Derivations, Automorphisms, and Representations of Complex $\omega$-Lie Algebras

TL;DR

This paper investigates the derivations, automorphisms, and representations of complex non-Lie $oldsymbol{ extomega}$-Lie algebras, providing explicit classifications for low-dimensional cases and exploring their structural properties.

Contribution

It introduces $oldsymbol{ extomega}$-derivations and automorphisms, computes derivation and automorphism groups for 3- and 4-dimensional cases, and studies their representation theory.

Findings

$oldsymbol{ extomega}$-derivations form a Lie subalgebra

All 3-dimensional $oldsymbol{ extomega}$-Lie algebras are multiplicative

Any irreducible representation of simple $oldsymbol{ extomega}$-Lie algebra $C_{oldsymbol{ extalpha}}$ is 1-dimensional

Abstract

Let $(\mathfrak{g},\omega)$ be a finite-dimensional non-Lie complex $\omega$-Lie algebra. We study the derivation algebra $Der(\mathfrak{g})$ and the automorphism group $Aut(\mathfrak{g})$ of $(\mathfrak{g},\omega)$. We introduce the notions of $\omega$-derivations and $\omega$-automorphisms of $(\mathfrak{g},\omega)$ which naturally preserve the bilinear form $\omega$. We show that the set $Der_{\omega}(\mathfrak{g})$ of all $\omega$-derivations is a Lie subalgebra of $Der(\mathfrak{g})$ and the set $Aut_{\omega}(\mathfrak{g})$ of all $\omega$-automorphisms is a subgroup of $Aut(\mathfrak{g})$. For any 3-dimensional and 4-dimensional nontrivial $\omega$-Lie algebra $\mathfrak{g}$, we compute $Der(\mathfrak{g})$ and $Aut(\mathfrak{g})$ explicitly, and study some Lie group properties of $Aut(\mathfrak{g})$. We also study representation theory of $\omega$-Lie algebras. We show that all 3-dimensional nontrivial $\omega$-Lie algebras are multiplicative, as well as we provide a 4-dimensional example of $\omega$-Lie algebra that is not multiplicative. Finally, we show that any irreducible representation of the simple $\omega$-Lie algebra $C_{\alpha}(\alpha\neq 0,-1)$ is 1-dimensional.

Tables

Table 1. Table 1. Derivations of 3-dimensional ω ω \omegaup -Lie algebras 𝔤 𝔤 \mathfrak{g}

$𝔤$	Elements in $\mathrm{Der}\left(𝔤\right)$	${\mathrm{Der}}_{\omega }\left(𝔤\right)=\mathrm{Der}\left(𝔤\right)$	Properties of $\mathrm{Der}\left(𝔤\right)$	Remarks
$L_1$		True	Soluble (not nilpotent)	$(\simeq ){𝔤}_2$
$L_2$		True	Abelian	$(\simeq ){𝔤}_1$
$\begin{array}{c}{A}_{\alpha }\\ \left(\alpha \in ℂ\right)\end{array}$		True	Abelian	$(\simeq ){𝔤}_1$
$B$		True	Abelian	$(\simeq ){𝔤}_1$
$\begin{array}{c}{C}_{\alpha }\\ \left(\alpha \in ℂ-\{1,0,-1\}\right)\end{array}$		True	Abelian	$(\simeq ){𝔤}_1$
$C_1$		True	Simple	$(\simeq )𝔰{𝔩}_2\left(ℂ\right)$

Table 2. Table 2. Automorphisms of 3-dimensional ω ω \omegaup -Lie algebras 𝔤 𝔤 \mathfrak{g}

$𝔤$	Elements in $\mathrm{Aut}\left(𝔤\right)$	${\mathrm{Aut}}_{\omega }\left(𝔤\right)=\mathrm{Aut}\left(𝔤\right)$	Properties of $\mathrm{Aut}\left(𝔤\right)$	Unipotent?
$L_1$		True	Soluble	False
$L_2$		True	Abelian	False
$\begin{array}{c}{A}_{\alpha }\\ \left(\alpha \in ℂ\right)\end{array}$		True	Abelian	True
$B$		True	Abelian	True
$\begin{array}{c}{C}_{\alpha }\\ \left(\alpha \in ℂ-\{1,0,-1\}\right)\end{array}$		True	Abelian	False
$C_1$		True	$(\simeq ){\mathrm{SL}}_2\left(ℂ\right)$	False

Table 3. Table 3. Derivations of 4-dimensional ω ω \omegaup -Lie algebras 𝔤 𝔤 \mathfrak{g}

$𝔤$	Relations in $𝔤$	Elements in $\mathrm{Der}\left(𝔤\right)$	$dim\left(\mathrm{Der}\left(𝔤\right)\right)$
$L_{1,1}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=-y,\\ \omega (x,y)=1\end{array}$		6
$L_{1,2}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=z,[e,y]=-e,\\ \omega (x,y)=1\end{array}$		4
$L_{1,3}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=y,[e,y]=-e,\\ \omega (x,y)=1,\omega (e,x)=1\end{array}$		2
$L_{1,4}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=y+z,[e,y]=-e,\\ \omega (x,y)=1,\omega (e,x)=1\end{array}$		2
$L_{1,5}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=e,[e,y]=-e,\\ \omega (x,y)=1\end{array}$		4
$L_{1,6}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=e+y,[e,y]=-e,\\ \omega (x,y)=1=\omega (e,x)\end{array}$		3
$L_{1,7}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=e,[e,y]=z-e,\\ \omega (x,y)=1\end{array}$		3
$L_{1,8}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=e+y,[e,y]=z-e,\\ \omega (x,y)=1=\omega (e,x)\end{array}$		2
$L_{2,1}$	$\begin{array}{c}[x,z]=y,[y,z]=z,\\ [e,y]=-e,\\ \omega (x,z)=1\end{array}$		3
$L_{2,2}$	$\begin{array}{c}[x,z]=y,[y,z]=z,\\ [e,y]=-e,[e,x]=z,\\ \omega (x,z)=1\end{array}$		1
$L_{2,3}$	$\begin{array}{c}[x,z]=y,[y,z]=z,\\ [e,y]=-e,[e,x]=e,\\ \omega (x,z)=1\end{array}$		2
$L_{2,4}$	$\begin{array}{c}[x,z]=y,[y,z]=z,\\ [e,y]=-e,[e,x]=e+z,\\ \omega (x,z)=1\end{array}$		1
$\tilde{B}$	$\begin{array}{c}[x,y]=y,[x,z]=y+z,[y,z]=x,\\ [e,y]=-e,[e,x]=-2e,\\ \omega (y,z)=2\end{array}$		2
$\begin{array}{c}{E}_{1,\alpha }\\ \left(\alpha \ne 0,1\right)\end{array}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=-e,[e,x]=\alpha e,\\ \omega (x,y)=1\end{array}$		4
$\begin{array}{c}{F}_{1,\alpha }\\ \left(\alpha \ne 0,1\right)\end{array}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=-e,[e,x]=\alpha e+y,\\ \omega (x,y)=1=\omega (e,x)\end{array}$		2
$\begin{array}{c}{G}_{1,\alpha }\end{array}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=x-e,[e,x]=e+\alpha y,\\ \omega (x,y)=1,\omega (e,x)=\alpha \end{array}$		2
$\begin{array}{c}{H}_{1,\alpha }\end{array}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=x+z-e,[e,x]=e+\alpha y,\\ \omega (x,y)=1,\omega (e,x)=\alpha \end{array}$		2
$\begin{array}{c}{\tilde{A}}_{\alpha }\end{array}$	$\begin{array}{c}[x,y]=x,[x,z]=x+y,\\ [y,z]=z+\alpha x,\\ [e,z]=e,\omega (y,z)=-1\end{array}$		2
$\begin{array}{c}{\tilde{C}}_{\alpha }\\ \left(\alpha \ne 0,-1,1\right)\end{array}$	$\begin{array}{c}[x,y]=y,[x,z]=\alpha z,\\ [y,z]=x,[e,x]=-\left(1+\alpha \right)e,\\ \omega (y,z)=1+\alpha \end{array}$		2
$\begin{array}{c}{\tilde{C}}_1\end{array}$	$\begin{array}{c}[x,y]=y,[x,z]=z,\\ [y,z]=x,[e,x]=-2e,\\ \omega (y,z)=2\end{array}$		4

Table 4. Table 4. Automorphisms of 4-dimensional ω ω \omegaup -Lie algebras 𝔤 𝔤 \mathfrak{g}

$𝔤$	Relations in $𝔤$	Elements in $\mathrm{Aut}\left(𝔤\right)$
$L_{1,1}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=-y,\\ \omega (x,y)=1\end{array}$
$L_{1,2}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=z,[e,y]=-e,\\ \omega (x,y)=1\end{array}$	,   $c\ne 0$
$L_{1,3}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=y,[e,y]=-e,\\ \omega (x,y)=1,\omega (e,x)=1\end{array}$
$L_{1,4}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=y+z,[e,y]=-e,\\ \omega (x,y)=1,\omega (e,x)=1\end{array}$
$L_{1,5}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=e,[e,y]=-e,\\ \omega (x,y)=1\end{array}$
$L_{1,6}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=e+y,[e,y]=-e,\\ \omega (x,y)=1=\omega (e,x)\end{array}$
$L_{1,7}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=e,[e,y]=z-e,\\ \omega (x,y)=1\end{array}$
$L_{1,8}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,x]=e+y,[e,y]=z-e,\\ \omega (x,y)=1=\omega (e,x)\end{array}$
$L_{2,1}$	$\begin{array}{c}[x,z]=y,[y,z]=z,\\ [e,y]=-e,\\ \omega (x,z)=1\end{array}$
$L_{2,2}$	$\begin{array}{c}[x,z]=y,[y,z]=z,\\ [e,y]=-e,[e,x]=z,\\ \omega (x,z)=1\end{array}$
$L_{2,3}$	$\begin{array}{c}[x,z]=y,[y,z]=z,\\ [e,y]=-e,[e,x]=e,\\ \omega (x,z)=1\end{array}$
$L_{2,4}$	$\begin{array}{c}[x,z]=y,[y,z]=z,\\ [e,y]=-e,[e,x]=e+z,\\ \omega (x,z)=1\end{array}$	, $a\ne 0$
$\tilde{B}$	$\begin{array}{c}[x,y]=y,[x,z]=y+z,[y,z]=x,\\ [e,y]=-e,[e,x]=-2e,\\ \omega (y,z)=2\end{array}$
$\begin{array}{c}{E}_{1,\alpha }\\ \left(\alpha \ne 0,1\right)\end{array}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=-e,[e,x]=\alpha e,\\ \omega (x,y)=1\end{array}$
$\begin{array}{c}{F}_{1,\alpha }\\ \left(\alpha \ne 0,1\right)\end{array}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=-e,[e,x]=\alpha e+y,\\ \omega (x,y)=1=\omega (e,x)\end{array}$
$\begin{array}{c}{G}_{1,0}\end{array}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=x-e,[e,x]=e,\\ \omega (x,y)=1\end{array}$
$\begin{array}{c}{G}_{1,\alpha }\\ \left(\alpha \ne 0\right)\end{array}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=x-e,[e,x]=e+\alpha y,\\ \omega (x,y)=1,\omega (e,x)=\alpha \end{array}$
$\begin{array}{c}{H}_{1,0}\end{array}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=x+z-e,[e,x]=e,\\ \omega (x,y)=1\end{array}$
$\begin{array}{c}{H}_{1,\alpha }\\ \left(\alpha \ne 0\right)\end{array}$	$\begin{array}{c}[x,y]=y,[y,z]=z,\\ [e,y]=x+z-e,[e,x]=e+\alpha y,\\ \omega (x,y)=1,\omega (e,x)=\alpha \end{array}$
$\begin{array}{c}{\tilde{A}}_{\alpha }\end{array}$	$\begin{array}{c}[x,y]=x,[x,z]=x+y,\\ [y,z]=z+\alpha x,\\ [e,z]=e,\omega (y,z)=-1\end{array}$
$\begin{array}{c}{\tilde{C}}_{\alpha }\\ \left(\alpha \ne 0,-1,1\right)\end{array}$	$\begin{array}{c}[x,y]=y,[x,z]=\alpha z,\\ [y,z]=x,[e,x]=-\left(1+\alpha \right)e,\\ \omega (y,z)=1+\alpha \end{array}$
$\begin{array}{c}{\tilde{C}}_1\end{array}$	$\begin{array}{c}[x,y]=y,[x,z]=z,\\ [y,z]=x,[e,x]=-2e,\\ \omega (y,z)=2\end{array}$

Equations

$$[[x,y],z]+[[y,z],x]+[[z,x],y]=\text{omegaup}(x,y)z+\text{omegaup}(y,z)x+\text{omegaup}(z,x)y$$

$$d([x,y])=[d(x),y]+[x,d(y)]$$

$$\text{rhoup}([x,y])=[\text{rhoup}(x),\text{rhoup}(y)]$$

$$\text{omegaup}(d(x),y)+\text{omegaup}(x,d(y))=0$$

$$\text{omegaup}(x,y)=\text{omegaup}(\text{rhoup}(x),\text{rhoup}(y))$$

$$\mathrm{Der}(\mathfrak{g}),{\mathrm{Der}}_{\text{omegaup}}(\mathfrak{g}),\mathrm{Aut}(\mathfrak{g})\text{ and }{\mathrm{Aut}}_{\text{omegaup}}(\mathfrak{g})$$

$$\bigg{\{}L_{1,1},\dots,L_{1,8},L_{2,1},L_{2,2},L_{2,3},L_{2,4},\widetilde{B},E_{1,\alphaup}(\alphaup\neq 0,1),F_{1,\alphaup}(\alphaup\neq 0,1),G_{1,\alphaup},H_{1,\alphaup},\widetilde{A}_{\alphaup},\widetilde{C}_{\alphaup}(\alphaup\neq 0,-1)\bigg{\}},$$

$$\text{omegaup}([d,e](x),y)$$

$$\text{omegaup}(x,[d,e](y))$$

$$a_{12}=0=a_{32}.$$

$$a_{21}=0=a_{11}\text{ and }{a}_{23}=-a_{13}.$$

$$a_{31}=0=a_{22}\text{ and }{a}_{31}=-a_{32}=0,$$

$$d=\left(\begin{array}{ccc}0& 0& a_{13}\\ 0& 0& -a_{13}\\ 0& 0& a_{33}\end{array}\right)$$

$${\mathfrak{g}}_2:\{x,y\}\text{ is a basis with }[x,y]=y$$

$$\text{omegaup}(d(\text{alphaup}),\text{betaup})+\text{omegaup}(\text{alphaup},d(\text{betaup}))=0$$

$$a_{11}{a}_{32}=a_{12}{a}_{31}\text{ and }{a}_{12}{a}_{33}=a_{13}{a}_{32}.$$

$$a_{21}=0,a_{22}=a_{11}{a}_{22}-a_{12}{a}_{21}\text{ and }{a}_{23}=a_{12}{a}_{23}-a_{13}{a}_{22}.$$

$$a_{31}=0,a_{32}=a_{21}{a}_{32}-a_{22}{a}_{31}\text{ and }{a}_{33}=a_{22}{a}_{33}-a_{23}{a}_{32}.$$

$$\text{sigmaup}=\left(\begin{array}{ccc}1& 0& a_{13}\\ 0& 1& -a_{13}\\ 0& 0& a_{33}\end{array}\right),$$

$$\text{omegaup}(u,v)=\text{omegaup}(\text{sigmaup}(u),\text{sigmaup}(v))$$

$$X\mapsto \sum _{k=0}^{\infty }\frac{X^k}{k!},$$

$$\exp (\mathrm{Der}(L_1))\supseteq \mathrm{Aut}(L_1),$$

$$a_0:=\frac{a\cdot b_0}{e^{b_0}-1}$$

$$d:=\left(\begin{array}{ccc}0& 0& a_0\\ 0& 0& -a_0\\ 0& 0& b_0\end{array}\right).$$

$$\exp (d)$$

$$d:=\left(\begin{array}{ccc}0& 0& a\\ 0& 0& -a\\ 0& 0& 0\end{array}\right).$$

$$\exp (d)=\left(\begin{array}{ccc}1& 0& a\\ 0& 1& -a\\ 0& 0& 1\end{array}\right)=\text{sigmaup}.$$

$$a_{11}x+a_{12}y+a_{13}z$$

$$a_{11}$$

$$a_{12}$$

$$a_{13}$$

Full text

1. Introduction

In 2007, Nurowski [8] was motivated by the study of isoparametric hypersufaces in Riemannian geometry (Bobieński-Nurowski [1] and Nurowski [9]), and introduced the notion of $\omegaup$-Lie algebras which can be viewed as a kind of natural generalization of Lie algebras. Specifically, for a finite-dimensional vector space $\mathfrak{g}$ over a field $\mathbb{F}$ of characteristic zero equipped with a skew symmetric bracket $[-,-]:\mathfrak{g}\times\mathfrak{g}\longrightarrow\mathfrak{g}$ and a bilinear form $\omegaup:\mathfrak{g}\times\mathfrak{g}\longrightarrow\mathbb{F}$, we say that the triple $(\mathfrak{g},[-,-],\omegaup)$ is an $\omegaup$-Lie algebra if

[TABLE]

for all $x,y,z\in\mathfrak{g}$. Equation (1.1) is called the $\omegaup$-Jacobi identity. By this identity, we see that the bilinear form $\omegaup$ is skew-symmetric for any $\omegaup$-Lie algebra $(\mathfrak{g},\omegaup)$. Clearly, an $\omegaup$-Lie algebra $(\mathfrak{g},\omegaup)$ with $\omegaup=0$ is an ordinary Lie algebra, which is called a trivial $\omegaup$-Lie algebra. An $\omegaup$-Lie algebra $(\mathfrak{g},\omegaup)$ is called nontrivial (or non-Lie) if $\mathfrak{g}$ is not a Lie algebra.

In 2010, Zusmanovich [13] developed a fundamental structure theorem on finite-dimensional $\omegaup$-Lie algebras over an algebraically closed field of characteristic zero, showing that nontrivial finite-dimensional $\omegaup$-Lie algebras are either low-dimensional or have a very degenerate structure, see [13, Section 9, Theorem 1]. Two important and interesting results were derived: If $\mathfrak{g}$ is a finite-dimensional $\omegaup$-Lie algebra with non-degenerate $\omegaup$, then $\dim(\mathfrak{g})=2$ (see [13, Lemma 8.1]); A finite-dimensional semisimple $\omegaup$-Lie algebra is either a Lie algebra, or has dimension $\leqslant 4$ (see [13, Theorem 2]). These results indicate essentially the importance of low-dimensional $\omegaup$-Lie algebras.

By the definition of $\omegaup$-Lie algebras, there are no nontrivial $\omegaup$-Lie algebras in the cases of dimensions 1 and 2. The first example of nontrivial 3-dimensional $\omegaup$-Lie algebra was given by Nurowski [8] in which the author gave a classification of 3-dimensional $\omegaup$-Lie algebras over the field of real numbers, under the action of 3-dimensional orthogonal group. In 2014, we extended a method for classifying 3-dimensional complex Lie algebras appeared in Fulton-Harris [4, Sections 10.2-10.4] to obtain a classification of 3-dimensional complex $\omegaup$-Lie algebras, see Chen-Liu-Zhang [2, Theorem 2]. This classification, together with a result of Zusmanovich [13, Lemma 8.2] which asserts that any 4-dimensional $\omegaup$-Lie algebra over an algebraically closed field contains a 3-dimensional subalgebra, has led us to complete a classification of 4-dimensional complex Lie algebras in a recent paper Chen-Zhang [3, Theorem 1.5]. In particular, we obtained a classification of nontrivial finite-dimensional complex simple $\omegaup$-Lie algebras, see [3, Theorem 1.7].

The first purpose of this paper is to study derivations and automorphisms of low-dimensional complex $\omegaup$-Lie algebras, which are two kinds of classical objects in the study of nonassociative algebra and representation theory. In what follows we fix the ground field to be $\mathbb{C}$, the field of complex numbers. Let’s recall the definitions of classical derivation and automorphism of any nonassociative algebra. Let $\mathfrak{g}$ be a finite-dimensional $\omegaup$-Lie algebra (as a nonassociative algebra) and $d:\mathfrak{g}\longrightarrow\mathfrak{g}$ be a linear map. We say that $d$ is a derivation of $\mathfrak{g}$ if

[TABLE]

for any $x,y\in\mathfrak{g}$. We write $\mathfrak{gl}(\mathfrak{g})$ for the general linear Lie algebra on $\mathfrak{g}$. Then the set ${\rm Der}(\mathfrak{g})$ of all derivations of $\mathfrak{g}$ forms a Lie subalgebra of $\mathfrak{gl}(\mathfrak{g})$, which is called the derivation algebra of $\mathfrak{g}$. A linear isomorphism $\rhoup:\mathfrak{g}\longrightarrow\mathfrak{g}$ is called an automorphism of $\mathfrak{g}$ if

[TABLE]

for any $x,y\in\mathfrak{g}$. The set ${\rm Aut}(\mathfrak{g})$ of all automorphisms of $\mathfrak{g}$ forms a closed Lie subgroup of the general linear group ${\rm GL}(\mathfrak{g})$, which means that ${\rm Aut}(\mathfrak{g})$ is a matrix Lie group, see Sagle-Walde [11, Proposition 7.1]. We call ${\rm Aut}(\mathfrak{g})$ the automorphism group of $\mathfrak{g}$. It is well-known that the Lie algebra of ${\rm Aut}(\mathfrak{g})$ is just the derivation algebra ${\rm Der}(\mathfrak{g})$, see Sagle-Walde [11, Proposition 7.3 (b)]. However, $\omegaup$-Lie algebras as a kind of special nonassociative algebras and a generalization of Lie algebras, deserve to have their feature and properties. Comparing with Hom-Lie algebras and their derivations (automorphisms), see Sheng [10, Section 3] and Jin-Li [6, Definition 1.2], we introduce the following two notions: $\omegaup$-derivation and $\omegaup$-automorphism, which both preserve naturally the skew-symmetric bilinear form $\omegaup$.

Definition 1.1**.**

Let $\mathfrak{g}$ be a finite-dimensional $\omegaup$-Lie algebra. A derivation $d\in{\rm Der}(\mathfrak{g})$ is called an $\omegaup$-derivation of $\mathfrak{g}$ if

[TABLE]

for any $x,y\in\mathfrak{g}$. An automorphism $\rhoup\in{\rm Aut}(\mathfrak{g})$ is called an $\omegaup$-automorphism of $\mathfrak{g}$ if

[TABLE]

for any $x,y\in\mathfrak{g}$.

We write ${\rm Der}_{\omegaup}(\mathfrak{g})$ for the set consisting of all $\omegaup$-derivations of $\mathfrak{g}$, and ${\rm Aut}_{\omegaup}(\mathfrak{g})$ for the set of all $\omegaup$-automorphisms of $\mathfrak{g}$. Clearly, ${\rm Der}_{\omegaup}(\mathfrak{g})\subseteq{\rm Der}(\mathfrak{g})$ and ${\rm Aut}_{\omegaup}(\mathfrak{g})\subseteq{\rm Aut}(\mathfrak{g})$. However, Proposition 5.1 below indicates that the two equalities do not necessary hold here. In this paper we calculate

[TABLE]

for 3-dimensional and 4-dimensional $\omegaup$-Lie algebras. We also study some Lie algebra properties of ${\rm Der}(\mathfrak{g})$ and Lie group properties of ${\rm Aut}(\mathfrak{g})$ for 3-dimensional $\omegaup$-Lie algebras.

The second purpose of this paper is to study representation theory of $\omegaup$-Lie algebras. Given an $\omegaup$-Lie algebra $\mathfrak{g}$ and a $\mathfrak{g}$-module $V$, we construct a semi-direct product on $\mathfrak{g}\oplus V$ such that $(\mathfrak{g}\oplus V,[-,-],\Omega)$ becomes an $\Omega$-Lie algebra. We clarify some assertions in Zusmanovich [13, Section 2] and give detailed proofs. We also study multiplicative $\omegaup$-Lie algebras, showing that all 3-dimensional nontrivial $\omegaup$-Lie algebras are multiplicative while giving an example of non-multiplicative $\omegaup$-Lie algebra in 4-dimensional case. Furthermore, we study irreducible representations of $\omegaup$-Lie algebras. Recall that a $\mathfrak{g}$-module $V$ is called irreducible if $V$ has no nontrivial submodules, i.e., if $V_{0}\subseteq V$ is any submodule of $V$, then $V_{0}$ is either $\{0\}$ or equal to $V$. We characterize all irreducible representations of $C_{\alphaup}(\alphaup\neq 0,-1)$ in detail, showing that there exist only 1-dimensional irreducible representations for $C_{\alphaup}(\alphaup\neq 0,-1)$.

This paper is organized as follows. Section 2 is a preliminary section which contains the classification lists of 3-dimensional and 4-dimensional nontrivial complex $\omegaup$-Lie algebras. We show that ${\rm Der}_{\omegaup}(\mathfrak{g})\subseteq{\rm Der}(\mathfrak{g})$ is a Lie subalgebra and ${\rm Aut}_{\omegaup}(\mathfrak{g})\leqslant{\rm Aut}(\mathfrak{g})$ is a subgroup for any finite-dimensional $\omegaup$-Lie algebra $\mathfrak{g}$. In Section 3 we calculate ${\rm Der}(\mathfrak{g})$ and ${\rm Der}_{\omegaup}(\mathfrak{g})$, and study their structures, for any 3-dimensional $\omegaup$-Lie algebra $\mathfrak{g}$. We give a detailed proof for the case $\mathfrak{g}=L_{1}$ (Proposition 3.1), omitting the proofs of other cases because of the similarity. Section 4 is devoted to calculating ${\rm Aut}_{\omegaup}(\mathfrak{g})$ and ${\rm Aut}(\mathfrak{g})$ when $\mathfrak{g}$ is a 3-dimensional $\omegaup$-Lie algebra. Usually, the automorphism group of a nonassociative algebra is more complicated than the corresponding derivation algebra. Here we study ${\rm Aut}(L_{1})$ and ${\rm Aut}(A_{\alphaup})$ in detail, see Propositions 4.1 and 4.3. In particular, we use a “trick” coming from Linear Algebra to characterize ${\rm Aut}(A_{\alphaup})$. We prove that ${\rm Aut}(L_{1})$ and ${\rm Aut}(A_{\alphaup})$ are connected matrix Lie groups, and moreover $\exp({\rm Der}(L_{1}))={\rm Aut}(L_{1})$ and $\exp({\rm Der}(A_{\alphaup}))={\rm Aut}(A_{\alphaup})$. In Section 5, we apply similar arguments to calculate derivations and automorphisms for 4-dimensional complex $\omegaup$-Lie algebras, without detailed proofs for saving space. The main results on derivations and automorphisms are summarized in Tables LABEL:3d-LABEL:4a. Section 6 is devoted to clarifying some discussions in Zusmanovich [13, Section 2]. We give a sufficient and necessary condition (essentially due to Zusmanovich) for that an $\omegaup$-Lie algebra has 1-dimensional module, which as an application shows that all nontrivial 3-dimensional $\omegaup$-Lie algebras are multiplicative (Proposition 6.5). In Section 7, we study irreducible representations of $C_{\alphaup}$ where $\alphaup\neq 0,-1$. We show that any irreducible representation of $C_{\alpha}$ is 1-dimensional (Theorem 7.1). Note that if we extend the parameter $\alphaup$ to contain $-1$, then $C_{-1}$ is just isomorphic to the special linear Lie algebra $\mathfrak{sl}_{2}(\mathbb{C})$ which has a unique irreducible representation in any finite dimension. These results indicate that structure and representation of $\omegaup$-Lie algebras might be very “degenerate” in sense of Zusmanovich [13, Section 1].

Throughout this paper, we assume that the ground field is the field of complex numbers $\mathbb{C}$, $W^{*}:=\operatorname{Hom}(W,\mathbb{C})$ denotes the dual space of a vector space $W$, all vector spaces (modules) are finite-dimensional, and all $\omegaup$-Lie algebras are nontrivial, unless stated otherwise.

2. Preliminaries

Let’s begin with the classifications of 3-dimensional and 4-dimensional nontrivial $\omegaup$-Lie algebras over $\mathbb{C}$.

Theorem 2.1** (Chen-Liu-Zhang [2]).**

Let $\mathfrak{g}$ be a 3-dimensional nontrivial $\omegaup$-Lie algebra over $\mathbb{C}$, then it must be isomorphic to one of the following algebras:

1. (1)

$L_{1}:\quad[x,z]=0,[y,z]=z,[x,y]=y\textrm{ and }\omegaup(y,z)=\omegaup(x,z)=0,\omegaup(x,y)=1.$** 2. (2)

$L_{2}:\quad[x,y]=0,[x,z]=y,[y,z]=z\textrm{ and }\omegaup(x,y)=0,\omegaup(x,z)=1,\omegaup(y,z)=0.$** 3. (3)

$A_{\alphaup}:\quad[x,y]=x,[x,z]=x+y,[y,z]=z+\alphaup x\textrm{ and }\omegaup(x,y)=\omegaup(x,z)=0,$**

$\omegaup(y,z)=-1,\textrm{ where }\alphaup\in\mathbb{C}.$** 4. (4)

$B:\quad[x,y]=y,[x,z]=y+z,[y,z]=x\textrm{ and }\omegaup(x,y)=\omegaup(x,z)=0,$* $\omegaup(y,z)=2.$* 5. (5)

$C_{\alphaup}:\quad[x,y]=y,[x,z]=\alphaup z,[y,z]=x\textrm{ and }\omegaup(x,y)=\omegaup(x,z)=0,$**

$\omegaup(y,z)=1+\alphaup,\textrm{ where }0,-1\neq\alphaup\in\mathbb{C}.$**

Theorem 2.2** (Chen-Zhang [3]).**

Any 4-dimensional nontrivial $\omegaup$-Lie algebra over $\mathbb{C}$ must be isomorphic to one of the following algebras:

[TABLE]

where the parameter $\alphaup\in\mathbb{C}$. For the nontrivial generating relations of these $\omegaup$-Lie algebras see [3, Sections 2-6] or Table LABEL:4d below.

Let $\mathfrak{g}$ be an $\omegaup$-Lie algebra. The following two properties are elementary.

Proposition 2.3**.**

${\rm Aut}_{\omegaup}(\mathfrak{g})$* is a subgroup of ${\rm Aut}(\mathfrak{g})$.*

Proof.

Clearly, the identity map $I$ belongs to ${\rm Aut}_{\omegaup}(\mathfrak{g})$. For any $\rhoup,\sigmaup\in{\rm Aut}_{\omegaup}(\mathfrak{g})$, it suffices to show that $\rhoup^{-1}\cdot\sigmaup\in{\rm Aut}_{\omegaup}(\mathfrak{g})$. Suppose $x,y\in\mathfrak{g}$ are any two elements. Since $\omegaup(x,y)=\omegaup(I(x),I(y))=\omegaup(\rhoup\cdot\rhoup^{-1}(x),\rhoup\cdot\rhoup^{-1}(y))=\omegaup(\rhoup^{-1}(x),\rhoup^{-1}(y))$, we have $\rhoup^{-1}\in{\rm Aut}_{\omegaup}(\mathfrak{g})$. Thus $\omegaup(\rhoup^{-1}\cdot\sigmaup(x),\rhoup^{-1}\cdot\sigmaup(y))=\omegaup(\sigmaup(x),\sigmaup(y))=\omegaup(x,y)$ and so $\rhoup^{-1}\cdot\sigmaup\in{\rm Aut}_{\omegaup}(\mathfrak{g})$. ∎

Proposition 2.4**.**

${\rm Der}_{\omegaup}(\mathfrak{g})$* is a Lie subalgebra of ${\rm Der}(\mathfrak{g})$.*

Proof.

We need only to show that $[d,e]=d\cdot e-e\cdot d\in{\rm Der}_{\omegaup}(\mathfrak{g})$ for any $d,e\in{\rm Der}_{\omegaup}(\mathfrak{g})$. Indeed, for any $x,y\in\mathfrak{g}$, we have

[TABLE]

and

[TABLE]

Thus $\omegaup([d,e](x),y)+\omegaup(x,[d,e](y))=0$, which implies that $[d,e]\in{\rm Der}_{\omegaup}(\mathfrak{g})$. ∎

3. Derivations in Dimension 3

We first study the derivations of $L_{1}$, where $L_{1}$ has a basis $\{x,y,z\}$ and is defined as in Theorem 2.1 (1). We denote by $E_{ij}$ the $n\times n$ matrix in which the $(i,j)$-entry is 1 and other entries are zero. It is well-known that the $\{E_{ij}\mid 1\leqslant i,j\leqslant n\}$ is a basis for the general linear Lie algebra $\mathfrak{gl}_{n}(\mathbb{C})$.

Proposition 3.1**.**

1. (1)

${\rm Der}(L_{1})$* is a 2-dimensional soluble (but not nilpotent) Lie algebra.* 2. (2)

${\rm Der}_{\omegaup}(L_{1})={\rm Der}(L_{1})$.

Proof.

(1) For any $d\in{\rm Der}(L_{1})$, we assume that $d=(a_{ij})\in\mathfrak{gl}_{3}(\mathbb{C})$. Suppose $d(x)=a_{11}x+a_{12}y+a_{13}z,d(y)=a_{21}x+a_{22}y+a_{23}z,$ and $d(z)=a_{31}x+a_{32}y+a_{33}z.$ Since $[x,z]=0$ in $L_{1}$, $0=d([x,z])=[d(x),z]+[x,d(z)]=[a_{11}x+a_{12}y+a_{13}z,z]+[x,a_{31}x+a_{32}y+a_{33}z]=a_{12}z+a_{32}y$. The last equality follows from the relations that $[y,z]=z$ and $[x,y]=y$ in $L_{1}$. Since $y,z$ are linearly independent, we have

[TABLE]

Similarly, $a_{21}x+a_{22}y+a_{23}z=d(y)=d([x,y])=[d(x),y]+[x,d(y)]=[a_{11}x+a_{12}y+a_{13}z,y]+[x,a_{21}x+a_{22}y+a_{23}z]=(a_{11}+a_{22})y-a_{13}z$. Thus $a_{21}x-a_{11}y+(a_{23}+a_{13})z=0$. Since $x,y,z$ are linearly independent,

[TABLE]

Finally, $a_{31}x+a_{32}y+a_{33}z=d(z)=d([y,z])=[d(y),z]+[y,d(z)]=[a_{21}x+a_{22}y+a_{23}z,z]+[y,a_{31}x+a_{32}y+a_{33}z]=-a_{31}y+(a_{22}+a_{33})z$. Thus

[TABLE]

and

[TABLE]

where $a_{13},a_{33}\in\mathbb{C}.$ Clearly, $\{E_{13}-E_{23},E_{33}\}$ is a basis for ${\rm Der}(L_{1})$. Therefore, ${\rm Der}(L_{1})$ is a 2-dimensional Lie algebra.

Note that $[E_{13}-E_{23},E_{33}]=E_{13}-E_{23}$, so ${\rm Der}(L_{1})$ is not abelian. Since all 2-dimensional nonabelian Lie algebras are isomorphic to

[TABLE]

(see Fulton-Harri [4, page 135]), we have ${\rm Der}(L_{1})\simeq\mathfrak{g}_{2}$. Since $\mathfrak{g}_{2}^{(1)}=[\mathfrak{g}_{2},\mathfrak{g}_{2}]$ is the ideal generated by $y$ in $\mathfrak{g}_{2}$, $\mathfrak{g}_{2}^{(2)}=[\mathfrak{g}_{2}^{(1)},\mathfrak{g}_{2}^{(1)}]=0$ and so $\mathfrak{g}_{2}$ is soluble. To see that $\mathfrak{g}_{2}$ is not nilpotent, we assume by way of contradiction that there exists some $m\in\mathbb{N}^{+}$ such that $\mathfrak{g}_{2}^{m+1}=[\mathfrak{g}_{2}^{m},\mathfrak{g}_{2}]=0$. However, since $[x,y]=y$, the element $[\cdots[y,\underbrace{x],x],\cdots,x]}_{m}\in\mathfrak{g}_{2}^{m+1}$ is not zero, contradiction.

(2) For any $d\in{\rm Der}(L_{1})$, since $\omegaup$ is bilinear, we need only to show that

[TABLE]

for $\alphaup,\betaup\in\{x,y,z\}$. If $\alphaup=\betaup$, then Equation (3.2) holds from the fact that $\omegaup$ is anti-symmetric. So we may assume that $\alphaup\neq\betaup$.

Subcase 1: $(\alphaup,\betaup)=(x,y)$ or $(y,x)$. Note that $\omegaup(y,z)=0=\omegaup(x,z)$, so $\omegaup(d(x),y)+\omegaup(x,d(y))=\omegaup(a_{13}z,y)+\omegaup(x,-a_{13}z)=a_{13}\omegaup(z,y)-a_{13}\omegaup(x,z)=0$.

Subcase 2: $(\alphaup,\betaup)=(x,z)$ or $(z,x)$. In this case, $\omegaup(d(x),z)+\omegaup(x,d(z))=\omegaup(a_{13}z,z)+\omegaup(x,a_{33}z)=a_{13}\omegaup(z,z)+a_{33}\omegaup(x,z)=0.$

Subcase 3: $(\alphaup,\betaup)=(y,z)$ or $(z,y)$. Similarly, $\omegaup(d(y),z)+\omegaup(y,d(z))=\omegaup(-a_{13}z,z)+\omegaup(y,a_{33}z)=0.$

Therefore, Equation (3.2) follows, completing the proof of the second assertion. ∎

Similar arguments can be applied to the remaining 3-dimensional $\omegaup$-Lie algebras, so we summarize the result in the following Table LABEL:3d, without detailed proofs. Note that in Table LABEL:3d, $\mathfrak{g}_{1}$ is the unique 1-dimensional Lie algebra, $\mathfrak{g}_{2}$ is defined as Equation (3.1), and $\mathfrak{sl}_{2}(\mathbb{C})$ denotes the 3-dimensional simple Lie algebra.

4. Automorphisms in Dimension 3

In this section we calculate automorphisms of $3$-dimensional $\omegaup$-Lie algebras. We only give the detailed proofs for the cases of $L_{1}$ and $A_{\alpha}$, omitting the proofs for the remaining cases because of similarity of arguments.

We begin with the computations of ${\rm Aut}(L_{1})$.

For any $\sigmaup\in{\rm Aut}(L_{1})$, we assume that $\sigmaup=(a_{ij})\in\mathfrak{gl}_{3}(\mathbb{C})$ with $\det(\sigmaup)\neq 0.$ Then $\sigmaup(x)=a_{11}x+a_{12}y+a_{13}z,\sigmaup(y)=a_{21}x+a_{22}y+a_{23}z$ and $\sigmaup(z)=a_{31}x+a_{32}y+a_{33}z$. Since $[x,z]=0$, we have $0=\sigmaup([x,z])=[\sigmaup(x),\sigmaup(z)]=[a_{11}x+a_{12}y+a_{13}z,a_{31}x+a_{32}y+a_{33}z]=(a_{11}a_{32}-a_{12}a_{31})y+(a_{12}a_{33}-a_{13}a_{32})z$. Thus

[TABLE]

Since $[x,y]=y$, we expand the equation $\sigmaup(y)=[\sigmaup(x),\sigmaup(y)]$ and derive

[TABLE]

Similarly, since $[y,z]=z$, we have $[\sigmaup(y),\sigmaup(z)]=\sigmaup(z)$. Expanding this equation we get

[TABLE]

A direct computation shows that

[TABLE]

where $a_{33}\neq 0$. This completes the characterization of arbitrary element in ${\rm Aut}(L_{1})$.

To describe ${\rm Aut}_{\omegaup}(L_{1})$, we suppose $\sigmaup=\begin{pmatrix}1&0&a\\ 0&1&-a\\ 0&0&b\end{pmatrix}\in{\rm Aut}(L_{1})$. Then $\sigmaup(x)=x+az,\sigmaup(y)=y-az$ and $\sigmaup(z)=bz$. Recall that $\omegaup(x,z)=0=\omegaup(y,z)$ and $\omegaup(x,y)=1$, we see that for any $u,v\in L_{1}$,

[TABLE]

which shows that ${\rm Aut}_{\omegaup}(L_{1})={\rm Aut}(L_{1})$. Moreover, we can prove

Proposition 4.1**.**

1. (1)

${\rm Aut}(L_{1})$* is a matrix Lie group.* 2. (2)

${\rm Aut}(L_{1})=\exp({\rm Der}(L_{1}))$, where $\exp(-)$ denotes the matrix exponential. 3. (3)

${\rm Aut}(L_{1})$* is a connected Lie group, i.e., path-connected as a topological space.* 4. (4)

${\rm Aut}(L_{1})$* is a soluble Lie group.*

Proof.

(1) We have seen that ${\rm Aut}(L_{1})$ is a matrix Lie group because it is closed in the general linear group ${\rm GL}(3,\mathbb{C})$, see Sagle-Walde [11, Proposition 7.1].

(2) Recall that the matrix exponential exp$:\mathfrak{gl}_{3}(\mathbb{C})\longrightarrow{\rm GL}(3,\mathbb{C})$ was given by

[TABLE]

for any $3\times 3$ complex matrix $X$. By Sagle-Walde [11, Proposition 7.3 (a)], we have $\exp(tX)\in{\rm Aut}(L_{1})$ for any $t\in\mathbb{R}$ and any derivation $X\in{\rm Der}(L_{1})$. Thus the exponential mapping $\exp:{\rm Der}(L_{1})\longrightarrow{\rm Aut}(L_{1})$ does make sense, i.e., the image $\exp({\rm Der}(L_{1}))\subseteq{\rm Aut}(L_{1})$ as a subset. To show $\exp({\rm Der}(L_{1}))={\rm Aut}(L_{1})$, we need only to show that

[TABLE]

which means that for any $\sigmaup=\begin{pmatrix}1&0&a\\ 0&1&-a\\ 0&0&b\end{pmatrix}\in{\rm Aut}(L_{1})$, it is sufficient to show that there exists some derivation $d\in{\rm Der}(L_{1})$ such that $\exp(d)=\sigmaup.$

Subcase 1: $b\neq 1$. Since nonzero number $b\neq 1$, it follows from elementary analysis that $e^{x}=b$ has a nonzero solution in $\mathbb{C}$, i.e., there exists a complex number $0\neq b_{0}\in\mathbb{C}$ such that $e^{b_{0}}=b.$ We define

[TABLE]

and consider the derivation

[TABLE]

We make the convention that $d^{0}:=I_{3}$, the identity matrix. By induction on $k$, we can show that $d^{k}=b_{0}^{k-1}\cdot d$ for all $k\in\mathbb{N}^{+}$. Thus

[TABLE]

Subcase 2: $b=1$. In this case we define

[TABLE]

Then $d$ is nilpotent since $d^{2}=0$. Thus

[TABLE]

The proof of the second assertion is completed.

(3) To see that ${\rm Aut}(L_{1})$ is connected, we first note that the Lie algebra of ${\rm Aut}(L_{1})$ is just ${\rm Der}(L_{1})$, see [11, Proposition 7.3 (b)]. Since ${\rm GL}(3,\mathbb{C})$ is a matrix Lie group with $\mathfrak{gl}_{3}(\mathbb{C})$ as its Lie algebra, ${\rm Aut}(L_{1})$ is an analytic subgroup of ${\rm GL}(3,\mathbb{C})$, according to [7, Definition 3.12], the definition of analytic subgroup. It follows from [7, Proposition 3.13] that ${\rm Aut}(L_{1})$ is path-connected, i.e., ${\rm Aut}(L_{1})$ is a connected Lie group.

(4) Since ${\rm Aut}(L_{1})$ is connected and its Lie algebra is soluble, it follows from Sagle-Walde [11, Theorem 10.9 (b)] that ${\rm Aut}(L_{1})$ is soluble. ∎

Remark 4.2**.**

We remark that there also exists an approach to compute the exponential of any matrix, using the SN-Decomposition, see Hall [7, Sections 2.2.1-2.2.3] for the details.

Similar arguments can be applied to the cases of $L_{2},B,$ and $C_{\alphaup}$. We will see below that the computation of ${\rm Aut}(A_{\alphaup})$ is more complicated. Here a “trick” coming from Linear Algebra will be useful.

Suppose $\sigmaup=(a_{ij})_{3\times 3}\in{\rm Aut}(A_{\alphaup})$ and $\sigmaup(x)=a_{11}x+a_{12}y+a_{13}z,\sigmaup(y)=a_{21}x+a_{22}y+a_{23}z$ and $\sigmaup(z)=a_{31}x+a_{32}y+a_{33}z$. Since $[x,y]=x$, $\sigmaup(x)=\sigmaup([x,y])=[\sigmaup(x),\sigmaup(y)]$. Note that the relations $[y,z]=z+\alphaup x$ and $[x,z]=x+y$. It follows that

[TABLE]

The algebraic independence of $x,y,z$ implies that

[TABLE]

where $A_{ij}$ denotes the minor determinant obtained by deleting the $i$-th row and $j$-th column in $\det(\sigmaup)$.

Similarly, it follows from $[\sigmaup(x),\sigmaup(z)]=\sigmaup(x+y)$ that

[TABLE]

We expand $\sigmaup([y,z])=\sigmaup(z+\alphaup x)$ to get

[TABLE]

We denote by $\sigmaup^{*}$ the matrix for which $\sigmaup\cdot\sigmaup^{*}=\det(\sigmaup)\cdot I_{3}=\sigmaup^{*}\cdot\sigmaup$. Then

[TABLE]

where $A_{13}=a_{31}+\alpha a_{11}-a_{32}-\alpha a_{12}-\alpha(a_{33}+\alpha a_{13})$ and $A_{23}=a_{11}+a_{21}-a_{12}-a_{22}-\alpha a_{13}-\alpha a_{23}$. This equality gives rise to 9 equations involving 9 unknowns: $\{a_{ij}\mid 1\leqslant i,j\leqslant 3\}$. It follows from these 9 equations that

[TABLE]

Note that $\det(\sigmaup)\neq 0$, so $a_{11},a_{22},a_{33}$ are not zero. A direct calculation shows that $a_{11}=a_{22}=a_{33}=1,a_{21}=a_{32},$ and $2a_{31}-a_{32}^{2}-a_{32}=0.$ Therefore,

[TABLE]

which is exactly what we want. Furthermore, from the generating relations of $A_{\alphaup}$ we observe that

[TABLE]

With a similar argument as Proposition 4.1 we obtain

Proposition 4.3**.**

1. (1)

${\rm Aut}(A_{\alphaup})$* is a connected, abelian matrix Lie group.* 2. (2)

${\rm Aut}(A_{\alphaup})=\exp({\rm Der}(A_{\alphaup}))$.

Proof.

(1) The proof is similar to Proposition 4.1. It is immediate to see that ${\rm Aut}(A_{\alphaup})$ is a connected matrix Lie group. For any elements $\sigmaup,\sigmaup^{\prime}\in{\rm Aut}(A_{\alphaup})$ we see that $\sigmaup\cdot\sigmaup^{\prime}=\sigmaup^{\prime}\cdot\sigmaup.$ Thus ${\rm Aut}(A_{\alphaup})$ is abelian.

(2) Since ${\rm Aut}(A_{\alphaup})\supseteq\exp({\rm Der}(A_{\alphaup}))$, we need only to show that ${\rm Aut}(A_{\alphaup})\subseteq\exp({\rm Der}(A_{\alphaup}))$. For any derivation

[TABLE]

it follows that

[TABLE]

and $d^{3}=0$, so $d$ is nilpotent. Thus

[TABLE]

which means that we can find a derivation $d\in{\rm Der}(A_{\alphaup})$ such that $\exp(d)=\sigmaup$ for any automorphism $\sigmaup\in{\rm Aut}(A_{\alphaup}).$ ∎

We summarize a characterization on automorphisms of 3-dimensional $\omegaup$-Lie algebras as following Table LABEL:3a.

By Propositions 4.1 (2) and 4.3 (2), one might ask whether $\exp({\rm Der}(\mathfrak{g}))={\rm Aut}(\mathfrak{g})$ always holds for all 3-dimensional $\omegaup$-Lie algebra $\mathfrak{g}$? Unfortunately, the answer to this question is “no” in general, as the following example shows.

Example 4.4**.**

Consider $\exp({\rm Der}(B))$ and ${\rm Aut}(B)$. For any $d\in{\rm Der}(B)$, we may take

[TABLE]

Then $d^{2}=0$ is nilpotent. Thus $\exp(d)=d^{0}+d^{1}/1!=I_{3}+d=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&a&1\end{pmatrix}.$ On the other hand,

[TABLE]

but $\sigmaup$ doesn’t belong to $\exp({\rm Der}(B))$. Hence $\exp({\rm Der}(\mathfrak{g}))\subset{\rm Aut}(\mathfrak{g})$ but $\exp({\rm Der}(\mathfrak{g}))\neq{\rm Aut}(\mathfrak{g})$. **

5. Derivations and Automorphisms in Dimension 4

Suppose $\mathfrak{g}$ is a 4-dimensional nontrivial $\omegaup$-Lie algebra over $\mathbb{C}$ and $\{x,y,z,e\}$ is a basis for $\mathfrak{g}.$ The characterizations of ${\rm Der}(\mathfrak{g})$ and ${\rm Aut}(\mathfrak{g})$ are summarized in the following Tables LABEL:4d and LABEL:4a respectively. Note that all parameters in Tables LABEL:4d and LABEL:4a belong to $\mathbb{C}$.

As in the case of 3-dimensional $\omegaup$-Lie algebras, direct computations show that

Proposition 5.1**.**

${\rm Der}_{\omegaup}(\mathfrak{g})={\rm Der}(\mathfrak{g})$* for any 4-dimensional nontrivial $\omegaup$-Lie algebra $\mathfrak{g}$, except for $\mathfrak{g}=L_{1,6}$ and $L_{1,8}$.*

Proof.

Here we only prove that ${\rm Der}_{\omegaup}(L_{1,6})\neq{\rm Der}(L_{1,6})$. Consider a derivation

[TABLE]

with $a\neq 0.$ Then

[TABLE]

Thus $D$ is not an $\omegaup$-derivation.

Similar arguments show that ${\rm Der}_{\omegaup}(L_{1,8})\neq{\rm Der}(L_{1,8})$. ∎

Remark 5.2**.**

It seems that our method in previous sections might be applied to discuss the Lie group structures of the automorphism groups in Table LABEL:4a. **

6. Representations of $\omegaup$-Lie Algebras

Definition 6.1**.**

Let $\mathfrak{g}$ be a finite-dimensional $\omegaup$-Lie algebra over $\mathbb{C}$ and $V$ a finite-dimensional vector space over $\mathbb{C}$. We say that $V$ is a $\mathfrak{g}$-module if there exists a bilinear map $\mathfrak{g}\times V\longrightarrow V,(x,v)\mapsto x\cdot v$ such that

[TABLE]

for all $x,y\in\mathfrak{g}$ and $v\in V.$

For example, let $\mathfrak{g}=B$ denote the complex 3-dimensional $\omegaup$-Lie algebra in Theorem 2.1 and $V=\mathbb{C}$. By Propositions 6.3 and 6.5 below, we see that $\mathbb{C}$ is an one-dimensional $B$-module.

Note that if $x\cdot v=0$ for all $x\in\mathfrak{g}$ and $v\in V$, then it does not make $V$ to be a $\mathfrak{g}$-module, unless $\mathfrak{g}$ is a Lie algebra.

Proposition 6.2**.**

Let $\mathfrak{g}$ be a finite-dimensional $\omegaup$-Lie algebra and $V$ a $\mathfrak{g}$-module. Consider the direct sum $\mathfrak{g}\oplus V$ of vector spaces. Define a skew-symmetric bilinear bracket operation $[-,-]:(\mathfrak{g}\oplus V)\times(\mathfrak{g}\oplus V)\longrightarrow\mathfrak{g}\oplus V$ by

[TABLE]

and define a bilinear form $\Omega:(\mathfrak{g}\oplus V)\times(\mathfrak{g}\oplus V)\longrightarrow\mathbb{C}$ by

[TABLE]

for any $x,y\in\mathfrak{g}$ and $u,v\in V$. Then $(\mathfrak{g}\oplus V,[-,-],\Omega)$ is an $\Omega$-Lie algebra, which is called the semi-direct product of $\mathfrak{g}$ and $V$.

Proof.

For any $x,y,z\in\mathfrak{g}$ and $v,u,w\in V$ we have

[TABLE]

Thus

[TABLE]

This shows that $(\mathfrak{g}\oplus V,[-,-],\Omega)$ is an $\Omega$-Lie algebra. ∎

Proposition 6.3**.**

Let $\mathfrak{g}$ be a finite-dimensional nontrivial $\omegaup$-Lie algebra over $\mathbb{C}$. Then $\mathbb{C}$ is a 1-dimensional $\mathfrak{g}$-module if and only if there exists a linear functional $\tau\in\mathfrak{g}^{*}$ such that $\omegaup(x,y)=\tau([x,y])$ for all $x,y\in\mathfrak{g}$.

Proof.

$(\Longrightarrow)$ Suppose $c_{0}\in\mathbb{C}$ is a nonzero complex number. Since $\mathbb{C}$ is a 1-dimensional $\mathfrak{g}$-module, then for any $x\in\mathfrak{g}$, we can assume that $x\cdot c_{0}=\tau(x)\cdot c_{0}$ for some function $\tau:\mathfrak{g}\longrightarrow\mathbb{C}$. Since the action of $\mathfrak{g}$ on $\mathbb{C}$ is bilinear, $\tau\in\mathfrak{g}^{*}$ is a linear functional. Thus for any $y\in\mathfrak{g}$, we have

[TABLE]

Hence, $\omegaup(x,y)=\tau([x,y])$ for all $x,y\in\mathfrak{g}$.

$(\Longleftarrow)$ Define a map $\rhoup:\mathfrak{g}\times\mathbb{C}\longrightarrow\mathbb{C},(x,c)\mapsto x\cdot c$ by $x\cdot c:=\tau(x)\cdot c$ for any $x\in\mathfrak{g}$ and $c\in\mathbb{C}$. Clearly $\rhoup$ is bilinear. Moreover,

[TABLE]

which means that $\rhoup$ makes $\mathbb{C}$ to be a $\mathfrak{g}$-module. ∎

Remark 6.4**.**

This result is essentially due to Zusmanovich [13, Section 2]. An $\omegaup$-Lie algebra having a 1-dimensional module is called multiplicative. **

Theorem 6.5**.**

All 3-dimensional nontrivial $\omegaup$-Lie algebras over $\mathbb{C}$ are multiplicative.

Proof.

According to the classification of 3-dimensional $\omegaup$-Lie algebras (Theorem 2.1), we need only to show that every $\omegaup$-Lie algebra $\mathfrak{g}$ in the list is multiplicative. By Proposition 6.3, it suffices to find a linear functional $\tau\in\mathfrak{g}^{*}$ such that $\omegaup(x,y)=\tau([x,y])$ for all $x,y\in\mathfrak{g}$. In fact, suppose $\{x^{*},y^{*},z^{*}\}$ is a dual basis for $\mathfrak{g}^{*}$. For $L_{1}$ and $L_{2}$, such $\tau$ could be $y^{*}$. For $A_{\alphaup}$, we can take $\tau=-z^{*}$. We can take $\tau=2x^{*}$ for $B$, and $\tau=(1+\alphaup)x^{*}$ for $C_{\alphaup}$. ∎

Example 6.6**.**

The 4-dimensional $\omegaup$-Lie algebra $L_{1,1}$ is not multiplicative. Assume by the way of contradiction that $L_{1,1}$ is multiplicative, then it follows from Proposition 6.3 that there is a linear functional $\tau\in L_{1,1}^{*}$ such that $\omegaup(x,y)=\tau([x,y])$ for all $x,y\in L_{1,1}$. Suppose $\{x,y,z,e\}$ is a basis for $L_{1,1}$. By the generating relations in $L_{1,1}$ (see Table LABEL:4d previously), we have $\tau(y)=\tau([x,y])=\omegaup(x,y)=1$ and $-\tau(y)=\tau(-y)=\tau([e,y])=\omegaup(e,y)=0$, contradiction. **

Remark 6.7**.**

We can use the method in Proposition 6.5 and Example 6.6 to classify all 4-dimensional multiplicative $\omegaup$-Lie algebras over $\mathbb{C}$. **

Remark 6.8** (Zusmanovich [13]).**

By the definition of $\mathfrak{g}$-module, one can check that an $\omegaup$-Lie algebra $\mathfrak{g}$ is a module over itself under the adjoint action if and only if $\mathfrak{g}$ is trivial, i.e., $\mathfrak{g}$ is a Lie algebra.

7. Irreducible Representations of $C_{\alphaup}$

In this section we study the irreducible representations of $C_{\alphaup}$. Theorem 7.1 is our main result in this section, showing that any irreducible $C_{\alphaup}$-module is 1-dimensional, whenever $0,-1\neq\alphaup\in\mathbb{C}.$

Theorem 7.1**.**

Any irreducible $C_{\alphaup}$-module is 1-dimensional, where $\alphaup\neq 0,-1$.

Proof.

Recall that $C_{\alphaup}:[x,y]=y,[x,z]=\alphaup z,[y,z]=x$ with $\omegaup(x,y)=\omegaup(x,z)=0,\omegaup(y,z)=1+\alphaup$, where $0,-1\neq\alphaup\in\mathbb{C}.$ Suppose that $V$ is a finite-dimensional irreducible $C_{\alphaup}$-module. We may view every element in $C_{\alphaup}$ as a linear map from $V$ to itself. Since the linear map $x$ is over $\mathbb{C}$, it must have an eigenvector $v_{0}\neq 0$, say. Suppose $\eta$ is the corresponding eigenvalue, then $x\cdot v_{0}=\eta v_{0}$.

It follows from Equation (6.1) that $x\cdot(y\cdot v_{0})=[x,y]\cdot v_{0}+y\cdot(x\cdot v_{0})-\omegaup(x,y)v_{0}=y\cdot v_{0}+y\cdot(\eta v_{0})=(\eta+1)(y\cdot v_{0})$. Thus $y\cdot v_{0}$ is also an eigenvector of $x$ with eigenvalue $\eta+1$. Furthermore, we observe that

[TABLE]

In fact,

[TABLE]

Thus Equation (7.1) holds. This means that the vectors $\{y^{k}\cdot v_{0}\mid k=0,1,2,3,\dots\}$ are either zero or eigenvectors of $x$. Note that any two eigenvalues in $\{\eta+k\mid k=0,1,2,3,\dots\}$ are distinct, and since the eigenvectors corresponding to different eigenvalues are linear independent, we may find a minimal integer $k_{0}$ such that $y^{k_{0}}\cdot v_{0}\neq 0$ and $y^{k_{0}+1}\cdot v_{0}=0$. Let $v_{1}:=y^{k_{0}}\cdot v_{0}$ and $\eta_{1}:=\eta+k_{0}$, we have seen that

[TABLE]

In order to analyze $x\cdot(z^{r}\cdot v_{1})$, we use the induction on $r$ to see that

[TABLE]

Namely, $\{z^{r}\cdot v_{1}\mid r=0,1,2,3,\dots\}$ are also eigenvectors of $x$. Since $\alphaup\neq 0$, any two eigenvalues in $\{r\alphaup+\eta_{1}\mid r=0,1,2,3,\dots\}$ are distinct. Thus we may find a minimal integer $n$ such that

[TABLE]

Let $V_{0}$ be the subspace generated by $\{z^{j}\cdot v_{1}\mid 0\leqslant j\leqslant n\}$, then $V_{0}\neq\{0\}$. To see that $V_{0}$ is a $C_{\alphaup}$-submodule of $V$, it suffices to show that $V_{0}$ is stable under the actions of $x,y,$ and $z$. Clearly, $x(V_{0})\subseteq V_{0}$ and $z(V_{0})\subseteq V_{0}$. Moreover, we claim that

[TABLE]

for all $j>0$. In fact, the induction on $j$ shows that

[TABLE]

Thus, this claim holds. Equation (7.2), together with $y\cdot v_{1}=0$ implies that $y(V_{0})\subseteq V_{0}$. Hence, $V_{0}$ is a nonzero $C_{\alphaup}$-submodule of $V$. Since $V$ is irreducible, we have $V=V_{0}$ and $\dim(V)=\dim(V_{0})=n+1$.

Now we want to seek when the irreducible representation $V$ exists. We are going to show that there are no irreducible representations in dimension $>1$. We still follow the previous notations, and suppose $V=\langle v_{1},z^{1}\cdot v_{1},\dots,z^{n}\cdot v_{1}\rangle$ is an irreducible $C_{\alphaup}$-module of dimension $n+1$.

Taking $j=n+1$ in Equation (7.2), we obtain

[TABLE]

which, together with the fact that $n\in\mathbb{N}$ implies

[TABLE]

To prove that $n=0$, we assume by the way of contradiction that $n>0.$ Then $z\cdot v_{1}\neq 0$ in $V$, and

[TABLE]

Since $[x,y]=y$ in $C_{\alphaup}$, then $[x,y]\cdot v=y\cdot v$ for any element in $V$. Thus

[TABLE]

gives rise to $\eta_{1}-1-\alphaup=-\alphaup(\eta_{1}-1-\alphaup)$, i.e.,

[TABLE]

Since $\alphaup\neq-1$, it follows that

[TABLE]

which, combining with Equation (7.3) obtains

[TABLE]

Recall that $\alphaup\neq 0$. Therefore, $n=0$. The proof is completed. ∎

Remark 7.2**.**

If we extend the parameter $\alphaup$ in $C_{\alphaup}$ to contain the case of $\alphaup=-1$, then it is easy to see that $C_{-1}=\mathfrak{sl}_{2}(\mathbb{C})$, the special linear Lie algebra, which is trivial as an $\omegaup$-Lie algebra. It is well-known that up to equivalence, $C_{-1}$ has one unique irreducible representation in any dimension $n<\infty$, see for example, Fulton-Harris [4, Section 11.1]. This result, comparing with Theorem 7.1 shows that representation theory of nontrivial $\omegaup$-Lie algebras might be very “degenerate” . **

Finally, we give some remarks for more potential research in the theory of $\omegaup$-Lie algebras.

Remark 7.3**.**

Note that our many results might be valid over other fields of characteristic zero, especially over the field $\mathbb{R}$ of real numbers. Real $\omegaup$-Lie algebras should deserve more attention because the motivation and the first application of introducing $\omegaup$-Lie algebras were presented over $\mathbb{R}$ in mathematical physics, see Nurowski [8]. Based on the links between physics and low-dimensional nonassociative algebras have been found, for example, see Zhang-Bai [12] and reference therein, we also expect that our computations might have more applications in mathematical physics.

In terms of pure mathematics, there are several possible directions concerning the theory of $\omegaup$-Lie algebras that might be worth investigating. The first one was to develop a theory of infinite dimensional $\omegaup$-Lie algebras. In the last section of Chen-Zhang [3], we have already constructed an example of nontrivial infinite dimensional $\omegaup$-Lie algebras. But it seems that there are no more results even examples appeared so far. The second one was the study of solvable and nilpotent $\omegaup$-Lie algebras. In Chen-Liu-Zhang [2] and [3], we have completed the classification of complex simple finite-dimensional $\omegaup$-Lie algebras. It is natural to consider the structures of solvable and nilpotent $\omegaup$-Lie algebras. In this paper we did not discuss the solvability and nilpotence of $\omegaup$-Lie algebras but according to the experience we gained, the present calculations probably provide a concrete guide for more research in this direction. The third direction was the restricted version of $\omegaup$-Lie algebras, i.e., $\omegaup$-Lie algebra over a field of characteristic $p>0$ satisfying some restricted conditions in sense of Jacobson [5, Chapter V, Section 7]. However, we do not know how to give a correct definition of restricted $\omegaup$-Lie algebras and whether there is a big difference between modular $\omegaup$-Lie algebras and modular Lie algebras. **

Acknowledgments

This research was partially supported by NNSF of China (No. 11301061, 11401087) and Ministry of Educations of China (No. 201510200029). We would like to thank the referee for careful reading and useful comments.