Cohomology Characterizations of Diagonal Non-Abelian Extensions of Regular Hom-Lie Algebras
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The notion of a Hom-Lie algebra was introduced by Hartwig, Larsson, and Silvestrov in [1] as part of a study of deformations of the Witt and the Virasoro algebras. In a Hom-Lie algebra, the Jacobi identity is twisted by a linear map, called the Hom-Jacobi identity. The set of $(\sigma ,\sigma )$-derivations of an associative algebra and some q-deformations of the Witt and the Virasoro algebras have the structure of a Hom-Lie algebra [1,2,3]. Because of the close relation to discrete and deformed vector fields and differential calculus [1,4,5], more people have started paying attention to this algebraic structure. In particular, representations and deformations of Hom-Lie algebras are studied in [6,7,8]; extensions of Hom-Lie algebras are studied in [4,9,10]. Some split regular Hom-structures are studied in [11,12].

The notion of a Hom-Lie 2-algebra, which is the categorification of a Hom-Lie algebra, is given in [13]. The category of Hom-Lie 2-algebras and the category of 2-term $H{L}_{\infty }$-algebras are equivalent. Skeletal Hom-Lie 2-algebras can be classified by the third cohomology group of a Hom-Lie algebra. Many known Hom-structures, such as Hom-pre-Lie algebras and symplectic Hom-Lie algebras, lead to skeletal or strict Hom-Lie 2-algebras. In [14], we give the notion of a derivation of a regular Hom-Lie algebra $(\mathfrak{g},{[·,·]}_{\mathfrak{g}},{\varphi }_{\mathfrak{g}})$. The set of derivations $\mathrm{Der}(\mathfrak{g})$ is a Hom-Lie subalgebra of the regular Hom-Lie algebra $(\mathfrak{gl}(\mathfrak{g}),{[·,·]}_{{\varphi }_{\mathfrak{g}}},{\mathrm{Ad}}_{{\varphi }_{\mathfrak{g}}})$, which is given in [15]. We constructed the derivation Hom-Lie 2-algebra $\mathrm{DER}(\mathfrak{g})$, by which we characterize non-abelian extensions of regular Hom-Lie algebras as Hom-Lie 2-algebra morphisms. More precisely, we characterize a diagonal non-abelian extension of a regular Hom-Lie algebra $\mathfrak{g}$ by a regular Hom-Lie algebra $\mathfrak{h}$ using a Hom-Lie 2-algebra morphism from $\mathfrak{g}$ to the derivation Hom-Lie 2-algebra $\mathrm{DER}(\mathfrak{h})$. Associated to a non-abelian extension of a regular Hom-Lie algebra $\mathfrak{g}$ by a regular Hom-Lie algebra $\mathfrak{h}$, there is a Hom-Lie algebra morphism from $\mathfrak{g}$ to $\mathrm{Out}(\mathfrak{h})$ naturally. However, given an arbitrary Hom-Lie algebra morphism from $\mathfrak{g}$ to $\mathrm{Out}(\mathfrak{h})$, whether there is a non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ that induces the given Hom-Lie algebra morphism and what the obstruction is are not known yet.

The aim of this paper is to solve the above problem. It turns out that the result is not totally parallel to the case of Lie algebras [16,17,18,19,20,21]. We need to add some conditions on the short exact sequence related to derivations of Hom-Lie algebras. Under these conditions, first we show that under the assumption of the center of $\mathfrak{h}$ being zero, there is a one-to-one correspondence between diagonal non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ and Hom-Lie algebra morphisms from $\mathfrak{g}$ to $\mathrm{Out}(\mathfrak{h})$. Then for the general case, we show that the obstruction of the existence of a non-abelian extension is given by an element in the third cohomology group.

The paper is organized as follows. In Section 2, we recall some basic notions of Hom-Lie algebras, representations of Hom-Lie algebras, their cohomologies and derivations of Hom-Lie algebras. In Section 3, we study non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ in the case that the center of $\mathfrak{h}$ is zero. We show that if the center of $\mathfrak{h}$ is zero and the short exact sequence related to derivations of Hom-Lie algebras is also diagonal, then diagonal non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ correspond bijectively to Hom-Lie algebra morphisms from $\mathfrak{g}$ to $\mathrm{Out}(\mathfrak{h})$ (Theorem 2). In Section 4, we give a cohomology characterization of the existence of general non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$. We show that the obstruction of the existence of a diagonal non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ that induces a given Hom-Lie algebra morphism from $\mathfrak{g}$ to $\mathrm{Out}(\mathfrak{h})$ is given by a cohomology class in ${\mathcal{H}}^3(\mathfrak{g};\mathrm{Cen}(\mathfrak{h}))$ (Theorem 3). Moreover, isomorphism classes of diagonal non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ are parameterized by ${\mathcal{H}}^2(\mathfrak{g};\mathrm{Cen}(\mathfrak{h}))$ (Theorem 4). In Section 5, we give a conclusion of the paper.

In this paper, we work over an algebraically closed field $\mathbb{K}$ of characteristic 0, and all the vector spaces are over $\mathbb{K}$. We only work on finite-dimensional vector spaces.

In this section, we classify diagonal non-abelian extensions of Hom-Lie algebras for the case that $\mathrm{Cen}(\mathfrak{h})=0.$

Let $(\mathfrak{g}\oplus \mathfrak{h},{[·,·]}_{({\rho }^{\prime },{\omega }^{\prime })},\varphi )$ and $(\mathfrak{g}\oplus \mathfrak{h},{[·,·]}_{(\rho ,\omega )},\varphi )$ be isomorphic diagonal non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$. By Proposition 2, we have

Thus, we obtain that isomorphic diagonal non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ correspond to the same Hom-Lie algebra homomorphism from $\mathfrak{g}$ to $\mathrm{Out}(\mathfrak{h})$.

Conversely, let $\overline{\rho }$ be a Hom-Lie algebra homomorphism from $\mathfrak{g}$ to $\mathrm{Out}(\mathfrak{h})$. Because the short exact sequence of Hom-Lie algebras (Equation (25)) is a diagonal non-abelian extension of $\mathrm{Out}(\mathfrak{h})$ by $\mathrm{Inn}(\mathfrak{h})$, we can choose a diagonal section s of $\pi :\mathrm{Der}(\mathfrak{g})\to \mathrm{Out}(\mathfrak{h})$. Moreover, we define $\rho :\mathfrak{g}\to \mathfrak{gl}(\mathfrak{h})$ by

We have ${\rho }_x\in \mathrm{Der}(\mathfrak{h})$. Thus we obtain Equation (18). Because s is a diagonal section, we have

Thus, we obtain Equation (17). Because $\pi$ and $\overline{\rho }$ are Hom-Lie algebra homomorphisms, for all $x,y\in \mathfrak{g}$, we have which implies that ${[{\rho }_x,{\rho }_y]}_{{\varphi }_{\mathfrak{h}}}-{\rho }_{{[x,y]}_{\mathfrak{g}}}\in \mathrm{Inn}(\mathfrak{h})$. Because we have the following short exact sequence of Hom-Lie algebra morphisms: and $\mathrm{Cen}(\mathfrak{h})=0$, there exists a unique linear map $\omega :\mathfrak{g}\wedge \mathfrak{g}\to \mathfrak{h}$ such that

Furthermore, we claim that

In fact, for all $u\in \mathfrak{h}$, we have which implies that ${\varphi }_{\mathfrak{h}}(\omega (x,y))-\omega ({\varphi }_{\mathfrak{g}}(x),{\varphi }_{\mathfrak{g}}(y))=0$ because $\mathrm{Cen}(\mathfrak{h})=0.$ Thus, we obtain Equations (19) and (20). For all $x,y,z\in \mathfrak{g},u\in \mathfrak{h}$, by ${\rho }_x\in \mathrm{Der}(\mathfrak{h})$ and Equations (27) and (28), we have

Thus, we have

Because $\mathrm{Cen}(\mathfrak{h})=0$, we have Equation (21). Therefore, we deduce that Equations (17)–(21) hold. By Proposition 1, $(\mathfrak{g}\oplus \mathfrak{h},{[·,·]}_{(\rho ,\omega )},\varphi )$ is a diagonal non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$.

If we choose another section $s^{\prime }$ of $\pi :\mathrm{Der}(\mathfrak{g})\to \mathrm{Out}(\mathfrak{h})$, we obtain another diagonal non-abelian extension $(\mathfrak{g}\oplus \mathfrak{h},{[·,·]}_{({\rho }^{\prime },{\omega }^{\prime })},\varphi )$. Clearly, we have which implies that ${\rho }_x^{\prime }-{\rho }_x\in \mathrm{Inn}(\mathfrak{h})$. Because $\mathrm{Cen}(\mathfrak{h})=0$, there is a unique linear map $\xi :\mathfrak{g}\to \mathfrak{h}$ such that

We claim that

In fact, for all $u\in \mathfrak{h}$, we have which implies that ${\varphi }_{\mathfrak{h}}(\xi (x))=\xi ({\varphi }_{\mathfrak{g}}(x))$ because $\mathrm{Cen}(\mathfrak{h})=0$. Thus, we obtain Equations (22) and (23). By Lemma 1 and Equation (28), for all $x,y\in \mathfrak{g}$ we have

By $\mathrm{Cen}(\mathfrak{h})=0$, we obtain Equation (24). Thus, we have Equations (22)–(24). Therefore, we deduce that $(\mathfrak{g}\oplus \mathfrak{h},{[·,·]}_{(\rho ,\omega )},\varphi )$ and $(\mathfrak{g}\oplus \mathfrak{h},{[·,·]}_{({\rho }^{\prime },{\omega }^{\prime })},\varphi )$ are isomorphic diagonal non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$. The proof is finished. ☐

In this section, we always assume that the following short exact sequences of Hom-Lie algebra morphisms: are diagonal non-abelian extensions. Given a Hom-Lie algebra morphism $\overline{\rho }:\mathfrak{g}\to \mathrm{Out}(\mathfrak{h})$, where $\mathrm{Cen}(\mathfrak{h})\ne 0$, we consider the obstruction of existence of non-abelian extensions. By choosing a diagonal section s of $\pi :\mathrm{Der}(\mathfrak{h})\to \mathrm{Out}(\mathfrak{h})$, we can still define $\rho$ by Equation (26) such that Equation (27) holds. Moreover, we can choose a linear map $\omega :\mathfrak{g}\wedge \mathfrak{g}\to \mathfrak{h}$ such that Equations (28) and (29) hold. Thus, $(\mathfrak{g}\oplus \mathfrak{h},{[·,·]}_{(\rho ,\omega )},\varphi )$ is a diagonal non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ if and only if

Let ${\mathrm{d}}_{\rho }$ be the formal coboundary operator associated to $\rho$. Then we have

Therefore, $(\mathfrak{g}\oplus \mathfrak{h},{[·,·]}_{(\rho ,\omega )},\varphi )$ is a diagonal non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ if and only if ${\mathrm{d}}_{\rho }\omega =0.$

For all $u\in \mathrm{Cen}(\mathfrak{h})$, it is clear that ${\varphi }_{\mathfrak{h}}(u)\in \mathrm{Cen}(\mathfrak{h})$. For $v\in \mathfrak{h}$, we have