_{1} : V_{1} \to L_{1}

(resp.

R_{2} : V_{2} \to L_{2}

) be a generalized Reynolds operators on a Hom-Lie triple system

(L_{1}, {[-, -, -]}_{1}, α_{1})

(resp.

(L_{2}, {[-, -, -]}_{2}, α_{2})

) associated to

(V_{1}, β_{1}; θ_{1})

(resp.

(V_{2}, β_{2}; θ_{2})

) and 3-cocycle

H_{1}

(resp.

H_{2}

), and

(η, ζ)

be a homomorphism from

R_{1}

R_{2}

. Let

(V_{1}, {-, -, -}_{θ_{1}}, {[-, -, -]}_{H_{1}}, β_{1})

and

(V_{2}, {-, -, -}_{θ_{2}}, {[-, -, -]}_{H_{2}}, β_{2})

be the induced Hom-NS-Lie triple systems respectively. Then, ζ is a homomorphism from the Hom-NS-Lie triple system

(V_{1}, {-, -, -}_{θ_{1}}, {[-, -, -]}_{H_{1}}, β_{1})

(V_{2}, {-, -, -}_{θ_{2}}, [-, -,

{-]}_{H_{2}}, β_{2})

Proof.

For any

u, v, w \in V

, by Eqs. (3.3)-(3.6), we have

\begin{matrix} ζ ({u, v, w}_{θ_{1}}) = & ζ (θ_{1} (R_{1} v, R_{1} w) u) = θ_{2} (η (R_{1} v), η (R_{1} w)) ζ (u) \\ = & θ_{2} (R_{2} ζ (v), R_{2} ζ (w)) ζ (u) \\ = & {ζ (u), ζ (v), ζ (w)}_{θ_{2}}, \\ ζ ({[u, v, w]}_{H_{1}}) = & ζ (H_{1} (R_{1} u, R_{1} v, R_{1} w)) = H_{2} (η (R_{1} u), η (R_{1} v), η (R_{1} w)) \\ = & H_{2} (R_{2} ζ (u), R_{2} ζ (v), R_{2} ζ (w)) \\ = & {[ζ (u), ζ (v), ζ (w)]}_{H_{2}} . \end{matrix}

Hence,

φ_{h}

is a homomorphism from

(V_{1}, {-, -, -}_{θ_{1}}, {[-, -, -]}_{H_{1}}, β_{1})

(V_{2}, {-, -, -}_{θ_{2}}, [-,

{-, -]}_{H_{2}}, β_{2})

. □

Acknowledgments

The paper is supported by the Foundation of Science and Technology of Guizhou Province(Grant No. [2018]1020 ).

References

A. Baklouti. Quadratic Hom-Lie triple systems, J. Geometry and Physics, 2017, 121(18), 166–175.
I. Bakayoko. Hom-post-Lie modules, O-operators and some functors on Hom-algebras, arXiv preprint, 2016, arXiv:1610.02845.
Baxter, G. An analytic problem whose solution follows from a simple algebraic identity Pac. J. Math., 1960, 10, 731–742.
E. Cartan. Oeuvres completes. Part 1. Gauthier-Villars. Paris. (1952) vol. 2. nos. 101–138.
S. Chen, Q. Lou, Q. Sun. Cohomologies of Rota-Baxter Lie triple systems and applications, Commun. Algebra, 2023, 51(1957), 1–17.
T. Chtioui, A. Hajjaji, S. Mabrouk, A. Makhlouf. Cohomologies and deformations of O-operators on Lie triple systems, J. Math. Phys., 2023, 64, 081701.
T. Chtioui, A. Hajjaji, S. Mabrouk, A. Makhlouf. Twisted O-operators on 3-Lie algebras and 3-NS-Lie algebras, arXiv preprint, 2021, arXiv:2107.10890v1.
A. Das. Cohomology and deformations of twisted Rota-Baxter operators and NS-algebras, J. Homotopy Relat. Struct., 2022, 17, 233–262.
A. Das. Twisted Rota-Baxter operators and Reynolds operators on Lie algebras and NS-Lie algebras, J. Math. Phys., 2021, 62, 091701.
R. Gharbi, S. Mabrouk, A. Makhlouf. Maurer-Cartan type cohomology on generalized Reynolds operators and NS-structures on Lie triple systems, arXiv preprint, 2023, arXiv:2309.01385.
J.Hartwig, D. Larsson, S. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra, 2006, 295, 321–344.
S. Hou, Y. Sheng. Generalized Reynolds operators on 3-Lie algebras and NS-3-Lie algebras, Int. J. Geom. Methods Mod. Phys., 2021, 18, 2150223.
Y. Hou, Y. Ma, L. Chen. Product and complex structures on Hom-Lie triple systems. https://www.researchgate.net/publication/341360022.
N. Jacobson. Lie and Jordan triple Systems, Amer. J. Math. Soc., 1949, 71(1), 49–170.
N. Jacobson, General representation theory of Jordan algebras, Trans. Amer. Math. Soc., 1951, 70, 509–530.
Kampé de Fériet. J. Introduction to the Statistical Theory of Turbulence, Correlation and Spectrum; The Institute for Fluid Dynamics and Applied Mathematics University of Maryland: College Park, MD, USA, 1951; pp. iv+162.
Kupershmidt, B.A. What a classical r-matrix really is, J. Nonlinear Math. Phys., 1999, 6, 448–488.
Y. Li, D. Wang. Relative Rota-Baxter operators on Hom-Lie triple systems, Communications in Algebra, DOI: 10.1080/00927872.2023.2258223.
Y. Li, D. Wang. Twisted Rota-Baxter operators on 3-Hom-Lie algebras, Communications in Algebra, 2023, 51(1), 1–14.
Y. Ma, L. Chen, J. Lin. Central extensions and deformations of Hom-Lie triple systems, Communications in Algebra, 46(3),(2018), 1212–1230.
S. Mishra, A. Naolekar. O-operators on hom-Lie algebras, J. Math. Phys., 61 (2020), 121701.
Uchino, K. Quantum analogy of Poisson geometry, related dendriform algebras and Ro-ta-Baxter operators, Lett. Math. Phys., 2008, 85, 91–109.
O. Reynolds. On the dynamical theory of incompressible viscous fluids and the determina-tion of the criterion, Philos. Trans. Roy. Soc. A, 1895, 136, 123–164.
W. Teng, F. Long, H. Zhang, J. Jin. On compatible Hom-Lie triple systems, arXiv preprint, 2023, arXiv:2311.07531.
W. Teng, J. Jin, F. Long. Generalized reynolds operators on Lie-Yamaguti Algebras, Axioms, 2023, 12(10), 934.
W. Teng, J. Jin, F. Long. Relative Rota-Baxter operators on Hom-Lie-Yamaguti algebras, Journal of Mathematical Research with Applications, 2023, 43(6),648–664.
X. Wang, Y. Ma, L. Chen. Generalized Reynolds operators on Lie supertriple systems, 2023, https://www.researchgate.net/publication/371835147.
K. Yamaguti. On the cohomology space of Lie triple system, Kumamoto J. Sci. Ser. A, 1960,5, 44–52.
D. Yau. On n-ary Hom-Nambu and Hom-Nambu-Lie algebras, J. Geom. Phys., 2012, 62(2),506–522.
J. Zhou, L, Chen, Y. Ma. Generalized Derivations of Hom-Lie triple systems, Bull.Malaysian Math. Sciences Society, 2018, 41, 637–656.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.