Lie algebras with differential operators of any weights

1.   Introduction

Derivations play a crucial role in studying deformation formulas ^[1], differential Galois theory ^[2] and homotopy algebras ^[3]. They also are useful of control systems theory ^[4,5] and gauge algebras ^[6]. The authors studied the operad of associative algebras with derivation in ^[7]. Recently, the authors introduced Lie algebras with derivations, and studied their cohomology and deformations, extensions in ^[8]. Later, Das^[9] considered the similar results for Leibniz algebras with derivations. The authors studied cohomology of Leibniz triple systems with derivations in ^[10].

More and more scholars have begun to pay close attention to the structures of any weight thanks to the result of outstanding work ^[12], all kinds of Rota-Baxter algebras of any weight ^{[13,14,15,16,17,18]} appear successively. In order to study non-abelian extensions of Lie algebras. The notion of crossed homomorphisms of Lie algebras was introduced by Lue ^[19], which was applied to study the representations of Cartan Lie algebras ^[20]. For $\lambda\in {\textbf k}$, the notion of a differential algebra of weight $\lambda$ was first introduced by Guo and Keigher ^[21], which generalizes simultaneously the concept of the classical differential algebra and difference algebra ^[22]. Applying the same method as for differential Lie algebras of weight $\lambda$. Later, the authors defined the cohomology of relative difference Lie algebras, and studied some properties in ^[23]. Our aim in this paper is to consider Lie algebras with differential operators of weight $\lambda$ (also known as differential Lie algebras). More precisely, we define a cohomology theory for differential Lie algebras and consider some properties.

The paper is organized as follows. In Section 2, we consider the representations of differential Lie algebras of any weight. In Section 3, we define a cohomology theory for differential Lie algebras of any weight. In Section 4, we study central extensions of differential Lie algebras of any weight. In Section 5, we study formal deformations of differential Lie algebras of any weight. In Section 6, we consider homotopy differential operators on $\mathrm{L}_{\infty}$ algebras and 2-differential operators of any weight on Lie 2-algebras. In Section 7, we prove that the category of 2-term $\mathrm{L}_{\infty}$ algebras with homotopy differential operator of any weight and the category of Lie 2-algebras with 2-differential operators of any weight are equivalent.

Throughout this paper, ${\textbf k}$ denotes a field of characteristic zero. All the vector spaces, algebras, linear maps and tensor products are taken over ${\textbf k}$ unless otherwise specified.

2.   Representations of differential Lie algebras of any weight

For $\lambda\in {\textbf k}$. A differential operator of weight $\lambda$ on a Lie algebra $\mathfrak{g}$ is a linear operator $d_\mathfrak{g} : \mathfrak{g}\rightarrow \mathfrak{g}$ such that

We denote by $\mathrm{Der}_\lambda(\mathfrak{g})$ the set of differential operators of weight $\lambda$ of the Lie algebra $\mathfrak{g}$.

Definition 2.1. Denote a Lie algebra $\mathfrak{g}$ with a differential operator $d_\mathfrak{g} \in \mathrm{Der}_\lambda(\mathfrak{g})$ by $(\mathfrak{g}, d_\mathfrak{g})$ and we call it a differential Lie algebra.

Definition 2.2. Given two differential Lie algebras $(\mathfrak{g}, d_\mathfrak{g}), \, (\mathfrak{h}, d_\mathfrak{h})$, a homomorphism of differential Lie algebras is a Lie algebra homomorphism $\varphi:\mathfrak{g}\rightarrow \mathfrak{h}$ such that $\varphi\circ d_\mathfrak{g} = d_\mathfrak{h}\circ\varphi$. We denote by ${\bf{LieD}}_\lambda$ the category of differential Lie algebras and their morphisms.

To simply notations, for all the above notions, we will often suppress the mentioning of the weight $\lambda$ unless it needs to be specified.

Definition 2.3. ${\rm (i)}$ A representation over the differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ is a pair $(V, d_V)$, where $d_V\in \mathrm{End}_{\textbf k}(V)$, and $(V, \rho)$ is a representation over the Lie algebra $\mathfrak{g}$, such that $\forall x\in \mathfrak{g}, v\in V,$ the following identity holds:

${\rm (ii)}$ Given two representations $(U, \rho_U, d_U), \, (V, \rho_V, d_V)$ over $(\mathfrak{g}, d_\mathfrak{g})$, a linear map $f:U\rightarrow V$ is called a homomorphism of representations, if $f\circ d_U = d_V\circ f$ and

One denotes by $(\mathfrak{g}, d_\mathfrak{g})$-Rep the category of representations over the differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$.

Example 2.4. Any differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ is a representation over itself with

It is called the adjoint representation over the differential Lie algebras $(\mathfrak{g}, d_\mathfrak{g})$.

Example 2.5. Let $(V, \rho)$ be a representation of a Lie algebra $\mathfrak{g}$. Then the pair $(V,$ Id$_V)$ is a representation of the differential Lie algebra $(\mathfrak{g},$Id$_\mathfrak{g})$ of weight $-1$.

Example 2.6. Let $(\mathfrak{g}, d_\mathfrak{g})$ be a differential Lie algebra of weight $\lambda$ and $(V, d_V)$ be a representation of it. Then for $\kappa\neq 0\in {\textbf k}$, the pair $(V, \kappa d_V)$ is a representation of the differential Lie algebra $(\mathfrak{g}, \kappa d_\mathfrak{g})$ of any weight $\frac{1}{\kappa}\lambda$.

The following result is easily to check and we omit it.

Proposition 2.7. Let $(V, d_V)$ be a representation of the differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ of weight $\lambda$. Then $(\mathfrak{g}\oplus V, d_\mathfrak{g}\oplus d_V)$ is a differential Lie algebra, where

3.   Cohomology of differential Lie algebras of any weight

Recall that the cochain complex of Lie algebra $\mathfrak{g}$ with coefficients in representation $V$ is the cochain complex

and the coboundary operator

is given by

for all $f\in \mathrm{C}^n_{\mathrm{Lie}}(\mathfrak{g}, V), \; x_1, \dots, x_{n+1}\in \mathfrak{g}$. The corresponding cohomology is denoted by $\mathrm{H}^*_{\mathrm{Lie}}(\mathfrak{g}, V)$. When $V$ is the adjoint representation, we write $\mathrm{H}^n_{\mathrm{Lie}}(\mathfrak{g}) = \mathrm{H}^n_{\mathrm{Lie}}(\mathfrak{g}, V), n\geq 0$.

In the following, we will define the cohomology of the differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ of weight $\lambda$ with coefficients in the representation $(V, d_V)$.

Define

and define a linear map $\delta:C^n_{\mathrm{Lie}}(\mathfrak{g}, V)\rightarrow C^n_{\mathrm{Lie}}(\mathfrak{g}, V)$ $(n \geq 1)$ by

for any $f_n\in C^n_{\mathrm{LieD_\lambda}}(\mathfrak{g}, V)$ and

Lemma 3.1. We have $\partial_{\mathrm{Lie}} \circ \delta = \delta \circ \partial_{\mathrm{Lie}}$.

Theorem 3.2. The pair $(C_{\mathrm{LieD_\lambda}}^*(\mathfrak{g}, V), \partial_{\mathrm{LieD_\lambda}})$ is a cochain complex. So $\partial_{\mathrm{LieD_\lambda}}^2 = 0.$

Proof. For any $v\in C^0_{\mathrm{LieD_\lambda}}(\mathfrak{g}, V)$, we have

Given any $f\in C^n_{\mathrm{Lie}}(\mathfrak{g}, V), \, g\in C^{n-1}_{\mathrm{Lie}}(\mathfrak{g}, V)$ with $n\geq1$, we have

Hence, the proof is finished.

Definition 3.3. The cohomology of the cochain complex $(C_{\mathrm{LieD_\lambda}}^*(\mathfrak{g}, V), \partial_{\mathrm{LieD_\lambda}})$, denoted by $H_{\mathrm{LieD_\lambda}}^*(\mathfrak{g}, V)$, is called the cohomology of the differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ of weight $\lambda$.

4.   Abelian extensions of differential Lie algebras of any weight

In this section, we show that abelian extensions of differential Lie algebras are classified by the second cohomology.

Definition 4.1. An abelian extension of differential Lie algebras is a short exact sequence of homomorphisms of differential Lie algebras

such that $[u, v]_V = 0$ for all $u, v\in V.$

$(\hat{\mathfrak{g}}, d_{\hat{\mathfrak{g}}})$ is called an abelian extension of $(\mathfrak{g}, d_\mathfrak{g})$ by $(V, d_V)$.

Definition 4.2. Let $(\hat{\mathfrak{g}}_1, d_{\hat{\mathfrak{g}}_1})$ and $(\hat{\mathfrak{g}}_2, d_{\hat{\mathfrak{g}}_2})$ be two abelian extensions of $(\mathfrak{g}, d_\mathfrak{g})$ by $(V, d_V)$. They are said to be isomorphic if there exists $\zeta:(\hat{\mathfrak{g}}_1, d_{\hat{\mathfrak{g}}_1})\leftrightarrow (\hat{\mathfrak{g}}_2, d_{\hat{\mathfrak{g}}_2})$ is an isomorphism of differential Lie algebras such that:

A section of an abelian extension $(\hat{\mathfrak{g}}, d_{\hat{\mathfrak{g}}})$ of $(\mathfrak{g}, d_\mathfrak{g})$ by $(V, d_V)$ is a linear map $s:\mathfrak{g}\rightarrow \hat{\mathfrak{g}}$ satisfying $p\circ s = \mathrm{Id}_\mathfrak{g}$.

Given a section $s:\mathfrak{g}\rightarrow \hat{\mathfrak{g}}$, define $\rho: \mathfrak{g}\to \mathrm{End}_{\textbf k}(V)$ by

Proposition 4.3. With the above notations, $(V, d_V)$ is a representation over the differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$.

Proof. Firstly, we prove that $\rho$ is a Lie algebra homomorphism, in fact, for any $x, y\in \mathfrak{g}, \, v\in V$, we have

Moreover, we obtain

Hence, $(V, d_V)$ is a representation over $(\mathfrak{g}, d_\mathfrak{g})$.

We further consider linear maps $\psi:\mathfrak{g}\otimes \mathfrak{g}\rightarrow V$ and $\chi:\mathfrak{g}\rightarrow V$ by

The differential Lie algebra structure on $\mathfrak{g}\oplus V$ with a multiplication $[\cdot, \cdot]_\psi$ and the differential operator $d_\chi$ defined by

Proposition 4.4. The triple $(\mathfrak{g}\oplus V, [\cdot, \cdot]_\psi, d_\chi)$ is a differential Lie algebra if and only if $(\psi, \chi)$ is a 2-cocycle.

Proof. For any $x, y, z\in \mathfrak{g}$, By (4.1), we have

Since $d_\chi$ satisfies Eq (2.1), we deduce that

Therefore, $(\psi, \chi)$ is a 2-cocycle.

Conversely, if $(\psi, \chi)$ satisfies Eqs (4.3) and (4.4), direct verification that $(\mathfrak{g}\oplus V, [\cdot, \cdot]_\psi, d_\chi)$ is a differential Lie algebra.

In the following, we will classify abelian extensions of differential Lie algebras.

Theorem 4.5. Let $V$ be a vector space and $d_V\in \mathrm{End}_{{\textbf k}}(V)$. Then abelian extensions of a differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ by $(V, d_V)$ are classified by $H_{\mathrm{LieD_{\lambda}}}^2(\mathfrak{g}, V)$ of $(\mathfrak{g}, d_\mathfrak{g})$.

Proof. Let $(\hat{\mathfrak{g}}, d_{\hat{\mathfrak{g}}})$ be an abelian extension of $(\mathfrak{g}, d_\mathfrak{g})$ by $(V, d_V)$. We choose a section $s:\mathfrak{g}\rightarrow \hat{\mathfrak{g}}$ to obtain a 2-cocycle $(\psi, \chi)$ and let $s_1$ and $s_2$ be two distinct sections providing 2-cocycles $(\psi_1, \chi_1)$ and $(\psi_2, \chi_2)$ respectively. Define $\phi:\mathfrak{g}\rightarrow V$ by $\phi(x) = s_1(x)-s_2(x)$, we have

and

That is, $(\psi_1, \chi_1) = (\psi_2, \chi_2)+\partial_{\mathrm{LieD_{\lambda}}}(\phi)$. Thus $(\psi_1, \chi_1)$ and $(\psi_2, \chi_2)$ are in the same cohomological class in $H_{\mathrm{LieD_{\lambda}}}^2(\mathfrak{g}, V)$.

Next, we prove that isomorphic abelian extensions give rise to the same element in $H_{\mathrm{LieD_{\lambda}}}^2(\mathfrak{g}, V)$. Assume that $(\hat{\mathfrak{g}}_1, d_{\hat{\mathfrak{g}}_1})$ and $(\hat{\mathfrak{g}}_2, d_{\hat{\mathfrak{g}}_2})$ are two isomorphic abelian extensions of $(\mathfrak{g}, d_\mathfrak{g})$ by $(V, d_V)$ with the associated homomorphism $\zeta:(\hat{\mathfrak{g}}_1, d_{\hat{\mathfrak{g}}_1})\rightarrow (\hat{\mathfrak{g}}_2, d_{\hat{\mathfrak{g}}_2})$. Let $s_1$ be a section of $(\hat{\mathfrak{g}}_1, d_{\hat{\mathfrak{g}}_1})$. As $p_2\circ\zeta = p_1$, we have

Therefore, $\zeta\circ s_1$ is a section of $(\hat{\mathfrak{g}}_2, d_{\hat{\mathfrak{g}}_2})$. Denote $s_2: = \zeta\circ s_1$. Since $\zeta$ is a homomorphism of differential Lie algebras such that $\zeta|_V = \mathrm{Id}_V$, we have

and

Therefore, the result can be obtained.

Conversely, given two 2-cocycles $(\psi_1, \chi_1)$ and $(\psi_2, \chi_2)$, we can construct two abelian extensions $(\mathfrak{g}\oplus V, [\cdot, \cdot]_{\psi_1}, d_{\chi_1})$ and $(\mathfrak{g}\oplus V, [\cdot, \cdot]_{\psi_2}, d_{\chi_2})$ via Eqs (4.1) and (4.2), and then there exists a linear map $\phi:\mathfrak{g}\to V$ such that

Define $\zeta:\mathfrak{g}\oplus V\rightarrow \mathfrak{g}\oplus V$ by

Then $\zeta$ is an isomorphism of these two abelian extensions.

5.   Deformations of differential Lie algebras of any weight

In this section, we show that if $H^2_{\mathrm{LieD_{\lambda}}}(\mathfrak{g}, \mathfrak{g}) = 0$, then the differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ is rigid.

Let $(\mathfrak{g}, d_\mathfrak{g})$ be a differential Lie algebra. Denote by $\mu_\mathfrak{g}$ the multiplication of $\mathfrak{g}$. Consider the 1-parameterized family

Definition 5.1. A 1-parameter formal deformation of a differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ is a pair $(\mu_t, d_t)$ which endows the ${\textbf k}[[t]]$-module $(\mathfrak{g}[[t]], \mu_t, d_t)$ with the differential Lie algebra over ${\textbf k}[[t]]$ such that $(\mu_0, d_0) = (\mu_\mathfrak{g}, d_\mathfrak{g})$.

Given any differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$, interpret $\mu_\mathfrak{g}$ and $d_\mathfrak{g}$ as the formal power series $\mu_t$ and $d_t$ with $\mu_i = \delta_{i, 0}\mu_\mathfrak{g}$ and $d_i = \delta_{i, 0}d_\mathfrak{g}$ respectively for all $i\geq0$. Then $(\mathfrak{g}[[t]], \mu_\mathfrak{g}, d_\mathfrak{g})$ is a 1-parameter formal deformation of $(\mathfrak{g}, d_\mathfrak{g})$.

The pair $(\mu_t, d_t)$ generates a 1-parameter formal deformation of the differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ if and only if the following identities hold:

Expanding these identities and collecting coefficients of $t^n$, we see that Eqs (5.1) and (5.2) are equivalent to the systems of identities:

Remark 5.2. For $n = 0$, Eq (5.3) is equal to the Jabobi identity of $\mu_\mathfrak{g}$, and Eq (5.4) is equal to the fact that $d_\mathfrak{g}$ is a differential operator of weight $\lambda$.

Proposition 5.3. Let $(\mathfrak{g}[[t]], \mu_t, d_t)$ be a $1$-parameter formal deformation of a differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$. Then $(\mu_1, d_1)$ is a 2-cocycle of the differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ with the coefficient in the adjoint representation $(\mathfrak{g}, d_\mathfrak{g})$.

Proof. For $n = 1$, Eq (5.3) is equal to $\partial_{\mathrm{Lie}} \mu_1 = 0$, and Eq (5.4) is equal to

Thus for $n = 1$, Eqs (5.3) and (5.4) imply that $(\mu_1, d_1)$ is a 2-cocycle.

If $\mu_t = \mu_\mathfrak{g}$ in the above $1$-parameter formal deformation of the differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$, we obtain a $1$-parameter formal deformation of the differential operator $d_\mathfrak{g}$. Consequently, we have

Corollary 5.4. Let $d_t$ be a $1$-parameter formal deformation of the differential operator $d_\mathfrak{g}$. Then $d_1$ is a 1-cocycle of the differential operator $d_\mathfrak{g}$ with coefficients in the adjoint representation $(\mathfrak{g}, d_\mathfrak{g})$.

Proof. In the special case when $n = 1$, Eq (5.4) is equal to $\partial_{\mathrm{Lie}} d_1 = 0$, which implies that $d_1$ is a 1-cocycle of the differential operator $d_\mathfrak{g}$ with coefficients in the adjoint representation $(\mathfrak{g}, d_\mathfrak{g})$.

Definition 5.5. The $2$-cocycle $(\mu_1, d_1)$ is called the infinitesimal of the $1$-parameter formal deformation $(\mathfrak{g}[[t]], \mu_t, d_t)$ of $(\mathfrak{g}, d_\mathfrak{g})$.

Definition 5.6. Two $1$-parameter formal deformations $(\mathfrak{g}[[t]], \mu_t, d_t)$ and $(\mathfrak{g}[[t]], \bar{\mu}_t, \bar{d}_t)$ of $(\mathfrak{g}, d_\mathfrak{g})$ are said to be equivalent if there exists a formal isomorphism from $(\mathfrak{g}[[t]], \bar{\mu}_t, \bar{d}_t)$ to $(\mathfrak{g}[[t]], \mu_t, d_t)$ is a power series $\Phi_t = \sum_{i\geq0}\phi_it^i:\mathfrak{g}[[t]]\rightarrow \mathfrak{g}[[t]]$, where $\phi_i:\mathfrak{g}\to \mathfrak{g}$ are linear maps with $\phi_0 = \mathrm{Id}_\mathfrak{g}$, such that

Theorem 5.7. The infinitesimals of two equivalent $1$-parameter formal deformations of $(\mathfrak{g}, d_\mathfrak{g})$ are in the same cohomology class $H_{\mathrm{LieD_{\lambda}}}^2(\mathfrak{g}, \mathfrak{g})$.

Proof. Let $\Phi_t:(\mathfrak{g}[[t]], \bar{\mu}_t, \bar{d}_t)\rightarrow (\mathfrak{g}[[t]], \mu_t, d_t)$ be a formal isomorphism. For all $x, y\in \mathfrak{g}$, we have

Furthermore, we obtain

Thus, we have

which implies that $[(\bar{\mu}_1, \bar{d}_1)] = [(\mu_1, d_1)]$ in $H_{\mathrm{LieD_{\lambda}}}^2(\mathfrak{g}, \mathfrak{g})$.

Definition 5.8. A $1$-parameter formal deformation $(\mathfrak{g}[[t]], \mu_t, d_t)$ of $(\mathfrak{g}, d_\mathfrak{g})$ is said to be trivial if it is equal to the deformation $(\mathfrak{g}[[t]], \mu_\mathfrak{g}, d_\mathfrak{g})$, that is, there exists $\Phi_t = \sum_{i\geq0}\phi_it^i:\mathfrak{g}[[t]]\rightarrow \mathfrak{g}[[t]]$, where $\phi_i:\mathfrak{g}\to \mathfrak{g}$ are linear maps with $\phi_0 = \mathrm{Id}_\mathfrak{g}$, such that

Definition 5.9. A differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ is said to be rigid if every $1$-parameter formal deformation is trivial.

Theorem 5.10. Regarding $(\mathfrak{g}, d_\mathfrak{g})$ as the adjoint representation over itself, if $H_{\mathrm{LieD_{\lambda}}}^2(\mathfrak{g}, \mathfrak{g}) = 0$, the differential Lie algebra $(\mathfrak{g}, d_\mathfrak{g})$ is rigid.

Proof. Let $(\mathfrak{g}[[t]], \mu_t, d_t)$ be a $1$-parameter formal deformation of $(\mathfrak{g}, d_\mathfrak{g})$. By Proposition 5.3, $(\mu_1, d_1)$ is a 2-cocycle. By $H_{\mathrm{LieD_{\lambda}}}^2(\mathfrak{g}, \mathfrak{g}) = 0$, there exists a 1-cochain $\phi_1\in C^1_{\mathrm{Lie}}(\mathfrak{g}, \mathfrak{g})$ such that

Then setting $\Phi_t = \mathrm{Id}_\mathfrak{g}+\phi_1 t$, we have a deformation $(\mathfrak{g}[[t]], \bar{\mu}_t, \bar{d}_t)$, where

Thus, $(\mathfrak{g}[[t]], \bar{\mu}_t, \bar{d}_t)$ is equivalent to $(\mathfrak{g}[[t]], \mu_t, d_t)$. Furthermore, we have

Therefore,

By Eq (5.9), we have

Then by repeating the argument, we can show that $(\mathfrak{g}[[t]], \mu_t, d_t)$ is equivalent to $(\mathfrak{g}[[t]], \mu_\mathfrak{g}, d_\mathfrak{g})$. Thus, $(\mathfrak{g}, d_\mathfrak{g})$ is rigid.

6.   Homotopy differential operators of any weight on 2-term L$_{\infty}$-algebras

In this section, we pay our attention to the homotopy differential operator of any weight on 2-term L$_{\infty}$-algebras introduced by ^[24].

Definition 6.1. A 2-term L$_{\infty}$-algebra consists of

● a complex of vector spaces $L_1\stackrel{d}{\longrightarrow} L_0$,

● bilinear maps $l_2: L_i\otimes L_j\rightarrow L_{i+j}$, where $i+j\leq 1$,

● a skew-symmetric trilinear map $l_3: L_0\otimes L_0\otimes L_0\rightarrow L_1$, satisfying:

for any $a, b, c, w \in L_0$ and $u, v \in L_1$.

One denotes a 2-term L$_{\infty}$-algebra as above by $(L_1\stackrel{d}{\longrightarrow} L_0, l_2, l_3)$. A 2-term L$_{\infty}$-algebra is called skeletal if $d = 0$.

Definition 6.2. Let $L = (L_1\stackrel{d}{\longrightarrow} L_0, l_2, l_3)$ and $L' = (L'_1\stackrel{d'}{\longrightarrow} L'_0, l'_2, l'_3)$ be two 2-term L$_{\infty}$-algebras. A morphism $f: L \rightarrow L'$ consists of

● a chain map $f: L \rightarrow L'$ (which consists of linear maps $f_0: L_0 \rightarrow L_0'$ and $f_1: L_1 \rightarrow L_1'$ with $f_0 \circ d = d'\circ f_1$),

● a bilinear map $f_2 : L_0\otimes L_0 \rightarrow L'_1$ satisfying

for any $a, b, c\in L_0$ and $u \in L_1$.

If $f = (f_0, f_1, f_2): L\rightarrow L'$ and $g = (g_0, g_1, g_2): L'\rightarrow L''$ are two morphism of 2-term L$_{\infty}$-algebras, their composition $g\circ f: L \rightarrow L''$ is defined by $(g\circ f)_0 = g_0 \circ f_0, (g\circ f)_1 = g_1\circ f_1$ and

For any 2-term L$_{\infty}$-algebra $L$, the identity morphism $\mathrm{Id}_L: L \rightarrow L$ is given by the identity chain map $L \rightarrow L$ together with $(\mathrm{Id}_L)_2 = 0$.

The collection of 2-term L$_{\infty}$-algebras and morphisms between them form a category. We denote this category by ${\bf{2Lie}}_{\infty}$.

Definition 6.3. Let $L = (L_1\stackrel{d}{\longrightarrow} L_0, l_2, l_3)$ be a 2-term L$_{\infty}$-algebra. A homotopy differential operator of weight $\lambda$ on it consists of a chain map of the underlying chain complex (i.e., linear maps $\theta_0: L_0\rightarrow L_0$ and $\theta_1: L_1 \rightarrow L_1$ with $\theta_0 \circ d = d\circ \theta_1$) and a bilinear map $\theta_2 : L_0\otimes L_0 \rightarrow L_1$ such that for any $a, b, c\in L_0$ and $u \in L_1$, the following identities are hold

A 2-term L$_{\infty}$-algebra with a homotopy differential operator of weight $\lambda$ as above denoted by the pair $((L_1\stackrel{d}{\longrightarrow} L_0, l_2, l_3), (\theta_0, \theta_1, \theta_2))$. A 2-term L$_{\infty}$-algebra with a homotopy differential operator of weight $\lambda$ is said to be skeletal if the underlying 2-term L$_{\infty}$-algebra is skeletal, i.e., $d = 0$.

Definition 6.4. Let $((L_1\stackrel{d}{\longrightarrow} L_0, l_2, l_3), (\theta_0, \theta_1, \theta_2))$ and $((L'_1\stackrel{d'}{\longrightarrow} L'_0, l'_2, l'_3), (\theta'_0, \theta'_1, \theta'_2))$ be two 2-term L$_{\infty}$-algebras with homotopy differential operators of weight $\lambda$. A morphism between them consists of a morphism $(f_0, f_1, f_2)$ between the underlying 2-term L$_{\infty}$-algebras and a linear map $\Psi: L_0\rightarrow L_1'$ satisfying

We denote the category of 2-term L$_{\infty}$-algebras with homotopy differential operators of weight $\lambda$ and morphisms between them by ${\bf{2LieD}}_{\lambda\infty}$.

Theorem 6.5. There is a one-to-one correspondence between skeletal 2-term L$_{\infty}$-algebras with homotopy differential operators with weight $\lambda$ and tuples $((\mathfrak{g}, d_\mathfrak{g}), (V, d_V), (\theta, \overline{\theta}))$, where $(\mathfrak{g}, d_\mathfrak{g})$ is a differential Lie algebra of weight $\lambda$, $(V, d_V)$ is a representation and $(\theta, \overline{\theta})$ is a 3-cocycle of the differential Lie algebra of weight $\lambda$ with coefficients in the representation.

Proof. Let $(L_1\stackrel{0}{\longrightarrow} L_0, l_2, l_3, (\theta_0, \theta_1, \theta_2))$ be a skeletal 2-term L$_{\infty}$-algebra with a homotopy differential operator of weight $\lambda$. Then $\theta_0$ is a differential operator of weight $\lambda$ for the Lie algebra $(L_0, l_2)$. We have that $(L_1, \theta_1)$ is a representation of the differential Lie algebra $(L_0, \theta_0)$ of weight $\lambda$ from Definition 6.3. According to the condition (c) in Definition 6.3, we have $\partial_{\mathrm{LieD_{\lambda}}}(\theta_2)+\delta(l_3) = 0$. Therefore $(l_3, -\theta_2)$ is a 3-cocycle.

Conversely, define $L_0 = L, L_1 = V$ and $\theta_0 = d_\mathfrak{g}, \theta_1 = d_V, \theta_2 = -\overline{\theta}$. We define multiplications $l_2: L_i \otimes L_j \rightarrow L_{i+j}$ and $l_3: L_0 \otimes L_0 \otimes L_0 \rightarrow L_1$ by

for $a, b, c\in L_0 = L$ and $u \in L_1 = V$. Then it is easy to verify that $((L_1\stackrel{0}{\longrightarrow} L_0, l_2, l_3), (\theta_0, \theta_1, \theta_2))$ is a skeletal 2-term L$_{\infty}$-algebra with a homotopy differential operator of weight $\lambda$. Hence, the proof is finished.

A 2-term L$_{\infty}$-algebra with a homotopy differential operator of weight $\lambda$ is said to be strict if the underlying 2-term L$_{\infty}$-algebra is strict, i.e., $\theta_2 = 0$. Next we introduce crossed modules of differential Lie algebras of weight $\lambda$ and show that strict 2-term L$_{\infty}$-algebra with a homotopy differential operator of weight $\lambda$ are in one-to-one correspondence with crossed module of differential Lie algebras of weight $\lambda$.

Definition 6.6. A crossed module of differential Lie algebras of weight $\lambda$ consist of $((\mathfrak{g}, d_\mathfrak{g}), (\mathfrak{h}, d_\mathfrak{h}), dt, \Lambda)$ where $(\mathfrak{g}, d_\mathfrak{g})$ and $(\mathfrak{h}, d_\mathfrak{h})$ are differential Lie algebras of weight $\lambda$, $dt: \mathfrak{g} \rightarrow \mathfrak{h}$ is a differential Lie algebra morphism and

such that for $u, v \in \mathfrak{g}, a, b\in \mathfrak{h}$,

Theorem 6.7. There is a one-to-one correspondence between strict 2-term L$_{\infty}$-algebras with homotopy differential operators of weight $\lambda$ and crossed module of differential Lie algebras of weight $\lambda$.

Proof. Let $(L_1\stackrel{d}{\longrightarrow} L_0, l_2, l_3 = 0, (\theta_0, \theta_1, \theta_2))$ be a strict 2-term L$_{\infty}$-algebra with a homotopy differential operator of weight $\lambda$. Then $\theta_0$ is a differential operator of weight $\lambda$ for the Lie algebra $(L_0, l_2)$ and $\theta_1$ is a differential operator of weight $\lambda$ for the Lie algebra $(L_1, l_2)$ from Definition 6.3. Thus $(L_0, \theta_0)$ and $(L_1, \theta_1)$ are both differential Lie algebras of weight $\lambda$. Since $\theta_0 \circ d = d\circ \theta_1$, the map $dt = d : L_1\rightarrow L_0$ is a morphism of differential Lie algebras of weight $\lambda$. Finally, the condition (b) of Definition 6.3 is equal to the condition (d) of Definition 6.6. Hence, the results are obtained.

7.   Categorification of differential Lie algebras of any weight

In this section, we study categorified differential operators of any weight (also called 2-differential operator) on Lie 2-algebras.

Definition 7.1. A Lie 2-algebra is a 2-vector space $L$ equipped with

● a bilinear functor $[\cdot, \cdot]: L \otimes L \rightarrow L$,

● a trilinear natural isomorphism, called the Jacobiator

satisfying

where $\Theta, R, P$ and $Q$ are given by

Definition 7.2. Let $(L, [\cdot, \cdot], \mathcal{J})$ and $(L', [\cdot, \cdot]', \mathcal{J}')$ be two Lie 2-algebras. A Lie 2-algebra morphism consists of

● a linear functor $(F_0, F_1)$ from the underlying 2-vector space $L$ to $L'$;

● a bilinear natural transformation

satisfying

Let $L, L'$ and $L''$ be three Lie 2-algebras and $F : L\rightarrow L', G: L'\rightarrow L''$ be Lie 2-algebra morphisms. Their composition $G\circ F : L\rightarrow L''$ is a Lie 2-algebra morphism whose components are given by $(G\circ F)_0 = G_0\circ F_0, (G\circ F)_1 = G_1\circ F_1$ and $(G\circ F)_2$ is given by

For any Lie 2-algebra $L$, the identity morphism $\mathrm{Id}_L: L\rightarrow L$ is given by the identity functor as its linear functor together with the identity natural transformation as $(\mathrm{Id}_L)_2$.

Lie 2-algebras and Lie 2-algebra morphisms form a category. We denote this category by Lie2.

In the next, we define 2-differential operators of weight $\lambda$ on Lie 2-algebras. They are categorification of differential operators on Lie algebras.

Definition 7.3. Let $(L, [\cdot, \cdot], \mathcal{J})$ be a Lie 2-algebra. A 2-differential operator of weight $\lambda$ on it consists of a linear map functor $D: L\rightarrow L$ and a natural isomorphism

satisfying

where

Definition 7.4. Let $(L, [\cdot, \cdot], \mathcal{J}, D, \mathcal{D})$ and $(L', [\cdot, \cdot]', \mathcal{J'}, D', \mathcal{D'})$ be two Lie 2-algebras with 2-differential operators of weight $\lambda$. A morphism between them consists of a Lie 2-algebras mophism $(F = (F_0, F_1), F_2)$ and a natural isomorphism

satisfying

We denote the category of Lie 2-algebras with 2-differential operators of weight $\lambda$ and morphisms between them by ${{\bf LieD2}_{\lambda}}$.

In the following, we will give our main result of this section.

Theorem 7.5. The categories $\mathrm{{\bf 2LieD}}_{\lambda\infty}$ and $\mathrm{{\bf LieD}}2_{\lambda}$ are equivalent.

Proof. First we construct a functor $T:{{\bf{2LieD}}_{\lambda\infty}}\rightarrow {{\bf{LieD2}}_{\lambda}}$ as follows. Given a 2-term L$_{\infty}$-algebra with a homotopy differential operator of weight $\lambda$ $((L_1\stackrel{d}{\longrightarrow} L_0, l_2, l_3), (\theta_0, \theta_1, \theta_2))$, we obtain the 2-vector space $C = (L_0 \oplus L_1 \rightrightarrows L_0)$. Define a bilinear functor $[\cdot, \cdot] : C \otimes C \rightarrow C$ by

for $(a, u), (b, v) \in C_1 = L_0 \oplus L_1$. Define

According to the identities (a)–(e), we can check that $(C, [\cdot, \cdot], \mathcal{J})$ is a Lie 2-algebra. Moreover, we define a 2-differential operator of weight $\lambda$ $(D, \mathcal{D}')$ by

Given any 2-term L$_{\infty}$-algebra with a homotopy differential operator of weight $\lambda$ morphism $(f_0, f_1, f_2, \Psi)$ from $L$ to $L'$, for any $F_0 = f_0, F_1 = f_1$ and

Direct verification that $F$ is a morphism from $C$ to $C'$. Furthermore, we can check that $T$ preserve the identity morphisms and composition of morphisms. Hence, $T$ is a functor from ${\bf{2LieD}}_{\lambda\infty}$ to ${\bf{LieD2}}_\lambda$.

Conversely. Given a Lie 2-algebra $C = (C_1 \oplus C_0, \mathcal{J}, D, \mathcal{D})$ with a 2-differential operator of weight $\lambda$, we have the 2-term chain complex

Define $l_2 : L_i \otimes L_j \rightarrow L_{i+j}$ by

The map $l_3 : L_0 \otimes L_0 \otimes L_0 \rightarrow L_1$ is defined by

where $pr$ denote the projection on $ker(s)$. Moreover, we define a homotopy differential operator by

For any Lie 2-algebra morphism $(F_0, F_1, F_2, \Theta) : C\rightarrow C'$, then $f_0 = F_0$, $f_1 = F_1 |_{L_1} = ker s$ with a 2-differential operator of weight $\lambda$ and define $f_2$ by

Moreover, $S$ preserve the identity morphisms and composition of morphisms. Therefore, $S$ is a functor from ${\bf{LieD2}}_{\lambda}$ to ${\bf{2LieD}}_{\lambda\infty}$.

Finally, it is easy to prove that $T \circ S \cong 1_{{{\bf LieD2} _{\lambda}}}$, and the composite $S\circ T \cong 1_{{{\bf 2LieD }_{\lambda\infty} }}$ and we omit them.

Acknowledgments

The paper is supported by the NSF of China (No. 12271292).

Conflict of interest

The authors declare no conflict of interest in this paper.