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# A classification of low dimensional multiplicative Hom-Lie superalgebras

Chunyue Wang
• Department of Mathematics, Jilin University, 130012, Changchun, China and School of Applied Sciences, Jilin Engineering Normal University, Changchun, 130052, China
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Qingcheng Zhang
/ Zhu Wei
Published Online: 2016-09-20 | DOI: https://doi.org/10.1515/math-2016-0056

## Abstract

We study a twisted generalization of Lie superalgebras, called Hom-Lie superalgebras. It is obtained by twisting the graded Jacobi identity by an even linear map. We give a complete classification of the complex multiplicative Hom-Lie superalgebras of low dimensions.

MSC 2010: 17B99; 55U15

## 1 Introduction

In recent years, Hom-Lie algebras and other Hom-algebras have been extensively studied. They first arised in quasideformation of Lie algebras of vector fields. The Hom-Lie and quasi-Hom-Lie structures are obtained from twisted derivations of discrete modifications of vector fields. For those algebras, the Jacobi condition is twisted [1]. Homalgebras were studied in [210]. Furthermore, Hom-Lie algebras on some q-deformations of Witt and Virasoro algebras were considered in [1113].

Hom-Lie algebras were generalized to Hom-Lie superalgebras by Ammar and Makhlouf [14] and to Hom-Lie color algebras by Yuan [15]. In particular, Cao and Luo [16] proved that there were only the trivial Hom-Lie superalgebras on the finite-dimensional simple Lie superalgebras. The representation theory and the cohomology theory of Hom-Lie superalgebras were studied in [17]. In [18], some infinite-dimensional Hom-Lie superalgebras were constructed, induced by affinizations of the Hom-Balinskii-Novikov superalgebras and Hom-Novikov superalgebras. Furthermore, the central extensions of the infinite-dimensional Hom-Lie superalgebras were studied.

During the past few years, the theories of Hom-Lie superalgebras make great advances. But, only few people study the classification of low dimensional Hom-Lie superalgebras. There are few results about it. Zhang [19] and Li [20] studied the classifications of low dimensional Lie superalgebras and low dimensional multiplicative Hom-Lie algebras, respectively. In this paper, we will discuss the classification of low dimensional multiplicative Hom-Lie superalgebras. For the pure even Hom-Lie superalgebras, they are Hom-Lie algebras. Their classifications have been obtained. It can be found in [20], so we only discuss the odd part, which is nontrivial. We know that derived algebras of non-abelian Hom-Lie superalgebras are non-zero. We note that the properties of the derived algebras can be hold up to isomorphisms. Thus, it is crucial to determine the multiplication tables by using the dimension and properties of derived algebras. Finally, we can obtain the classifications of the low dimensional multiplicative Hom-Lie superalgebras.

The paper is organized as follows: In Section 2, we give some notations and some basic properties of Hom-Lie superalgebras. We give a method to determine whether two different multiplicative Hom-Lie superalgebras are isomorphism or not, and then we obtain a complete classification of multiplicative Hom-Lie superalgebras up to isomorphism. In Section 3 and 4, we get the classifications of the low dimensional multiplicative Hom-Lie superalgebras.

In this paper, we discuss all algebras and vector spaces over a complex field ℂ. The parity of the homogeneous element x is denoted by |x|. hg(L) denotes the set of all homogeneous elements of Hom-Lie superalgebras (L, [, ], σ). ${L}_{\overline{0}}$ and ${L}_{\overline{1}}$ denote the even part and the odd part of Hom-Lie superalgebras (L, [, ], σ), respectively.

## 2 Preliminaries

In this section, we recall the notions of Hom-Lie superalgebras. Then we can get some basic properties of Hom-Lie superalgebras. We aim to give a method to determine whether two different multiplicative Hom-Lie superalgebras are isomorphism or not.

A Hom-Lie superalgebra is a triple (L, [, ], σ) consisting of a super space L, an even bilinear [, ] : L × LL and an even linear map σ : LL satisfying $[x, y]=−(−1)|x||y|[y, x],$(1) $(−1)|x||z|[σ(x), [y, z]]+(−1)|z||y|[σ(z), [x, y]]+(−1)|y||x|[σ(y), [z, x]]=0.$(2) for all homogeneous elements x, y, z in L.

Let (L, [, ], σ) and (L′, [, ]′, σ′) be two Hom-Lie superalgebras. An even linear map ϕ : LL′ is said to be a homomorphism of Hom-Lie superalgebras if the following conditions are satisfied: $ϕ([x, y])=[ϕ(x), ϕ(y)]′,$(3) $ϕoσ=σ′oϕ,$(4) for any x, yhg(L). If ϕ is a bijection, then ϕ is called an isomorphism of Hom-Lie superalgebras.

A Hom-Lie superalgebra (L, [, ], σ) is called a multiplicative Hom-Lie superalgebra if the even linear map σ satisfies σ[x, y] = [σ(x, σ(y)] for any x, y ∈ hg(L). A Hom-Lie superalgebra (L, [, ], σ) is called a classical Hom-Lie superalgebra if (L, [, ], σ) is a Lie superalgebra.

In the following paper, Hom-Lie superalgebras always denote to be the multiplicative Hom-Lie superalgebras.

Let (L, [, ], σ) be a Hom-Lie superalgebra and σ be an even invertible linear map. Define [, ]1 : L × LL by [x, y]1 = [σ-1(x), σ-1(y)], for any x, y ∈ hg(L). Then (L, [, ]1) is a Lie superalgebra and σ is a Lie superalgebra endomorphism.

It is clear that [, ]1 is an even bilinear map and is super skew-symmetric. Since $σ−1[x, y]=σ−1(σ[σ−1(x), σ−1(y)])=[σ−1(x), σ−1(y)]=[x, y]1,$ we have, for any x, y, z ∈ hg(L), $(−1)|x||z|[[x, y]1, z]1+(−1)|x||y|[[y, z]1, x]1+(−1)|y||z|[[z, x]1, y]1(−1)|x||z|[σ−1[x, y], z]1+(−1)|x||y|[σ−1[y, z], x]1+(−1)|y||z|[σ−1[z, x], y]1+(−1)|x||z|σ−1[σ−1[x, y], z]+(−1)|x||y|σ−1[σ−1[y, z], x]+(−1)|y||z|σ−1[σ−1[z, x], y]=σ−2((−1)|x||z|[[x, y], σ(z)]+(−1)|x||y|[[y, z], σ(x)]+(−1)|y||z|[[z, x], σ(y)])= 0.$ Obviously, σ is an even endomorphism of the Lie superalgebra (L, [, ]1).

Let L be a linear super space and [, ] : L × LL be an even bilinear map. Mor(L, [, ]) denotes the set of all even linear maps σ such that (L, [, ], σ) is a multiplicative Hom-Lie superalgebra.

Let L be a linear super space, [, ]1 and [, ]2 be two even bilinear maps on L. If there exists an even invertible linear map ϕ of L satisfying $ϕ[x, y]1=[ϕ(x), ϕ(y)]2,$ then $Mor(L, [, ]2)={ϕσϕ−1|σ∈Mor(L, [, ]1)}.$

Since ϕ is invertible, we have, for any σ1 Mor(L, [, ]1), x, y, z ∈ hg(L) $ϕσ1ϕ−1[x, y]2=ϕσ1[ϕ−1(x), ϕ−1(y)]1=[ϕσ1ϕ−1(x), ϕσ1ϕ−1(y)]2,$ and $(−1)|x||z|[[x,y]2,ϕσ1ϕ−1(z)]2+(−1)|x||y|[[y,z]2,ϕσ1ϕ−1(x)]2+(−1)|y||z|[[z,x]2,ϕσ1ϕ−1(y)]2=(−1)|x||z|[ϕ[ϕ−1(x),ϕ−1(y)]1,ϕσ1ϕ−1(z)]2+(−1)|x||y|[ϕ[ϕ−1(y),ϕ−1(z)]1,ϕσ1ϕ−1(x)]2+(−1)|y||z|[ϕ[ϕ−1(z),ϕ−1(x)]1,ϕσ1ϕ−1(y)]2=(−1)|x||z|ϕ[[ϕ−1(x),ϕ−1(y)]1,σ1ϕ−1(z)]1+(−1)|x||y|ϕ[[ϕ−1(y),ϕ−1(z)]1,σ1ϕ−1(x)]1+(−1)|y||z|ϕ[[ϕ−1(z),ϕ−1(x)]1,σ1ϕ−1(y)]1= 0.$ So $\varphi {\sigma }_{1}{\varphi }^{-1}\in Mor\left(L,\left[,{\right]}_{2}\right)$.

Conversely, for any σ2Mor(L, [, ]2), similar to above, we have ϕ-1σ2ϕMor(L, [, ]1). Set ϕ-1σ2ϕ = σ1. Then σ2 = ϕσ1ϕ-1. Thus, we have $Mor\left(L,\left[,{\right]}_{2}\right)=\left\{\varphi \sigma {\varphi }^{-1}|\sigma \in Mor\left(L,{\left[,\right]}_{1}\right)\right\}$.

By Lemma 2.5, we know that there exists a one-to-one correspondence between the multiplicative Hom-Lie superalgebras (L, [, ]1, σ1) and the multiplicative Hom-Lie superalgebras (L, [, ]2, σ2). In order to simplify the classification of multiplicative Hom-Lie superalgebras, we can simplify [, ]1 to [, ]2 by an even invertible linear map ϕ. Then we determine the multiplicative Hom-Lie superalgebra structure by means of [, ]2.

Let (L, [, ]1, σ1) and (L, [, ]2, σ2) be two Hom-Lie superalgebras with the same underlying vector super space L. (L, [, ]1, σ1) is isomorphic to (L, [, ]2, σ2) if and only if there exists an invertible matrix T such that T[ei, ej]1 = [T(ei), T(ej)]2 and A2 = TA1T-1, where A1, A2 and T denote matrices corresponding to σ1, σ2 and the homomorphism ϕ : LL with respect to the basis {e1, e2, ..., en}, respectively.

We denote the derived algebra of Hom-Lie superalgebras (L, [, ], σ) by L′ = [L, L]. It consists of all linear combinations of commutators [x, y] for x, yhg(L). The Hom-Lie superalgebra (L, [, ], σ) is called an abelian Hom-Lie superalgebra if [L, L] = 0.

Let (L1, [, ]1, σ1) and (L2, [, ]2, σ2) be two abelian Hom-Lie superalgebras. If (L1, [, ]1, σ1) is isomorphic to (L2, [, ]2, σ2), then ${\mathrm{dim}}_{ℂ}{L}_{1\overline{0}}={\mathrm{dim}}_{ℂ}{L}_{2\overline{0}}$ and ${\mathrm{dim}}_{ℂ}{L}_{1\overline{1}}={\mathrm{dim}}_{ℂ}{L}_{2\overline{1}}$.

An isomorphism from (L1, [, ]1, σ1) to (L2, [, ]2, σ2) is an isomorphism of their underlying vector super space. So there exists an even invertible linear map ϕ : ${L}_{1\overline{0}}\to {L}_{2\overline{0}}\to {L}_{1\overline{1}}\to {L}_{2\overline{1}}$.

Let (L, [, ]1, σ1) and (L, [, ]2, σ2) be two abelian Hom-Lie superalgebras with the same underlying vector super space L. (L, [, ]1, σ1) is isomorphic to (L, [, ]2, σ2) if and only if the matrices corresponding to σ1 and σ2 have the same elementary divisors.

It is straightforward.

Let (L1, [, ]1, σ1) and (L2, [, ]2, σ2) be two Hom-Lie superalgebras. If $\varphi :{L}_{1}\to {L}_{2}$ is a surjective homomorphism of Hom-Lie superalgebras, then $\varphi \left({{L}^{\prime }}_{1}\right)={{L}^{\prime }}_{2}$.

For any x, yhg(L1), we have $\varphi \left[x,y{\right]}_{1}=\left[\varphi \left(x\right),\varphi \left(y\right){\right]}_{2}$. So by linearity $\varphi \left({{L}^{\prime }}_{1}\right)\subseteq {{L}^{\prime }}_{2}$. Now L2 is spanned by commutators [y1, y2] with y1, y2hg(L2). Set ϕ(x1) = y1, ϕ(x2) = y2 It follows that ϕ[x1, x2]1 = ϕ[y1, y2]2.

## 3 A classification of 2-dimensional Hom-Lie superalgebras

In this section, we are going to give the classification of 2-dimensional Hom-Lie superalgebras.

Every 2-dimensional Hom-Lie superalgebra is isomorphic to one of the following nonisomorphic Hom-Lie superalgebra: each algebra is denoted by ${L}_{i,j}^{k}$, where i is the dimension of ${L}_{\overline{0}}$, j is is the dimension of ${L}_{\overline{1}}$, k is the number.

(1) ${L}_{0,2}^{1}$ is an abelian Hom-Lie superalgebra.

(2) ${L}_{1,1}^{2}$ is an abelian Hom-Lie superalgebra.

(3) ${L}_{1,1}^{3,b}:\left[{e}_{0},{e}_{1}\right]={e}_{1},\left[{e}_{1},{e}_{1}\right]=0,\sigma =\left(\begin{array}{cc}1& 0\\ 0& b\end{array}\right)$.${L}_{1,1}^{3,b}$ is isomorphic to ${L}_{1,1}^{3,{b}^{\prime }}$ if and only if b = b′.

(4) ${L}_{1,1}^{4,a}:\left[{e}_{0},{e}_{1}\right]={e}_{1},\left[{e}_{1},{e}_{1}\right]=0,\sigma =\left(\begin{array}{cc}a& 0\\ 0& 0\end{array}\right)\left(a\ne 0,1\right)$. ${L}_{1,1}^{4,a}$ is isomorphic to ${L}_{1,1}^{4,{a}^{\prime }}$ if and only if a = a′.

(5) ${L}_{1,1}^{5,b}:\left[{e}_{0},{e}_{1}\right]=0,\left[{e}_{1},{e}_{1}\right]={e}_{0},\sigma =\left(\begin{array}{cc}{b}^{2}& 0\\ 0& 0\end{array}\right)\left(b\ne 0\right)$. ${L}_{1,1}^{5,b}$ is isomorphic to ${L}_{1,1}^{5,{b}^{\prime }}$ if and only if b = b′.

In order to prove the above theorem, we give some lemmas.

Let (L, [, ]1, σ) be a 2-dimensional Hom-Lie superalgebra and {e0, e1} be a basis of (L, [, ], σ).

If e0, e1${L}_{\overline{1}}$ and ${L}_{\overline{0}}$ = 0, then (L, [, ], σ) is an abelian Hom-Lie superalgebra.

If e0${L}_{\overline{0}}$, e1${L}_{\overline{1}}$ and [L, L]=0, then (L, [, ], σ) is also an abelian Hom-Lie superalgebra.

In the following, we will discuss the situation: ${e}_{0}\in {L}_{\overline{0}},{e}_{1}\in {L}_{\overline{1}}$ and [L, L] ≠ 0. Obviously, (L, [, ], σ) is a non-abelian Hom-Lie superalgebra.

Let $\left[{e}_{0},{e}_{1}\right]=\alpha {e}_{1},\left[{e}_{1},{e}_{1}\right]=\beta {e}_{0},{\alpha }^{2}+{\beta }^{2}\ne 0$.

If α ≠ 0, β = 0, by Lemma 2.5, then there exists an even invertible linear map$\varphi =\left(\begin{array}{cc}\alpha & 0\\ 0& \frac{1}{\alpha }\end{array}\right)$ such that the bracket can be simplified as $[e0, e1]=e1, [e1, e1]=0.$(5)

Similarly, if α = 0, β ≠ 0, the bracket can be simplified as $[e0, e1]=0, [e1, e1]=e0.$(6)

If α ≠ 0, β ≠ 0, the bracket can be simplified as $[e0, e1]=e1, [e1, e1]=e0.$(7)

However, there does not exist an even invertible linear map such that formulas (5), (6) and (7) are equivalent.

For the bracket (5), Hom-Lie superalgebras are (L, [, ], σi)(i = 1, 2) up to isomorphism, where ${\sigma }_{{}^{1}}=\left(\begin{array}{cc}1& 0\\ 0& b\end{array}\right),{\sigma }_{2}\left(\begin{array}{cc}a& 0\\ 0& 0\end{array}\right)\left(a\ne 0,1\right)$.

Let σ : LL be an even linear map of Hom-Lie superalgebras and σ(e0) = ae0, σ(e1) = be1. The Hom-Jacobi identity is satisfied for any a, b ∈ ℂ. Since σ is multiplicative, we have a = 1 or b = 0.

For the bracket (6), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where $\sigma =\left(\begin{array}{cc}{b}^{2}& 0\\ 0& b\end{array}\right)\left(b\ne 0\right)$.

It is similar to Lemma 3.2.

For the bracket (7), there is not a non-zero even linear map σ such that it is multiplicative and satisfies Hom-Jacobi identity.

From Lemmas 3.2 and 3.3, we obtain Theorem 3.1.

(1) All 2-dimensional Hom-Lie superalgebras are classical Lie superalgebras.

(2) From Lemma 2.8, Hom-Lie superalgebras ${L}_{0,2}^{1}$ and ${L}_{1,1}^{2}$ are not isomorphism.

## 4 A classification of 3-dimensional Hom-Lie superalgebras

By Lemma 2.10, we note that it can keep the structures of the derived algebras with the isomorphism. Thus, we can use the properties of the derived algebras to determine the structures of 3-dimensional Hom-Lie superalgebras.

Every 3-dimensional Hom-Lie superalgebra is isomorphic to one of the following non-isomorphic Hom-Lie superalgebra: each algebra is denoted by ${L}_{i,j}^{k}$, where i is the dimension of ${L}_{\overline{0}}$, j is is the dimension of ${L}_{\overline{1}}$, k is the number.

(1) ${L}_{0,3}^{1}$ is an abelian Hom-Lie superalgebra.

(2) ${L}_{2,1}^{2}$ is an abelian Hom-Lie superalgebra.

(3) ${L}_{2,1}^{3,a,b,c}:\left[{e}_{1},{e}_{2}\right]=0,\left[{e}_{1},{e}_{3}\right]=0,\left[{e}_{2},{e}_{3}\right]=0,\left[{e}_{3},{e}_{3}\right]={e}_{1},\sigma =\left(\begin{array}{ccc}{c}^{2}& a& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right)\left({a}^{2}+{b}^{2}+{c}^{2}\ne 0\right)$. ${L}_{2,1}^{3,a,b,c}$ is isomorphic to ${L}_{2,1}^{3,{a}^{\prime },{b}^{\prime },{c}^{\prime }}$ if and only if b = b′, c = cand a = x(c2 - b) + ya′(y ≠ 0).

(4) ${L}_{2,1}^{4,a,b,c}:\left[{e}_{1},{e}_{2}\right]={e}_{1},\left[{e}_{1},{e}_{3}\right]=0,\left[{e}_{2},{e}_{3}\right]=0,\left[{e}_{3},{e}_{3}\right]=0,\sigma =\left(\begin{array}{ccc}0& a& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right)\left({a}^{2}+{b}^{2}+{c}^{2}\ne 0\right)$. ${L}_{2,1}^{4,a,b,c}$ is isomorphic to ${L}_{2,1}^{4,{a}^{\prime },{b}^{\prime },{c}^{\prime }}$ if and only if b = b′, c = c′ and a′ = xa + yb(x ≠ 0).

(5) ${L}_{2,1}^{5,a,b,c}:\left[{e}_{1},{e}_{2}\right]={e}_{1},\left[{e}_{1},{e}_{3}\right]=0,\left[{e}_{2},{e}_{3}\right]=0,\left[{e}_{3},{e}_{3}\right]=0,\sigma =\left(\begin{array}{ccc}{a}_{1}& {a}_{2}& 0\\ 0& 1& 0\\ 0& 0& b\end{array}\right)\left({a}_{1}\ne 0\right)$. ${L}_{2,1}^{5,{a}_{1},{a}_{2},b}$ is isomorphic to ${L}_{2,1}^{5,{{a}^{\prime }}_{1},{{a}^{\prime }}_{2},{b}^{\prime }}$ if and only if b = b′, a1 = a1 and a2x + y = ya1 + a2(x ≠ 0).

(6) ${L}_{2,1}^{6}:\left[{e}_{1},{e}_{2}\right]={e}_{1},\left[{e}_{1},{e}_{3}\right]=0,\left[{e}_{2},{e}_{3}\right]=0,\left[{e}_{3},{e}_{3}\right]={e}_{1},\sigma =\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$

(7) ${L}_{{}^{2,1}}^{7,a,b,c}:\left[{e}_{1},{e}_{2}\right]=0\left[{e}_{1},{e}_{3}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}{\text{]=e}}_{3},\left[{e}_{3},{e}_{3}\right]=0,\sigma =\left(\begin{array}{ccc}a& b& 0\\ 0& c& 0\\ 0& 0& 0\end{array}\right)\text{\hspace{0.17em}(}{a}^{2}\text{+}{b}^{2}\text{+}{c}^{2}\ne \text{0)}\text{.\hspace{0.17em}}{L}_{{}^{2,1}}^{7,a,b,c}$ is isomorphic to $\text{\hspace{0.17em}}{L}_{{}^{2,1}}^{7,{a}^{\prime },{b}^{\prime },{c}^{\prime }}$ if and only if a = a, c = c and b = xb + yc – yc (x ≠ 0).

(8) ${L}_{{}^{2,1}}^{8,a,b,c}:\left[{e}_{1},{e}_{2}\right]=0,\left[{e}_{1},{e}_{3}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}{\text{]=e}}_{3},\left[{e}_{3},{e}_{3}\right]=0,\sigma =\left(\begin{array}{ccc}a& b& 0\\ 0& 1& 0\\ 0& 0& c\end{array}\right)\text{\hspace{0.17em}(}c\ne \text{0)}\text{.\hspace{0.17em}}{L}_{{}^{2,1}}^{8,a,b,c}$ isomorphic to $\text{\hspace{0.17em}}{L}_{{}^{2,1}}^{8,{a}^{\prime },{b}^{\prime },{c}^{\prime }}$ if and only if a = a, c = c and b = xb - ya + yc(x ≠ 0).

(9) ${L}_{{}^{2,1}}^{9,a}:\left[{e}_{1},{e}_{2}\right]={e}_{1},\left[{e}_{1},{e}_{3}\right]=0,\left[{\text{e}}_{2},{\text{e}}_{3}\right]\text{=}0,\left[{e}_{3},{e}_{3}\right]={e}_{2},\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& {a}^{2}& 0\\ 0& 0& a\end{array}\right)\left(a\ne \text{0}\right).\text{\hspace{0.17em}}{L}_{{}^{2,1}}^{9,a}$ is isomorphic to ${L}_{{}^{2,1}}^{9,{a}^{\prime }}$ if and only if a = a.

(10) ${L}_{{}^{2,1}}^{10,a}:\left[{e}_{1},{e}_{2}\right]={e}_{1},\left[{e}_{1},{e}_{3}\right]={e}_{3},{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}0,\left[{e}_{3},{e}_{3}\right]=0,\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& a\end{array}\right)\text{\hspace{0.17em}}\text{.}$

(11) ${L}_{{}^{2,1}}^{11,a}:\left[{e}_{1},{e}_{2}\right]={e}_{1},\left[{e}_{1},{e}_{3}\right]={e}_{3},{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}0,\left[{e}_{3},{e}_{3}\right]={e}_{2},\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right)\text{\hspace{0.17em}(}a\ne \text{1)}\text{.\hspace{0.17em}}{L}_{{}^{2,1}}^{11,a}$ is isomorphic to ${L}_{{}^{2,1}}^{11,{a}^{\prime }}$ if and only if a.

(12) ${L}_{{}^{2,1}}^{12,a}:\left[{e}_{1},{e}_{2}\right]={e}_{1},\left[{e}_{1},{e}_{3}\right]={e}_{3},{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}0,\left[{e}_{3},{e}_{3}\right]=0,\sigma =\left(\begin{array}{ccc}a& 0& 0\\ 0& 1& 0\\ 0& 0& a\end{array}\right)\text{\hspace{0.17em}(}a\ne \text{1)}\text{.\hspace{0.17em}}{L}_{{}^{2,1}}^{12,a}$ is isomorphic to ${L}_{{}^{2,1}}^{12,{a}^{\prime }}$ if and only if a = a.

(13) ${L}_{{}^{2,1}}^{13,a}:\left[{e}_{1},{e}_{2}\right]=0,\left[{e}_{1},{e}_{3}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}{e}_{3},\left[{e}_{3},{e}_{3}\right]={e}_{{}_{1}},\sigma =\left(\begin{array}{ccc}0& a& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)\text{\hspace{0.17em}}\text{.\hspace{0.17em}}{L}_{{}^{2,1}}^{13,a}$ is isomorphic to ${L}_{{}^{2,1}}^{13,{a}^{\prime }}$ if and only if a x2a + y(x ≠ 0).

(14) ${L}_{{}^{2,1}}^{14,{a}_{1},{a}_{2}}:\left[{e}_{1},{e}_{2}\right]=0,\left[{e}_{1},{e}_{3}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}{e}_{3},\left[{e}_{3},{e}_{3}\right]={e}_{{}_{1}},\sigma =\left(\begin{array}{ccc}0& {a}_{1}& 0\\ 0& {a}_{2}& 0\\ 0& 0& 0\end{array}\right)\text{\hspace{0.17em}(}a\ne 1\right)\text{.\hspace{0.17em}}{L}_{{}^{2,1}}^{14,{a}_{1},{a}_{2}}$ is isomorphic to ${L}_{{}^{2,1}}^{14,{{a}^{\prime }}_{1},{{a}^{\prime }}_{2}}$ if and only if ${a}_{2}={{a}^{\prime }}_{2},\text{\hspace{0.17em}}{{a}^{\prime }}_{1}={x}^{2}{a}_{1}+y{a}_{2}\left(x\ne 0\right)$.

(15) ${L}_{{}^{2,1}}^{15,a,a}:\left[{e}_{1},{e}_{2}\right]=0,\left[{e}_{1},{e}_{3}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}{e}_{3},\left[{e}_{3},{e}_{3}\right]={e}_{{}_{1}},\sigma =\left(\begin{array}{ccc}{a}^{2}& b& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)\text{\hspace{0.17em}(}a\ne 1\right)\text{.\hspace{0.17em}}{L}_{{}^{2,1}}^{15,a,b}$ is isomorphic to ${L}_{{}^{2,1}}^{15,{a}^{\prime },{b}^{\prime }}$ if and only if $a={a}^{\prime },\text{\hspace{0.17em}}{b}^{\prime }={x}^{2}b-y{a}^{2}+y\left(x\ne 0\right)$.

(16) ${L}_{{}^{2,1}}^{16,a}:\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{1},\left[{e}_{1},{e}_{3}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}\text{λ}{e}_{3},\left[{e}_{3},{e}_{3}\right]=0,\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& a\end{array}\right)\text{\hspace{0.17em}(}a\ne 0\right)\text{.\hspace{0.17em}}{L}_{{}^{2,1}}^{16,a}$ is isomorphic to ${L}_{{}^{2,1}}^{16,{a}^{\prime }}$ if and only if a = a.

(17) ${L}_{{}^{2,1}}^{17}:\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{1},\left[{e}_{1},{e}_{3}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}\text{λ}{e}_{3},\left[{e}_{3},{e}_{3}\right]=0,\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& a\end{array}\right).$

(18) ${L}_{{}^{2,1}}^{18,a,b}:\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{1},\left[{e}_{1},{e}_{3}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}\text{λ}{e}_{3},\left[{e}_{3},{e}_{3}\right]=0,\sigma =\left(\begin{array}{ccc}0& a& 0\\ 0& b& 0\\ 0& 0& 0\end{array}\right)\text{\hspace{0.17em}(}b\ne 1\right)\text{.\hspace{0.17em}}{L}_{2,1}^{18,a,b}$ is isomorphic to ${L}_{2,1}^{18,{a}^{\prime },{b}^{\prime }}$ if and only if $b={b}^{\prime },{a}^{\prime }=xa+yb\left(x\ne 0\right)$.

(19) ${L}_{{}^{2,1}}^{19,a,b}:\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{1},\left[{e}_{1},{e}_{3}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}\text{λ}{e}_{3},\left[{e}_{3},{e}_{3}\right]=0,\sigma =\left(\begin{array}{ccc}a& b& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)\text{\hspace{0.17em}(}a\ne 0\right)\text{.\hspace{0.17em}}{L}_{2,1}^{19,a,b}$ is isomorphic to ${L}_{2,1}^{19,{a}^{\prime },{b}^{\prime }}$ if and only if $a={a}^{\prime },{b}^{\prime }=xb-y{a}^{\prime }+y\left(x\ne 0\right)$.

(20) ${L}_{{}^{2,1}}^{20,a,b,c}:\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{1},\left[{e}_{1},{e}_{3}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}\text{λ}{e}_{3},\left[{e}_{3},{e}_{3}\right]=0,{\sigma }_{5}=\left(\begin{array}{ccc}a& b& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)\text{\hspace{0.17em}(}a\ne 0,c\ne 0\right)\text{.\hspace{0.17em}}{L}_{2,1}^{20,a,b,c}$ is isomorphic to ${L}_{2,1}^{20,{a}^{\prime },{b}^{\prime },{c}^{\prime }}$ if and only if $a={a}^{\prime },c={c}^{\prime },{b}^{\prime }=xb-y{a}^{\prime }+y\left(x\ne 0\right)$.

(21) ${L}_{{}^{2,1}}^{21,a}:\left[{e}_{1},{e}_{2}\right]=0,\left[{e}_{1},{e}_{3}\right]={e}_{3},{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}0,\left[{e}_{3},{e}_{3}\right]={e}_{1},\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& a& 0\\ 0& 0& 0\end{array}\right)\text{\hspace{0.17em}(}a\ne 0\right)\text{.\hspace{0.17em}}{L}_{2,1}^{21,a}$ is isomorphic to ${L}_{2,1}^{21,{a}^{\prime }}$ if and only if a = a.

(22) ${L}_{{}^{2,1}}^{22,a}:\left[{e}_{1},{e}_{2}\right]=0,\left[{e}_{1},{e}_{3}\right]={e}_{3},{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}0,\left[{e}_{3},{e}_{3}\right]={e}_{1},\sigma =\left(\begin{array}{ccc}1& 0& 0\\ 0& a& 0\\ 0& 0& ±1\end{array}\right)\text{.\hspace{0.17em}}{L}_{2,1}^{22,a}$ is isomorphic to ${L}_{2,1}^{22,{a}^{\prime }}$ if and only if a = a.

(23) ${L}_{{}^{2,1}}^{23}:\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{1},\left[{e}_{1},{e}_{3}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=}\text{λ}{\text{e}}_{3},\left[{e}_{3},{e}_{3}\right]=\mu {e}_{1},\sigma =\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$

(24) ${L}_{1,2}^{24}:\left(L,\left[,\right],\sigma \right)$ is an abelian Hom-Lie superalgebra.

(25) ${L}_{{}^{2,1}}^{25,a,b}:\left[{e}_{2},{e}_{2}\right]={e}_{1},{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=0, \hspace{0.17em}}\left[{e}_{3},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& a& b\end{array}\right)\left({a}^{2}+{b}^{2}\ne 0\right).{L}_{1,2}^{25,a,b}$ is isomorphic to ${L}_{1,2}^{25,{a}^{\prime },{b}^{\prime }}$ if and only if $b={b}^{\prime },a=x{a}^{\prime }+y{b}^{\prime }\left(x\ne 0\right)$.

(26) ${L}_{{}^{2,1}}^{26,a,b,c}:\left[{e}_{2},{e}_{2}\right]={e}_{1},{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}\text{]=0, \hspace{0.17em}}\left[{e}_{3},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}{a}^{2}& 0& 0\\ 0& a& 0\\ 0& b& c\end{array}\right)\left(a\ne 0\right).\text{\hspace{0.17em}}{L}_{1,2}^{26,a,b,c}$ is isomorphic to ${L}_{1,2}^{26,{a}^{\prime },{b}^{\prime },{c}^{\prime }}$ if and only if $a={a}^{\prime },{b}^{\prime }=xa+yb-x{c}^{\prime }\left(x\ne 0\right)$.

(27) ${L}_{{}^{2,1}}^{27,a}:\left[{e}_{2},{e}_{2}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}{\text{]=e}}_{1}\text{, \hspace{0.17em}}\left[{e}_{3},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& 0& 0\end{array}\right)\left(a\ne 0\right).\text{\hspace{0.17em}}{L}_{1,2}^{27,a}$ is isomorphic to ${L}_{1,2}^{27,{a}^{\prime }}$ if and only if ${a}^{\prime }=xa\left(x\ne 0\right)$.

(28) ${L}_{{}^{2,1}}^{28,a}:\left[{e}_{2},{e}_{2}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}{\text{]=e}}_{1}\text{, \hspace{0.17em}}\left[{e}_{3},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& a\end{array}\right)\left(a\ne 0\right).\text{\hspace{0.17em}}{L}_{1,2}^{28,a}$ is isomorphic to ${L}_{1,2}^{28,{a}^{\prime }}$ if and only if a = a.

(29) ${L}_{{}^{2,1}}^{29,a,b}:\left[{e}_{2},{e}_{2}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}{\text{]=e}}_{1}\text{, \hspace{0.17em}}\left[{e}_{3},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& a& b\end{array}\right)\left(a\ne 0,b\ne 0\right).\text{\hspace{0.17em}}{L}_{1,2}^{29,a,b}$ is isomorphic to ${L}_{1,2}^{29,{a}^{\prime },{b}^{\prime }}$ if and only if ${b}^{\prime }=b,{a}^{\prime }=xa\left(x\ne 0\right)$.

(30) ${L}_{{}^{2,1}}^{30,a,b}:\left[{e}_{2},{e}_{2}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}{\text{]=e}}_{1}\text{, \hspace{0.17em}}\left[{e}_{3},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}ab& 0& 0\\ 0& 0& b\\ 0& a& 0\end{array}\right)\left(a\ne 0,b\ne 0\right).\text{\hspace{0.17em}}{L}_{1,2}^{30,a,b}$ is isomorphic to ${L}_{1,2}^{30,{a}^{\prime },{b}^{\prime }}$ if and only if $a=xa,{b}^{\prime }=\frac{1}{x}b\left(x\ne 0\right)$.

(31) ${L}_{{}^{2,1}}^{31,a,b}:\left[{e}_{2},{e}_{2}\right]=0,{\text{\hspace{0.17em}[e}}_{2}{\text{, e}}_{3}{\text{]=e}}_{1}\text{, \hspace{0.17em}}\left[{e}_{3},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}ab& 0& 0\\ 0& 0& b\\ 0& a& 0\end{array}\right)\left(a\ne 0,b\ne 0\right).\text{\hspace{0.17em}}{L}_{1,2}^{31,a,b}$ is isomorphic to ${L}_{1,2}^{31,{a}^{\prime },{b}^{\prime }}$ if and only if a = a, b = b.

(32) ${L}_{1,2}^{32,a}:\left[{e}_{1},{e}_{2}\right]={e}_{2},\left[{e}_{1},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& a\end{array}\right).{L}_{1,2}^{32,a}$ is isomorphic to ${L}_{1,2}^{32,{a}^{\prime }}$ if and only if a′ = a.

(33) ${L}_{1,2}^{33,a}:\left[{e}_{1},{e}_{2}\right]={e}_{2},\left[{e}_{1},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& a\end{array}\right).\left(a\ne 0\right).{L}_{1,2}^{33,a}$ is isomorphic to ${L}_{1,2}^{33,{a}^{\prime }}$ if and only if a′ = a.

(34) ${L}_{1,2}^{34,a,b}:\left[{e}_{1},{e}_{2}\right]={e}_{2},\left[{e}_{1},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& 0& b\end{array}\right).\left(a\ne 0\right).{L}_{1,2}^{34,a,b}$ is isomorphic to ${L}_{1,2}^{34,{a}^{\prime },{b}^{\prime }}$ if and only if b′ = b, a′ = |a.

(35) ${L}_{1,2}^{35,a,b}:\left[{e}_{1},{e}_{2}\right]={e}_{2},\left[{e}_{1},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}a& 0& 0\\ 0& 0& 0\\ 0& 0& b\end{array}\right).\left(a\ne 0,1\right).{L}_{1,2}^{35,a,b}$ is isomorphic to ${L}_{1,2}^{35,{a}^{\prime },{b}^{\prime }}$ if and only if b′ = b, a′ = |a.

(36) ${L}_{1,2}^{36,a,b}:\left[{e}_{1},{e}_{2}\right]={e}_{2},\left[{e}_{1},{e}_{3}\right]=0,\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}1& 0& 0\\ 0& a& 0\\ 0& 0& b\end{array}\right)\left(a\ne 0\right).{L}_{1,2}^{36,a,b}$ is isomorphic to ${L}_{1,2}^{36,{a}^{\prime },{b}^{\prime }}$ if and only if b′ = b, a′ = a.

(37) ${L}_{1,2}^{37,a,b}:\left[{e}_{1},{e}_{2}\right]=0,\left[{e}_{1},{e}_{3}\right]={e}_{2},\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& 0& b\end{array}\right)\left({a}^{2}+{b}^{2}\ne |0\right).{L}_{1,2}^{37,a,b}$ is isomorphic to ${L}_{1,2}^{37,{a}^{\prime },{b}^{\prime }}$ if and only if b′ = b, a′ = xa + yb(x ≠ 0).

(38) ${L}_{1,2}^{38,a,b,c}:\left[{e}_{1},{e}_{2}\right]=0,\left[{e}_{1},{e}_{3}\right]={e}_{2},\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}a& 0& 0\\ 0& ac& b\\ 0& 0& c\end{array}\right)\left(a\ne 0,c\ne |0\right).{L}_{1,2}^{38,a,b,c}$ is isomorphic to ${L}_{1,2}^{38,{a}^{\prime },{b}^{\prime },{c}^{\prime }}$ if and only if a′ = a, c = c′, b′ = xb + yc - ya′c′(x≠0).

(39) ${L}_{1,2}^{39,{a}_{1},{a}_{2},{b}_{1},{b}_{2}}:\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{2},\left[{e}_{1},{e}_{3}\right]=\mu {e}_{3},\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}1& 0& 0\\ 0& {a}_{1}& {b}_{1}\\ 0& {a}_{2}& {b}_{2}\end{array}\right).{L}_{1,2}^{39,{a}_{1},{a}_{2},{b}_{1},{b}_{2}}$ is isomorphic to ${L}_{1,2}^{39,{{a}^{\prime }}_{1},{{a}^{\prime }}_{2},{{b}^{\prime }}_{1},{{b}^{\prime }}_{2}}$ if and only if ${{a}^{\prime }}_{1}={a}_{1},{b}_{2}={{b}^{\prime }}_{2},{{b}^{\prime }}_{1}=x{b}_{1},{{a}^{\prime }}_{2}=|\frac{1}{x}{a}_{2}\left(x\ne 0\right).$

(40) ${L}_{1,2}^{40,a,b}:\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{2},\left[{e}_{1},{e}_{3}\right]=\mu {e}_{3},\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}1& 0& 0\\ 0& a& 0\\ 0& 0& b\end{array}\right).{L}_{1,2}^{40,a,b}$ is isomorphic to $\begin{array}{l}{L}_{1,2}^{40,{a}^{\prime },{b}^{\prime }}\\ \end{array}$ if and only if a = a′, b = b′.

(41) ${L}_{1,2}^{41,a}:\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{2},\left[{e}_{1},{e}_{3}\right]=\mu {e}_{3},\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}a& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\left(a\ne 0,1\right).{L}_{1,2}^{41,a}$ is isomorphic to $\begin{array}{l}{L}_{1,2}^{41,{a}^{\prime }}\\ \end{array}$ if and only if a = a′.

(42) ${L}_{1,2}^{42,a}:\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{2},\left[{e}_{1},{e}_{3}\right]=\mu {e}_{3},\text{\hspace{0.17em}}\sigma =\left(\begin{array}{ccc}\frac{\alpha }{\mu }& 0& 0\\ 0& 0& 0\\ 0& a& 0\end{array}\right)\left(a\ne 0\right).{L}_{1,2}^{42,a}$ is isomorphic to ${L}_{1,2}^{42,{a}^{\prime }}$ if and only if a′ = xa(x ≠ 0).

(43) ${L}_{1,2}^{43,a}:\left[{e}_{1},{e}_{2}\right]=0,\left[{e}_{1},{e}_{3}\right]=\beta {e}_{2},\text{\hspace{0.17em}}\left[{e}_{2},{e}_{2}\right]=0,\left[{e}_{2},{e}_{3}\right]=0,\left[{e}_{3},{e}_{3}\right]=\gamma {e}_{1},\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& a& 0\end{array}\right)\left(a\ne 0\right).{L}_{1,2}^{43,a}$ is isomorphic to $\begin{array}{l}{L}_{1,2}^{43,{a}^{\prime }}\\ \end{array}$ if and only if a′ = a.

(44) ${L}_{1,2}^{44,a,b}:\left[{e}_{1},{e}_{2}\right]=0,\left[{e}_{1},{e}_{3}\right]=\beta {e}_{2},\text{\hspace{0.17em}}\left[{e}_{2},{e}_{2}\right]=0,\left[{e}_{2},{e}_{3}\right]=0,\left[{e}_{3},{e}_{3}\right]=\gamma {e}_{1},\sigma =\left(\begin{array}{ccc}{a}^{2}& 0& 0\\ 0& {a}^{3}& b\\ 0& 0& a\end{array}\right)\left(a\ne 0\right).{L}_{1,2}^{44,a,b}$ is isomorphic to $\begin{array}{l}{L}_{1,2}^{44,{a}^{\prime },{b}^{\prime }}\\ \end{array}$ if and only if a′ = a, b′ = xb + ya − ya′3(x ≠ 0).

${L}_{1,2}^{45,a}:\text{\hspace{0.17em}[}{e}_{1},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}0, [}{e}_{1},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\beta {e}_{2},\text{[}{e}_{2},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}0, [}{e}_{2},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\mu {e}_{1},\text{[}{e}_{3},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\gamma {e}_{1},\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& 0& 0\end{array}\right)\text{\hspace{0.17em}(}a\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}0)}$. ${L}_{1,2}^{45,a}$ is isomorphic to ${L}_{1,2}^{45,{a}^{\prime }}$ if and only if a = a.

(46) ${L}_{1,2}^{46,a,b}:\text{\hspace{0.17em}[}{e}_{1},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}0, [}{e}_{1},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\beta {e}_{2},\text{[}{e}_{2},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}0, [}{e}_{2},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\mu {e}_{1},\text{[}{e}_{3},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}0},\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& 0& b\end{array}\right)\text{\hspace{0.17em}(}{a}^{2}\text{\hspace{0.17em}=\hspace{0.17em}}{b}^{2}\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}0)}$. ${L}_{1,2}^{46,a,b}$ is isomorphic to ${L}_{1,2}^{46,{a}^{\prime },{b}^{\prime }}$ if and only if b′ = b, a′ = xa(x ≠ 0).

(47) ${L}_{1,2}^{47,a}:\text{\hspace{0.17em}[}{e}_{1},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}0, [}{e}_{1},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\beta {e}_{2},\text{[}{e}_{2},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}0, [}{e}_{2},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\mu {e}_{1},\text{[}{e}_{3},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}0},\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}a& 0& 0\\ 0& ±a& 0\\ 0& 0& ±1\end{array}\right)\text{\hspace{0.17em}(}a\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}0)}$. ${L}_{1,2}^{47,a}$ is isomorphic to ${L}_{1,2}^{47,{a}^{\prime }}$ if and only if a = a′.

(48) ${L}_{1,2}^{48,a}:\text{\hspace{0.17em}[}{e}_{1},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\alpha {e}_{2}\text{, [}{e}_{1},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}0},\text{[}{e}_{2},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\text{λ}{e}_{1}\text{, [}{e}_{2},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}0},\text{[}{e}_{3},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}0},\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& a\end{array}\right)\text{\hspace{0.17em}(}a\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}0)}$. ${L}_{1,2}^{48,a}$ is isomorphic to ${L}_{1,2}^{48,{a}^{\prime }}$ if and only if a = a′.

(49) ${L}_{1,2}^{49,a}:\text{\hspace{0.17em}[}{e}_{1},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\alpha {e}_{2}\text{, [}{e}_{1},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}0},\text{[}{e}_{2},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}0, [}{e}_{2},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}0},\text{[}{e}_{3},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\gamma {e}_{1},\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}{a}^{2}& 0& 0\\ 0& 0& 0\\ 0& 0& a\end{array}\right)\text{\hspace{0.17em}(}a\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}0)}$. ${L}_{1,2}^{49,a}$ is isomorphic to ${L}_{1,2}^{49,{a}^{\prime }}$ if and only if a = a′.

(50) ${L}_{1,2}^{50,a}:\text{\hspace{0.17em}[}{e}_{1},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\alpha {e}_{2}\text{, [}{e}_{1},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\beta {e}_{2},\text{[}{e}_{2},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}0, [}{e}_{2},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}0},\text{[}{e}_{3},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\gamma {e}_{1},\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}{a}^{2}& 0& 0\\ 0& 0& -\frac{\beta }{\alpha }a\\ 0& 0& a\end{array}\right)\text{\hspace{0.17em}(}a\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}0)}$. ${L}_{1,2}^{50,a}$ is isomorphic to ${L}_{1,2}^{50,{a}^{\prime }}$ if and only if a = a′.

In order to prove the above theorem, we give some lemmas.

Let (L, [, ], σ) be a 3-dimensional Hom-Lie superalgebra and {e1e2e3} be a basis of (L, [, ], σ).

If ${e}_{1},{e}_{2},{e}_{3}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{L}_{\overline{1}}$ and ${L}_{\overline{0}}\text{\hspace{0.17em}=\hspace{0.17em}0}$, then (L, [, ], σ) is an abelian Hom-Lie superalgebra.

Now we consider the 3-dimensional Hom-Lie superalgebra (L, [, ], σ), where dim ${L}_{\overline{0}}\text{\hspace{0.17em}=\hspace{0.17em}2}$ and dim ${L}_{\overline{1}}\text{\hspace{0.17em}=\hspace{0.17em}1}$. Set ${e}_{1},{e}_{2}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{L}_{\overline{0}},\text{\hspace{0.17em}}{e}_{3}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{L}_{\overline{1}}$.

(1) If [L, L] = 0, then (L, [, ], σ) is an abelian Hom-Lie superalgebra.

(2) If dim [L, L] = 1, we consider two cases as follows.

Case I: $\left[L,L\right]\text{\hspace{0.17em}=\hspace{0.17em}}ℂ{e}_{1}$. Let $\text{[}{e}_{1},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\alpha {e}_{1}\text{, [}{e}_{1},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}0},\text{[}{e}_{2},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}0, [}{e}_{3},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\beta {e}_{1}$, where α2 + β2 ≠ 0.

If α = 0, β ≠ 0, by Lemma 2.5, there exists an even invertible linear map $\varphi \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}\frac{1}{\beta }& 1& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$ such that the bracket can be simplified as $[e1, e2]=0, [e1, e3]=0, [e2, e3]=0, [e3, e3]=e1.$(8)

Similarly, if α ≠ 0, β = 0, the bracket can be simplified as $[e1, e2]=e1, [e1, e3]=0, [e2, e3]=0, [e3, e3]=0.$(9)

If α ≠ 0, β = 0, the bracket can be simplified as $[e1, e2]=e1, [e1, e3]=0, [e2, e3]=0, [e3, e3]=e1.$(10)

However, there is not an even invertible linear map such that formulas (8), (9) and (10) are equivalent.

For the bracket (8), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where $\sigma \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}{c}^{2}& a& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right)\text{\hspace{0.17em}(}{a}^{2}\text{\hspace{0.17em}+\hspace{0.17em}}{b}^{2}\text{\hspace{0.17em}}{c}^{2}\ne \text{\hspace{0.17em}0)}$.

Let σ : LL be an even linear map of the Hom-Lie superalgebras and σ(e1) = a1 e1 + a2 e2, σ(e2) = b1 e1 + b2 e2, σ(e3) = ce3. For any ${a}_{1},{a}_{2},{b}_{1}\text{,}{b}_{2}\text{,}c\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}ℂ$ Hom-Jacobi identity is satisfied. Since σ is multiplicative, we have a1 = c2 and a2 = 0. Set b1 = a, b2 = b.

For the bracket (9), the Hom-Lie superalgebra is (L, [, ], σi)(i = 1, 2) up to isomorphism, where ${\sigma }_{1}\text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}0& a& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right)\text{\hspace{0.17em}(}{a}^{2}\text{\hspace{0.17em}+\hspace{0.17em}}{b}^{2}\text{\hspace{0.17em}}{c}^{2}\ne \text{\hspace{0.17em}0), \hspace{0.17em}}{\sigma }_{2}\text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}{a}_{1}& {a}_{2}& 0\\ 0& 1& 0\\ 0& 0& b\end{array}\right)\text{\hspace{0.17em}(}{a}_{1}\ne \text{\hspace{0.17em}0)}$.

It is similar to Lemma 4.2.

For the bracket (10), the Hom-Lie superalgebra is (L, [, ], σi up to isomorphism, where $\sigma \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$.

Let σ : LL be an even linear map of the Hom-Lie superalgebras and σ(e1) = a1 e1 + a2 e2, σ(e2) = b1 e1 + b2 e2, σ(e3) = ce3. Since σ is multiplicative, we have a1 = c2 = a1b2 - a2b1 and a2 = 0. By Hom-Jacobi identity, we obtain b2 = 0, then c = 0. Set ${\sigma }^{\prime }\text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}0& {b}_{1}& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\text{\hspace{0.17em}(}{b}_{1}\ne \text{\hspace{0.17em}0)}$. There exists an even invertible linear map $\varphi \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}\frac{1}{{b}_{1}}& 1& 0\\ 0& 1& 0\\ 0& 0& ±\frac{1}{\sqrt{{b}_{1}}}\end{array}\right)$ satisfying $[ϕ(e1), ϕ(e2)]=ϕ(e1), [ϕ(e1), ϕ(e3)]=0, [ϕ(e2), ϕ(e3)]=0, [ϕ(e3), ϕ(e3)]=ϕ(e1)$ such that $\varphi {\sigma }^{\prime }{\varphi }^{-1}\text{\hspace{0.17em}=\hspace{0.17em}}\sigma \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$.

Case II: $\left[L,L\right]\text{\hspace{0.17em}=\hspace{0.17em}}ℂ{e}_{3}$ Let $\text{[}{e}_{1},{e}_{2}\right]\text{\hspace{0.17em}=\hspace{0.17em}0, [}{e}_{1},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\alpha {e}_{3},\text{[}{e}_{2},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}}\beta {e}_{3}\text{, [}{e}_{3},{e}_{3}\right]\text{\hspace{0.17em}=\hspace{0.17em}0}$, where α2 + β2 ≠ 0.

If α = 0, β ≠ 0, by Lemma 2.5, there exists an even invertible linear map $\varphi \text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}1& 1& 0\\ 0& \beta & 0\\ 0& 0& 1\end{array}\right)$ such that the bracket can be simplified as $[e1, e2]=0, [e1, e3]=0, [e2, e3]=e3, [e3, e3]=0.$(11)

For the bracket (11), the Hom-Lie superalgebras are (L, [, ], σi)(i = 1, 2) up to isomorphism, where ${\sigma }_{1}\text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}a& b& 0\\ 0& c& 0\\ 0& 0& 0\end{array}\right)\text{\hspace{0.17em}(}{a}^{2}\text{\hspace{0.17em}+\hspace{0.17em}}{b}^{2}\text{\hspace{0.17em}}{c}^{2}\ne \text{\hspace{0.17em}0), \hspace{0.17em}}{\sigma }_{2}\text{\hspace{0.17em}=\hspace{0.17em}}\left(\begin{array}{ccc}a& b& 0\\ 0& 1& 0\\ 0& 0& c\end{array}\right)\text{\hspace{0.17em}(}c\ne \text{\hspace{0.17em}0)}$.

It is similar to Lemma 4.4.

If α ≠ 0, β = 0 or α ≠ 0, β ≠, 0 by Lemma 2.5, we know that Hom-Lie superalgebras are isomorphic to those Hom-Lie superalgebras in Lemma 4.5.

All Hom-Lie superalgebras in Theorem 4.2-4.5 are classical Lie superalgebras.

(3) If dim [L, L] = 2, we will consider two cases as follows.

Case I: $\left[L,L\right]=ℂ{e}_{1}+ℂ{e}_{2}={L}_{\overline{0}}$. Let [e1, e2] = α1e1+α2e2, [e1, e3] = 0, [e2, e3] = 0, [e3, e3] = β1e1+α2e2.

The matrix $A=\left(\begin{array}{cc}{\alpha }_{1}& {\beta }_{1}\\ {\alpha }_{2}& {\beta }_{2}\end{array}\right)$ is non-degenerate, otherwise, it is contradictory to dim [L, L] = 2.

If β1 = α2 = 0, we have α1 ≠ 0, β2 ≠ 0. By Lemma 2.5, there exists an even invertible linear map $\varphi =\left(\begin{array}{ccc}1& 0& 0\\ 0& {\alpha }_{1}& 0\\ 0& 0& ±\sqrt{{\alpha }_{1}{\beta }_{2}}\end{array}\right)$ such that the bracket can be simplified as $[e1, e2]=e1, [e1, e3]=0, [e2, e3]=0, [e3, e3]=e2.$(12)

Other cases can be simplified as (12) by Lemma 2.5.

For the bracket (12), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where σ = $\left(\begin{array}{ccc}1& 0& 0\\ 0& {a}^{2}& 0\\ 0& 0& a\end{array}\right)\left(a\ne 0\right).$

It is similar to Lemma 4.4.

Case II: [L, L] = ℂe1 + ℂe3. Let [e1, e2] = αe1, [e1, e3] = βe3, [e2, e3] = λe3, [e3, e3] = μe1. We will consider situations as follows.

If α ≠ 0, β ≠ 0, λ = 0, μ = 0, by Lemma 2.5, there exists an even invertible linear map $\varphi =\left(\begin{array}{ccc}\beta & 0& 0\\ 0& \alpha & 0\\ 0& 0& 1\end{array}\right)$ such that the bracket can be simplified as $[e1, e2]=e1, [e1, e3]=e3, [e2, e3]=0, [e3, e3]=0.$(13)

Moreover, for α ≠ 0, β ≠ 0, λ ≠ 0, μ = 0, the bracket can be simplified as (13).

For the bracket (13), Hom-Lie superalgebras are (L, [, ], σi) (1 ≥ i ≥ 3) up to isomorphism, where ${\sigma }_{1}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right),\text{\hspace{0.17em}}{\sigma }_{2}=\left(\begin{array}{ccc}0& 0& 0\\ 0& a& 0\\ 0& 0& 0\end{array}\right)\left(a\ne 1\right),\text{\hspace{0.17em}}{\sigma }_{3}=\left(\begin{array}{ccc}a& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)\left(a\ne 1\right)$.

If α = 0, β = 0, λ ≠ 0, μ ≠ 0, then the bracket can be simplified as $[e1, e2]=0, [e1, e3]=0, [e2, e3]=e3, [e3, e3]=e1.$(14)

For the bracket (14), Hom-Lie superalgebras are (L, [, ], σi) (1 ≥ i ≥ 3) up to isomorphism, where ${\sigma }_{1}=\left(\begin{array}{ccc}0& a& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right),\text{\hspace{0.17em}}{\sigma }_{2}=\left(\begin{array}{ccc}0& {a}_{1}& 0\\ 0& {a}_{2}& 0\\ 0& 0& 0\end{array}\right)\left({a}_{2}\ne 1\right),\text{\hspace{0.17em}}{\sigma }_{3}=\left(\begin{array}{ccc}{a}^{2}& b& 0\\ 0& 1& 0\\ 0& 0& a\end{array}\right)\left(a\ne 0\right)$.

If α ≠ 0, β = 0, λ ≠ 0, μ = 0, then $[e1, e2]=αe1, [e1, e3]=0, [e2, e3]=λe3, [e3, e3]=0.$(15).

For the bracket (15), Hom-Lie superalgebras are (L, [, ], σi) (1 ≥ i ≥ 5) up to isomorphism, where ${\sigma }_{1}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)\left(a\ne 0\right),\text{\hspace{0.17em}}{\sigma }_{2}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right),\text{\hspace{0.17em}}{\sigma }_{3}=\left(\begin{array}{ccc}0& a& 0\\ 0& b& 0\\ 0& 0& 0\end{array}\right)\left(b\ne 1\right),\text{\hspace{0.17em}}{\sigma }_{4}=\left(\begin{array}{ccc}a& b& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)\left(a\ne 0\right),$ ${\sigma }_{5}=\left(\begin{array}{ccc}a& b& 0\\ 0& 1& 0\\ 0& 0& c\end{array}\right)\left(a\ne 0,\text{\hspace{0.17em}}c\ne 0\right)$.

If α = 0, β ≠ 0, λ = 0, μ ≠ 0, then the bracket can be simplified as $[e1, e2]=0, [e1, e3]=e3, [e2, e3]=0, [e3, e3]=e1.$(16)

Moreover, for α = 0, β ≠ 0, λ ≠ 0, μ ≠ 0, the bracket can be simplified as (16).

For the bracket (16), Hom-Lie superalgebras are (L, [, ], σi) (1 ≥ i ≥ 2) up to isomorphism, where ${\sigma }_{1}=\left(\begin{array}{ccc}0& 0& 0\\ 0& a& 0\\ 0& 0& 0\end{array}\right)\left(a\ne 0\right),\text{\hspace{0.17em}}{\sigma }_{2}=\left(\begin{array}{ccc}1& 0& 0\\ 0& a& 0\\ 0& 0& ±1\end{array}\right).$

It is similar to Lemma 4.4.

If α ≠ 0, β ≠ 0, λ = 0, μ ≠ 0, then the bracket can be simplified as $[e1, e2]=e1, [e1, e3]=e3, [e2, e3]=0, [e3, e3]=e1.$(17)

Moreover, for α ≠ 0, β ≠ 0, λ ≠ 0, μ ≠ 0, the bracket can be simplified as (17). However, for formula (17), there are not non-zero even linear maps σ satisfying σ[ei, ej] = [σ(ei), σ(ej)] such that (L, [, ], σ) is a Hom-Lie superalgebra.

If α ≠ 0, β = 0, λ ≠ 0, μ ≠ 0, then $[e1, e2]=αe1, [e1, e3]=0, [e2, e3]=λe3, [e3, e3]=μe1.$(18)

For the bracket (17), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where σ = $\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$

It is similar to Lemma 4.4.

(4) If dim [L, L] = 3, then [L, L] = L. Let [e1, e2] = α1e1 + α2e2, [e1, e3] = μe3, [e3, e3] = β1e1 + β2e2. From dim [L, L] = 3, it follows that λ2 + μ2 ≠ 0 and the matrix $A=\left(\begin{array}{cc}{\alpha }_{1}& {\beta }_{1}\\ {\alpha }_{2}& {\beta }_{2}\end{array}\right)$ is non-degenerate. So we only consider three cases as follows, the other cases are equivalent to those.

If μ ≠ 0, α1 ≠ 0, β2 ≠ 0, λ = β1 = α2 = 0, then [e1, e2] = α1e1, [e1, e3] = 0, [e2, e3] = μe3, [e3, e3] = β2e2.

If λ ≠ 0, α1 ≠ 0, β2 ≠ 0, μ = β1 = α2 = 0, then [e1, e2] = α1e1, [e1, e3] = λe3, [e2, e3] = 0, [e3, e3] = β2e2.

If λ ≠ 0, α1 ≠ 0, β2 ≠ 0, μ ≠ 0, β1 = α2 = 0, then [e1, e2] = α1e1, [e1, e3] = λe3, [e2, e3] = μe3, [e3, e3] = β2e2.

For the above formulas, there are not non-zero even linear maps σ satisfying σ[ei, ej] = [σ(ei), σ(ej)] such that (L, [, ], σ) is a Hom-Lie superalgebra.

In the following paragraph, we will consider the 3-dimensional Hom-Lie superalgebra (L, [, ], σ), where dim ${L}_{\overline{0}}=1,$ dim ${L}_{\overline{1}}=2$. Set e1${L}_{\overline{0}}$, e2, e3${L}_{\overline{1}}$.

(1) If dim [L, L] = 0, then (L, [, ], σ) is an abelian Hom-Lie superalgebra.

(2) If dim [L, L] = 1, we consider the two cases as follows.

Case I: [L, L] = ℂe1 = ${L}_{\overline{0}}$. Let [e1, e2] = 0, [e1, e3] = 0, [e2, e2] = αe1, [e2, e3] = βe1, [e3, e3] = γe1, where α2 + β2 + γ2 ≠ 0.

Similar to above, we have $[e2, e2]=e1, [e2, e3]=0, [e3, e3]=0.$(19) $[e2, e2]=0, [e2, e3]=e1, [e3, e3]=0.$(20)

For the bracket (19), Hom-Lie superalgebras are (L, [, ], σi) (1 ≥ i ≥ 2) up to isomorphism, where ${\sigma }_{1}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& a& b\end{array}\right)\left({a}^{2}+{b}^{2}\ne 0\right),\text{\hspace{0.17em}}{\sigma }_{2}=\left(\begin{array}{ccc}{a}^{2}& 0& 0\\ 0& a& 0\\ 0& b& c\end{array}\right)\left(a\ne 0\right).$

For the bracket (20), Hom-Lie superalgebras are (L, [, ], σi)(1 ≥ i ≥ 5) up to isomorphism, where ${\sigma }_{1}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& 0& 0\end{array}\right)\left(a\ne 0\right),\text{\hspace{0.17em}}{\sigma }_{2}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& a\end{array}\right)\left(a\ne 0\right),\text{\hspace{0.17em}}{\sigma }_{3}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& a& b\end{array}\right)\left(a\ne 0,\text{\hspace{0.17em}b}\ne 0\right),\text{\hspace{0.17em}}{\sigma }_{4}=\left(\begin{array}{ccc}ab& 0& 0\\ 0& 0& b\\ 0& a& 0\end{array}\right)\left(a\ne 0,\text{\hspace{0.17em}b}\ne 0\right),\text{\hspace{0.17em}}{\sigma }_{5}=\left(\begin{array}{ccc}ab& 0& 0\\ 0& a& 0\\ 0& 0& b\end{array}\right)\left(a\ne 0,\text{\hspace{0.17em}b}\ne 0\right).$

Case II: [L, L] = ℂe2${L}_{\overline{1}}$. Let [e1, e2] = αe2, [e1, e3] = βe2, where α2 + β2 ≠ 0.

Similar to above, we have $[e1, e2]=e2, [e1, e3]=0.$(21) $[e1, e2]=0, [e1, e3]=e2$(22).

For the bracket (21), Hom-Lie superalgebras are (L, [, ], σi)(1 ≥ i ≥ 5) up to isomorphism, where ${\sigma }_{1}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& a\end{array}\right),\text{\hspace{0.17em}}{\sigma }_{2}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& a\end{array}\right)\left(a\ne 0\right),\text{\hspace{0.17em}}{\sigma }_{3}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& 0& b\end{array}\right)\left(a\ne 0\right),\text{\hspace{0.17em}}{\sigma }_{4}=\left(\begin{array}{ccc}a& 0& 0\\ 0& 0& 0\\ 0& 0& b\end{array}\right)\left(a\ne 0,\text{\hspace{0.17em}1}\right),\text{\hspace{0.17em}}{\sigma }_{5}=\left(\begin{array}{ccc}1& 0& 0\\ 0& a& 0\\ 0& 0& b\end{array}\right)\left(a\ne 0\right).$

For the bracket (22), Hom-Lie superalgebras are (L, [, ], σi)(1 ≥ i ≥ 2) up to isomorphism, where ${\sigma }_{1}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& 0& b\end{array}\right)\left({a}^{2}+{b}^{2}\ne 0\right),\text{\hspace{0.17em}}{\sigma }_{3}=\left(\begin{array}{ccc}a& 0& 0\\ 0& ac& b\\ 0& 0& c\end{array}\right)\left(a\ne 0,\text{\hspace{0.17em}}c\ne 0\right).$

Hom-Lie superalgebras in Lemma 4.15 and Lemma 4.16 are classical Lie superalgebras.

(3) If dim [L, L] = 2, we consider the two cases as follows.

Case I: $\left[L,L\right]=\mathbb{C}{e}_{2}+\mathbb{C}{e}_{3}={L}_{\overline{1}\cdot }\mathrm{L}\mathrm{e}\mathrm{t}\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{2}+\beta {e}_{3},\left[{e}_{1},{e}_{3}\right]=\text{λ}{e}_{2}+\mu {e}_{3}$ , where the matrix $A=\left(\begin{array}{cc}\alpha & \text{λ}\\ \beta & \mu \end{array}\right)$ is non-degenerate. Then we only consider α ≠ 0, μ ≠ 0, β = λ = 0, other cases are equivalent to this one. By the direct calculation, we can get $e1, e2=αe2, e1, e3=μe3$(23)

For the bracket (23), Hom-Lie superalgebras are (L, [, ], σi) (1 ≤ i ≤ 4) up to isomorphism, where ${\sigma }_{1}=\left(\begin{array}{ccc}1& 0& 0\\ 0& {a}_{1}& {b}_{1}\\ 0& {a}_{2}& {b}_{2}\end{array}\right),{\sigma }_{2}=\left(\begin{array}{ccc}1& 0& 0\\ 0& a& 0\\ 0& 0& b\end{array}\right),{\sigma }_{3}=\left(\begin{array}{ccc}a& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\left(a\ne 0,1\right),{\sigma }_{4}=\left(\begin{array}{ccc}\frac{\alpha }{\mu }& 0& 0\\ 0& 0& 0\\ 0& a& 0\end{array}\right)\left(a\ne 0\right).$

It is similar to Lemma 4.4.

Case II: $\left[L,L\right]={\mathbb{C}}_{e}1+\mathbb{C}{e}_{2\cdot }\mathrm{L}\mathrm{e}\mathrm{t}\phantom{\rule{thinmathspace}{0ex}}\left[{e}_{1},{e}_{2}\right]=\alpha {e}_{2},\left[{e}_{1},{e}_{3}\right]=\beta {e}_{2},\left[{e}_{2},{e}_{2}\right]=\text{λ}{e}_{1},\left[{e}_{2},{e}_{3}\right]=\mu {e}_{1},\left[{e}_{3},{e}_{3}\right]=\gamma {e}_{1},$

where α2 + β2 ≠ 0, λ2 + μ2 + γ2 ≠ 0.

Similar to above, we have $e1, e2=0, e1, e3=βe2, e2, e2=0, e2, e3=0, e3, e3=γe1.$(24) $e1, e2=0, e1, e3=βe2, e2, e2=0, e2, e3=μe1, e3, e3=γe1.$(25) $e1, e2=0, e1, e3=βe2, e2, e2=0, e2, e3=μe1, e3, e3=0.$(26) $e1, e2=αe2, e1, e3=0, e2, e2=λe1, e2, e3=0, e3, e3=0.$(27) $e1, e2=αe2, e1, e3=0, e2, e2=0, e2, e3=0, e3, e3=γe1.$(28) $e1, e2=αe2, e1, e3=βe2, e2, e2=0, e2, e3=0, e3, e3=γe1.$(29)

For the bracket (24), Hom-Lie superalgebras are (L, [, ], σi)(1 ≤ i ≤ 2) up to isomorphism, where ${\sigma }_{1}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& 0& 0\end{array}\right)\left(a\ne 0\right),{\sigma }_{2}=\left(\begin{array}{ccc}{a}^{2}& 0& 0\\ 0& {a}^{3}& b\\ 0& 0& a\end{array}\right)\left(a\ne 0\right).$

It is similar to Lemma 4.4.

For the bracket (25), the Hom-Lie superalgebra is (L, [, ]σ) up to isomorphism, where σ = $\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& 0& 0\end{array}\right)\left(a\ne 0\right).$

It is similar to Lemma 4.4.

For the bracket (26), Hom-Lie superalgebras are (L, [, ], σi)(1 ≤ i ≤ 2) up to isomorphism, where ${\sigma }_{1}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& a\\ 0& 0& b\end{array}\right)\left({a}^{2}+{b}^{2}\ne 0\right),{\sigma }_{2}=\left(\begin{array}{ccc}a& 0& 0\\ 0& ±a& 0\\ 0& 0& ±1\end{array}\right)\left(a\ne 0\right).$

It is similar to Lemma 4.4.

For the bracket (27), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where σ = $\sigma =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& a\end{array}\right)\left(a\ne 0\right).$

It is similar to Lemma 4.4.

For the bracket (28), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where σ = $\sigma =\left(\begin{array}{ccc}{a}^{2}& 0& 0\\ 0& 0& 0\\ 0& 0& a\end{array}\right)\left(a\ne 0\right).$

It is similar to Lemma 4.4.

For the bracket (29), Hom-Lie superalgebras are (L, [, ], σ) up to isomorphism,, where $\sigma =\left(\begin{array}{ccc}{a}^{2}& 0& 0\\ 0& 0& -\frac{\beta }{\alpha }a\\ 0& 0& a\end{array}\right)\left(a\ne 0\right).$

It is similar to Lemma 4.4.

(4) If dim [L, L] = 3, then [L, L] = L. Let [e1, e2] = α1e2 + β1e3, [e1, e3] = α2e2 + β2e3, [e2, e2] = λe1, [e2, e3] = μe1, [e3, e3] = γe1. From dim[L, L] = 3, it follows that λ2 + μ2 + γ2 ≠ 0 and the matrix $A=\left(\begin{array}{cc}{\alpha }_{1}& {\alpha }_{2}\\ {\beta }_{1}& {\beta }_{2}\end{array}\right)$ is non-degenerate. So we only consider three cases as follows, other cases are equivalent to those. $e1, e2=α1e2, e1, e3=β2e3, e2, e2=e1, e2, e3=0, e3, e3=0.$ $e1, e2=α1e2, e1, e3=β2e3, e2, e2=e1, e2, e3=0, e3, e3=e1.$ $e1, e2=α1e2, e1, e3=β2e3, e2, e2=0, e2, e3=e1, e3, e3=0.$

For the above formulas, there are not non-zero even linear maps σ satisfying σ [ei, ej] = [σ(ei), σ(ej)] such that (L, [, ], σ) is a Hom-Lie superalgebra.

From Lemmas 4.2-4.24, we obtain Theorem 4.1.

By Lemma 2.8, Hom-Lie superalgebras ${L}_{{0}_{;}3}^{1}$ , ${L}_{2,1}^{2}$ and ${L}_{1,2}^{24}$ are not isomorphism.

## Acknowledgement

This work was supported by National Natural Science Foundation of China(11471090) and Science and Technology Research Fund of Jilin Province (2016111).

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## About the article

Accepted: 2016-08-01

Published Online: 2016-09-20

Published in Print: 2016-01-01

Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 613–628, ISSN (Online) 2391-5455,

Export Citation


text:
# Simple Return, Log Return and Volatility easy intro

Following \$100k Optiver Realized Volatility Prediction Competition on Kaggle, interest in financial time-series and concepts has drastically increased. Besides, in this website we have not travelled past the very introductory concepts in time-series ecnnometrics. Therefore in this blog post we are going to introduce the concepts of simple returns, log returns and realized volatility.

In the following article we have already explored time-series and simple operations in pandas and numpy.

To illustrate the log returns and more complex concepts we are going to demonstrate them with the help of python and pandas and numpy libraries.

## Simple returns vs log returns

To begin with, benefit of using financial returns instead of prices is normalization which allows us to measure and compare all financial instruments and assets. This is also a requirement for many multidimensional statistical analysis and machine learning techniques.

The simple return on an asset is the amount we receive or lose after holding an asset for a certain time frame.

Meanwhile we calculate log returns as

where P is price of an asset and t is a time period.

In pandas we calculate the simple returns with an in-built pandas function `pct_change` that calculates percentage change and use numpy `log` to calculate log returns in addition to pandas `diff` that differences the series.

``````data['simple_returns'] = data.close.pct_change()
def log_return(list_stock_prices):
return np.log(list_stock_prices).diff()
data['log_return'] = log_return(data.close)``````

The main advantage of log returns is that we can easily aggregate them across time unlike simple returns. For instance, the log return for a year is the sum of the log returns of the days within the year. Additionally, log returns are symmetric around 0 and log return values can range from minus infinity to plus infinity. Whereas, simple returns’ downside is limited at -100% and a negative movement of -25% (movement from 100\$ to 75\$) does not reverse the losses by going +25% (75\$ to 93.75\$).

## Realized Volatility

Volatility plays an important role in trading and financial model forecasting. Next to directly modeling high–frequency returns, intra–day returns are also used to construct lower–frequency daily volatility measures, known as realized volatility. The application of volatility has tremendously risen due to the availability of high-frequency data and computational resources.

Layman’s definition:  Realized volatility is the magnitude of daily price movements, regardless of direction, of some underlying, over a specific period.

Technical definition:  Realized volatility is the daily standard deviation of log returns of an underlying asset, index, instrument, security, or ETF, over a defined period, with an assumed mean of zero.

https://www.realvol.com/Realizedvolatilitydefinition.htm

Traders and data scientist use realized volatility to assess the predictive performance and adequacy of existing models and strategies due to our ability to observe the metric (non-latent character). Realized volatility is a particularly powerful indicator of the price risk and its dynamics.

The formula for realized volatility:

In Python we create a function that calculates realized volatility with the help of numpy functions `sqrt` and `sum` and pandas `groupby` and `agg`.

``````def realized_volatility(series):
return np.sqrt(np.sum(series**2))
df.groupby(df.index.date).agg(realized_volatility)``````

## Realized Variance

Andersen and Bollerslev (1998) have suggested an ex-post intra-daily volatility measure based on the sum of the squared intra-day returns. This concept is now known as realized variance, and this is an ex-post observable measure.

``````def realized_variance(series):
return np.sum(series**2)
df.groupby(df.index.date).agg(realized_variance)``````

## References

You can contact me for professional inquires via my social media:

### 8 thoughts on “Simple Return, Log Return and Volatility easy intro”

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# Chapter 1 - Section 1.1 - Real Numbers - 1.1 Exercises: 51

2$\le$x$\lt$$\infty #### Work Step by Step [2,\infty) Find the set description {x\mida\lex\ltb} Plug in {x\mid2\lex\lt$$\infty$} = 2$\le$x$\lt$$\infty$ Create the number line (Note: the arrow is more bolded to indicate numbers going to infinity are shaded)

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
2016 Dec 22

Analysis and PDE's Seminar -- Percy Deift (Courant)

1:00pm to 2:00pm

Location:

Ross 70
On the Asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential.
T.Bothner, P.Deift, A.Its and I.Krasovsky
Abstract: We study the partition function Z of a Coulomb gas of particles with an external potential 2v applied to the
particles in an interval of length L. When v is infinite, Z describes the gap probability for GUE eigenvalues in the bulk scaling limit,
and has been well-studied for many years. Here we study the the behavior of Z in the full (v,L) plane.
2017 May 17

Mark Rudelson: Delocalization of the eigenvectors of random matrices.

2:00pm to 3:00pm

Location:

רוס 63
Abstract: Consider a random matrix with i.i.d. normal entries. Since its distribution is invariant under rotations, any normalized eigenvector is uniformly distributed over the unit sphere. For a general distribution of the entries, this is no longer true. Yet, if the size of the matrix is large, the eigenvectors are distributed approximately uniformly. This property, called delocalization, can be quantified in various senses. In these lectures, we will discuss recent results on delocalization for general random matrices.
2017 Dec 13

Jerusalem Analysis Seminar: "Exponential concentration of zeroes of Gaussian stationary functions" Naomi Feldheim (Weizmann Institute)

12:00pm to 1:00pm

Location:

Ross 70
A Gaussian stationary function (GSF) is a random f: R --> R whose
distribution is shift-invariant and all its finite marginals have
centered multi-normal distribution. It is a simple and popular model
for noise, for which the mean number of zeroes was computed already
in the 1940's by Kac and Rice. However, it is far more complicated
to estimate the probability of a significant deficiency or abundance
in the number of zeroes in a long interval (compared to the expectation).
2016 Nov 17

Analysis and PDEs- A. Logunov "The zero set of a nonconstant harmonic function in R^3 has infinite area"

1:00pm to 2:00pm

Location:

Ross 70
Abstract. We will give a sketch of the proof of the fact formulated in the title.
2017 Mar 23

Xiaolin Zeng (TAU)

1:00pm to 2:00pm

Location:

Ross 70
Title: a random Schroedinger operator stemming from reinforced process
Abstract:
We will explain the relationship between a toy model to Anderson localization, called the H^{2|2} model (according to Zirnbauer) and edge reinforced random walk.
2017 Jun 29

Barry Simon (Caltech)

1:00pm to 2:00pm

Title: Asymptotics for Chebyshev Polynomials of Infinite Gap Sets on the Real Line
Abstract: The Chebyshev Polynomials of a compact subset, e, of the complex plane are the monic polynomials minimizing the sup over e. We prove Szego--Widom asymptotics for the Chebyshev Polynomials of a compact subset of R which is regular for potential theory and obeys the Parreau--Widom and DCT conditions. We give indications why these sufficient conditions may also be necessary.
2016 Nov 03

Analysis and PDEs - Baptiste Devyver (Technion) - "Heat kernel estimates of Schrodinger-type operators"

1:00pm to 2:00pm

Location:

Ross 70
Let us consider the heat equation:
$u_t+Lu=0$ in a domain $\Omega$.
Here, $L$ will be a self-adjoint Schrodinger-type operator of the form
abla^*
2017 Nov 01

Jerusalem Analysis Seminar "When do the spectra of self-adjoint operators converge?" Siegfried Beckus (Technion)

12:00pm to 1:00pm

Location:

Ross 63

Abstract:
Given a self-adjoint bounded operator, its spectrum is a compact subset of the real numbers. The space of compact subsets of the real numbers is naturally equipped with the Hausdorff metric. Let $T$ be a topological (metric) space and $(A_t)$ be a family of self-adjoint, bounded operators. In the talk, we study the (Hölder-)continuity of the map assigning to each $t\in T$ the spectrum of the operator $A_t$.
2017 Mar 09

Analysis and PDEs: Leonid Parnovski (London) - Local Density of states and the spectral function for almost periodic operators

1:00pm to 2:00pm

Location:

Ross 70
I will discuss the asymptotic behaviour (both on and off the diagonal) of the spectral function of a Schroedinger operator with smooth bounded potential when energy becomes large. I formulate the conjecture that the local density of states (i.e. the spectral function on the diagonal) admits the complete asymptotic expansion and discuss the known results, mostly for almost-periodic potentials.
2017 May 25

Mark Rudelson: Delocalization of the eigenvectors of random matrices.

1:00pm to 2:00pm

Location:

Ross 70
Abstract: Consider a random matrix with i.i.d. normal entries. Since its distribution is invariant under rotations, any normalized eigenvector is uniformly distributed over the unit sphere. For a general distribution of the entries, this is no longer true. Yet, if the size of the matrix is large, the eigenvectors are distributed approximately uniformly. This property, called delocalization, can be quantified in various senses. In these lectures, we will discuss recent results on delocalization for general random matrices.
2016 Jun 16

Jerusalem Analysis and PDEs - Gilbert Weinstein (Ariel)

1:00pm to 2:00pm

Location:

Ross 70
Title: Harmonic maps with prescribed singularities and applications to general relativity
Abstract: We will present a general theory of existence and uniqueness for harmonic maps with prescribed singularities into Riemannian manifolds with non-positive curvature. The singularities are prescribed along submanifolds of co-dimension 2. This result generalizes one from 1996, and is motivated by a number of recent applications in general relativity including:
* a lower bound on the ADM mass in terms of charge and angular momentum for multiple black holes;
2017 Nov 08

Jerusalem Analysis Seminar: "To infinity and back (a bit)", Jonathan Breuer (HUJI)

12:00pm to 1:00pm

Location:

Ross 63

Let H be a self-adjoint operator defined on an infinite dimensional Hilbert space. Given some
spectral information about H, such as the continuity of its spectral measure, what can be said about
the asymptotic spectral properties of its finite dimensional approximations? This is a natural (and
general) question, and can be used to frame many specific problems such as the asymptotics of zeros of
orthogonal polynomials, or eigenvalues of random matrices. We shall discuss some old and new results
2016 Dec 29

Analysis and PDEs Seminar: Svetlana Jitomirskaya (UC Irvine)

1:00pm to 2:00pm

Location:

Ross 70
Title: Sharp arithmetic spectral transitions and universal hierarchical structure of quasiperiodic eigenfunctions.
Abstract: A very captivating question in solid state physics
is to determine/understand the hierarchical structure of spectral features
of operators describing 2D Bloch electrons in perpendicular magnetic
fields, as related to the continued fraction expansion of the magnetic
flux. In particular, the hierarchical behavior of the eigenfunctions of
the almost Mathieu operators, despite signifi cant numerical studies and
2017 May 24

Mark Rudelson: Delocalization of the eigenvectors of random matrices.

2:00pm to 3:00pm

Location:

רוס 63
Abstract: Consider a random matrix with i.i.d. normal entries. Since its distribution is invariant under rotations, any normalized eigenvector is uniformly distributed over the unit sphere. For a general distribution of the entries, this is no longer true. Yet, if the size of the matrix is large, the eigenvectors are distributed approximately uniformly. This property, called delocalization, can be quantified in various senses. In these lectures, we will discuss recent results on delocalization for general random matrices.
2017 Dec 27

Jerusalem Analysis Seminar: "Optimal Gaussian Partitions", Elchanan Mossel (MIT)

12:00pm to 1:00pm

Location:

Ross 70

How should we partition the Gaussian space into k parts in a way that minimizes Gaussian surface area, maximize correlation or simulate a specific distribution. The problem of Gaussian partitions was studied since the 70s first as a generalization of the isoperimetric problem in the context of the heat equation. It found a renewed interest in the context of the double bubble theorem proven in geometric measure theory and due to connection to problems in theoretical computer science and social choice theory.


text:
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

End October I tweeted this:

and it generated some discussion. Some people believe that this is the right approach, and some others think that one should first present base R and then show how the tidyverse complements it. This year, I taught three classes; a 12-hour class to colleagues that work with me, a 15-hour class to master’s students and 3 hours again to some of my colleagues. Each time, I decided to focus on the tidyverse(almost) entirely, and must say that I am not disappointed with the results!

The 12 hour class was divided in two 6 hours days. It was a bit intense, especially the last 3 hours that took place Friday afternoon. The crowd was composed of some economists that had experience with STATA, some others that were mostly using Excel and finally some colleagues from the IT department that sometimes need to dig into some data themselves. Apart from 2 people, all the other never had any experience with R.

We went from 0 to being able to do the plot below after the end of the first day (so 6 hours in). Keep in mind that practically none of them even had opened RStudio before. I show the code so you can see the progress made in just a few hours:

library(Ecdat)
library(tidyverse)
library(ggthemes)
data(Bwages)
bwages = Bwages %>%
mutate(educ_level = case_when(educ == 1 ~ "Primary School",
educ == 2 ~ "High School",
educ == 3 ~ "Some university",
educ == 4 ~ "Master's degree",
educ == 5 ~ "Doctoral degree"))

ggplot(bwages) +
geom_smooth(aes(exper, wage, colour = educ_level)) +
theme_minimal() +
theme(legend.position = "bottom", legend.title = element_blank())
## geom_smooth() using method = 'loess'

Of course some of them needed some help here and there, and I also gave them hints (for example I told them about case_when() and try to use it inside mutate() instead of nested ifs) but it was mostly due to lack of experience and because they hadn’t had the time to fully digest R’s syntax which was for most people involved completely new.

On the second day I showed purrr::map() and purrr::reduce() and overall it went quite well too. I even showed list-columns, and this is where I started losing some of them; I did not insist too much on it though, only wanted to show them the flexibility of data.frame objects. Some of them were quite impressed by list-columns! Then I started showing (for and while) loops and writing functions. I even showed them tidyeval and again, it went relatively well. Once they had the opportunity to play a bit around with it, I think it clicked (plus they have lots of code examples to go back too).

At the end, people seemed to have enjoyed the course, but told me that Friday was heavy; indeed it was, but I feel that it was mostly because 12 hours spread on 2 days is not the best format for this type of material, but we all had time constraints.

The 15 hour Master’s course was spread over 4 days, and covered basically the same. I just used the last 3 hours to show the students some basic functions for model estimation (linear, count, logit/probit and survival models). Again, the students were quite impressed by how easily they could get descriptive statistics by first grouping by some variables. Through their questions, I even got to show them scoped versions of dplyr verbs, such as select_if() and summarise_at(). I was expecting to lose them there, but actually most of them got these scoped versions quite fast. These students already had some experience with R though, but none with the tidyverse.

Finally the 3 hour course was perhaps the most interesting; I only had 100% total beginners. Some just knew R by name and had never heard/seen/opened RStudio (with the exception of one person)! I did not show them any loops, function definitions and no plots. I only showed them how RStudio looked and worked, what were (and how to install) packages (as well as the CRAN Task Views) and then how to import data with rio and do descriptive statistics only with dplyr. They were really interested and quite impressed by rio (“what do you mean I can use the same code for importing any dataset, in any format?”) but also by the simplicity of dplyr.

In all the courses, I did show the $ primitive to refer to columns inside a data.frame. First I showed them lists which is where I introduced $. Then it was easy to explain to them why it was the same for a column inside a data.frame; a data.frame is simply a list! This is also the distinction I made from the previous years; I simply mentioned (and showed really quickly) matrices and focused almost entirely on lists. Most participants, if not all, had learned to program statistics by thinking about linear algebra and matrices. Nothing wrong with that, but I feel that R really shines when you focus on lists and on how to work with them.

Overall as the teacher, I think that focusing on the tidyverse might be a very good strategy. I might have to do some adjustments here and there for the future courses, but my hunch is that the difficulties that some participants had were not necessarily due to the tidyverse but simply to lack of time to digest what was shown, as well as a total lack of experience with R. I do not think that these participants would have better understood a more traditional, base, matrix-oriented course.


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## fit

The key part is to show that the submanifolds $U_k$ fit together to form a complex submanifold.