# Capability of Nilpotent Lie Superalgebras of Small Dimension

Rudra Narayan Padhan  Centre for Data Science, Institute of Technical
Education and Research
Siksha ‘O’ Anusandhan (A Deemed to be University)
Bhubaneswar-751030
Odisha, India<EMAIL_ADDRESS><EMAIL_ADDRESS>,
IBRAHEM YAKZAN HASAN  Centre for Data Science, Institute of Technical
Education and Research
Siksha ‘O’ Anusandhan (A Deemed to be University)
Bhubaneswar-751030
Odisha, India<EMAIL_ADDRESS>and  Saudamini Nayak  Department of
Mathematics, National Institute of Technology Calicut, NIT Campus,
Kozhikode-673601, India
India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we define partially capable Lie superalgebra. As an application
we classify all capable nilpotent Lie superalgebras of dimension less than
equal to five.

###### Key words and phrases:

Heisenberg Lie superalgebra, multiplier, capability, partially capable

###### 2020 Mathematics Subject Classification:

Primary 17B10, Secondary 17B01.

##  1\. Introduction

Lie superalgebras have applications in many areas of Mathematics and
Theoretical Physics as they can be used to describe supersymmetry. Kac [21]
gives a comprehensive description of mathematical theory of Lie superalgebras,
and establishes the classification of all finite dimensional simple Lie
superalgebras over an algebraically closed field of characteristic zero. In
the last few years the theory of Lie superalgebras has evolved remarkably,
obtaining many results in representation theory and classification. Most of
the results are extension of well known facts of Lie algebras. But the
classification of all finite dimensional nilpotent Lie superalgebras upto
isomorphism is still an open problem like that of finite dimensional nilpotent
Lie algebras.

In 1904, I. Schur introduced the Schur multiplier and cover of a group in his
work on projective representation [22]. Batten [6] introduced and studied
Schur multiplier and cover of a Lie algebra and later on, studied by several
authors [8, 7]. For a finite dimensional Lie algebra $L$ over a field
$\mathbb{F}$ the free presentation of $L$ is the exact sequence,
$0\longrightarrow R\longrightarrow F\longrightarrow L$, where $F$ is a free
Lie algebra and $R$ is an ideal of $F$. Then the Schur multiplier
$\mathcal{M}(L)$ is isomorphic to $F^{\prime}\cap R/[F,R]$. The notion of
multiplier and cover for Lie algebras is extended to the case of Lie
superalgebras and studied in [27, 28, 30, 29, 23, 34].

Baer [4] defined the notion of capable group. Beyl et. al., [9] introduced the
epicenter $Z^{*}(G)$ of a group $G$, and they proved that a group $G$ is
capable if and only if $Z^{*}(G)=1$. Exterior center $Z^{\wedge}(G)$ of a
group $G$ is defined as $Z^{\wedge}(G)=\\{g\in G|~{}g\wedge
h=1,~{}\forall~{}h\in G\\}$. It was studied for the first time in [11], and
Ellis in [14] proved that $Z^{\wedge}(G)=Z^{*}(G)$. A Lie algebra $L$ is
capable if and only if $L\cong\frac{H}{Z(H)}$ for some Lie algebra $H$. The
notion of epicenter $Z^{*}(L)$ of a Lie algebra $L$ is given by Alamian et.
al., [1]. The non-abelian tensor product, and exterior product of Lie algebras
are defined, and some of the properties are studied by Ellis [12, 13, 14].
Recently, Niroomand et. al., [32] investigated the connection between
epicenter and exterior center of a finite dimensional Lie algebra and have
classified all capable Heisenberg Lie algebras. As an application they have
shown that there exists at least one capable Lie algebra of arbitrary corank.

A complete classification of nilpotent Lie superalgebras $L$ with $\dim L\leq
5$ over real and complex fields are known [2, 3, 20, 24]. All Lie
superalgebras up to dimension four over real field are known since 1978 [3],
and in this list there are no simple Lie superalgebras. Then nilpotent Lie
superalgebras up to dimension five over both real and complex fields are
classified by Hegazi [20]. Further, classification of all five dimensional Lie
supealgebras are done over complex field [24]. Recently Isabbel et. al., have
given classification of nilpotent Lie superalgebras $L$ with $\dim L\leq 5$
[2], which includes some missing five dimensional superalgebras in [20, 24].
Capable Lie superalgebras are defined and studied by Padhan et. al. [36]. In
this paper we find out all capable nilpotent Lie superalgebras over
$\mathbb{C}$ whose dimension are less than equal to five.

##  2\. Preliminaries

Let $\mathbb{Z}_{2}=\\{\bar{0},\bar{1}\\}$ be a field. A
$\mathbb{Z}_{2}$-graded vector space $V$ is simply a direct sum of vector
spaces $V_{\bar{0}}$ and $V_{\bar{1}}$, i.e., $V=V_{\bar{0}}\oplus
V_{\bar{1}}$. It is also referred as a superspace. We consider all vector
superspaces and superalgebras are over field $\mathbb{F}$ (characteristic of
$\mathbb{F}\neq 2,3$). Elements in $V_{\bar{0}}$ (resp. $V_{\bar{1}}$) are
called even (resp. odd) elements. Non-zero elements of $V_{\bar{0}}\cup
V_{\bar{1}}$ are called homogeneous elements. For a homogeneous element $v\in
V_{\sigma}$, with $\sigma\in\mathbb{Z}_{2}$ we set $|v|=\sigma$ as the degree
of $v$. A subsuperspace (or, subspace) $U$ of $V$ is a $\mathbb{Z}_{2}$-graded
vector subspace where $U=(V_{\bar{0}}\cap U)\oplus(V_{\bar{1}}\cap U)$. We
adopt the convention that whenever the degree function appears in a formula,
the corresponding elements are supposed to be homogeneous.

A Lie superalgebra (see [21, 26, 16, 5]) is a superspace $L=L_{\bar{0}}\oplus
L_{\bar{1}}$ with a bilinear mapping $[.,.]:L\times L\rightarrow L$ satisfying
the following identities:

  1. (1)

$[L_{\alpha},L_{\beta}]\subset L_{\alpha+\beta}$, for
$\alpha,\beta\in\mathbb{Z}_{2}$ ($\mathbb{Z}_{2}$-grading),

  2. (2)

$[x,y]=-(-1)^{|x||y|}[y,x]$ (graded skew-symmetry),

  3. (3)

$(-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0$
(graded Jacobi identity),

for all $x,y,z\in L$. Clearly $L_{\bar{0}}$ is a Lie algebra, and
$L_{\bar{1}}$ is a $L_{\bar{0}}$-module. If $L_{\bar{1}}=0$, then $L$ is just
Lie algebra, but in general, a Lie superalgebra is not a Lie algebra. A Lie
superalgebra $L$ is called abelian if $[x,y]=0$ for all $x,y\in L$. Lie
superalgebras without even part, i.e., $L_{\bar{0}}=0$, are abelian. A
subsuperalgebra (or subalgebra) of $L$ is a $\mathbb{Z}_{2}$-graded vector
subspace that is closed under bracket operation. The graded subalgebra
$[L,L]$, of $L$, is known as the derived subalgebra of $L$. A
$\mathbb{Z}_{2}$-graded subspace $I$ is a graded ideal of $L$ if
$[I,L]\subseteq I$. Consider $Z(L)=\\{z\in L:[z,x]=0\;\mbox{for all}\;x\in
L\\},$ is a graded ideal and it is called the center of $L$. A homomorphism
between superspaces $f:V\rightarrow W$ of degree $|f|\in\mathbb{Z}_{2}$, is a
linear map satisfying $f(V_{\alpha})\subseteq W_{\alpha+|f|}$ for
$\alpha\in\mathbb{Z}_{2}$. In particular, if $|f|=\bar{0}$, then the
homomorphism $f$ is called the homogeneous linear map of even degree. A Lie
superalgebra homomorphism $f:L\rightarrow M$ is a homogeneous linear map of
even degree such that $f([x,y])=[f(x),f(y)]$ holds for all $x,y\in L$. If $I$
is an ideal of $L$, the quotient Lie superalgebra $L/I$ inherits a canonical
Lie superalgebra structure such that the natural projection map becomes a
homomorphism. The notions of epimorphisms, isomorphisms, and automorphisms
have obvious meaning. Consider the descending central sequence $L^{1}\supset
L^{2}\supset\cdots\supset L^{k}\supset\cdots$ of a Lie superalgebra
$L=L_{\bar{0}}\oplus L_{\bar{1}}$ where $L^{1}=L,L^{k+1}=[L^{k},L]$ for all
$k\geq 1$. If $L^{k+1}=\\{0\\}$ for some $k+1$, with $L^{k}\neq 0$, then $L$
is called nilpotent with nilpotency class $k$. Note that the derived
subalgebra is $[L,L]=L^{2}.$

We simply write $\dim L=(m\mid n)$ for the superdimension of Lie superalgebra
$L$ throughout this article, where $\dim L_{\bar{0}}=m$ and $\dim
L_{\bar{1}}=n$. Additionally, $A(m\mid n)$ denotes an abelian Lie superalgebra
with $\dim A=(m\mid n)$. In this article Heisenberg Lie superalgebra is a Lie
superalgebra $L$ with $L^{2}=Z(L)$ and $\dim Z(L)=1$. According to the
homogeneous generator of $Z(L)$, Heisenberg Lie superalgebras can further
split into even or odd Heisenberg Lie superalgebras [38]. Further $L$ is
called a generalized Heisenberg Lie superalgebra of rank $(r\mid s)$ if
$L^{2}=Z(L)$ and $\dim Z(L)=(r\mid s)$. So a Heisenberg Lie super algebra is a
generalized Heisenberg superalgebra of rank one, i.e., $(1\mid 0)$ or $(0\mid
1)$. For more details on Heisenberg Lie superalgebra and their multipliers see
[18, 27, 29, 36, 33, 35, 37, 19]. Also, for more details on the Schur
multiplier and capability of Lie algebras, see [1, 6, 7, 8, 10, 12, 13, 14,
15, 17, 22, 25, 31, 32]. Now we list some useful known results for further
use.

######  Theorem 2.1.

[27, See Theorem 3.4]

$\dim\mathcal{M}(A(m\mid n))=\big{(}\frac{1}{2}(m^{2}+n^{2}+n-m)\mid
mn\big{)}.$

######  Theorem 2.2.

[27, See Theorem 4.2, 4.3] Every Heisenberg Lie superalgebra with an even
center has dimension $(2m+1\mid n)$, is isomorphic to
$H(m,n)=H_{\overline{0}}\oplus H_{\overline{1}}$, where

$H_{\overline{0}}=\left\langle x_{1},\ldots,x_{2m},z\mid[x_{i},x_{m+i}]=z,\
i=1,\ldots,m\right\rangle$

and

$H_{\overline{1}}=\left\langle y_{1},\ldots,y_{n}\mid[y_{j},y_{j}]=z,\
j=1,\ldots,n\right\rangle.$

Further,

$\dim\mathcal{M}(H(m,n))=\begin{cases}(2m^{2}-m+n(n+1)/2-1\mid
2mn)\quad\mbox{if}\;m+n\geq 2\\\ (0\mid 0)\quad\mbox{if}\;m=0,n=1\\\ (2\mid
0)\quad\mbox{if}\;m=1,n=0.\end{cases}$

Similarly the multiplier and cover for Heisenberg Lie superalgebra of odd
center is known.

######  Theorem 2.3.

[29, See Theorem 2.8] Every Heisenberg Lie superalgebra, with an odd center
has dimension $(m\mid m+1)$, is isomorphic to $H_{m}=H_{\overline{0}}\oplus
H_{\overline{1}}$, where

$H_{m}=\left\langle
x_{1},\ldots,x_{m},y_{1},\ldots,y_{m},z\mid[x_{j},y_{j}]=z,j=1,\ldots,m\right\rangle.$

Further,

$\dim\mathcal{M}(H_{m})=\begin{cases}(m^{2}\mid m^{2}-1)\quad\mbox{if}\;m\geq
2\\\ (1\mid 1)\quad\quad\mbox{if}\;m=1.\end{cases}$

For any two Lie superalgebras $H$ and $K$ the Lie superalgebra direct sum
$H\oplus K$ is a Lie superalgebra with natural grading $(H\oplus
K)_{\alpha}=H_{\alpha}\oplus K_{\alpha}$ where $\alpha\in\mathbb{Z}_{2}$. If
$\mathcal{M}(H)$ and $\mathcal{M}(K)$ are known, then $\mathcal{M}(H\oplus K)$
is given by the following result.

######  Theorem 2.4.

[27, See Theorem 3.9] For Lie superalgebras $H$ and $K$,

$\mathcal{M}(H\oplus
K)\cong\mathcal{M}(H)\oplus\mathcal{M}(K)\oplus(H/H^{2}\otimes K/K^{2}).$

A Lie superalgebra $L$ is capable if there exists a Lie superalgebra $H$ such
that $L\cong H/Z(H)$. The epicenter $Z^{*}(L)$ of $L$ is the smallest graded
ideal in $L$ such that $L/Z^{*}(L)$ is capable [36]. Infact $Z^{*}(L)$ is a
central ideal.

######  Lemma 2.5.

[36, Lemma 5.2] A Lie superalgebra $L$ is capable if and only if $Z^{*}(L)=0$.

######  Lemma 2.6.

[36, Theorem 5.8] $N\subseteq Z^{*}(L)$ if and only if the natural map
$\mathcal{M}(L)\longrightarrow\mathcal{M}(L/N)$ is a monomorphism.

######  Theorem 2.7.

[36, Theorem 6.7] $H_{m}$ is capable if and only if $m=1$.

######  Theorem 2.8.

[36, Theorem 6.3] $A(m\mid n)$ is capable if and only if $m=0,n=1$ or $m+n\geq
2$.

######  Theorem 2.9.

[36, Theorem 6.4] $H(m,n)$ is capable if and only if $m=1,n=0$.

##  3\.  The Main Results

In this section we define the partially capable Lie superalgebra, as a result
we classify all capable nilpotent Lie superalgebras $L$ from [20, 2] with
$\dim L\leq 5$. We organise all nilpotent Lie superalgebras from [20, 2]
according to the derived algebras dimension.

######  Theorem 3.1.

Let $L$ be a finite-dimensional nilpotent Lie superalgebra over $\mathbb{C}$.

  1. (1)

The following are the Lie superalgebras with $\dim L^{2}=1:$

$L^{1}_{(1,1)}=\\{\left\langle x_{1}\right\rangle\oplus\left\langle
x_{2}\right\rangle|~{}[x_{2},x_{2}]=x_{1}\\}.$
$L^{2}_{(3,0)}=\\{\left\langle
x_{1},x_{2},x_{3}\right\rangle\oplus\left\langle
0\right\rangle|~{}[x_{1},x_{2}]=x_{3}\\}.$
$(L^{1}_{(1,1)})^{\prime}$ Derived from $L^{1}_{(1,1)}=\\{\left\langle
x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{1},x_{2}]=0\\}$.
$L^{3}_{(1,2)}=\\{\left\langle x_{1}\right\rangle\oplus\left\langle
x_{2},x_{3}\right\rangle|~{}[x_{1},x_{2}]=x_{3}\\}.$
$L^{4}_{(1,2)}=\\{\left\langle x_{1}\right\rangle\oplus\left\langle
x_{2},x_{3}\right\rangle|~{}[x_{2},x_{2}]=x_{1},[x_{3},x_{3}]=x_{1}\\}.$
$L^{7}_{(3,1)}=\\{\left\langle
x_{1},x_{2},x_{3}\right\rangle\oplus\left\langle
x_{4}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{4},x_{4}]=x_{3}\\}.$
$L^{14}_{(1,3)}=\\{\left\langle x_{1}\right\rangle\oplus\left\langle
x_{2},x_{3},x_{4}\right\rangle|~{}[x_{2},x_{2}]=x_{1},[x_{3},x_{3}]=x_{1}],[x_{4},x_{4}]=x_{1}\\}.$
$L^{16}_{(5,0)}=\\{\left\langle
x_{1},x_{2},x_{3},x_{4},x_{5}\right\rangle\oplus\left\langle
0\right\rangle|~{}[x_{1},x_{2}]=x_{5},[x_{3},x_{4}]=x_{5}\\}.$
$L^{25}_{(3,2)}=\\{\left\langle
x_{1},x_{2},x_{3}\right\rangle\oplus\left\langle
x_{4},x_{5}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{4},x_{4}]=x_{3},[x_{5},x_{5}]=x_{3}\\}.$
$L^{26}_{(3,2)}=\\{\left\langle
x_{1},x_{2},x_{3}\right\rangle\oplus\left\langle
x_{4},x_{5}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{4},x_{4}]=x_{3},[x_{5},x_{5}]=-x_{3}\\}.$
$L^{40}_{(1,4)}=\\{\left\langle x_{1}\right\rangle\oplus\left\langle
x_{2},x_{3},x_{4},x_{5}\right\rangle|~{}[x_{2},x_{2}]=x_{1},[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{1},[x_{5},x_{5}]=x_{1}\\}.$
$L^{41}_{(1,4)}=\\{\left\langle x_{1}\right\rangle\oplus\left\langle
x_{2},x_{3},x_{4},x_{5}\right\rangle|~{}[x_{2},x_{2}]=x_{1},[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{1},[x_{5},x_{5}]=-x_{1}\\}.$
$L^{42}_{(1,4)}=\\{\left\langle x_{1}\right\rangle\oplus\left\langle
x_{2},x_{3},x_{4},x_{5}\right\rangle|~{}[x_{2},x_{2}]=x_{1},[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=-x_{1},[x_{5},x_{5}]=-x_{1}\\}.$

  2. (2)

The following are the Lie superalgebras with $\dim L^{2}=2:$

$L^{6}_{(4,0)}=\\{\left\langle
x_{1},x_{2},x_{3},x_{4}\right\rangle\oplus\left\langle
0\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{1},x_{3}]=x_{4}\\}.$
$L^{8}_{(2,2)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4}\right\rangle|~{}[x_{1},x_{3}]=x_{4},[x_{3},x_{3}]=x_{2}\\}.$
$L^{9}_{(2,2)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2},[x_{3},x_{4}]=\frac{1}{2}(x_{1}+x_{2})\\}.$
$L^{10}_{(2,2)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2}\\}.$
$L^{11}_{(2,2)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2},[x_{3},x_{4}]=x_{1}-x_{2}\\}.$
$L^{12}_{(2,2)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2},[x_{3},x_{4}]=x_{1}\\}.$
$L^{13}_{(1,3)}=\\{\left\langle x_{1}\right\rangle\oplus\left\langle
x_{2},x_{3},x_{4}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{1},x_{3}]=x_{4}\\}.$
$L^{17}_{(5,0)}=\\{\left\langle
x_{1},x_{2},x_{3},x_{4},x_{5}\right\rangle\oplus\left\langle
0\right\rangle|~{}[x_{1},x_{2}]=x_{4},[x_{1},x_{3}]=x_{5}\\}.$
$L^{18}_{(5,0)}=\\{\left\langle
x_{1},x_{2},x_{3},x_{4},x_{5}\right\rangle\oplus\left\langle
0\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{1},x_{3}]=x_{4},[x_{2},x_{5}]=x_{4}\\}.$
$L^{22}_{(4,1)}=\\{\left\langle
x_{1},x_{2},x_{3},x_{4}\right\rangle\oplus\left\langle
x_{5}\right\rangle|~{}[x_{4},x_{2}]=x_{1},[x_{4},x_{3}]=x_{3},[x_{5},x_{5}]=x_{1}\\}.$
$L^{23}_{(4,1)}=\\{\left\langle
x_{1},x_{2},x_{3},x_{4}\right\rangle\oplus\left\langle
x_{5}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{4},x_{2}]=x_{1},[x_{5},x_{5}]=x_{3}\\}.$
$L^{24}_{(3,2)}=\\{\left\langle
x_{1},x_{2},x_{3}\right\rangle\oplus\left\langle
x_{4},x_{5}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{1},x_{5}]=x_{4}\\}.$
$L^{27}_{(3,2)}=\\{\left\langle
x_{1},x_{2},x_{3}\right\rangle\oplus\left\langle
x_{4},x_{5}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{1},x_{5}]=x_{4},[x_{5},x_{5}]=x_{3}\\}.$
$L^{28}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{1},x_{4}]=x_{3},[x_{4},x_{4}]=x_{2},[x_{5},x_{5}]=x_{2}\\}.$
$L^{29}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{1},x_{4}]=x_{3},[x_{4},x_{4}]=x_{2},[x_{5},x_{5}]=-x_{2}\\}.$
$L^{30}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2},[x_{5},x_{5}]=x_{1}+x_{2}\\}.$
$L^{31}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2},[x_{5},x_{5}]=-(x_{1}+x_{2})\\}.$
$L^{32}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2},[x_{5},x_{5}]=x_{1}-x_{2}\\}.$
$L^{33}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2},[x_{3},x_{5}]=x_{2}\\}.$
$L^{34}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2},[x_{3},x_{5}]=x_{1}+x_{2}\\}.$
$L^{35}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2},[x_{3},x_{5}]=x_{1}-x_{2}\\}.$
$L^{36}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{3},x_{5}]=x_{2},[x_{4},x_{5}]=x_{1}\\}.$
$L^{37}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{3},x_{4}]=x_{1},[x_{4},x_{5}]=x_{2}\\}.$
$L^{38}_{(1,4)}=\\{\left\langle x_{1}\right\rangle\oplus\left\langle
x_{2},x_{3},x_{4},x_{5}\right\rangle|~{}[x_{1},x_{3}]=x_{2},[x_{1},x_{5}]=x_{4}\\}.$
$L^{43}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{1},x_{5}]=x_{3},[x_{4},x_{5}]=x_{2}\\}.$

  3. (3)

The following are the Lie superalgebras with $\dim L^{2}=3:$

$L^{19}_{(5,0)}=\\{\left\langle
x_{1},x_{2},x_{3},x_{4}\right\rangle\oplus\left\langle
x_{5}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{1},x_{3}]=x_{4},[x_{2},x_{3}]=x_{5}\\}.$
$L^{20}_{(5,0)}=\\{\left\langle
x_{1},x_{2},x_{3},x_{4}\right\rangle\oplus\left\langle
x_{5}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{1},x_{3}]=x_{4},[x_{1},x_{4}]=x_{5}.\\}$
$L^{21}_{(5,0)}=\\{\left\langle
x_{1},x_{2},x_{3},x_{4}\right\rangle\oplus\left\langle
x_{5}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{1},x_{3}]=x_{4},[x_{1},x_{4}]=x_{5}=[x_{2},x_{3}]\\}.$
$L^{39}_{(1,4)}=\\{\left\langle x_{1}\right\rangle\oplus\left\langle
x_{2},x_{3},x_{4},x_{5}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{1},x_{3}]=x_{4},[x_{1},x_{4}]=x_{5}=[x_{3},x_{4}]\\}.$
$L^{44}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{1},x_{5}]=x_{3},[x_{2},x_{4}]=x_{3},[x_{4},x_{5}]=-x_{1},[x_{5},x_{5}]=2x_{2}\\}.$
$L^{45}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{1},x_{4}]=x_{3},[x_{1},x_{5}]=x_{4},[x_{5},x_{5}]=x_{2}\\}.$
$L^{46}_{(2,3)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4},x_{5}\right\rangle|~{}[x_{1},x_{4}]=x_{3},[x_{1},x_{5}]=x_{4},[x_{3},x_{5}]=-x_{2},[x_{4},x_{4}]=x_{2}\\}.$

Consider the Heisenberg Lie superalgebra $H_{2}$ with a homogeneous basis
$\\{x_{1},x_{2};y_{1},y_{2},z\\}$ and the only non-vanishing brackets are
$[x_{1},y_{1}]=z=[x_{2},y_{2}]$. By Theorem $2.7$ it is not capable but the
even part of this Lie supealgebra is capable Lie algebra as even part is
isomorphic to $H(1,0)/Z(H)$. This leads to following definition.

######  Definition 3.2.

We call a Lie superalgebra $L=L_{\bar{0}}\oplus L_{\bar{1}}$ partially capable
if the even part $L_{\bar{0}}$ is a capable Lie algebra.

######  Theorem 3.3.

Let $L=L_{\bar{0}}\oplus L_{\bar{1}}$ be a Lie superalgebra. Then
$Z^{*}(L_{\bar{0}})\subseteq Z^{*}(L)$.

###### Proof.

Consider the two homomorphism,
$\phi:L_{\bar{0}}/Z^{*}(L_{\bar{0}})\longrightarrow L$ which is given by
$\phi(l+Z^{*}(L_{\bar{0}}))=l$, and the canonical projection
$\pi:L\longrightarrow L/Z^{*}(L)$. The composition map
$\psi=\pi\phi:L_{\bar{0}}/Z^{*}(L_{\bar{0}})\longrightarrow L/Z^{*}(L)$
defined by $\psi(l+Z^{*}(L_{\bar{0}}))=l+Z^{*}(L)$ is also a homomorphism.
Thus $Z^{*}(L_{\bar{0}})\subseteq Z^{*}(L)$. ∎

The following is the immediate consequence of the above theorem.

######  Corollary 3.4.

If $L=L_{\bar{0}}\oplus L_{\bar{1}}$ is a capable Lie superalgebra, then the
even part is also capable.

######  Remark 1.

Every capable Lie superalgebra $L$ is partially capable but converse is not
true. However if $L$ is not partially capable then $L$ is not capable.

From now onwards $L$ denote a nilpotent Lie superalgebra with $\dim L\leq 5$.

###  3.1. The Capability of $L$ with $\dim L^{2}=1$

In this case it has been proved that $L$ is direct sum of Heisenberg Lie
superalgebra and abelian Lie superalgebra. Also all capable Lie superalgebras
having derived subalgebras dimension at most one have been classified (see
[36]).

######  Proposition 3.5.

[29, Proposition 3.4][36, Theorem 6.9] Let $L$ be a nilpotent Lie superalgebra
of dimension $(k\mid l)$ with $\dim L^{2}=(r\mid s)$, where $r+s=1$. If
$r=1,s=0$ then $L\cong H(m,n)\oplus A(k-2m-1\mid l-n)$ for $m+n\geq 1$. If
$r=0,s=1$ then $L\cong H_{m}\oplus A(k-m\mid l-m-1)$. Moreover, $L$ is capable
if and only if either $L\cong H(1,0)\oplus A(k-3\mid l)$ or $L\cong
H_{1}\oplus A(k-1\mid l-2)$.

######  Proposition 3.6.

$L^{1}_{(1,1)},L^{3}_{(1,2)},L^{4}_{(1,2)},L^{14}_{(1,3)},L^{40}_{(1,4)},L^{41}_{(1,4)},L^{42}_{(1,4)}$
are non-capable Lie superalgebras.

###### Proof.

Even part of all these Lie superalgebras are $A(1\mid 0)$. So by Theorem 2.8,
all these superalgebras are not partially capable. ∎

######  Proposition 3.7.

$L^{7}_{(3,1)},~{}L^{16}_{(5,0)},~{}L^{25}_{(3,2)},(L^{1}_{(1,1)})^{2}$ are
non-capable Lie superalgebras.

###### Proof.

We have $L^{7}_{(3,1)}\cong H(1,1),~{}L^{16}_{(5,0)}\cong H(2,0)$ and
$L^{25}_{(3,2)}\cong H(1,2)$. Now, it follows from Theorem 2.9 that
$L^{7}_{(3,1)},~{}L^{16}_{(5,0)},~{}L^{25}_{(3,2)}$ are non-capable. Finally
as $(L^{1}_{(1,1)})^{2}\cong H(0,1)\oplus A(1\mid 0)$, using Proposition 3.5
$(L^{1}_{(1,1)})^{2}$ is non-capable. ∎

From the above, we can draw the following conclusion.

######  Theorem 3.8.

Let $L$ be a nilpotent Lie superalgebra with $\dim L\leq 5$ and $\dim
L^{2}=1$. Then $L$ is capable if and only if $L\cong L^{2}_{(3,0)}=H(1\mid
0)$.

###### Proof.

The proof follows from Theorem 2.9, Propositio 3.6 and Proposition 3.7. ∎

###  3.2. The Capability $L$ with $\dim L^{2}=2$.

The following results will be useful in this section.

######  Proposition 3.9.

[31, Proposition 2.6] Let $L$ be a finite dimensional capable nilpotent Lie
algebra with nilpotency class two. Then $5\leq\dim L\leq 7.$

######  Proposition 3.10.

[35, Corollary 3.11] Let $L$ be a Lie superalgebra of dimension $(m\mid n)$
with nilpotency class two such that $\dim(L/Z(L))=(r\mid s)$ and
$\dim(L^{2})=\frac{1}{2}[(r+s)^{2}+(s-r)]$, then $L$ is capable.

In [27] Nayak calculated the dimension of Schur multiplier of some nilpotent
Lie superalgebras.

######  Proposition 3.11.

[27, Theorems 3.2, 3.6] $\dim\mathcal{M}(L^{24}_{(3,2)})=6$,
$\dim\mathcal{M}(L^{l}_{(2,2)})=1$ (with $l=9,~{}10,~{}11,~{}12)$ and
$\dim\mathcal{M}(L^{k}_{(2,3)})=4$ (with
$k=28,~{}30,~{}31,~{}32,~{}33,~{}34,~{}35)$.

The dimensions of Schur multipliers of the nilpotent Lie superalgebras
$L^{26}_{(3,2)},L^{27}_{(3,2)},L^{29}_{(2,3)},L^{36}_{(2,3)},L^{37}_{(2,3)}$
are not computed in [27]. We give the rest of the computations in the
following lemma by using the method described by Hardy and Stitzinger in [17].

######  Proposition 3.12.

$\dim\mathcal{M}(L^{26}_{(3,2)})=7,\dim\mathcal{M}(L^{27}_{(3,2)})=\dim\mathcal{M}(L^{29}_{(2,3)})=\dim\mathcal{M}(L^{36}_{(2,3)})=\dim\mathcal{M}(L^{44}_{(2,3)})=4,\dim\mathcal{M}(L^{37}_{(2,3)})=\dim\mathcal{M}(L^{43}_{(2,3)})=5,\dim\mathcal{M}(L^{45}_{(2,3)})=\dim\mathcal{M}(L^{46}_{(2,3)})=3.$

###### Proof.

Consider

$L^{27}_{(3,2)}=\\{\left\langle
x_{1},x_{2},x_{3}\right\rangle\oplus\left\langle
x_{4},x_{5}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{1},x_{5}]=x_{4},[x_{5},x_{5}]=x_{3}\\}.$

To compute the multiplier of $L^{27}_{(3,2)},$ we start with the following
setting

$[x_{1},x_{2}]=x_{3}+m_{1}$ | $[x_{1},x_{3}]=m_{2}$ | $[x_{1},x_{4}]=m_{3}$
---|---|---
$[x_{1},x_{5}]=x_{4}+m_{4}$ | $[x_{2},x_{3}]=m_{5}$ | $[x_{2},x_{4}]=m_{6}$
$[x_{2},x_{5}]=m_{7}$ | $[x_{3},x_{4}]=m_{8}$ | $[x_{3},x_{5}]=m_{9}$
$[x_{4},x_{5}]=m_{10}$ | $[x_{4},x_{4}]=m_{11}$ |  $[x_{5},x_{5}]=x_{3}+m_{12}$.

Putting $x^{\prime}_{3}=x_{3}+m_{1},~{}x^{\prime}_{4}=x_{4}+m_{4}$, then
$m_{1}=m_{4}=0$. By applying the graded Jacobi identity on all possible
triples, we have

$\displaystyle m_{2}-2m_{10}$
$\displaystyle=(-1)^{|x_{1}||x_{5}|}[x_{1},[x_{5},x_{5}]]+(-1)^{|x_{1}||x_{5}|}[x_{5},[x_{5},x_{1}]]+(-1)^{|x_{5}||x_{5}|}[x_{5},[x_{1},x_{5}]]=0$
$\displaystyle m_{5}$
$\displaystyle=(-1)^{|x_{2}||x_{5}|}[x_{2},[x_{5},x_{5}]]+(-1)^{|x_{2}||x_{5}|}[x_{5},[x_{5},x_{2}]]+(-1)^{|x_{5}||x_{5}|}[x_{5},[x_{2},x_{5}]]=0$
$\displaystyle-m_{6}-m_{9}$
$\displaystyle=(-1)^{|x_{1}||x_{5}|}[x_{1},[x_{2},x_{5}]]+(-1)^{|x_{1}||x_{2}|}[x_{2},[x_{5},x_{1}]]+(-1)^{|x_{5}||x_{2}|}[x_{5},[x_{1},x_{2}]]=0$
$\displaystyle-m_{8}$
$\displaystyle=(-1)^{|x_{1}||x_{5}|}[x_{1},[x_{3},x_{5}]]+(-1)^{|x_{3}||x_{1}|}[x_{3},[x_{5},x_{1}]]+(-1)^{|x_{5}||x_{3}|}[x_{5},[x_{1},x_{3}]]=0$
$\displaystyle-m_{11}$
$\displaystyle=(-1)^{|x_{1}||x_{5}|}[x_{1},[x_{4},x_{5}]]+(-1)^{|x_{4}||x_{1}|}[x_{4},[x_{5},x_{1}]]+(-1)^{|x_{4}||x_{5}|}[x_{5},[x_{1},x_{4}]]=0$
$\displaystyle m_{9}$ $\displaystyle=[x_{5},[x_{5},x_{5}]]=0.$

Hence $\mathcal{M}(L^{27}_{(3,2)})=\left\langle
m_{2},m_{3},m_{7},m_{12}\right\rangle$ and
$\dim\mathcal{M}(L^{27}_{(3,2)})=4$. Similarly the rest can be proven. ∎

######  Proposition 3.13.

[35, Proposition 3.1] Let $L$ be a finite dimensional nilpotent Lie
superalgebra of nilpotency class two. Then $L=H\oplus K$ and
$Z^{*}(L)=Z^{*}(H)$, where $K$ is abelian and $H$ is a generalized Heisenberg
Lie superalgebra.

######  Lemma 3.14.

If $L$ is finite dimensional, then $N\subseteq Z^{*}(L)$ if and only if
$\dim\mathcal{M}(L/N)=\dim\mathcal{M}(L)+\dim(N\cap L^{2})$.

###### Proof.

As we know
$\mathcal{M}(L)\xrightarrow{\sigma}\mathcal{M}(L/N)\xrightarrow{\varphi}N\cap
L^{2}\longrightarrow 0$ is exact sequence, thus $\mathcal{M}(L/N)\cong
ker\varphi\oplus Im\varphi=Im\sigma\oplus N\cap L^{2}$. We can observe that if
$\sigma$ is a monomorphism, then
$\dim\mathcal{M}(L/N)=\dim\mathcal{M}(L)+\dim(N\cap L^{2})$. Therefore, the
proof follows from Lemma 2.6. ∎

######  Corollary 3.15.

Let $L=H(t,q)\oplus H(m,n)$. If $t+q\geq 2$, or $t=0,q=1$ then $L$ is non-
capable for all $m+n\geq 1$.

###### Proof.

If we use Proposition 2.4, then

$\mathcal{M}(L)=\mathcal{M}(H(t,q))\oplus\mathcal{M}(H(m,n))\oplus(H(t,q)/H^{2}(t,q)\otimes
H(m,n)/H^{2}(m,n)).$

Now,

$\mathcal{M}(L/H^{2}(t,q))=\mathcal{M}(H(t,q)/H^{2}(t,q))\oplus\mathcal{M}(H(m,n))\oplus(H(t,q)/H^{2}(t,q)\otimes
H(m,n)/H^{2}(m,n)).$

Now from Theorem 2.1 and Theorem 2.2, we have

$\mathcal{M}(L)=\mathcal{M}(L/H^{2}(t,q))-(1\mid 0).$

Thus, using Proposition 3.14, $H^{2}(t,q)\subseteq Z^{*}(L)$, and $L$ is non-
capable. ∎

######  Example 3.16.

Let $L=L^{10}_{(2,2)}=\\{\left\langle
x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2}\\}.$
Clearly $L\cong H(0,1)\oplus H(0,1)$ and hence $L$ is non-capable by corollary
3.15.

######  Theorem 3.17.

$L^{6}_{(4,0)},L^{13}_{(1,3)},L^{17}_{(5,0)},L^{18}_{(5,0)},L^{22}_{(4,1)},L^{23}_{(4,1)},L^{38}_{(1,4)}$
are non-capable Lie superalgebras.

###### Proof.

One can observe that $L^{6}_{(4,0)},L^{22}_{(4,1)},L^{23}_{(4,1)}$ are not
partially capable by Proposition 3.9. Fom Theorem 2.8,
$L^{38}_{(1,4)},L^{13}_{(1,3)}$ are not partially capable . ∎

######  Theorem 3.18.

$L^{9}_{(2,2)},L^{11}_{(2,2)},L^{12}_{(2,2)}$ are non-capable Lie
superalgebras.

###### Proof.

Consider $L=L^{9}_{(2,2)}=\\{\left\langle
x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4}\right\rangle|~{}[x_{3},x_{3}]=x_{1},[x_{4},x_{4}]=x_{2},[x_{3},x_{4}]=\frac{1}{2}(x_{1}+x_{2})\\}$,
then $Z(L)=L^{2}=\left\langle x_{1},x_{2}\right\rangle$. Consider $N=L^{2}$,
then by Theorem 2.1, $\dim\mathcal{M}(L/N)=\dim\mathcal{M}(A(0\mid 2))=(3\mid
0)=3$. From Proposition3.11, $\dim\mathcal{M}(L)=1$, then
$3=\dim\mathcal{M}(L/N)=\dim\mathcal{M}(L)+\dim N\cap L^{2}=2+1$, so by using
Lemma 3.14, we have $L^{2}\subseteq Z^{*}(L)$. Hence $L=L^{9}_{(2,2)}$ is non-
capable. Similarly, we one can check that $L^{11}_{(2,2)},L^{12}_{(2,2)}$ are
non-capable. ∎

######  Theorem 3.19.

$L^{30}_{(2,3)},L^{31}_{(2,3)},L^{32}_{(2,3)},L^{33}_{(2,3)},L^{34}_{(2,3)},$
$L^{26}_{(3,2)},L^{27}_{(3,2)},L^{29}_{(2,3)},L^{35}_{(2,3)}$ and
$L^{36}_{(2,3)}$ are non-capable Lie superalgebras.

###### Proof.

The proof is similar to Theorem 3.18. ∎

######  Theorem 3.20.

The only capable Lie superalgebras with two dimension derived subalgebras are
$L^{8}_{(2,2)},L^{24}_{(3,2)},L^{28}_{(2,3)},L^{17}_{(5,0)},L^{18}_{(5,0)},L^{37}_{(2,3)},L^{43}_{(2,3)}$.

###### Proof.

As $L^{8}_{(2,2)}=\\{\left\langle x_{1},x_{2}\right\rangle\oplus\left\langle
x_{3},x_{4}\right\rangle|~{}[x_{1},x_{3}]=x_{4},[x_{3},x_{3}]=x_{2}\\}$ is
nilpotent Lie superalgebra of class two with $\dim(L/Z(L))=(1\mid 1)$ and
$\frac{1}{2}[(1+1)^{2}+(1-1)]=2=\dim(L^{2})$, then $L^{8}_{(2,2)}$ is capable
by Proposition 3.10. For $L=L^{24}_{(3,2)}=\\{\left\langle
x_{1},x_{2},x_{3}\right\rangle\oplus\left\langle
x_{4},x_{5}\right\rangle|~{}[x_{1},x_{2}]=x_{3},[x_{1},x_{5}]=x_{4}\\},$ we
have $Z(L)=L^{2}=\\{\left\langle x_{3},x_{4}\right\rangle\\}$, and by
Proposition 3.11, $\dim\mathcal{M}(L)=6.$ Now, let $N$ be a $(1\mid
0)$-dimensional central graded ideal generated by $x_{3}$. Then using Theorems
2.1,2.3 and 2.4, we have

$\displaystyle\dim\mathcal{M}(L/N)$ $\displaystyle=\dim\mathcal{M}(H_{1}\oplus
A(1\mid 0))$ $\displaystyle=\dim\mathcal{M}(H_{1})+\dim\mathcal{M}(A(1\mid
0))+\dim(H_{1}/H^{2}_{1}\otimes A(1\mid 0))$ $\displaystyle=(1\mid 1)+(0\mid
0)+(1\mid 1)\times(1\mid 0)$ $\displaystyle=4.$

This shows that $\dim\mathcal{M}(L/N)=4\neq\dim\mathcal{M}(L)+\dim(N\cap
L^{2})=6+1$. Hence from Lemma 3.14, $N\nsubseteq Z^{*}(L)$. Similarly, if we
consider a $(0\mid 1)$-dimensional central graded ideal $K$ which is generated
by $x_{4}$, then $\dim\mathcal{M}(L/K)=5\neq\dim\mathcal{M}(L)+\dim(K\cap
L^{2})=6+1$. Thus $K\nsubseteq Z^{*}(L)$. Therefore, by Lemma 2.5,
$L=L^{24}_{(3,2)}$ is capable. Using same technique one can check that
$L^{28}_{(2,3)},~{}L^{37}_{(2,3)},L^{43}_{(2,3)}$ are capable. Finally, from
[39, Theorems 2.6, 1.1], we have
$L^{17}_{(5,0)}=L_{5,8},~{}L^{18}_{(5,0)}=L_{5,5}$ are capable Lie algebras. ∎

######  Remark 2.

The technique used to prove $L^{8}_{(2,2)}$ is capable, does not work for
$L^{24}_{(3,2)},L^{28}_{(2,3)},L^{37}_{(2,3)}$, so we use a different one.

###  3.3. The Capability of all nilpotent Lie superalgebras $L$ with $\dim
L\leq 5$ and $\dim L^{2}\leq 3.$

######  Lemma 3.21.

$L^{39}_{(1,4)}$ is non-capable Lie superalgebra.

###### Proof.

As the even part of $L^{39}_{(1,4)}$ is $A(1\mid 0)$, then by Theorem 2.8,
$L^{39}_{(1,4)}$ is not partially capable. ∎

######  Lemma 3.22.

$L^{44}_{(2,3)},L^{45}_{(2,3)},~{}L^{46}_{(2,3)}$ are capable Lie
superalgebras.

###### Proof.

Using Proposition 3.11, and proceeding as inTheorem 3.20, for every central
ideal $N$ of $L^{i}_{(2,3)}$ we have,
$\dim\mathcal{M}(L^{i}_{(2,3)}/N)\neq\dim\mathcal{M}(L^{i}_{(2,3)})+\dim N\cap
L^{i^{2}}_{(2,3)},$ for $i=44,45,46.$ Thus, $L^{45}_{(2,3)},~{}L^{46}_{(2,3)}$
are capable Lie superalgebras.. ∎

######  Lemma 3.23.

$L^{19}_{(5,0)},~{}L^{20}_{(5,0)},~{}L^{21}_{(5,0)}$ are capable Lie
superalgebras.

###### Proof.

From [39, Theorems 2.6, 1.1],
$L^{19}_{(5,0)}=L_{5,9},~{}L^{20}_{(5,0)}=L_{5,7},~{}L^{21}_{(5,0)}=L_{5,6}$
are capable Lie algebras. ∎

Finally following is the list all capable nilpotent Lie superalgebras $L$ with
$\dim L\leq 5.$

######  Theorem 3.24.

The only capable nilpotent Lie superalgebras with dimension at most 5 are

$L^{2}_{(3\mid
0)},~{}L^{8}_{(2,2)},~{}L^{17}_{(5,0)},~{}L^{18}_{(5,0)},~{}L^{19}_{(5,0)},~{}L^{20}_{(5,0)},~{}L^{21}_{(5,0)},~{}L^{24}_{(3,2)},~{}L^{28}_{(2,3)},~{}L^{37}_{(2,3)},~{}L^{43}_{(2,3)},~{}L^{44}_{(2,3)},~{}L^{45}_{(2,3)},~{}L^{46}_{(2,3)}.$

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# Learning Multiple Categories on Deep Convolution Networks
Why deep convolution networks are effective in solving large recognition
problems

Mohamed Hajaj Duncan Gillies
Department of Computing, Imperial College London

## Abstract

Deep convolution networks have proved very successful with big datasets such
as the 1000-classes ImageNet. Results show that the error rate increases
slowly as the size of the dataset increases. Experiments presented here may
explain why these networks are very effective in solving big recognition
problems. If the big task is made up of multiple smaller tasks, then the
results show the ability of deep convolution networks to decompose the complex
task into a number of smaller tasks and to learn them simultaneously. The
results show that the performance of solving the big task on a single network
is very close to the average performance of solving each of the smaller tasks
on a separate network. Experiments also show the advantage of using task
specific or category labels in combination with class labels.

##  1 Introduction

Since 2012 and starting with the introduction of Alex Krizhevsky model
[Krizhevsky et al., 2012], all the winners [Simonyan and Zisserman, 2014,
Szegedy et al., 2015, He et al., 2016a, Hu et al., 2017] of the classification
competition part of the ImageNet challenge were deep convolution networks. The
ImageNet dataset is made up of 1000-classes which is much bigger in size
compared to earlier image recognition benchmarks such as the MNIST dataset,
CIFAR10 dataset, CIFAR100 dataset etc. The experiments presented here
investigate why these networks are very efficient in solving such big image
recognition problems.

The first set of experiments measure how the performance of deep convolution
networks changes as the number of classes increases. Multiple datasets with
different sizes were randomly sampled from ImageNet, and the test error rate
was measured at each size. For each size, multiple datasets were sampled and
tested to reduce variance in the results. The results reveal that the error
rate increases at a much lower rate compared to the increase in the number of
classes. It is interesting to ask why the performance of these networks is
resilient against increasing the number of classes.

The main experiments in this study may provide some insight into why deep
convolution networks are very effective in solving large recognition problems.
In this experiment a dataset made up of multiple categories (sampled from
ImageNet) were used to train a deep convolution network, and the results were
compared to the results of learning each small category on a separate network.
The performance of the single network trained on all categories was very close
to the average performance of the other networks trained on single categories.
This means the network was able to break down the main task into smaller tasks
and learn them simultaneously with very small drop in performance. The network
has the remarkable inherent ability to recognize that a certain complex task
(like ImageNet) is made up of multiple smaller tasks, and without any hint, is
able to discover those smaller tasks and learn them simultaneously to solve
the large main task.

Finally, using the data available from the main experiment, an extra
experiment showed that using both the class and category labels of the image
outperformed the standard labeling scheme of only using the class labels.

##  2 Experiments

###  2.1 Performance vs number of classes

The first part of this study tries to measure the performance of deep
convolution networks in relation to the number of classes in the image
dataset. Datasets with 5 different sizes were randomly sampled from the
1000-classes ImageNet. The number of classes of these datasets were 10, 50,
100, 500, and 1000 classes. To reduce variance in the results, multiple
datasets were sampled at each size (except the last one) and the average
performance is reported. The number of datasets sampled at each size were 10,
10, 5, 2, 1 respectively. The ImageNet dataset used here is the one used in
the ILSVRC 2015 competition, and out of the 1000 classes, 891 of them had the
maximum number of training images of 1300 images per class. All the classes
sampled here belong to these 891, and therefore all the classes used in this
experiment had 1300 training images (expect the one using the entire ImageNet
of 1000 classes). The ImageNet validation set was used to sample the different
test sets, and each class has 50 test images.

Table (1) shows the structure of the 34-layer residual network used in this
experiment. All datasets used the same structure with the only difference
being the number of neurons in the output layer. Instead of using the standard
data augmentation technique [Simonyan and Zisserman, 2014] that is usually
used with deep residual networks, the more aggressive augmentation method
(usually used with the inception model [Szegedy et al., 2015]) is used here.
The reason for this choice is because it performs slightly better with smaller
datasets (probably because it is more effective in reducing overfitting). The
size of the cropped square is chosen randomly to be between 8% and 100% of the
size of the maximum square in the image, and the aspect ratio is changed
randomly to be between $3/4$ and $4/3$.

The method in [He et al., 2015] was used to initialize the network weights,
and the standard color augmentation method in [Krizhevsky et al., 2012] was
used to simulate variance in illumination and intensity that exists in natural
images. The RMSProp optimization method was used instead of gradient decent
with momentum to update the network parameters, using a decay value of 0.999
to calculate the running average per parameter. RMSProp produces similar
results to ADAM [Kingma and Ba, 2014] with the advantage of using a single
running average per parameter instead of 2.

$\mathrm{\genfrac{}{}{0.0pt}{}{output}{size}}$ | 34 Layers
---|---
$112\times 112$ |  $conv,\ 7\times 7$, stride 2 64
$56\times 56$ |  $max\ pool\ 3\times 3$, stride 2
$\left[\genfrac{}{}{0.0pt}{}{conv,\ 3\times 3,\hskip 5.69046pt64}{conv,\
3\times 3,\hskip 5.69046pt64}\right]\times 3$
$28\times 28$ |  $max\ pool\ 3\times 3$, stride 2
$\left[\genfrac{}{}{0.0pt}{}{conv,\ 3\times 3,\hskip 5.69046pt128}{conv,\
3\times 3,\hskip 5.69046pt128}\right]\times 4$
$14\times 14$ |  $max\ pool\ 3\times 3$, stride 2
$\left[\genfrac{}{}{0.0pt}{}{conv,\ 3\times 3,\hskip 5.69046pt256}{conv,\
3\times 3,\hskip 5.69046pt256}\right]\times 6$
$7\times 7$ |  $max\ pool\ 3\times 3$, stride 2
$\left[\genfrac{}{}{0.0pt}{}{conv,\ 3\times 3,\hskip 5.69046pt512}{conv,\
3\times 3,\hskip 5.69046pt512}\right]\times 3$
$1\times 1$ | $\genfrac{}{}{0.0pt}{}{global\ avg\ pool\ 7\times 7}{10-\mathrm{d}\ fc,\ softmax}$
Table 1: Network Structure

Table (2) shows the results for all dataset sizes. The results show the multi-
crop error rate for all 5 sizes. As the number of classes increases by a
factor of 10 from 10 to 100 to 1000 classes, the error rate only increases by
a factor close to 2 from 4.81% to 10.1% to 21.2%. This shows how effective
these networks are in solving very large problems. The exact values of these
results may vary based on the makeup of the datasets sampled from ImageNet,
and to reduce this variance multiple datasets are sampled at each size.
However, despite the small variance the error rate always grows at a much
lower rate compared to the increase in the number of classes.

$\mathrm{\genfrac{}{}{0.0pt}{}{Data}{Size}}$ | $\mathrm{\genfrac{}{}{0.0pt}{}{10}{Classes}}$ | $\mathrm{\genfrac{}{}{0.0pt}{}{50}{Classes}}$ | $\mathrm{\genfrac{}{}{0.0pt}{}{100}{Classes}}$ | $\mathrm{\genfrac{}{}{0.0pt}{}{500}{Classes}}$ | $\mathrm{\genfrac{}{}{0.0pt}{}{1000}{Classes}}$
---|---|---|---|---|---
$\mathrm{\genfrac{}{}{0.0pt}{}{Test}{Error}}$ | 4.81% | 7.7% | 10.1% | 16% | 21.8%
Table 2: Results for datasets with different sizes sampled from ImageNet.

Figure (1) shows the relative increase in the error rate compared to the
relative increase in the number of classes. The relative increase in the error
and in the number of classes are obtained by dividing all entries in table (2)
by the entries in the first column. As the number of classes is increased up
100 times, the error rate only increases 4.5 times.

Figure 1: the relative increase in error rate compared to the relative
increase in the number of classes.

###  2.2 Classifying Multiple Categories

As the previous experiment showed, one of the strengths of deep convolution
networks is their ability to effectively solve large recognition problems such
as the 1000-classes ImageNet. It also showed that the performance drops slowly
as the size of the task increases significantly. Looking at the classes that
make up the ImageNet dataset, there are many similar classes that can be
divided into categories (e.g. multiple species of dogs, cats, birds, multiple
types of cars etc.). It is often likely for very large datasets to contain
similar classes that can be put into categories, and the next experiment tries
to measure how deep convolution networks react to such similarities, and if
that might explain the success of these networks with large datasets.

Category |  Cars |  Bugs |  Cats |  China |  Birds |  Fruits |  Furniture |  Lizards |  Monkeys |  Snakes |  Avg.
---|---|---|---|---|---|---|---|---|---|---|---
$\mathrm{\genfrac{}{}{0.0pt}{}{Network\ Per}{Category}}$ | 16.6% | 12.6% | 19.0% | 23.4% | 2.43% | 10.4% | 16.4% | 22.7% | 23.4% | 24.9% | 17.18%
$\mathrm{\genfrac{}{}{0.0pt}{}{Shared}{Network}}$ | 17.4% | 12.5% | 20.4% | 27.0% | 2.37% | 10.9% | 18.9% | 23.6% | 24.5% | 25.3% | 18.28%
Table 3: Results per category for the shared network vs the results obtained
using separate network per category.

A dataset made up of 100 classes was constructed from ImageNet, where the
chosen classes belong to 10 different categories. These categories are, birds,
bugs, cars, cats, china and cookware, fruits and vegetables, furniture,
lizards, monkeys, and snakes. Each of these naturally divided categories
consists of 10 classes, for example the cars category is divided into
ambulances, jeep (four wheel) cars, family cars, convertible cars, police
cars, taxis, sports cars, small buses, large family cars, pickup cars. Each
class has 1300 training images, and 50 test images (sampled from the ImageNet
validation set). For reproducing the results, table (6) shows the folder names
of the classes that make up all the categories.

In the first part of this experiment all 10 categories are considered as a
single dataset and used to train a deep convolution network. Images were
labeled regularly using a vector of 100 numbers, with only one of them is ON
to reflect a specific class. This is the regular way of coding image labels
when softmax is used as the activation function of the output layer.
Therefore, no hint is given to the network to treat these 100 classes as 10
separate categories.

In the second part of the experiment, each category was considered as a
separate dataset, and used to train a separate network. Therefore, 10 separate
networks will be trained using the 10 different categories, and the size and
structure of these networks is the same as the size and structure of the
shared network used to learn all categories. The only difference is the size
of the output layer.

The accuracy of classifying a category using a separate network will be
compared to the accuracy of classifying that category on the shared network
used to learn all categories. This comparison will measure the drop in
performance per category for the shared network. The drop should reflect the
added confusion caused by learning all the categories on the same network.

The same network structure shown in table (1) will be used here, with the same
setup to the hyper-parameters used in the previous experiment. Table (3) shows
the results per category for both parts of the experiment. The top row shows
the results per category for the 10 separate networks, while the bottom row
shows the results per category for the shared network. The last column shows
the average results for all 10 categories. For most categories the results
were very close with only a small drop in the performance of the shared
network, as the average error increases from 17.18% to 18.28%.

From the results in table (3), the shared network utilized the fact that the
100 classes belong to 10 different categories, and was able to learn all of
them with accuracy very close to learning each one on a separate network. The
network was able to break down the main task into multiple smaller tasks, and
learn them simultaneously. Therefore, for a big task made up of multiple
smaller tasks, what dictates the difficulty of learning the main task is not
its size (number of classes), but rather the difficulty of learning each of
the smaller tasks. From table (3), the performance of the network used to
solve the main task was very close to the average performance of solving each
of the subtasks separately.

If a big dataset is made of multiple groups where each group contains classes
that are similar and hard to distinguish, then the difficulty of
distinguishing between the members of these groups will probably decide the
performance of the network. In reality however, big datasets such as ImageNet
have a mixed bag of classes that can be separated into categories (cars, cats,
dogs etc.), and classes that have common features with many other classes, and
cannot be put into a specific group or category. Therefore, the performance of
deep convolution networks on such big datasets, will be affected by both, the
difficulty of the subtasks within the big dataset, as well as the size of the
dataset. If most of the classes belong to well separated categories, then the
difficulty of learning those categories will probably decide the performance
of the network, while if most of the classes belong to a big vague group that
cannot be broken down to smaller categories, then the size of the task will
probably decide the performance of the network. The results of the previous
experiment on datasets with different sizes sampled randomly from ImageNet
shows something in between, where the performance of convolution networks
drops slowly as the number of classes increases.

In order to put these results into perspective, they will be compared with
results obtained using another dataset with the same size, that is randomly
sampled from ImageNet, and randomly divided into 10 groups. Figure (2) shows
the results per category (or group) for both cases. The yellow bars show the
error rates per category (group) when each category is learned separately, and
the blue bars show the error rates per category (group) for the shared
network. The left figure shows the results for the naturally divided dataset,
and the right figure shows the results for the randomly divided dataset. By
visually comparing the two figures, we see that the shared network succeeded
in learning all naturally divided groups, while it failed to do the same for
the randomly divided groups.

Figure 2: The blue bars show the error rate per category obtained using the shared network, the yellow bars were obtained using a separate network per category. left:- for the naturally divided dataset, right:- for the randomly divided dataset. Super-Class |  Cars |  Bugs |  Cats |  China |  Birds |  Fruits |  Furniture |  Lizards |  Monkeys |  Snakes |  Avg.
---|---|---|---|---|---|---|---|---|---|---|---
Error Rate | 0.56% | 1.95% | 1.86% | 5.2% | 0.34% | 1.4% | 2.86% | 4.2% | 0.83% | 4.46% | 2.36%
Table 4: results obtained by merging each category into a super-class. The
results show the amount of leakage between categories.

####  2.2.1 Inter-category Leakage

Inter-category leakage measures the ratio of images in each category that have
been misclassified as classes from other categories, when all categories are
learned using a shared network. The leakage can be measured using the
confusion matrix, by merging the results of all the classes that belong to a
single category to form a single superclass that represents that category. The
merger process goes like this: - if an image is misclassified as a class from
the same category then this is considered a correct classification, while if
an image is misclassified as a class from a different category then this is
considered a wrong classification. When combining all the classes in each
category into a single superclass, then the inter-category error rate or
leakage is equal to 2.36% as shown in table (4). The low inter-category
leakage of 2.36% shows that the network rarely misclassifies an image as one
from a different category. Out of the 18.28% misclassified images in table (4)
only 2.36% happened between categories, while 18.28 – 2.36% = 15.92% happened
locally within each category.

Table (3) showed a drop in performance for the shared network compared to
using a separate network per category equal to 18.28 – 17.18 = 1.1%, while
table (4) shows a bigger leakage between categories for the shared network
equal to 2.36%. If 2.36% of the images has suffered from being learned with
other categories on the same network, then 2.36 - 1.1 = 1.26% of the images
must have benefited by being learned with other categories. Figure (3)
explains this and shows a histogram for the difference in accuracy per class.
It shows that about two third of the classes lost some performance by being
learned on the shared network, while about one third has actually gained some
performance. The shape of the histogram is close to a normal distribution with
a negative average close to zero that reflects the 1.1% drop in performance
for the shared network, and a small variance that reflects the similarity
between the results obtained using the shared network and the 10 separate
networks.

Figure 3: Histogram of the difference in performance per class, between using
the shared network, and using a separate network per category. 100 values for
100 classes.

###  2.3 Using Class and Category labels

Using the dataset from the previous experiment (made up of 100 classes divided
into 10 categories), a combined labeling scheme, that uses both class and
category labels, will be tested against the standard labeling scheme that uses
only class labels. The adopted method of adding the category label is straight
forward and can easily be implemented. The combined class/category label is a
vector made of 110 numbers, 10 numbers for each of the 10 categories, and 100
numbers for each of the 100 classes. For each image, 2 numbers will be ON, one
represents the class of the image, and the other one represents the category
of that image. In order to use a single softmax in the output layer, the two
ON numbers per label are set to 0.5.

Labels scheme | Error Rate
---|---
class label | 18.28%
$\mathrm{\genfrac{}{}{0.0pt}{}{class/category}{label}}$ | 17.3%
Table 5: results using class/category labels vs using only class labels.

Table (5) shows the results of using the combined class/category labeling
scheme, vs using only the class label. There is about 1% (about 5.3% relative
reduction) improvement in accuracy when the category label is added. This
shows that the standard labeling method of using only the class label is very
basic and can be improved. The construction of ImageNet is done by hand, and a
closer inspection of the cars category shows some sports cars labeled as
convertible cars and vice versa. This way of randomly labeling cars that share
both attributes causes confusion to the network. If the image label was
constructed using only the class number, then two convertible cars can have
two completely different labels (one as a convertible car and the other as a
sports car), while if the category number is added, then these two cars will
at least share the same category number, and that will make their labels 50%
similar, rather than 0% similar. Figure (4) shows some convertible cars from
ImageNet that have been labeled as sports cars, because they share both
attributes. In fact, both classes were among those that benefited from adding
the category label, with the error rate for the convertible cars class
dropping from 14.2% to 10.2%, and for the sports cars class dropping from
15.6% to 12.8%, which is much better than 1% average improvement.

Figure 4: convertible cars that were labeled as sports cars, In ImageNet.

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Table 6: Folder names of 100 Classes sampled from ImageNet ILSVRC 2015 to form 10 categories, each category has 10 classes. Cars | n02701002 | n02814533 | n02930766 | n03100240 | n03594945 | n03769881 | n03770679 | n03930630 | n03977966 | n04285008
---|---|---|---|---|---|---|---|---|---|---
Bugs | n01773797 | n01774384 | n01775062 | n02165105 | n02167151 | n02168699 | n02174001 | n02177972 | n02229544 | n02233338
Cats | n02123045 | n02123159 | n02123394 | n02123597 | n02124075 | n02127052 | n02128385 | n02128757 | n02128925 | n02130308
China | n03063599 | n03063689 | n03443371 | n03775546 | n03786901 | n03950228 | n04398044 | n04522168 | n07920052 | n07930864
Birds | n01530575 | n01531178 | n01532829 | n01534433 | n01537544 | n01558993 | n01560419 | n01580077 | n01592084 | n01828970
Fruits | n07720875 | n07742313 | n07745940 | n07747607 | n07749582 | n07753113 | n07753592 | n07768694 | n12620546 | n12768682
Furniture | n02870880 | n03014705 | n03016953 | n03018349 | n03125729 | n03131574 | n03201208 | n03337140 | n04099969 | n04550184
Lizards | n01629819 | n01630670 | n01631663 | n01632458 | n01675722 | n01682714 | n01685808 | n01687978 | n01689811 | n01693334
Monkeys | n02483362 | n02486261 | n02486410 | n02487347 | n02488291 | n02492035 | n02492660 | n02493509 | n02493793 | n02494079
Snakes | n01728572 | n01728920 | n01729322 | n01734418 | n01735189 | n01740131 | n01742172 | n01753488 | n01755581 | n01756291