Open AccessArticle

Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras

Hui Wu

¹,

Shuangjian Guo

² and

Xiaohui Zhang

^1,*

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China

Author to whom correspondence should be addressed.

Axioms 2024, 13(10), 685; https://doi.org/10.3390/axioms13100685

Submission received: 12 August 2024 / Revised: 19 September 2024 / Accepted: 30 September 2024 / Published: 2 October 2024

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

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Abstract

In this paper, we introduce two-term differential

L e i b_{\infty}

-conformal algebras and give characterizations of some particular classes of such two-term differential

L e i b_{\infty}

-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms of non-Abelian cohomology groups. Finally, we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of differential Leibniz conformal algebras.

Keywords:

differential Leibniz conformal algebra; cohomology; crossed module; non-Abelian extension; Wells exact sequences

MSC:

17B10; 17B38; 17B56; 18G45

1. Introduction

Kac in [1] has proposed Lie conformal algebras, usually considered as an axiomatic description of the singular part of the operator product expansion of chiral fields in conformal field theory. The past few years have witnessed considerable scholarly attention to this algebraic structure in the past few years because they are closely related to vertex algebras [2]. Many more properties and structures of Lie conformal algebras have been developed; see [3,4,5,6,7] and references cited therein.

Leibniz conformal algebras were introduced in [8], which are closely related to field algebras [9] and vertex algebras. Later, the author further elaborated upon and elucidated the concept of a conformal representation of a Leibniz algebra in [10]. After that, Zhang introduced the cohomology of Leibniz conformal algebras in [11] and Wu articulated the notion of a Leibniz pseudoalgebra, which is a multivariable generalization of the concept of Leibniz conformal algebras in [12]. Recently, Feng and Chen studied

O

-operators, also known as relative Rota–Baxter operators on Leibniz conformal algebras with respect to representations in [13]. Subsequently, the first author and Wang investigated some properties of relative Rota–Baxter operators on Leibniz conformal algebras with respect to representations and their connections with Leibniz dendriform conformal algebras in [14]. For further details on Leibniz conformal algebras, see [15,16]. Recently, the authors [17] introduced

L e i b_{\infty}

-conformal algebras where the Leibniz conformal identity holds up to homotopy. Additionally, they presented equivalent descriptions of

L e i b_{\infty}

-conformal algebras and identified certain characteristics of some particular classes of

L e i b_{\infty}

-conformal algebras in terms of the cohomology of Leibniz conformal algebras and crossed modules of Leibniz conformal algebras as a generalization of [18]. This study would introduce two-term differential

L e i b_{\infty}

-conformal algebra and generalize the common characteristics of some particular classes of such homotopy differential Leibniz conformal algebras, which constitute the major academic focus of this paper.

The extension problem has persisted and incurred scholarly dispute. Non-Abelian extensions were first developed in [19], which induces cohomology to the low dimensional non-Abelian group. The authors examined non-Abelian extensions of Leibniz algebras in [20]. See [21] and references cited therein. Naturally, we look into non-Abelian extensions of a differential Leibniz conformal algebra by another differential Leibniz conformal algebras. Another interesting study linked to extensions of algebraic structures is given by the inducibility of a pair of automorphisms, which, after all, is intimately connected with extensions of algebras. Such a study was first initiated by Wells in extensions of abstract groups in [22]. Later, the authors investigated extending automorphism in [23]. In [24], the authors studied the inducibility of a pair of automorphisms about a non-Abelian extension of Lie algebras. The results of [20,24] have been extended to Rota–Baxter Leibniz algebras in [25]. Naturally, we study the inducibility of a pair of differential Leibniz conformal algebra automorphisms and characterize them by equivalent conditions. This forms the second research focus of this paper.

The paper is organized as follows. In Section 2, we recall some basic definitions of differential Leibniz conformal algebras. In Section 3, we introduce homotopy differential operators on two-term

L e i b_{\infty}

-conformal algebras. A two-term

L e i b_{\infty}

-conformal algebra equipped with a homotopy differential operators is called a two-term differential

L e i b_{\infty}

-conformal algebra, and we give characterizations of some particular classes of such two-term differential

L e i b_{\infty}

-conformal algebras. In Section 4, we introduce non-Abelian cohomology groups and classify the non-Abelian extensions in terms of non-Abelian cohomology groups. In Section 5, we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of differential Leibniz conformal algebras.

2. Preliminaries

Throughout the paper, all algebraic systems are supposed to be over a field

C

. We denote by

Z

the set of all integers and

Z_{+}

the set of all nonnegative integers. We now recall some useful definitions in [8,11,26].

Definition 1.

A Leibniz conformal algebra is a

C [\partial]

-module R endowed with a λ-bracket

{[\cdot_{λ} \cdot]}_{R}

, which defines a

C

-bilinear map from

R \otimes R

R [λ] = C [λ] \otimes R

such that the following axioms hold:

\begin{matrix} {[\partial x_{λ} y]}_{R} = - λ {[x_{λ} y]}_{R}, {[x_{λ} \partial y]}_{R} = (\partial + λ) {[x_{λ} y]}_{R}, (conformal sesquilinearity) \\ {[x_{λ} {[y_{μ} z]}_{R}]}_{R} = {[{[x_{λ} y]}_{R λ + μ} z]}_{R} + {[y_{μ} {[x_{λ} z]}_{R}]}_{R}, (Jacobi identity) \end{matrix}

for any

x, y, z \in R

Definition 2.

A representation of a Leibniz conformal algebra R is a

C [\partial]

-module R endowed with left and right λ-actions, which are two

C

-linear maps

\begin{matrix} \cdot_{λ} : R \otimes V \to V [λ],_{λ} \cdot : V \otimes R \to V [λ] \end{matrix}

that satisfy the following conditions:

\begin{matrix} {(\partial x)}_{λ} u = - λ x_{λ} u, x_{λ} (\partial u) = (\partial + λ) x_{λ} u, \\ {(\partial u)}_{λ} x = - λ u_{λ} x, u_{λ} (\partial x) = (\partial + λ) u_{λ} x, \\ u_{λ} [x_{μ} y] = {(u_{λ} x)}_{λ + μ} y + x_{μ} (u_{λ} y), \\ x_{μ} (u_{λ} y) = {(x_{μ} u)}_{λ + μ} y + u_{λ} {[x_{μ} y]}_{R}, \\ x_{λ} (y_{μ} u) = {[x_{λ} y]}_{R λ + μ} u + y_{μ} (x_{λ} u), \end{matrix}

for any

x, y \in R

and

u \in V

It follows that any Leibniz conformal algebra R is a representation of itself with

\begin{matrix} x \cdot_{λ} y = {(L_{x})}_{λ} (y) = {[x_{λ} y]}_{R} and y \cdot_{λ} x = {(R_{x})}_{λ} (y) = [y_{λ} x], for x, y \in R . \end{matrix}

Here,

L_{x}

and

R_{x}

denote the left and right

λ

-bracket on R by x, respectively. This is called the regular representation.

Let R be a Leibniz conformal algebra and V a representation of R. For

n \geq 1

, an n-

λ

-bracket on R with coefficients in V is a

C

-linear map

f_{λ_{1}, \dots, λ_{n - 1}} : R^{\otimes n} \to V [λ_{1}, . . ., λ_{n - 1}]

denoted by

\begin{matrix} x_{1} \otimes \dots \otimes x_{n} \mapsto f_{λ_{1}, \dots, λ_{n - 1}} (x_{1}, \dots, x_{n}), \end{matrix}

satisfying the following sesquilinearity conditions:

\begin{matrix} f_{λ_{1}, \dots, λ_{n - 1}} (x_{1}, \dots, \partial x_{i}, \dots x_{n}) = - λ_{i} f_{λ_{1}, \dots, λ_{n - 1}} (x_{1}, \dots, x_{n}), 1 \leq i < n, \\ f_{λ_{1}, \dots, λ_{n - 1}} (x_{1}, \dots, \partial x_{n}) = (λ_{1} + \dots + λ_{n - 1} + \partial) f_{λ_{1}, \dots, λ_{n - 1}} (x_{1}, \dots, x_{n}) . \end{matrix}

Let

C_{LeibC}^{0} = V / \partial V

. For

n \geq 1

, let

C_{LeibC}^{n} = C_{LeibC}^{n} (R, V)

be the space of all n-

λ

-brackets on R with coefficients in V. Define

C_{LeibC}^{*} = \oplus_{n \in N} C_{LeibC}^{n}

as the space of all poly

λ

-brackets.

For

n \geq 1

f \in C_{LeibC}^{n}

, define

\begin{matrix} {(\partial_{LeibC} f)}_{λ_{1}, \dots, λ_{n}} (x_{1}, \dots, x_{n + 1}) \\ = & \sum_{i = 1}^{n} {(- 1)}^{i + 1} a_{i_{λ_{i}}} f_{λ_{1}, \dots, λ_{n}} (x_{1}, \dots, x_{n + 1}) + {(- 1)}^{n + 1} f_{λ_{1}, \dots, λ_{n - 1}} {(x_{1}, \dots, x_{n})}_{λ_{1} + \dots + λ_{n}} x_{n + 1} \\ + \sum_{1 \leq i < j \leq n + 1} {(- 1)}^{i} f_{λ_{1}, \dots, λ_{i} + λ_{j}, \dots, λ_{n}} (x_{1}, \dots, {[x_{i λ_{i}} x_{j}]}_{R}, \dots, x_{n + 1}) . \end{matrix}

The cohomology of this complex denoted by

H_{LeibC}^{*} (R, V)

is called the cohomology of the Leibniz conformal algebra R with coefficients in a representation V.

Let R be a Leibniz conformal algebra. Recall that a

C [\partial]

-linear map

d_{R} : R \to R

is called a differential operator such that

d_{R} ([x_{λ} y]) = [d_{R} {(x)}_{λ} y] + [x_{λ} d_{R} (y)] + α [d_{R} {(x)}_{λ} d_{R} (y)], \forall x, y \in R .

(1)

One denotes by

Der (R)

the set of differential operators of the Leibniz conformal algebra R.

Definition 3.

A differential Leibniz conformal algebra is a Leibniz conformal algebra R with a differential operator

d_{R} \in Der (R)

. One denotes it by

(R, d_{R})

Definition 4.

Given two differential Leibniz conformal algebras

(R, d_{R}), (Q, d_{Q})

, a homomorphism of differential Leibniz conformal algebras from

(R, d_{R})

(Q, d_{Q})

is a Leibniz conformal algebra homomorphism

φ : R \to Q

such that

φ \circ d_{R} = d_{Q} \circ φ

Definition 5.

Let

(R, d_{R})

be a differential Leibniz conformal algebra.

(i): A representation over the differential Leibniz conformal algebra $(R, d_{R})$ is a pair $(V, d_{V})$ , where $d_{V} \in Cend (V)$ , and V is a representation over the Leibniz conformal algebra R, such that for all $x \in R, u \in V,$ the following equalities hold:

$\begin{matrix} d_{V} (x \cdot_{λ} u) = d_{R} (x) \cdot_{λ} u + x \cdot_{λ} d_{V} (u) + α d_{R} (x) \cdot_{λ} d_{V} (u), \\ d_{V} (u \cdot_{λ} x) = u \cdot_{λ} d_{R} (x) + d_{V} (u) \cdot_{λ} x + α d_{V} (u) \cdot_{λ} d_{R} (x) . \end{matrix}$
(ii): Given two representations $(U, d_{U}), (V, d_{V})$ over $(R, d_{R})$ , a conformal linear map $f : U \to V$ is called a homomorphism of representations, if $f \circ d_{U} = d_{V} \circ f$ and

$f (x \cdot_{λ} u) = x \cdot_{λ} f (u), f (u \cdot_{λ} x) = f (u) \cdot_{λ} x, \forall x \in R, u \in V .$

Define the set of n-cochains by

\begin{matrix} C_{DLeibC}^{n} (R, V) : = \{\begin{matrix} C_{LeibC}^{n} (R, V) \oplus C_{LeibC}^{n - 1} (R, V), n \geq 2, \\ C_{LeibC}^{1} (R, V) = Hom (R, V), n = 1, \\ C_{LeibC}^{0} (R, V) = V, n = 0 . \end{matrix} \end{matrix}

For

n \geq 1

, we define a linear map

δ : C_{LeibC}^{n} (R, V) \to C_{LeibC}^{n} (R, V)

\begin{matrix} δ f_{λ_{1}, \dots, λ_{n}} (x_{1}, \dots, x_{n}) : \\ = \sum_{k = 1}^{n} α^{k - 1} \sum_{1 \leq i_{1} < \dots < i_{k} \leq n} f_{λ_{1}, \dots, λ_{n}} (x_{1}, \dots, d_{R} (x_{i_{1}}), \dots, d_{R} (x_{i_{k}}), \dots, x_{n}) - d_{V} f_{λ_{1}, \dots, λ_{n}} (x_{1}, \dots, x_{n}), \end{matrix}

for any

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

and

δ v = - d_{V} (v), \forall v \in C_{LeibC}^{0} (R, V) = V / \partial V .

Define

\partial_{DLeibC} : C_{LeibC}^{1} (R, V) \to C_{LeibC}^{2} (R, V)

\begin{matrix} \partial_{DLeibC} (f) = (\partial_{LieC} (f), - δ f), \forall f \in Hom (R, V) . \end{matrix}

Then, for

n \geq 2

, we define

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

\begin{matrix} \partial_{DLeibC} (f_{n}, g_{n - 1}) = (\partial_{LeibC} (f_{n}), \partial_{LeibC} (g_{n - 1}) + {(- 1)}^{n} δ f_{n}), \end{matrix}

for any

f_{n} \in C_{LeibC}^{n} (R, V)

and

g_{n - 1} \in C_{LeibC}^{n - 1} (R, V)

. The cohomology of the cochain complex

(C_{DLeibC}^{*} (R, V), \partial_{DLeibC})

, denoted by

H_{DLeibC}^{*} (R, V)

, is called the cohomology of the differential Leibniz conformal algebra

(R, d_{R})

with coefficients in the representation

(V, d_{V})

3. Crossed Modules and Two-Term Differential ${Leib}_{\infty}$ -Conformal Algebras

In this section, we introduce homotopy differential operators on two-term

L e i b_{\infty}

-conformal algebras. A two-term

L e i b_{\infty}

-conformal algebra equipped with a homotopy differential operator is called a two-term differential

L e i b_{\infty}

-conformal algebra. We show that skeletal two-term differential

L e i b_{\infty}

-conformal algebras correspond to the third cocycles of differential Leibniz conformal algebras. Next, we introduce crossed modules of differential Leibniz conformal algebras and show that crossed modules of differential Leibniz conformal algebras correspond to strict two-term differential

L e i b_{\infty}

-conformal algebras.

Definition 6

([17]). A two-term

L e i b_{\infty}

-conformal algebra is a triple

(R_{1} \overset{π}{⟶} R_{0}, ρ_{2}, ρ_{3})

consisting of a complex

R_{1} \overset{π}{⟶} R_{0}

C [\partial]

-modules equipped with

a $C$ -linear conformal sesquilinear map $ρ_{2} : R_{i} \otimes R_{j} \to R_{i + j} [λ]$ , for $0 \leq i, j, i + j \leq 1$ ,
a $C$ -linear conformal sesquilinear map $ρ_{3} : R_{0} \otimes R_{0} \otimes R_{0} \to R_{1} [λ, μ]$

that satisfy the following set of identities: for all

x, y, z, w \in R_{0}

and

u, v \in R_{1}

\begin{matrix} (L e i b 1) {(ρ_{2})}_{λ} (u, v) = 0, \\ (L e i b 2) π ({(ρ_{2})}_{λ} (x, u)) = {(ρ_{2})}_{λ} (x, π u), \\ (L e i b 3) π ({(ρ_{2})}_{λ} (u, x)) = {(ρ_{2})}_{λ} (π u, x), \\ (L e i b 4) {(ρ_{2})}_{λ} (π u, v) = {(ρ_{2})}_{λ} (u, π v), \\ (L e i b 5) π (ρ_{3})_{λ, μ} (x, y, z)) = {(ρ_{2})}_{λ} (x, {(ρ_{2})}_{μ} (y, z)) - {(ρ_{2})}_{λ + μ} ({(ρ_{2})}_{λ} (x, y), z) - {(ρ_{2})}_{μ} (y, {(ρ_{2})}_{λ} (x, z)), \\ (L e i b 6) {(ρ_{3})}_{λ, μ} (x, y, π v) = {(ρ_{2})}_{λ} (x, {(ρ_{2})}_{μ} (y, v)) - {(ρ_{2})}_{λ + μ} ({(ρ_{2})}_{λ} (x, y), v) - {(ρ_{2})}_{μ} (y, {(ρ_{2})}_{λ} (x, v)), \\ (L e i b 7) {(ρ_{3})}_{λ, μ} (x, π v, y) = {(ρ_{2})}_{λ} (x, {(ρ_{2})}_{μ} (v, y)) - {(ρ_{2})}_{λ + μ} ({(ρ_{2})}_{λ} (x, v), y) - {(ρ_{2})}_{μ} (v, {(ρ_{2})}_{λ} (x, y)), \\ (L e i b 8) {(ρ_{3})}_{λ, μ} (π v, x, y) = {(ρ_{2})}_{λ} (v, {(ρ_{2})}_{μ} (x, y)) - {(ρ_{2})}_{λ + μ} ({(ρ_{2})}_{λ} (v, x), y) - {(ρ_{2})}_{μ} (x, {(ρ_{2})}_{λ} (v, y)), \\ (L e i b 9) {(ρ_{2})}_{λ} (x, {(ρ_{3})}_{μ, ν} (y, z, w)) - {(ρ_{2})}_{μ} (y, {(ρ_{3})}_{λ, ν} (x, z, w)) + {(ρ_{2})}_{ν} (z, {(ρ_{3})}_{λ, μ} (x, y, w)) \\ + {(ρ_{2})}_{λ + μ + ν} ({(ρ_{3})}_{λ, μ} (x, y, z), w) - {(ρ_{3})}_{λ + μ, ν} ({(ρ_{2})}_{λ} (x, y), z, w) - {(ρ_{3})}_{μ, λ + ν} (y, {(ρ_{2})}_{λ} (x, z), w) \\ - {(ρ_{3})}_{μ, ν} (y, z, {(ρ_{2})}_{λ} (x, w)) + {(ρ_{3})}_{λ, μ + ν} (x, {(ρ_{2})}_{μ} (y, z), w) + {(ρ_{3})}_{λ, ν} (x, z, {(ρ_{2})}_{μ} (y, w)) \\ - {(ρ_{3})}_{λ, μ} (x, y, {(ρ_{2})}_{ν} (z, w)) = 0 . \end{matrix}

Definition 7.

Let

R = (R_{1} \overset{π}{⟶} R_{0}, ρ_{2}, ρ_{3})

be a two-term

L e i b_{\infty}

-conformal algebra. A triple

d = (d_{0}, d_{1}, d_{2})

, where

d_{0} : R_{0} \to R_{0}

and

d_{1} : R_{1} \to R_{1}

are conformal linear maps and

d_{2} : \land^{2} R_{0} \to R_{1} [λ]

is a conformal bilinear map, is called a homotopy differential operator on

R

i f π \circ d_{1} = d_{0} \circ π

, and for all

x, y, z \in R_{0}

and

u \in R_{1}

\begin{matrix} (D 1) π ({(d_{2})}_{λ} (x, y)) = d_{0} ({(ρ_{2})}_{λ} (x, y)) - {(ρ_{2})}_{λ} (d_{0} (x), y) - {(ρ_{2})}_{λ} (x, d_{0} (y)) - α {(ρ_{2})}_{λ} (d_{0} (x), d_{0} (y)), \\ (D 2) {(d_{2})}_{λ} (x, π u) = d_{1} ({(ρ_{2})}_{λ} (x, u)) - {(ρ_{2})}_{λ} (d_{0} (x), u) - {(ρ_{2})}_{λ} (x, d_{1} (u)) - α {(ρ_{2})}_{λ} (d_{0} (x), d_{1} (u)), \\ (D 3) {(d_{2})}_{λ} (π u, x) = d_{1} ({(ρ_{2})}_{λ} (u, x)) - {(ρ_{2})}_{λ} (u, d_{0} (x)) - {(ρ_{2})}_{λ} (d_{1} (u), x) - α {(ρ_{2})}_{λ} (d_{1} (u), d_{0} (x)), \\ (D 4) {(ρ_{3})}_{λ, μ} (d_{0} (x), y, z) + α {(ρ_{3})}_{λ, μ} (x, d_{0} (y), z) + α^{2} {(ρ_{3})}_{λ, μ} (x, y, d_{0} (z)) - d_{1} {(ρ_{3})}_{λ, μ} (x, y, z) \\ = {(ρ_{2})}_{λ + μ} ({(d_{2})}_{λ} (x, y), z) - {(ρ_{2})}_{λ + ν} ({(d_{2})}_{λ} (x, z), y) - {(ρ_{2})}_{λ} (x, {(d_{2})}_{μ} (y, z)) + {(d_{2})}_{λ + μ} ({(ρ_{2})}_{λ} (x, y), z) \\ - {(d_{2})}_{λ + ν} ({(ρ_{2})}_{λ} (x, z), y) - {(d_{2})}_{λ} (x, {(ρ_{2})}_{μ} (y, z)) . \end{matrix}

A two-term differential

L e i b_{\infty}

-conformal algebra is a two-term

L e i b_{\infty}

-conformal algebra

R = (R_{1} \overset{π}{⟶} R_{0}, ρ_{2}, ρ_{3})

equipped with a homotopy differential operator

d = (d_{0}, d_{1}, d_{2})

. We denote a two-term differential

L e i b_{\infty}

-conformal algebra by

(R_{1} \overset{π}{⟶} R_{0}, ρ_{2}, ρ_{3}

d_{0}, d_{1}, d_{2})

or simply by

(R, d)

Definition 8.

Let

(R, d)

be a two-term differential

L e i b_{\infty}

-conformal algebra. It is said to be

(i): Skeletal if $π = 0$ ,
(ii): Strict if $ρ_{3} = 0$ and $d_{2} = 0$ .

Theorem 1.

There is a one-to-one correspondence between skeletal two-term differential

L e i b_{\infty}

-conformal algebras and triples of the form

(R_{T}, V_{S}, (f, θ))

, where

(R, d_{R})

is a differential Leibniz conformal algebra,

(V, d_{V})

is a representation and

(f, θ) \in C_{DLeibC}^{3} (R, V)

is a 3-cocycle.

Proof.

Let

(R_{1} \overset{π}{⟶} R_{0}, ρ_{2}, ρ_{3}, d_{0}, d_{1}, d_{2})

be a skeletal two-term differential

L e i b_{\infty}

-conformal algebra. Then, according to (Leib5) and (D1), we obtain

(R_{0}, ρ_{2})

and operator

d_{0}

is a differential Leibniz conformal algebra. On the other hand, by conditions (Leib6), (Leib7), (Leib8), (D2) and (D3), we obtain that

(R_{1}, d_{1})

is a representation of the differential Leibniz conformal algebra

(R_{0}, d_{0})

with the left and right

λ

-actions

\begin{matrix} \cdot_{λ} : R_{0} \otimes R_{1} \to R_{1} [λ], x \cdot_{λ} u = {(ρ_{2})}_{λ} (x, u), \\ λ^{\cdot} : R_{1} \otimes R_{0} \to R_{1} [λ], u \cdot_{λ} x = {(ρ_{2})}_{λ} (u, x), \forall x \in R_{0}, u \in R_{1} . \end{matrix}

The conditions (Leib9) and (D4) are, respectively, equivalent to

\begin{matrix} δ_{DLeibC}^{3} (ρ_{3}, - d_{2}) = (\partial_{LeibC} (ρ_{3}), - \partial_{LeibC} (d_{2}) - δ ρ_{3}) = 0 . \end{matrix}

Thus,

(ρ_{3}, - d_{2}) \in C_{DLeibC}^{3} (R, V)

is a 3-cocycle.

Conversely, given a triple

((R, d_{R}), (V, d_{V}), (ρ_{3}, d_{2}))

as in the statement, define conformal bilinear maps

ρ_{2}

\begin{matrix} l_{λ}^{2} (x, y) = {[x_{λ} y]}_{R}, {(ρ_{2})}_{λ} (x, u) = x \cdot_{λ} u, {(ρ_{2})}_{λ} (u, x) = u \cdot_{λ} x, \end{matrix}

for

x, y \in R, u \in V

. Then,

(V \overset{0}{⟶} R, ρ_{2}, f, d_{R}, d_{V}, - d_{2})

is a skeletal two-term differential

L e i b_{\infty}

-conformal algebra. □

Next, we introduce crossed modules of differential Leibniz-conformal algebras and characterize strict two-term differential

L e i b_{\infty}

-conformal algebras.

Definition 9.

A crossed module of differential Leibniz conformal algebras consists of

((R_{0}, d_{0}), (R_{1}, d_{1}),

π, ρ^{L}, ρ^{R})

, where

(R_{0}, d_{0})

and

(R_{1}, d_{1})

are differential Leibniz conformal algebras,

π : (R_{1}, d_{1}) \to (R_{0}, d_{0})

is a differential Leibniz conformal algebra homomorphism, and

ρ^{L} : R_{0} \otimes R_{1} \to R_{1} [λ]

ρ^{R} : R_{1} \otimes R_{0} \to R_{1} [λ]

, and make

(R_{1}, d_{1})

into a representation of the differential Leibniz conformal algebra

(R_{0}, d_{0})

satisfying

\begin{matrix} (C a) & π (ρ^{L} {(x)}_{λ} (u)) = {[x_{λ} π (u)]}_{R_{0}}, π (ρ^{R} {(x)}_{λ} (u)) = {[π {(u)}_{λ} x]}_{R_{0}}, \\ (C b) & ρ^{L} {(π (u))}_{λ} (v) = {[u_{λ} v]}_{R_{1}}, ρ^{R} {(π (u))}_{λ} (v) = {[v_{λ} u]}_{R_{1}}, \end{matrix}

for any

x \in R_{0}, u, v \in R_{1}

Proposition 1.

Let

((R_{0}, d_{0}), (R_{1}, d_{1}), π, ρ^{L}, ρ^{R})

be a crossed module of differential Leibniz conformal algebras. Then,

(R_{0} \oplus R_{1}, d_{0} \oplus d_{1})

is a differential Leibniz conformal algebra, where the bracket is

\begin{matrix} [{(x, u)}_{λ} (y, v)] = ({[x_{λ} y]}_{R_{0}}, ρ^{L} {(x)}_{λ} v + ρ^{R} {(y)}_{λ} u + {[u_{λ} v]}_{R_{1}}) \end{matrix}

for any

x, y \in R_{0}, u, v \in R_{1}

Proof.

Since

R_{0}, R_{1}

are both Leibniz conformal algebras and

(R_{1}, ρ^{L}, ρ^{R})

is a representation of

R_{0}

, then we have that

R_{0} \oplus R_{1}

is a Leibniz conformal algebra. Moreover, for any

(x, u), (y, v) \in R_{0} \oplus R_{1}

, we have

\begin{matrix} (d_{0} \oplus d_{1}) [{(x, u)}_{λ} (y, v)] \\ = (d_{0} \oplus d_{1}) ({[x_{λ} y]}_{R_{0}}, ρ^{L} {(x)}_{λ} v + ρ^{R} {(y)}_{λ} u + {[u_{λ} v]}_{R_{1}}) \\ = (d_{0} ({[x_{λ} y]}_{R_{0}}), d_{1} (ρ^{L} {(x)}_{λ} v) + d_{1} (ρ^{R} {(y)}_{λ} u) + d_{1} ({[u_{λ} v]}_{R_{1}})) \\ = ({[d_{0} {(x)}_{λ} y]}_{R_{0}} + {[x_{λ} d_{0} (y)]}_{R_{0}} + α {[d_{0} {(x)}_{λ} d_{0} (y)]}_{R_{0}}, ρ^{L} {(x)}_{λ} d_{1} (v) + ρ^{L} {(d_{0} (x))}_{λ} v \\ + α ρ^{L} {(d_{0} (x))}_{λ} d_{1} (v) + ρ^{R} {(y)}_{λ} d_{1} (u) + ρ^{R} {(d_{0} (y))}_{λ} u + α ρ^{R} {(d_{0} (y))}_{λ} d_{1} (u) \\ + {[d_{1} {(u)}_{λ} v]}_{R_{1}} + {[u_{λ} d_{1} (v)]}_{R_{1}} + α {[d_{1} {(u)}_{λ} d_{1} (v)]}_{R_{1}}) \\ = [(d_{0} \oplus d_{1}) {(x, u)}_{λ} (y, v)] + [{(x, u)}_{λ} (d_{0} \oplus d_{1}) (y, v)] + α [(d_{0} \oplus d_{1}) {(x, u)}_{λ} (d_{0} \oplus d_{1}) (y, v)] . \end{matrix}

This shows that the map

d_{0} \oplus d_{1} : R_{0} \oplus R_{1} \to R_{0} \oplus R_{1}

is a differential operator. And the proof is finished. □

Theorem 2.

There is a one-to-one correspondence between strict two-term differential

L e i b_{\infty}

-conformal algebras and crossed modules of differential Leibniz conformal algebras.

Proof.

Let

(R_{1} \overset{π}{⟶} R_{0}, ρ_{2}, l^{3} = 0, d_{0}, d_{1}, d_{2} = 0)

be a strict two-term differential

L e i b_{\infty}

-conformal algebra. Then, according to (Leib5) and (D1), we obtain

(R_{0}, ρ_{2})

, and operator

d_{0}

is a differential Leibniz conformal algebra. Next, we define

{[\cdot_{λ} \cdot]}_{R_{1}} : R_{1} \otimes R_{1} \to R_{1} [λ]

{[u_{λ} v]}_{R_{1}} = {(ρ_{2})}_{λ} (π u, v) = {(ρ_{2})}_{λ} (u, π v)

, for any

u, v \in R_{1}

. By conditions (Leib6) and (D3), we obtain that

(R_{1}, d_{1})

is a differential Leibniz conformal algebra. On the other hand, the condition (Leib2) implies that

π : (R_{1}, d_{1}) \to (R_{0}, d_{0})

is a differential Leibniz conformal algebra morphism. Finally, we define

\begin{matrix} ρ^{L} : R_{0} \to R_{1} \to R_{1} [λ], ρ^{L} {(x)}_{λ} u = {(ρ_{2})}_{λ} (x, u), \\ ρ^{R} : R_{1} \to R_{0} \to R_{1} [λ], ρ^{R} {(x)}_{λ} u = {(ρ_{2})}_{λ} (u, x), \forall \in R_{0}, u \in R_{1} . \end{matrix}

Then, we obtain that

((R_{1}, d_{1}), ρ^{L}, ρ^{R})

is a representation of the differential Leibniz conformal algebra

(R_{0}, d_{0})

; by the conditions (Leib9) and (D4), we also have

\begin{matrix} π (ρ^{L} {(x)}_{λ} u) = π {(ρ_{2})}_{λ} (x, u) = {(ρ_{2})}_{λ} (x, π u), π (ρ^{R} {(x)}_{λ} u) = π {(ρ_{2})}_{λ} (u, x) = {(ρ_{2})}_{λ} (π u, x), \\ ρ^{L} {(π u)}_{λ} v = π {(ρ_{2})}_{λ} (π u, v) = {[u_{λ} v]}_{R_{1}}, ρ^{R} {(π u)}_{λ} v = π {(ρ_{2})}_{λ} (v, π u) = {[v_{λ} u]}_{R_{1}}, \end{matrix}

for any

x \in R_{0}, u, v \in R_{1}

. Thus,

((R_{0}, d_{0}), (R_{1}, d_{1}), π, ρ^{L}, ρ^{R})

is a crossed module of differential Leibniz conformal algebras.

Conversely, let

((R_{0}, d_{0}), (R_{1}, d_{1}), π, ρ^{L}, ρ^{R})

be a crossed module of differential Leibniz conformal algebras. Define conformal bilinear maps

ρ_{2} : R_{i} \times R_{j} \to R_{i + j} [λ], i + j \leq 1

\begin{matrix} {(ρ_{2})}_{λ} (x, y) = {[x_{λ} y]}_{R_{0}}, {(ρ_{2})}_{λ} (x, u) = ρ^{L} {(x)}_{λ} u, \\ {(ρ_{2})}_{λ} (u, x) = ρ^{R} {(x)}_{λ} u, {(ρ_{2})}_{λ} (u, v) = 0, \end{matrix}

for

x, y \in R_{0}, u, v \in R_{1}

. Hence,

(R_{1} \overset{π}{⟶} R_{0}, ρ_{2}, ρ_{3} = 0, d_{0}, d_{1}, d_{2} = 0)

is a strict two-term differential

L e i b_{\infty}

-conformal algebra. □

Combining Proposition 1 and Theorem 2, we obtain the following result.

Proposition 2.

Let

(R, d)

be a strict two-term differential

L e i b_{\infty}

-conformal algebra. Then,

(R_{0} \oplus R_{1}, d_{0} \oplus d_{1})

is a differential Leibniz conformal algebra, where the bracket is

\begin{matrix} [{(x, u)}_{λ} (y, v)] = ({(ρ_{2})}_{λ} (x, y), {(ρ_{2})}_{λ} (x, v) + {(ρ_{2})}_{λ} (u, y) + {(ρ_{2})}_{λ} (u, v)), \end{matrix}

for any

(x, u), (y, v) \in R_{0} \oplus R_{1}

Example 1.

Let

(R_{0}, d_{0})

be a differential Leibniz conformal algebra. Then,

((R_{0}, d_{0}), (R_{0}, d_{0}), id, L_{x}, R_{x})

is a crossed module of differential Leibniz conformal algebras. Therefore, it follows that

\begin{matrix} (R_{0} \overset{id}{⟶} R_{0}, {[\cdot, \cdot]}_{R_{0}}, ρ_{3} = 0, d_{0}, d_{0}, d_{2} = 0) \end{matrix}

is a strict two-term differential

L e i b_{\infty}

-conformal algebra.

Example 2.

Let

(R_{0}, d_{0})

and

(R_{1}, d_{1})

be a differential Leibniz conformal algebras, let

f : (R_{0}, d_{0}) \to (R_{1}, d_{1})

be a differential Leibniz conformal algebra morphism and let

i : R_{1} \to R_{0}

be the inclusion map. Then,

(Ker f, R_{0}, i, L_{x}, R_{x})

is a crossed module of differential Leibniz conformal algebras.

4. Non-Abelian Extension of Differential Leibniz Conformal Algebras

In this section, we study non-Abelian extensions of a differential Leibniz conformal algebra by another differential Leibniz conformal algebra.

Definition 10.

Let

(R, d_{R})

and

(Q, d_{Q})

be two differential Leibniz conformal algebras. A non-Abelian extension of

(R, d_{R})

(Q, d_{Q})

is a differential Leibniz conformal algebra

(E, d_{E})

equipped with a short exact sequence of differential Leibniz conformal algebras

\begin{matrix} 0 \to (Q, d_{Q}) \overset{i}{\to} (E, d_{E}) \overset{p}{\to} (R, d_{R}) \to 0 . \end{matrix}

(2)

Definition 11.

Let

(E, d_{E})

and

(E^{'}, d_{E^{'}})

be two non-Abelian extensions of

(R, d_{R})

(Q, d_{Q})

. They are said to be equivalent if there is a morphism

τ : (E, d_{E}) \to (E^{'}, d_{E^{'}})

of differential Leibniz conformal algebras making the following diagram commutative:

\begin{matrix} \begin{matrix} 0 & \to & (Q, d_{Q}) & \overset{i_{1}}{\to} & (E, d_{E}) & \overset{p_{1}}{\to} & (R, d_{R}) & \to & 0 \\ id ↓ & τ ↓ & id ↓ \\ 0 & \to & (Q, d_{Q}) & \overset{i_{2}}{\to} & (E^{'}, d_{E^{'}}) & \overset{p_{2}}{\to} & (R, d_{R}) & \to & 0 . \end{matrix} \end{matrix}

(3)

The set of all equivalence classes of non-Abelian extensions of

(R, d_{R})

(Q, d_{Q})

is denoted by

{Ext}_{nab} ((R, d_{R}), (Q, d_{Q}))

Example 3.

Let

((R_{0}, d_{0}), (R_{1}, d_{1}), π, ρ^{L}, ρ^{R})

be a crossed module of differential Leibniz conformal algebras. Then, the exact sequence

\begin{matrix} 0 \to (R_{1}, d_{1}) \overset{i}{\to} (R_{0} \oplus R_{1}, d_{0} \oplus d_{1}) \overset{p}{\to} (R_{0}, d_{0}) \to 0, \end{matrix}

is a non-Abelian extension of

(R_{0}, d_{R_{0}})

(R_{1}, d_{R_{1}})

We denote the set of equivalence classes of non-Abelian 2-cocycles by

H_{nab}^{2} ((R, d_{R}), (Q, d_{Q}))

Let

(E, d_{E})

be a non-Abelian extension of the differential Leibniz conformal algebra

(R, d_{R})

(Q, d_{Q})

as of (2). A section of p is a linear map

s : R \to E

that satisfies

p \circ s = {id}_{R}

. We define conformal maps

ω : \land^{2} R \to Q [λ], \cdot_{λ} : R \otimes Q \to Q [λ],_{λ} \cdot : Q \otimes R \to Q [λ]

and

Ω : R \to Q

\begin{matrix} ω_{λ} (x, y) = {[s {(x)}_{λ} s (y)]}_{E} - s ({[x_{λ} y]}_{R}), \\ x \cdot_{λ} p = {[s {(x)}_{λ} p]}_{E}, \\ p \cdot_{λ} x = {[p_{λ} s (x)]}_{E}, \\ Ω (x) : = d_{E} (s (x)) - s (d_{R} (x)), \forall x, y \in R, p \in Q . \end{matrix}

Further, we define

R \oplus Q

by the bracket

\begin{matrix} [{(x, p)}_{λ} (y, q)] : = ({[x_{λ} y]}_{R}, x \cdot_{λ} q + p \cdot_{λ} y + ω_{λ} (x, y) + {[p_{λ} q]}_{Q}), \end{matrix}

with the conformal linear map

\begin{matrix} d_{Ω} (x, p) = (d_{R} (x), d_{Q} (p) + Ω (x)) . \end{matrix}

Lemma 1.

With the above notations,

R \oplus Q

is a Leibniz conformal algebra if and only if

ω, \cdot_{λ},_{λ} \cdot

satisfy the following conditions:

\begin{matrix} {[x_{λ} y]}_{R} \cdot_{λ + μ} p = x \cdot_{λ} (y \cdot_{μ} p) - y \cdot_{μ} (x \cdot_{λ} p) - {[ω_{λ} {(x, y)}_{λ + μ} p]}_{Q}, \end{matrix}

(4)

\begin{matrix} p \cdot_{μ} {[x_{λ} y]}_{R} = x \cdot_{λ} (p \cdot_{μ} y) - (x \cdot_{λ} p) \cdot_{λ + μ} y - {[p_{μ} ω_{λ} (x, y)]}_{Q}, \end{matrix}

(5)

\begin{matrix} p \cdot_{μ} {[x_{λ} y]}_{R} = (p \cdot_{μ} x) \cdot_{λ + μ} y + x \cdot_{λ} (p \cdot_{μ} y) - {[p_{μ} ω_{λ} (x, y)]}_{Q}, \end{matrix}

(6)

\begin{matrix} x \cdot_{λ} {[p_{μ} q]}_{Q} = {[{(x \cdot_{λ} p)}_{λ + μ} q]}_{Q} + {[p_{μ} (x \cdot_{λ} q)]}_{Q}, \end{matrix}

(7)

\begin{matrix} x \cdot_{λ} {[p_{μ} q]}_{Q} = {[p_{μ} (x \cdot_{λ} q)]}_{Q} - {[{(p \cdot_{μ} x)}_{λ + μ} q]}_{Q}, \end{matrix}

(8)

\begin{matrix} {[p_{λ} q]}_{Q} \cdot_{λ + μ} x = {[p_{λ} (q \cdot_{μ} x)]}_{Q} - {[q_{μ} (p \cdot_{λ} x)]}_{Q}, \end{matrix}

(9)

\begin{matrix} x \cdot_{λ} ω_{μ} (y, z) - y \cdot_{μ} ω_{λ} (x, z) - ω_{λ} (x, y) \cdot_{ν} z = ω_{λ + μ} ({[x_{λ} y]}_{R}, z) - ω_{λ} (x, {[y_{μ} z]}_{R}) + ω_{μ} (y, {[x_{λ} z]}_{R}) . \end{matrix}

(10)

Proof.

For any

x, y, z \in R, p, q \in Q

, we have

\begin{matrix} [\partial {(x + p)}_{λ} (y + q)] & = [{(\partial x + \partial p)}_{λ} (y + q)] \\ = ({[\partial x_{λ} y]}_{R}, (\partial x \cdot_{λ} q) + (\partial p \cdot_{λ} y) + ω_{λ} (\partial x, y) + {[\partial p_{λ} q]}_{Q}) \\ = (- λ {[x_{λ} y]}_{R}, - λ (x \cdot_{λ} q) - λ (p \cdot_{λ} y) - λ ω_{λ} (x, y) - λ {[p_{λ} q]}_{Q}) \\ = - λ [{(x + p)}_{λ} (y + q)] . \end{matrix}

Similar, we have

\begin{matrix} [{(x + p)}_{λ} \partial (y + q)] = (\partial + λ) [{(x + p)}_{λ} (y + q)] . \end{matrix}

Further, assume that

R \oplus Q

is a Leibniz conformal algebra. By

\begin{matrix} [x_{λ} [y_{μ} p]] = [{[x_{λ} y]}_{R λ + μ} p] + [y_{μ} [x_{λ} p]], \end{matrix}

we deduce that (4) holds. By

\begin{matrix} [x_{λ} [p_{μ} y]] = [{[x_{λ} p]}_{λ + μ} y] + [p_{μ} {[x_{λ} y]}_{R}], \end{matrix}

we deduce that (5) holds. Similar to deduce that (6) holds. By

\begin{matrix} [x_{λ} {[p_{μ} q]}_{Q}] = [{[x_{λ} p]}_{λ + μ} q] + [p_{μ} [x_{λ} q]], \end{matrix}

we deduce that (7) holds. Similarly, we deduce that (8)–(9) hold. By

\begin{matrix} {[x_{λ} {[y_{μ} z]}_{R}]}_{R} = {[{[x_{λ} y]}_{R λ + μ} z]}_{R} + {[y_{μ} {[x_{λ} z]}_{R}]}_{R}, \end{matrix}

we deduce that (10) holds.

Conversely, if (4)–(10) hold, it is straightforward to see that

R \oplus Q

is a Leibniz conformal algebra. The proof is finished. □

Lemma 2.

The maps

ω, \cdot_{λ},_{λ} \cdot, Ω

defined above satisfy the following compatible conditions: for all

x, y \in R

and

p \in Q

\begin{matrix} d_{Q} ω_{λ} (x, y) + Ω ({[x_{λ} y]}_{R}) = Ω (x) \cdot_{λ} y + ω_{λ} (d_{R} (x), y) + x \cdot_{λ} Ω (y) + ω_{λ} (x, d_{R} (y)) \end{matrix}

\begin{matrix} + α d_{R} (x) \cdot_{λ} Ω (y) + α Ω (x) \cdot_{λ} d_{R} (y) + α ω_{λ} (d_{R} (x), d_{R} (y)) + α {[Ω {(x)}_{λ} Ω (y)]}_{E}, \end{matrix}

(11)

\begin{matrix} {[Ω {(x)}_{λ} p]}_{E} + α d_{R} (x) \cdot_{λ} d_{Q} (p) + α {[Ω {(x)}_{λ} d_{Q} (p)]}_{E} - d_{Q} (x \cdot_{λ} p) + d_{R} (x) \cdot_{λ} p + x \cdot_{λ} d_{Q} (p) = 0, \end{matrix}

(12)

\begin{matrix} {[p_{λ} Ω (x)]}_{E} + α d_{Q} (p) \cdot_{λ} d_{R} (x) + α {[d_{Q} {(p)}_{λ} Ω (x)]}_{E} - d_{Q} (p \cdot_{λ} x) + d_{Q} (p) \cdot_{λ} x + p \cdot_{λ} d_{R} (x) = 0 . \end{matrix}

(13)

Proof.

For any

x, y \in R

, we have

\begin{matrix} Ω (x) \cdot_{λ} y + ω_{λ} (d_{R} (x), y) + x \cdot_{λ} Ω (y) + ω_{λ} (x, d_{R} (y)) - d_{Q} ω_{λ} (x, y) - Ω ({[x_{λ} y]}_{R}) \\ + α d_{R} (x) \cdot_{λ} Ω (y) + α Ω (x) \cdot_{λ} d_{R} (y) + α ω_{λ} (d_{R} (x), d_{R} (y)) + α {[Ω {(x)}_{λ} Ω (y)]}_{E} \\ = \underset{2 A}{\underset{︸}{{[d_{E} {(s (x))}_{λ} s (y)]}_{E}}} - \underset{1 D}{\underset{︸}{{[s {(d_{R} (x))}_{λ} s (y)]}_{E}}} + \underset{1 D}{\underset{︸}{{[s {(d_{R} (x))}_{λ} s (y)]}_{E}}} - \underset{2 B}{\underset{︸}{s ({[d_{R} {(x)}_{λ} y]}_{R})}} + \underset{2 A}{\underset{︸}{{[s {(x)}_{λ} d_{E} (s (y))]}_{E}}} \\ - \underset{1 C}{\underset{︸}{{[s {(x)}_{λ} s (d_{R} (y))]}_{E}}} + \underset{1 C}{\underset{︸}{{[s {(x)}_{λ} s (d_{R} (y))]}_{E}}} - \underset{2 B}{\underset{︸}{s ({[x_{λ} d_{R} (y)]}_{R})}} - \underset{2 A}{\underset{︸}{d_{Q} {[s {(x)}_{λ} s (y)]}_{E}}} + \underset{1 B}{\underset{︸}{d_{Q} s ({[x_{λ} y]}_{R})}} \end{matrix}

\begin{matrix} - \underset{1 B}{\underset{︸}{d_{E} (s ({[x_{λ} y]}_{R}))}} + \underset{2 B}{\underset{︸}{s (d_{R} ({[x_{λ} y]}_{R}))}} + \underset{1 E}{\underset{︸}{α {[s {(d_{R} (x))}_{λ} d_{E} (s (y))]}_{E}}} - \underset{1 A}{\underset{︸}{α {[s {(d_{R} (x))}_{λ} s (d_{R} (y))]}_{E}}} \\ + \underset{1 F}{\underset{︸}{α {[d_{E} {(s (x))}_{λ} s (d_{R} (y))]}_{E}}} - \underset{1 E}{\underset{︸}{α {[s {(d_{R} (x))}_{λ} s (d_{R} (y))]}_{E}}} + \underset{1 E}{\underset{︸}{α {[s {(d_{R} (x))}_{λ} s (d_{R} (y))]}_{E}}} - \underset{2 B}{\underset{︸}{α s ({[d_{R} {(x)}_{λ} d_{R} (y)]}_{R})}} \\ + \underset{2 A}{\underset{︸}{α {[d_{E} {(s (x))}_{λ} d_{E} (s (y))]}_{E}}} - \underset{1 F}{\underset{︸}{α {[d_{E} {(s (x))}_{λ} s (d_{R} (y))]}_{E}}} - \underset{1 E}{\underset{︸}{α {[s {(d_{R} (x))}_{λ} d_{E} (s (y))]}_{E}}} + \underset{1 A}{\underset{︸}{α {[s {(d_{R} (x))}_{λ} s (d_{R} (y))]}_{E}}} \\ = 0, \end{matrix}

and we deduce that (11) holds. Further, for any

x \in R

and

p \in Q

, we have

\begin{matrix} {[Ω {(x)}_{λ} p]}_{E} + α d_{R} (x) \cdot_{λ} d_{Q} (p) + α {[Ω {(x)}_{λ} d_{Q} (p)]}_{E} - d_{Q} (x \cdot_{λ} p) + d_{R} (x) \cdot_{λ} p + x \cdot_{λ} d_{Q} (p) \\ = {[d_{E} {(s (x))}_{λ} p]}_{E} - \underset{A}{\underset{︸}{{[s d_{R} {((x))}_{λ} p]}_{E}}} + \underset{B}{\underset{︸}{α {[s {(d_{R} (x))}_{λ} d_{Q} (p)]}_{E}}} + α {[d_{E} {(s (x))}_{λ} d_{Q} (p)]}_{E} \underset{B}{\underset{︸}{- α {[s {(d_{R} (x))}_{λ} d_{Q} (p)]}_{E}}} \\ - d_{Q} {[s {(x)}_{λ} p]}_{E} + \underset{A}{\underset{︸}{{[s {(d_{R} (x))}_{λ} p]}_{E}}} + {[s {(x)}_{λ} d_{Q} (p)]}_{E} \\ = {[d_{E} {(s (x))}_{λ} p]}_{E} + {[s {(x)}_{λ} d_{Q} (p)]}_{E} + α {[d_{E} {(s (x))}_{λ} d_{Q} (p)]}_{E} - d_{Q} {[s {(x)}_{λ} p]}_{E} \\ = 0 . \end{matrix}

This means Equation (12) is satisfied. Similarly, one can check that Equation (13) holds. □

Definition 12.

(i): Let $(R, d_{R})$ and $(Q, d_{Q})$ be two differential Leibniz conformal algebras. A non-Abelian 2-cocycle of $(R, d_{R})$ with values in $(Q, d_{Q})$ is a quadruple $(ω, \cdot_{λ},_{λ} \cdot, Ω)$ of conformal linear maps $ω : \land^{2} R \to Q [λ], \cdot_{λ} : R \otimes Q \to Q [λ],_{λ} \cdot : Q \otimes R \to Q [λ]$ and $Ω : R \to Q$ satisfying the conditions (4)–(13).
(ii): Let $(ω, \cdot_{λ},_{λ} \cdot, Ω)$ and $(ω^{'}, \cdot_{λ}^{'},_{λ} \cdot^{'}, Ω^{'})$ be two non-Abelian 2-cocycles of $(R, d_{R})$ with values in $(Q, d_{Q})$ . They are said to be equivalent if there exists a conformal linear map $η : R \to Q$ that satisfies

$\begin{matrix} ω_{λ} (x, y) - ω_{λ}^{'} (x, y) = x \cdot_{λ}^{'} η (y) + η (x) \cdot_{λ}^{'} y - η {[x_{λ} y]}_{R} + {[η {(x)}_{λ} η (y)]}_{Q}, \end{matrix}$

(14)

$\begin{matrix} x \cdot_{λ} p - x \cdot_{λ}^{'} p = {[η {(x)}_{λ} p]}_{E}, \end{matrix}$

(15)

$\begin{matrix} p \cdot_{λ} x - p \cdot_{λ}^{'} x = {[p_{λ} η (x)]}_{E}, \end{matrix}$

(16)

$\begin{matrix} Ω (x) - Ω^{'} (x) = d_{Q} (η (x)) - η (d_{R} (x)), \forall x, y \in R, p \in Q . \end{matrix}$

(17)

We denote the set of equivalence classes of non-Abelian 2-cocycles by

H_{nab}^{2} ((R, d_{R}), (Q, d_{Q}))

With the above notations, we obtain the following result.

Theorem 3.

Let

(R, d_{R})

and

(Q, d_{Q})

be two differential Leibniz conformal algebras. Then, the set of equivalence classes of non-Abelian extensions of

(R, d_{R})

(Q, d_{Q})

is classified by

H_{nab}^{2} ((R, d_{R}), (Q, d_{Q}))

. In other words,

\begin{matrix} {Ext}_{nab} ((R, d_{R}), (Q, d_{Q})) ≅ H_{nab}^{2} ((R, d_{R}), (Q, d_{Q})) . \end{matrix}

Proof.

Let

(E, d_{E})

and

(E^{'}, d_{E^{'}})

be two equivalent extensions of

(R, d_{R})

(Q, d_{Q})

. If

s : R \to E

is a section of the map p, then it is easy to observe that the map

s^{'} : = η \circ s

is a section of the map

p^{'}

. Let

(ω^{'}, \cdot_{λ}^{'},_{λ} \cdot^{'}, Ω^{'})

be the non-Abelian 2-cocycle corresponding to the non-Abelian extension

(E^{'}, d_{E^{'}})

with section

s^{'}

, for any

x, y \in R, q \in Q

, we have

\begin{matrix} ω_{λ}^{'} (x, y) & = {[s^{'} {(x)}_{λ} s^{'} (y)]}_{E} - s^{'} {[x_{λ} y]}_{R} \\ = {[η \circ s {(x)}_{λ} η \circ s (y)]}_{E} - η \circ s {[x_{λ} y]}_{R} \\ = η ({[s {(x)}_{λ} s (y)]}_{E} - s {[x_{λ} y]}_{R}) \\ = ω_{λ} (x, y) . \end{matrix}

Similarly,

x \cdot_{λ} q = x \cdot_{λ}^{'} q, q \cdot_{λ} x = q \cdot_{λ}^{'} x

and

Ω (x) = Ω^{'} (x)

. This shows that

(ω^{'}, \cdot_{λ}^{'},_{λ} \cdot^{'}, Ω^{'}) = (ω, \cdot_{λ},_{λ} \cdot, Ω)

. Hence they give rise to the same element in

H_{nab}^{2} ((R, d_{R}), (Q, d_{Q}))

. Therefore, there is a well-defined map

Π : {Ext}_{nab} ((R, d_{R}), (Q, d_{Q})) \to H_{nab}^{2} ((R, d_{R}), (Q, d_{Q}))

Conversely, let

(ω, \cdot_{λ},_{λ} \cdot, Ω)

be a non-Abelian 2-cocycle on

(R, d_{R})

with values in

(Q, d_{Q})

. Define

E : = R \oplus Q

with the bracket

\begin{matrix} [{(x, p)}_{λ} (y, q)] : = ({[x_{λ} y]}_{R}, x \cdot_{λ} q + p \cdot_{λ} y + ω_{λ} (x, y) + {[p_{λ} q]}_{Q}) . \end{matrix}

and the conformal linear map

\begin{matrix} d_{E}^{Ω} (x, p) = (d_{R} (x), d_{Q} (p) + Ω (x)) . \end{matrix}

According to the conditions (4)–(10), it can be easily verified that E is a Leibniz conformal algebra. Moreover, we observe that

\begin{matrix} d_{E}^{Ω} ([{(x, p)}_{λ} (y, q)]) \\ = d_{E}^{Ω} ({[x_{λ} y]}_{R}, x \cdot_{λ} q + p \cdot_{λ} y + ω_{λ} (x, y) + {[p_{λ} q]}_{Q}) \\ = (d_{R} ({[x_{λ} y]}_{R}), d_{Q} (x \cdot_{λ} q) + d_{Q} (p \cdot_{λ} y) + d_{Q} (ω_{λ} (x, y)) + d_{Q} ({[p_{λ} q]}_{Q}) + Ω ({[x_{λ} y]}_{R})) \\ = ({[d_{R} {(x)}_{λ} y]}_{R} + {[x_{λ} d_{R} (y)]}_{R} + α {[d_{R} {(x)}_{λ} d_{R} (y)]}_{R}, d_{Q} (x \cdot_{λ} q) + d_{Q} (p \cdot_{λ} y) + Ω ({[x_{λ} y]}_{R}) \\ + d_{Q} (ω_{λ} (x, y)) + {[d_{Q} {(p)}_{λ} q]}_{Q} + {[p_{λ} d_{Q} (q)]}_{Q} + α {[d_{Q} {(p)}_{λ} d_{Q} (q)]}_{Q}) \\ = (\underset{A}{\underset{︸}{{[d_{R} {(x)}_{λ} y]}_{R}}} + \underset{B}{\underset{︸}{{[x_{λ} d_{R} (y)]}_{R}}} + \underset{C}{\underset{︸}{α {[d_{R} {(x)}_{λ} d_{R} (y)]}_{R}}}, \underset{A}{\underset{︸}{Ω (x) \cdot_{λ} y}} + \underset{A}{\underset{︸}{ω_{λ} (d_{R} (x), y)}} + \underset{B}{\underset{︸}{x \cdot_{λ} Ω (y)}} \\ + \underset{B}{\underset{︸}{ω_{λ} (x, d_{R} (y))}} + \underset{C}{\underset{︸}{α d_{R} (x) \cdot_{λ} Ω (y) + α Ω (x) \cdot_{λ} d_{R} (y) + α ω_{λ} (d_{R} (x), d_{R} (y)) + α {[Ω {(x)}_{λ} Ω (y)]}_{E}}} \\ + \underset{A}{\underset{︸}{{[Ω {(x)}_{λ} q]}_{E}}} + \underset{C}{\underset{︸}{α d_{R} (x) \cdot_{λ} d_{Q} (q) + α {[Ω {(x)}_{λ} d_{Q} (q)]}_{E}}} + \underset{A}{\underset{︸}{d_{R} (x) \cdot_{λ} q}} + \underset{B}{\underset{︸}{x \cdot_{λ} d_{Q} (q)}} + \underset{C}{\underset{︸}{α d_{R} (x) \cdot_{λ} d_{Q} (q)}} \\ + \underset{B}{\underset{︸}{{[p_{λ} Ω (y)]}_{E}}} + \underset{C}{\underset{︸}{α d_{Q} (p) \cdot_{λ} d_{R} (y) + α {[d_{Q} {(p)}_{λ} Ω (y)]}_{E}}} + \underset{A}{\underset{︸}{d_{Q} (p) \cdot_{λ} y}} + p \cdot_{λ} d_{R} (y) + \underset{C}{\underset{︸}{α d_{Q} (p) \cdot_{λ} d_{R} (y)}} \end{matrix}

\begin{matrix} + \underset{A}{\underset{︸}{{[d_{Q} {(p)}_{λ} q]}_{Q}}} + \underset{B}{\underset{︸}{{[p_{λ} d_{Q} (q)]}_{Q}}} + \underset{C}{\underset{︸}{α {[d_{Q} {(p)}_{λ} d_{Q} (q)]}_{Q}}}) \\ = \underset{A}{\underset{︸}{[{(d_{R} (x), d_{Q} (p) + Ω (x))}_{λ} (y, q)]}} + \underset{B}{\underset{︸}{[{(x, p)}_{λ} (d_{R} (y), d_{Q} (q) + Ω (y))]}} \\ + \underset{C}{\underset{︸}{α [{(d_{R} (x), d_{Q} (p) + Ω (x))}_{λ} (d_{R} (y), d_{Q} (q) + Ω (y))]}} \\ = [d_{E}^{Ω} {(x, p)}_{λ} (y, q)] + [{(x, p)}_{λ} d_{E}^{Ω} (y, q)] + α [d_{E}^{Ω} {(x, p)}_{λ} d_{E}^{Ω} (y, q)] . \end{matrix}

This shows that

d_{E}^{Ω}

is a differential operator on the Leibniz conformal algebra E. In other words,

(E, d_{E}^{Ω})

is a differential Leibniz conformal algebra. Further, it is easy to see that

\begin{matrix} 0 \to (Q, d_{Q}) \overset{i}{\to} (E, d_{E}^{Ω}) \overset{p}{\to} (R, d_{R}) \to 0 \end{matrix}

is a non-Abelian extension of the differential Leibniz conformal algebra

(R, d_{R})

(Q, d_{Q})

Let

(ω^{'}, \cdot_{λ}^{'},_{λ} \cdot^{'}, Ω^{'})

and

(ω, \cdot_{λ},_{λ} \cdot, Ω)

be two equivalent 2-cocycles. Thus, there exists a conformal linear map

η : R \to Q

such that the identities (14)–(17) hold. Let

(E, d_{E}^{Ω})

be a differential Leibniz conformal algebra induced by the 2-cocycle

(ω^{'}, \cdot_{λ}^{'},_{λ} \cdot^{'}, Ω^{'})

. We define a map

τ : R \oplus Q \to R \oplus Q

τ (x, p) = (x, p + η (x))

for all

(x, p) \in R \oplus Q

. Then, we have

\begin{matrix} τ ({[{(x, p)}_{λ} (y, q)]}_{E}) \\ = τ ({[x_{λ} y]}_{R}, x \cdot_{λ} q + p \cdot_{λ} y + ω_{λ} (x, y) + {[p_{λ} q]}_{Q}) \\ = ({[x_{λ} y]}_{R}, x \cdot_{λ} q + p \cdot_{λ} y + ω_{λ} (x, y) + {[p_{λ} q]}_{Q} + η ({[x_{λ} y]}_{R})) \\ = ({[x_{λ} y]}_{R}, x \cdot_{λ}^{'} q + {[η {(x)}_{λ} q]}_{E} + p \cdot_{λ}^{'} y + {[p_{λ} η (y)]}_{E} + ω_{λ}^{'} (x, y) + x \cdot_{λ}^{'} η (y) \\ + η (x) \cdot_{λ}^{'} y - η {[x_{λ} y]}_{R} + {[η {(x)}_{λ} η (y)]}_{Q} + {[p_{λ} q]}_{Q} + η ({[x_{λ} y]}_{R}) \\ = ({[x_{λ} y]}_{R}, x \cdot_{λ}^{'} q + x \cdot_{λ}^{'} η (y) + p \cdot_{λ}^{'} y + η (x) \cdot_{λ}^{'} y + ω_{λ}^{'} (x, y) + {[{(p + η (x))}_{λ} (q + η (y))]}_{Q} \\ = {[{(x, p + η (x))}_{λ} (y, q + η (y))]}_{E^{'}} \\ = {[τ {(x, p)}_{λ} τ (y, q)]}_{E} . \end{matrix}

This is similar to checking that

τ \circ d_{E}^{Ω} = d_{E^{'}}^{Ω^{'}} \circ τ

. Hence, the map

τ : (E, d_{E}^{Ω}) \to (E^{'}, d_{E^{'}}^{Ω^{'}})

defines an equivalence between two non-Abelian extensions. Therefore, we obtain a well-defined map

Γ : H_{nab}^{2} ((R, d_{R}), (Q, d_{Q})) \to {Ext}_{nab} ((R, d_{R}), (Q, d_{Q}))

. Finally, it is straightforward to verify that the maps

Π

and

Γ

are inverse to each to each other. This completes the proof. □

5. Automorphisms of Differential Leibniz Conformal Algebras and the Wells Map

In this section, we study the inducibility of a pair of differential Leibniz conformal algebra automorphisms and characterize them by equivalent conditions.

Let

(R, d_{R})

and

(Q, d_{Q})

be two differential Leibniz conformal algebras, and let

0 ⟶ (Q, d_{Q}) \overset{i}{⟶} (E, d_{E}) \overset{p}{⟶} (R, d_{R}) ⟶ 0,

be a non-Abelian extension of

(R, d_{R})

(Q, d_{Q})

. Let

{Aut}_{Q} (E)

be the set of all differential automorphisms

Υ \in Aut (E, d_{E})

that satisfy

{Υ |}_{Q} \subset Q

. For any automorphism

Υ \in {Aut}_{Q} (E, d_{E})

, then

{Υ |}_{Q} \in Aut (Q, d_{Q})

. We define a conformal linear map

\bar{Υ} : R ⟶ R

\bar{Υ} (x) = p Υ s (x), \forall x \in R .

Assume that

s_{1}

and

s_{2}

are two distinct sections of E, since

p s_{1} (x) - p s_{2} (x) = 0

s_{1} (x) - s_{2} (x) \in Ker p ≅ Q

, it follows that

Υ (s_{1} (x) - s_{2} (x)) \in Q

. Thus,

p Υ s_{1} (x) = p Υ s_{2} (x)

, which indicates that

\bar{Υ}

is independent of the choice of a section.

For all

x, y \in R

, we have

\begin{matrix} \bar{Υ} ({[x_{λ} y]}_{R}) & = p Υ (s {[x_{λ} y]}_{R}) \\ = p Υ ({[s {(x)}_{λ} s (y)]}_{E} - ω (x, y)) \\ = p Υ ({[s {(x)}_{λ} s (y)]}_{E}) \\ = {[p Υ s {(x)}_{λ} p Υ s (y)]}_{R} \\ = {[\bar{Υ} {(x)}_{λ} \bar{Υ} (y)]}_{R} . \end{matrix}

Further,

\begin{matrix} (d_{R} \bar{Υ} - \bar{Υ} d_{R}) (x) & = (d_{R} p Υ s - p Υ s d_{R}) (x) \\ = (p d_{E} Υ s - p Υ s d_{R}) (x) \\ = p Υ (d_{E} s - s d_{R}) (x) \\ = 0, \end{matrix}

which yields that

\bar{Υ}

is a homomorphism of differential Leibni- conformal algebras. It is easy to check that

\bar{Υ}

is bijective. Thus,

\bar{Υ} \in Aut (R, d_{R})

. Then, we can define a group homomorphism

Λ : {Aut}_{Q} (E, d_{E}) ⟶ Aut (R, d_{R}) \times Aut (Q, d_{Q}), Λ (Υ) = (\bar{Υ}, Υ |_{Q}) .

Definition 13.

A pair

(Φ, Ψ) \in Aut (R, d_{R}) \times Aut (Q, d_{Q})

is said to be inducible if

(Φ, Ψ)

is an image of Λ.

Below, we investigate when a pair

(Φ, Ψ)

is inducible.

Let

0 ⟶ (Q, d_{Q}) \overset{i}{⟶} (E, d_{E}) \overset{p}{⟶} (R, d_{R}) ⟶ 0

be a non-Abelian extension of

(R, d_{R})

(Q, d_{Q})

and

(ω, \cdot_{λ},_{λ} \cdot, Ω)

be the corresponding non-Abelian 2-cocycle induced by a section s of E. Given any pair

(Φ, Ψ) \in Aut (R, d_{R}) \times Aut (Q, d_{Q})

. Define conformal maps

ω^{Φ, Ψ} : R \times R ⟶ Q [λ], \cdot_{λ}^{Φ, Ψ} : R \otimes Q ⟶ Q [λ], \cdot^{Φ, Ψ} : Q \otimes R ⟶ Q [λ], Ω^{Φ, Ψ} : R ⟶ Q

respectively, by

ω_{λ}^{Φ, Ψ} (x, y) = Ψ ω_{λ} (Φ^{- 1} (x), Φ^{- 1} (y)),

(18)

x \cdot_{λ}^{Φ, Ψ} q = Ψ (Φ^{- 1} (x) \cdot_{λ} Ψ^{- 1} (q)),

(19)

q \cdot_{λ}^{Φ, Ψ} x = Ψ (Ψ^{- 1} (q) \cdot_{λ} Φ^{- 1} (x)),

(20)

Ω^{Φ, Ψ} (x) = Ψ Ω (Φ^{- 1} (x)),

(21)

for all

x, y \in R, q \in Q

Proposition 3.

With the above notations,

(ω^{Φ, Ψ}, \cdot_{λ}^{Φ, Ψ},_{λ} \cdot^{Φ, Ψ}, Ω^{Φ, Ψ})

is a non-Abelian 2-cocycle.

Proof.

Using (18)–(21), we obtain

\begin{matrix} x \cdot_{λ}^{Φ, Ψ} (y \cdot_{μ}^{Φ, Ψ} q) - y \cdot_{μ}^{Φ, Ψ} (x \cdot_{λ}^{Φ, Ψ} q) - ({[x_{λ} y]}_{R}) \cdot_{λ + μ}^{Φ, Ψ} q \\ = Ψ (Φ^{- 1} (x) \cdot_{λ} (Φ^{- 1} {(y)}_{μ} Ψ^{- 1} q)) - Ψ (Φ^{- 1} {(y)}_{μ} (Φ^{- 1} {(x)}_{λ} Ψ^{- 1} q)) - Ψ ({[Φ^{- 1} {(x)}_{λ} Φ^{- 1} (y)]}_{R} Ψ^{- 1} (q)) \\ = Ψ (Φ^{- 1} {(x)}_{λ} (Φ^{- 1} {(y)}_{μ} Ψ^{- 1} (q)) - Φ^{- 1} {(y)}_{μ} (Φ^{- 1} {(x)}_{λ} Ψ^{- 1} (q)) - {[Φ^{- 1} {(x)}_{λ} Φ^{- 1} (y)]}_{R} Ψ^{- 1} (q)) \\ = Ψ {[ω_{λ} {(Φ^{- 1} (x), Φ^{- 1} (y))}_{λ + μ} Ψ^{- 1} (q)]}_{Q} \\ = {[Ψ ω_{λ} Φ^{- 1} (x), Φ^{- 1} (y))}_{λ + μ} {q]}_{Q} \\ = {[ω_{λ}^{Φ, Ψ} {(x, y)}_{λ + μ} q]}_{Q}, \end{matrix}

which implies that (5) holds. Similarly, (6)–(13) hold. The proof is finished. □

Let

0 ⟶ (Q, d_{Q}) \overset{i}{⟶} (E, d_{E}) \overset{p}{⟶} (R, d_{R}) ⟶ 0

be a non-Abelian extension of

(R, d_{R})

(Q, d_{Q})

. Suppose that

(ω, \cdot_{λ},_{λ} \cdot, Ω)

is the corresponding non-Abelian 2-cocycle induced by a section s. Define a linear map

W : A u t (R, d_{R}) \times A u t (Q, d_{Q}) \to H_{nab}^{2} ((R, d_{R}), (Q, d_{Q}))

\begin{matrix} W (Φ, Ψ) = [(ω^{Φ, Ψ}, \cdot_{λ}^{Φ, Ψ},_{λ} \cdot^{Φ, Ψ}, Ω^{Φ, Ψ}) - (ω, \cdot_{λ},_{λ} \cdot, Ω)] . \end{matrix}

It is remarkable that the map W is not a group homomorphism in general. The map W is also said to be the Wells map.

Theorem 4.

Let

0 ⟶ (Q, d_{Q}) \overset{i}{⟶} (E, d_{E}) \overset{p}{⟶} (R, d_{R}) ⟶ 0

be a non-Abelian extension of

(R, d_{R})

(Q, d_{Q})

and let

(ω, \cdot_{λ},_{λ} \cdot, Ω)

be the corresponding non-Abelian 2-cocycle induced by a section s. A pair

(Φ, Ψ) \in Aut (R, d_{R}) \times Aut (Q, d_{Q})

is inducible if and only if

W (Φ, Ψ) = 0

Proof.

Suppose that

(Φ, Ψ) \in Aut (R, d_{R}) \times Aut (Q, d_{Q})

is inducible; then, there is an automorphism

Υ \in {Aut}_{Q} (E, d_{E})

such that

{Υ |}_{Q} = Ψ

and

p Υ s = Φ

. For all

x \in R

, since s is a section of p, that is,

p s = id

p (Υ s Φ^{- 1} - s) (x) = x - x = 0,

which implies that

(Υ s Φ^{- 1} - s) (x) \in \ker p ≅ Q

. So we can define a conformal linear map

η : R ⟶ Q

η (x) = (Υ s Φ^{- 1} - s) (x), \forall x \in R .

For

x \in R, q \in Q

, we have

\begin{matrix} x \cdot_{λ}^{Φ, Ψ} q - x \cdot_{λ} q \\ = Ψ (Φ^{- 1} (x) \cdot_{λ} Ψ^{- 1} (q)) - x \cdot_{λ} q \\ = Ψ ({[s {(Φ^{- 1} (x))}_{λ} Ψ^{- 1} (q)]}_{E}) - {[s {(x)}_{λ} q]}_{E} \\ = {[Υ s {(Φ^{- 1} (x))}_{λ} Υ (Ψ^{- 1} (q))]}_{E} - {[s {(x)}_{λ} q]}_{E} \\ = {[Υ s {(Φ^{- 1} (x))}_{λ} q]}_{E} - {[s {(x)}_{λ} q]}_{E} \\ = {[η {(x)}_{λ} q]}_{Q} . \end{matrix}

Hence, we obtain (15). Similarly, by direct calculations, we observe that (14), (16), (17) hold. It follows from the above observation that the non-Abelian 2-cocycles

(ω^{Φ, Ψ}, \cdot_{λ}^{Φ, Ψ},_{λ} \cdot^{Φ, Ψ}, Ω^{Φ, Ψ})

and

(ω, \cdot_{λ},_{λ} \cdot, Ω)

are equivalent via the conformal linear map

η : R ⟶ Q

. Hence, we have

\begin{matrix} W (Φ, Ψ) = [(ω^{Φ, Ψ}, \cdot_{λ}^{Φ, Ψ},_{λ} \cdot^{Φ, Ψ}, Ω^{Φ, Ψ}) - (ω, \cdot_{λ},_{λ} \cdot, Ω)] = 0 . \end{matrix}

Conversely, suppose that

(Φ, Ψ) \in Aut (R, d_{R}) \times Aut (Q, d_{Q})

, Since

W (Φ, Ψ) = 0

, it follows that the non-Abelian 2-cocycles

(ω^{Φ, Ψ}, \cdot_{λ}^{Φ, Ψ},_{λ} \cdot^{Φ, Ψ}, Ω^{Φ, Ψ})

and

(ω, \cdot_{λ},_{λ} \cdot, Ω)

are equivalent, there is a conformal linear map

η : R ⟶ Q

satisfying (14)–(17). Due to s being a section of p, then for all

e \in E

can be written as

e = q + s (x)

for some

q \in Q, x \in R .

Define a conformal linear map

Υ : E ⟶ E

Υ (e) = Υ (q + s (x)) = Ψ (q) + η Φ (x) + s Φ (x) .

Υ (e) = 0,

then

s Φ (x) = 0

and

Ψ (q) + η Φ (x) = 0

. In view of s and

Φ

being injective, we obtain

x = 0

; it follows that

q = 0

. Thus,

e = q + s (x) = 0

; that is,

Υ

is injective. For any

e = q + s (x) \in E

Υ (Ψ^{- 1} (q) - Ψ^{- 1} η (x) + s Φ^{- 1} (x)) = q + s (x) = e,

which yields that

Υ

is surjective. In all,

Υ

is bijective.

Next, we show that

Υ

is a homomorphism of differential Leibniz conformal algebras. In fact, for all

e_{i} = q_{i} + s (x_{i}) \in E (i = 1, 2)

\begin{matrix} {[Υ {(e_{1})}_{λ} Υ (e_{2})]}_{E} \\ = {[{(Ψ (q_{1}) + η Φ (x_{1}) + s Φ (x_{1}))}_{λ} (Ψ (q_{2}) + η Φ (x_{2}) + s Φ (x_{2}))]}_{E} \\ = {[Ψ {(q_{1})}_{λ} Ψ (q_{2})]}_{E} + {[Ψ {(q_{1})}_{λ} η Φ (x_{2})]}_{E} + {[Ψ {(q_{1})}_{λ} s Φ (x_{2})]}_{E} + {[η Φ {(x_{1})}_{λ} Ψ (q_{2})]}_{E} \\ + {[η Φ {(x_{1})}_{λ} η Φ (x_{2})]}_{E} + {[η Φ {(x_{1})}_{λ} s Φ (x_{2})]}_{E} + {[s Φ {(x_{1})}_{λ} Ψ (q_{2})]}_{E} + {[s Φ {(x_{1})}_{λ} η Φ (x_{2})]}_{E} + {[s Φ {(x_{1})}_{λ} s Φ (x_{2})]}_{E} \\ = {[Ψ {(q_{1})}_{λ} Ψ (q_{2})]}_{E} + Ψ ((q_{1}) \cdot_{λ} x_{2}) \underset{︸}{- Ψ (q_{1}) \cdot_{λ} Φ (x_{2}) + Ψ (q_{1}) \cdot_{λ} Φ (x_{2})} + Ψ ((x_{1}) \cdot_{λ} q_{2}) \\ \underset{D}{\underset{︸}{- Φ (x_{1}) \cdot_{λ} Ψ (q_{2})}} + Ψ (ω_{λ} (x_{1}, x_{2})) \underset{B}{\underset{︸}{- ω_{λ} (Φ (x_{1}), Φ (x_{2}))}} \underset{A}{\underset{︸}{- Φ (x_{1}) \cdot_{λ} η Φ (x_{2})}} \underset{C}{\underset{︸}{- η Φ (x_{1}) \cdot_{λ} Φ (x_{2})}} \\ + η Φ ({[{(x_{1})}_{λ} x_{2}]}_{R}) + \underset{C}{\underset{︸}{η Φ (x_{1}) \cdot_{λ} Φ (x_{2})}} + \underset{D}{\underset{︸}{Φ (x_{1}) \cdot_{λ} Ψ (q_{2})}} + \underset{A}{\underset{︸}{Φ (x_{1}) \cdot_{λ} η Φ (x_{2})}} + \underset{B}{\underset{︸}{ω_{λ} (Φ (x_{1}), Φ (x_{2}))}} \\ + s {[Φ {(x_{1})}_{λ} Φ (x_{2})]}_{R} \end{matrix}

\begin{matrix} = Ψ ({[q_{1 λ} q_{2}]}_{E} + (q_{1}) \cdot_{λ} x_{2} + (x_{1}) \cdot_{λ} q_{2} + ω_{λ} (x_{1}, x_{2})) + η Φ ({[x_{1 λ} x_{2}]}_{R}) + s {[Φ {(x_{1})}_{λ} Φ (x_{2})]}_{R} \\ = Ψ ({[q_{1 λ} q_{2}]}_{E} + {[{(q_{1})}_{λ} s (x_{2})]}_{E} + {[s {(x_{1})}_{λ} q_{2}]}_{E} + ω_{λ} (x_{1}, x_{2})) + η Φ ({[x_{1 λ} x_{2}]}_{R}) + s Φ {[x_{1 λ} x_{2}]}_{R} \\ = Υ ({[q_{1 λ} q_{2}]}_{E} + {[q_{1 λ} s (x_{2})]}_{E} + {[s {(x_{1})}_{λ} q_{2}]}_{E} + {[s {(x_{1})}_{λ} s (x_{2})]}_{R}) \\ = Υ ({[{(q_{1} + s (x_{1}))}_{λ} (q_{2} + s (x_{2}))]}_{E}) = Υ ({[e_{1 λ} e_{2}]}_{E}) . \end{matrix}

Similarly, one can check that

Υ \circ d_{E} = d_{E} \circ Υ

. This proves that

Υ

is an automorphism of differential Leibniz-conformal algebras. Thus,

Υ \in {Aut}_{Q} (E, d_{E})

. Finally, we show that

{Υ |}_{Q} = Ψ

and

p Υ s = Φ

. In fact,

Υ (q) = Υ (q + s (0)) = Ψ (q), \forall q \in Q

and

(p Υ s) (x) = p Υ (0 + s (x)) = p (χ (x) + s Φ (x)) = p s Φ (x) = Φ (x), \forall x \in R .

Therefore,

{Υ |}_{Q} = Ψ

and

p Υ s = Φ

. Thus,

(Φ, Ψ) \in Aut (R, d_{R}) \times Aut (Q, d_{Q})

is inducible. □

Theorem 5.

Let

0 ⟶ (Q, d_{Q}) \overset{i}{⟶} (E, d_{E}) \overset{p}{⟶} (R, d_{R}) ⟶ 0

be a non-Abelian extension of

(R, d_{R})

(Q, d_{Q})

. Then there is an exact sequence

\begin{matrix} 1 ⟶ {Aut}_{Q}^{Q, R} (E, d_{E}) \overset{ι}{⟶} Aut (E, d_{E}) \overset{Λ}{⟶} Aut (R, d_{R}) \times Aut (Q_{S}) \overset{W}{⟶} H_{nab}^{2} ((R, d_{R}), (Q, d_{Q})), \end{matrix}

where

{Aut}_{Q}^{Q, R} (E, d_{E}) = {γ \in Aut (E_{,} d_{E}) | Λ (Υ) = ({id}_{R}, {id}_{Q})}

Proof.

Obviously,

Ker Λ = Im ι

and

ι

is injective. By Theorem 6.3, one can easily check that

Ker W = Im Λ

. This completes the proof. □

More generally, if we define

\begin{matrix} {Aut}_{Q}^{Q} (E, d_{E}) = {Υ \in {Aut}_{Q} (E, d_{E}) {| Υ |}_{Q} = {id}_{Q})}, \\ {Aut}_{Q}^{R} (E, d_{E}) = {γ \in {Aut}_{Q} (E, d_{E}) | \bar{Υ} : = p Υ s = {id}_{R}}, \end{matrix}

we obtain two morphisms of groups

Λ_{R} : {Aut}_{Q}^{Q} (E, d_{E}) \to Aut (R, d_{R}), Υ \mapsto \bar{Υ}

and

Λ_{Q} :

{Aut}_{Q}^{R} (E, d_{E}) \to Aut (Q, d_{Q}) {, Υ \mapsto Υ |}_{Q}

. Define the maps

W_{Q} : Aut (Q, d_{Q}) \to H_{nab}^{2} ((R, d_{R}), (Q, d_{Q}))

and

W_{R} : Aut (R, d_{R}) \to H_{nab}^{2} ((R, d_{R}), (Q, d_{Q}))

\begin{matrix} W_{R} (Φ) = [(ω^{Φ, id}, \cdot_{λ}^{Φ, id},_{λ} \cdot^{Φ, id}, Ω^{Φ, id}) - (ω, \cdot_{λ},_{λ} \cdot, Ω)], \\ W_{Q} (Ψ) = [(ω^{id, Ψ}, \cdot_{λ}^{id, Ψ},_{λ} \cdot^{id, Ψ}, Ω^{id, Ψ}) - (ω, \cdot_{λ},_{λ} \cdot Ω)] . \end{matrix}

Proposition 4.

Let

0 ⟶ (Q, d_{Q}) \overset{i}{⟶} (E, d_{E}) \overset{p}{⟶} (R, d_{R}) ⟶ 0

be a non-Abelian extension of

(R, d_{R})

(Q, d_{Q})

. Then, there are two exact sequences of groups

\begin{matrix} 1 ⟶ {Aut}_{Q}^{Q, R} (E, d_{E}) \overset{ι}{⟶} {Aut}_{Q}^{Q} (E, d_{E}) \overset{Λ_{R}}{⟶} Aut (R, d_{R}) \overset{W_{R}}{⟶} H_{nab}^{2} ((R, d_{R}), (Q, d_{Q})), \\ 1 ⟶ {Aut}_{Q}^{Q, R} (E, d_{E}) \overset{ι}{⟶} {Aut}_{Q}^{R} (E, d_{E}) \overset{Λ_{Q}}{⟶} Aut (Q, d_{Q}) \overset{W_{Q}}{⟶} H_{nab}^{2} ((R, d_{R}), (Q, d_{Q})) . \end{matrix}

Author Contributions

H.W. and S.G.: Writing—original draft (equal). X.Z.: Writing—review editing (equal). All authors have read and agreed to the published version of the manuscript.

Funding

The paper is supported the Guizhou Provincial Basic Research Program (Natural Science) (No. ZK[2021]006), the Natural Science Foundation of China (No. 12161013 and 12271292), and the Natural Science Foundation of Shandong Province (No. 2023MA008).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Conflicts of Interest

The authors declare no conflicts of interest.

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Wu, H.; Guo, S.; Zhang, X. Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras. Axioms 2024, 13, 685. https://doi.org/10.3390/axioms13100685

AMA Style

Wu H, Guo S, Zhang X. Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras. Axioms. 2024; 13(10):685. https://doi.org/10.3390/axioms13100685

Chicago/Turabian Style

Wu, Hui, Shuangjian Guo, and Xiaohui Zhang. 2024. "Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras" Axioms 13, no. 10: 685. https://doi.org/10.3390/axioms13100685

APA Style

Wu, H., Guo, S., & Zhang, X. (2024). Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras. Axioms, 13(10), 685. https://doi.org/10.3390/axioms13100685

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Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras

Abstract

1. Introduction

2. Preliminaries

3. Crossed Modules and Two-Term Differential ${Leib}_{\infty}$ -Conformal Algebras

4. Non-Abelian Extension of Differential Leibniz Conformal Algebras

5. Automorphisms of Differential Leibniz Conformal Algebras and the Wells Map

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Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras

Abstract

1. Introduction

2. Preliminaries

3. Crossed Modules and Two-Term Differential Leib ∞ -Conformal Algebras

4. Non-Abelian Extension of Differential Leibniz Conformal Algebras

5. Automorphisms of Differential Leibniz Conformal Algebras and the Wells Map

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

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3. Crossed Modules and Two-Term Differential ${Leib}_{\infty}$ -Conformal Algebras