# Spatial Moment Pooling Improves Neural Image Assessment

###### Abstract

In recent years, there has been widespread attention drawn to convolutional
neural network (CNN) based blind image quality assessment (IQA). A large
number of works start by extracting deep features from CNN. Then, those
features are processed through spatial average pooling (SAP) and fully
connected layers to predict quality. Inspired by full reference IQA and
texture features, in this paper, we extend SAP ($1^{st}$ moment) into spatial
moment pooling (SMP) by incorporating higher order moments (such as variance,
skewness). Moreover, we provide learning friendly normalization to circumvent
numerical issue when computing gradients of higher moments. Experimental
results suggest that simply upgrading SAP to SMP significantly enhances CNN-
based blind IQA methods and achieves state of the art performance.

Index Terms—  Image processing, image analysis, image quality, convolutional
neural networks

##  1 Introduction

Different from full reference image quality assessment (IQA) (e.g. PSNR,
SSIM), blind IQA computes quality directly from the distorted image without a
clean image, which makes it suitable to user generated content, communication
and other scenarios where clean sources are inaccessible. Traditional blind
IQA approach relies on the regularity of natural scene statistics. Hand-
crafted features [1] are designed to measure the distortion of image from
natural ones.

More recently, deep CNN based blind IQA approaches have shown promising
results as learned features are more powerful [2]. And a typical inference of
such IQA approach involves several steps: (1). a CNN backbone (e.g. VGG16) is
applied to extract deep features from distorted images. (2). the deep features
are spatially average pooled (SAP). (3). the averaged features are passed
through a regression head (e.g. fully connected layers) to predict image
quality. In some works, the CNN backbone is pretrained on other tasks (e.g.
ImageNet) and finetuned with regression head [3][4][5]. While for other works
auxiliary tasks such as predicting residual or noise type are added [6][5].
Despite variation in the training methods, the inference procedures are
similar.

Fig. 1: (a). CNN based full reference IQA, (b). CNN based blind IQA and (c).
our proposed SMP approach.

Formally, denote distorted image as $x^{\prime}\in R^{H\times W}$, the feature
from CNN as $f(x)\in R^{C\times H^{\prime}\times W^{\prime}}$, then the output
of spatially average pooling $SAP(f(x^{\prime}))$ can be seen as spatial mean
of feature map. And the pooled feature map is further processed by head $g$ to
obtain the predicted quality $d(x^{\prime})$.

$d(x^{\prime})=g(SAP(f(x^{\prime})))=\\\ g(E[f(x^{\prime})])$ (1)

On the other hand, CNN based full-reference IQA methods also show promising
results. As a representative method, LPIPS [7] has been widely adopted as
perceptual loss [8]. Similar to blind IQA, LPIPS first computes the feature
maps of reference image $x$ and distorted image $x^{\prime}$ from a CNN $f$.
Then the square difference of those feature maps are computed and spatially
averaged. Finally, the averaged results are processed by head $g$ to predict
the distance metric $d(x,x^{\prime})$.

$\begin{array}[]{l}d(x,x^{\prime})=g(SAP((f(x)-f(x^{\prime}))^{2}))\\\ \hskip
35.00005pt=g(Var[f(x)-f(x^{\prime})])\end{array}$ (2)

Though both Eq. 1 and Eq. 2 adopt SAP over features, the statistics computed
are different (See Fig. 1). For blind IQA, the SAP is used to compute $1^{st}$
moment (mean) of feature map. For LPIPS, by assuming the residual has $0$
mean, the SAP computes the $2^{nd}$ central moment (variance) of feature map
difference.

For long, variance has been an important part of texture feature [9, 10].
Higher moments (e.g. skewness, kurtosis) are also applied in texture analysis
[11]. So a natural question to ask is, is spatial variance and other moments
more representative than mean alone for image quality?

In this paper, we propose spatial moment pooling (SMP) as an extension of SAP
for blind IQA. Our approach of adding higher moments absorbs the arts of full
reference IQA methods and texture features. To be specific, for each channel
of feature map, those higher central moments (e.g. variance) are also computed
and concatenated with $1^{st}$ moment. It can be seen as a generalization of
SAP ($1^{st}$ moment). Moreover, we also propose a normalization approach to
avoid numeric issue when back-propagating through higher order moments.
Experimental results show that by simply replacing SAP with SMP, the
performance of many deep CNN based IQA methods is significantly improved. Our
method outperforms previous CNN approaches.

##  2 Related Works

###  2.1 Poolings and Deep CNN based blind IQA

Despite the pioneer of CNN based blind IQA [2] adopts min-max-pooling, the
majority of subsequent works [4][5][12][13] adopts SAP (include global average
pooling) as it has become popular in CNN. [14] adopts a mixture of SAP and ROI
Pooling. [3] and [15] resize input images or patches to fixed size instead,
and do not use pooling for final features.

On the other hand, covariance pooling [16] has been shown effective for high
level vision tasks. And the $i=2$ moment is the diagonal elements of a
covariance matrix. However, to the best of our knowledge, neither moment
pooling with $i\geq 3$ nor moment pooling with $i\geq 2$ for CNN based blind
IQA has been studied.

##  3 spatial Moment Pooling

###  3.1 Background

First, we will conduct a brief review of moments, spatial average pooling
(SAP) and its relationship to convolution.

Given a random variable $X\in R$, the $i^{th}$ moment is defined as
$E[X^{i}],i={1,...,n}$, and the $i^{th}$ central moment is defined as
$E[(X-E[X])^{i}],i={2,...,n}$. When $i=1$, the moment is the mean $\mu$ of
$X$. When $i=2$, the central moment is the variance $\sigma^{2}$ of $X$. When
$i=3$, the central moment is the unnormalized skewness of $X$, which is
defined as $E[(\frac{X-E[X]}{\sigma})^{3}]$. When $i=4$, the central moment is
the unnormalized kurtosis of $X$, which is defined as
$E[(\frac{X-E[X]}{\sigma})^{4}]$. In practice, higher order moments with $i>4$
are less commonly used.

On the other hand, given a feature map $f\in R^{C\times H\times W}$, the SAP
is the operation of computing the $i=1$ moment of values inside a pooling
window to output the pooled feature map $f_{SAP}\in R^{C\times
H^{\prime}\times W^{\prime}}$ (See Fig. 2). Identical to the standard
convolution operation, the outputted $H^{\prime}\times W^{\prime}$ are
determined by kernel size, stride, dilation and padding.

In fact, for any pooling, the pooling window is the same as convolution
windows given same settings. Besides, it can also be implemented as im2col.
The difference between pooling and convolution is after extracting windows to
row vectors. In convolution, the extracted rows $r_{i}^{T}$ is dot produced
with the flattened kernel. In SAP, the mean of $r_{i}^{T}$ is computed. This
makes SAP a special form of convolution with box filter kernel. However, the
spatial moment pooling-n in Section 3.2 can not be generalized by convolution
when $n\geq 2$, since the $i\geq 2$ moments are non-linear and convolution is
linear.

###  3.2 Spatial Moment Pooling-n / SMP(n)

Fig. 2: a. spatial average pooling/SAP illustrated. b. spatial moment
pooling-4/SMP(4) illustrated.

Similarly, we define the spatial moment pooling-n/SMP(n) as the operation of
computing the $i={1,...,n}$ central moments of a pooling window, concatenating
the moments in channel dimension and outputting the pooled feature map
$f_{SMP}\in R^{nC\times H^{\prime}\times W^{\prime}}$ (See Fig. 2). The
outputted spatial dimensions $H^{\prime}\times W^{\prime}$ are the same as
SAP. However, the channel size increases from $C$ to $nC$. SMP(n) is a
generalization of SAP. In fact, SMP(1) is exactly the same as SAP.

To intuitively show why SMP(n) might represent image quality better, we
provide a toy example in Fig. 3. Two $3\times 3$ feature maps with same mean
are shown. One is checkerboard pattern, and the other is solid color. Despite
the same means, the $i>1$ central moments differ.

Fig. 3: The $1^{st}$ moment, $2^{nd},3^{rd},4^{th}$ central moment of a
$3\times 3$ checkerboard feature map and solid color feature map.

###  3.3 Numerical Difficulties and Normalization

Simply elevating SMP(n) to $n\geq 3$ cases can cause severe difficulties when
optimizing the network with back-propagation. As shown in Tab. 1, the training
fails and produces NaN results without proper normalization. So we propose to
add layer normalization after $i\geq 3$ central moments (See Fig. 2). The
rationale of normalizing only $i\geq 3$ is: (1) $SMP(n),n\leq 2$ works well
without normalization. (2). only the statistical skewness and kurtosis are
normalized with $\sigma$, the $\mu$ and $\sigma$ themselves are defined as
unnormalized.

The reason why to choose layer normalization over others is purely empirical.
The comparison of normalization methods is detailed in Section 4.2.2. In the
following sections, SMP(n) with $n\geq 3$ implies that $i\geq 3$ moments are
layer normalized if no normalization method is specified.

##  4 Experiments

###  4.1 Experiment Setup

The SMP(n) is implemented in Pytorch 1.8, all the experiments are conducted on
a computer equipped with Intel Xeon E5-2620 v4 and 8 Nivida TitanXp GPU.

Basically, there are two types of datasets for blind IQA. One is large scale,
natural datasets without reference images (such as Koniq-10k [17]). The other
is small scale synthetic datasets with reference images and distortion
information (such as LIVE [18]). Usually the methods designed for small
dataset use reference and distortion as auxiliary task, which makes them not
applicable to large datasets. Consequently, the sota methods for large and
small dataset are different.

For large dataset, we choose Koniq-10k as dataset, and HyperIQA [4] as current
CNN-based sota method. For small dataset we choose LIVE [18] and CSIQ [19] as
dataset, and NemgIQA [5] as sota method. For ablation study we use Koniq-10k
as dataset, and a variant of NIMA [12][3] as base method. The dataset splits
are the same as [4] and [5].

###  4.2 Ablation Study

For baseline, we use an contemporary variant of classical NIMA [3]. As
proposed and described in [12], this variant incorporates the multi-scale
spatial average feature maps. To simplify notations in the tables below, we
denote this variant as NIMA. We only conduct experiments on VGG16 backbone, as
the selection of CNN backbone selection is beyond scope of this paper.

Table 1: Ablation study results. | Koniq-10k
---|---
| SRCC | PLCC
NIMA [3][12] / NIMA-SMP(1) | 0.864 | 0.879
NIMA-Vanilla-Large | 0.864 | 0.874
NIMA-SMP(2) | 0.886 | 0.896
NIMA-SMP(4)-w/o-Norm | NaN | NaN
NIMA-SMP(4)-BatchNorm | 0.880 | 0.892
NIMA-SMP(4)-MaxNorm | 0.888 | 0.899
NIMA-SMP(4)-LayerNorm | 0.890 | 0.900

####  4.2.1 Effects of spatial moment pooling

Tab. 1 shows that NIMA-SMP(2) evidently improves both PLCC and SRCC over NIMA.
NIMA-SMP(4)-LayerNorm outperforms SMP(2) despite the gain is not as
significant as NIMA-SMP(2) over NIMA. Therefore, we stop at $n=4$ considering
that the moments higher than $4^{th}$ are also uncommon in statistics.

####  4.2.2 Effects of high moments normalization

Simply expanding NIMA-SMP(2) to NIMA-SMP(4)-w/o-Norm brings optimization
difficulties. After several iterations of training, the network produces
constant results regardless of input images. And such constant output has $0$
variance, which further leads to NaN in PLCC and SRCC (See Tab. 1).

Although the NaNs can be eliminated by introducing batch normalization to
$3^{rd},4^{th}$ moments, it also brings performance decay. The NIMA-
SMP(4)-BatchNorm is outperformed by NIMA-SMP(2), this confirms the
observations that batch normalization negatively effects scale sensitive low
level computer vision tasks [20].

Empirically, we find that max and layer normalization enable NIMA-SMP(4) to
produce a reasonably superior performance over NIMA-SMP(2). Moreover, NIMA-
GMP(4)-LayerNorm outperforms all other methods.

####  4.2.3 Effects of network parameter increment

The SMP’s increment of parameters is of no significance. For NIMA-
SMP(4)-LayerNorm, the parameter is increased by $0.20\%$ compared with NIMA.
And for NIMA-SMP(2), the parameter increase is only $0.068\%$. Moreover, the
increment of MACs is also minimal. The base model NIMA has a MAC of 635.23G
Mac (multiply-accumulate) when input image size is 3x1920x1080. Replacing last
pooling with SMP-(2) brings 1.02M extra Macs. And replacing last pooling with
SMP-(4) brings 3.03M extra Macs.

To fully verify the performance improvement comes from SMP instead of
parameter increase, we build NIMA-Vanillia-Large with much wider and deeper
regression head. NIMA-Vanillia-Large increases the parameter by as much as
$14.74\%$. However, Tab. 1 shows that the performance does not benefit from
naïve model size increase.

###  4.3 Results on Large Dataset

Table 2: Results on Koniq-10K datasets. Blue and black numbers in bold represent the best and second best respectively. Reference methods’ results are sourced from [5][15]. | Koniq-10K
---|---
| SRCC | PLCC
DCNN based methods
BIECON [13] | 0.618 | 0.651
PQR [21] | 0.880 | 0.884
SFA [22] | 0.856 | 0.872
NIMA [3] | 0.864 | 0.879
DBCNN [23] | 0.875 | 0.884
HyperIQA [4] | 0.906 | 0.917
DCNN + SMP based methods
HyperIQA-SMP(2) | 0.909 | 0.924
HyperIQA-SMP(4) | 0.909 | 0.922
Transformer based methods
MUSIQ-Singlescale | 0.905 | 0.919
MUSIQ-Multiscale | 0.916 | 0.928

For HyperIQA on large dataset, we replace the global average pooling in local
distortion aware modules. Moreover, we set the learning rate to $1\times
10^{-5}$. The other settings are identical to original HyperIQA [4]. Results
in Tab. 2 show that such modification effectively improves HyperIQA, and a new
sota of deep CNN based approaches is made.

Notably, our HyperIQA-SMP(2) and SMP(4) are superior to the transformer with
single scale image patch input. Although the multi-scale transformer
outperforms SMP methods, it is not quite fair to draw comparison as MUSIQ-
Multiscale requires multi-scale image inputs. We also note that it is possible
to train HyperIQA-SMP(n) with multi-scale image crops, but the effects of
single-scale and multi-scale input is beyond the scope of this paper.

###  4.4 Results on Small Dataset

Table 3: Results on LIVE and CSIQ datasets. Blue and black numbers in bold represent the best and second best respectively. Reference methods’ results are sourced from [5][15]. | LIVE | CSIQ
---|---|---
| SRCC | PLCC | SRCC | PLCC
DCNN based methods
BIECON [13] | 0.963 | 0.965 | 0.837 | 0.858
PQR [21] | 0.965 | 0.971 | 0.873 | 0.901
SGDNet [24] | 0.961 | 0.964 | 0.878 | 0.909
HyperIQA [4] | 0.961 | 0.963 | 0.914 | 0.927
NemgIQA [5] | 0.971 | 0.975 | 0.923 | 0.934
DCNN + SMP based methods
NemgIQA-SMP(1) | 0.972 | 0.978 | 0.921 | 0.937
NemgIQA-SMP(2) | 0.973 | 0.980 | 0.929 | 0.942
NemgIQA-SMP(4) | 0.976 | 0.982 | 0.924 | 0.945

For NemgIQA on small dataset, it is not possible to directly replace last
global average pooling with SMP. Unlike other works with regression performed
after pooling, NemgIQA uses a single channel output point wise convolution as
regression head followed by pooling. To make SMP usable, we swap the order of
pooling and regression back. To be speficic, we change last layers to: point
wise regression with output channels 4, pooling, and an affine layer to
produce score. The other settings are the same as original NemgIQA [5].

Due to such modifications NemgIQA-SMP(1) is not strictly equivalent to
original NemgIQA. However, experimental results show that this does not
effects its performance. Moreover, the NemgIQA-SMP(2) and NemgIQA-SMP(4)
significantly improve PLCC and SRCC compared to both original NemgIQA and
NemgIQA-SMP(1), and achieve sota performance on LIVE and CSIQ datasets.

##  5 Conclusion

In this paper, we extend spatial average pooling (SAP) into spatial moment
pooling (SMP) by adding higher order moments. Moreover, we propose an
optimization friendly normalization trick that makes training of network with
SMP stable. Experimental results show that replacing SAP with SMP
significantly improves deep CNN-based blind IQA approaches.

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Bondi-Metzner-Sachs algebra as an extension of the

Poincaré symmetry in light-cone gravity

Sudarshan Ananth†, Lars Brink∗ and Sucheta Majumdar‡

$\dagger$ Indian Institute of Science Education and Research

Pune 411008, India

${}^{*}\,$Department of Physics, Chalmers University of Technology

S-41296 Göteborg, Sweden

$\ddagger$ Université Libre de Bruxelles and International Solvay Institutes,

Campus Plaine CP231, B-1050 Brussels, Belgium

Abstract

We analyze possible local extensions of the Poincaré symmetry in light-cone
gravity in four dimensions. We use a formalism where we represent the algebra
on the two physical degrees of freedom, one with helicity $2$ and the other
with helicity $-2$. The representation is non-linearly realized and one of the
light-cone momenta is the Hamiltonian, which is hence a non-linear generator
of the algebra. We find that this can be locally realized and the Poincaré
algebra extended to the BMS symmetry without any reference to asymptotic
limits.

##  1 Introduction

In a series of papers, over the years, we have studied gravity and Yang-Mills
theories in a formalism we have named the $lc_{2}$ formalism. In this
framework, the (super)Poincaré algebra is spanned on just the physical degrees
of freedom and one of the light-cone directions is identified as the evolution
(“time”) variable. By doing this, the conjugate light-cone momentum is the
Hamiltonian and can be read off by constructing the generators of the algebra
order by order in the coupling constant [1]. The formalism for gravity is set
up on a Minkowski background and the price to pay is that the Hamiltonian is
an infinite series of terms which after the four-point coupling becomes
unmanageable. This means that we never fully prove the symmetries we realize
but in our experience, checking symmetries up to the three-point coupling is
sufficient to be convinced that the theory in question satisfies that
particular symmetry. (We do know them all to the four-point coupling order.)
Since this is a perturbative expansion in the gravity fields we can only trust
it for weak enough fields. In a fairly recent paper we showed that (within the
limits alluded to above) the famous Ehlers symmetry [2, 3] can be shown to be
an additional symmetry, even in the four-dimensional theory. In this letter,
we ask a different question. Can any of the Poincaré generators be lifted to
be local? We will find that the Hamiltonian can indeed be made local and we
can identify it with the Bondi-van der Burg-Metzner-Sachs (BMS) group which is
an infinite-dimensional enhancement of the Poincaré group. In the usual
formulation, this arises as the asymptotic symmetry group at null infinity for
asymptotically flat spacetimes [4, 5, 6]. There has been a renewed interest in
the study of asymptotic symmetries following a recent body of work [7],
wherein these symmetries have been connected to soft theorems for gauge
theories. Here we will show the symmetry as a symmetry in the bulk avoiding
the usual sensitivity to the boundary conditions and gauge choices imposed on
the fields. At spatial infinity, for instance, under the standard boundary
conditions in the Hamiltonian formulation of gravity, only the Poincaré
algebra could be canonically realized, not the BMS algebra [8]. In a recent
work, however, the BMS group was recovered at spatial infinity by suitably
relaxing the boundary conditions, thereby, resolving a longstanding disparity
between the asymptotic structure at spatial and null infinity [9, 10]. In the
next section we will review shortly the formalism and in the following ones we
will show the invariance of the Hamiltonian under the extended symmetry and
connect the symmetry to the BMS one. In a subsequent paper we will discuss the
problem in more generality relating the symmetry to residual
reparameterization invariance and investigate if the symmetry can be further
extended.

##  2 Light-cone gravity and Poincaré symmetry

With the metric $(-,+,+,+)$, the light-cone coordinates are defined as

$\displaystyle x^{\pm}=\frac{1}{\sqrt{2}}(x^{0}\pm x^{3})\ ,$ $\displaystyle
x=\frac{1}{\sqrt{2}}\,(\,{x_{1}}\,+\,i\,{x_{2}}\,)\ ,\qquad\bar{x}=x^{*}\,,$
(2.1)

with $\partial_{\pm},\bar{\partial},\partial$ being the corresponding
derivatives.

The Poincaré algebra is spanned on the two physical degrees of freedom which
we call $h(x)$ with helicity $2$ and $\bar{h}(x)$ with helicity $-2$. We have
chosen $x^{+}$ as the evolution parameter, “time” according to Dirac’s
prescription [11]. One of the Poincaré generators, the conjugate momentum
$P_{+}$, is the light-cone Hamiltonian. The light-cone Poincaré generators
split into two kinds: the kinematical ones ($\mathbb{K}$) which do not involve
the time derivative $\partial_{+}$ and the dynamical ones that involve time
derivatives and hence get non-linear contributions in the interacting theory
[1]. Accordingly, we have the generators (the missing indices refer to the
transverse direction as in (2.1))

$\displaystyle\mathbb{K}:\quad\\{P,\bar{P},P_{-},J,J^{+},\bar{J}^{+},J^{+-}\\}\,,$
$\displaystyle\mathbb{D}:\quad\\{P_{+}\equiv H,J^{-},\bar{J}^{-}\\}\,.$ (2.2)

and the corresponding algebraic structure

$\displaystyle[\,\mathbb{K},\mathbb{K}\,]~{}=~{}\mathbb{K}\,,$
$\displaystyle[\,\mathbb{K},\mathbb{D}\,]~{}=~{}\mathbb{D}\,,$
$\displaystyle[\mathbb{D},\mathbb{D}\,]~{}=~{}0\,.$ (2.3)

For completeness, we present the light-cone Poincaré algebra in the appendix.
The canonical generators of the light-cone Poincaré algebra for gravity in
terms of the physical fields $h,\ \bar{h}$ can be found in [1].

### Gravity in light-cone gauge

The Hamiltonian for gravity reads [1, 12]

$\displaystyle H$ $\displaystyle=$ $\displaystyle\int
d^{3}x\,\left\\{\partial\bar{h}\,\bar{\partial}h\ +\
2\,\kappa\,\partial_{-}^{2}\bar{h}\,\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\
-\
\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)+c.c.\
+\ \mathcal{O}(\kappa^{2})\right\\}\,.$ (2.4)

The inverse derivative $\frac{1}{\partial_{-}}$, defined by
$\frac{1}{\partial_{-}}(\partial_{-}h)=h$, introduces a mild non-locality
along the light-cone which is easy to handle [13]. An alternative is to
Fourier transform to momentum space leading to a new kind of pole. Given the
following relation

$\displaystyle\delta_{H}h\,\equiv\,\\{h\,,\,H\\}\ ,$ (2.5)

where the r.h.s. is the Poisson bracket of the field with the
Hamiltonian111The fields $h$ and $\bar{h}$ satisfy the relations
$\\{h(x),\bar{h}(y)\\}\,=\,\frac{1}{\partial_{-}}\delta^{(3)}(x-y)\,,\quad\\{\,h(x),h(y)\,\\}\,=\,\\{\,\bar{h}(x),\bar{h}(y)\,\\}\,=\,0\,.$
, we obtain

$\displaystyle\delta_{H}h$ $\displaystyle=$
$\displaystyle\frac{\partial\bar{\partial}}{\partial_{-}}h\,+\,2\,\kappa\,\partial_{-}\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\,-\,\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)\,$
(2.6)
$\displaystyle+\,2\,\kappa\,\frac{1}{\partial_{-}^{3}}\left(\frac{\partial^{2}}{\partial_{-}^{2}}\bar{h}\,\partial_{-}^{2}h\,-\,2\,\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}\partial
h\,+\,\bar{h}\,\partial_{-}^{2}\partial^{2}h\right),$

with $\delta_{H}\bar{h}$ being the complex conjugate of the above expression.

Another interpretation of (2.6) is as a consequence of a shift in the time
coordinate

$\displaystyle x^{+}$ $\displaystyle\rightarrow$ $\displaystyle x^{+}\,+a\ ,$
(2.7)

with the infinitesimal constant, $a$ not explicitly shown.

### Extension of Poincare symmetry - step 1

It is natural to ask what happens if this constant is replaced by a function
of spacetime, i.e.

$\displaystyle x^{+}$ $\displaystyle\rightarrow$ $\displaystyle
x^{+}\,+f(x,\bar{x},x^{+},x^{-})\ .$ (2.8)

Under this, the fields transform non-linearly as

$\displaystyle\delta_{H_{f}}h$ $\displaystyle=$ $\displaystyle
f(x,\bar{x},x^{+},x^{-})\
\Bigg{\\{}\,\frac{\partial\bar{\partial}}{\partial_{-}}h\,+\,2\,\kappa\,\partial_{-}\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\,-\,\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)\,$
(2.9)
$\displaystyle+\,2\,\kappa\,\frac{1}{\partial_{-}^{3}}\left(\frac{\partial^{2}}{\partial_{-}^{2}}\bar{h}\,\partial_{-}^{2}h\,-\,2\,\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}\partial
h\,+\,\bar{h}\,\partial_{-}^{2}\partial^{2}h\right)\,+\,\mathcal{O}(\kappa^{2})\Bigg{\\}}\,,$

with $\delta_{H_{f}}\bar{h}$ given by the complex conjugate of the above
expression. It is now interesting to see under what circumstances, the
transformation in (2.8) represents a symmetry of the Hamiltonian. To this end,
we compute

$\displaystyle\delta_{H_{f}}H$ $\displaystyle=$
$\displaystyle\delta_{H_{f}}^{(0)}\,H^{(0)}\,+\,\delta_{H_{f}}^{(\kappa)}\,H^{(0)}\,+\,\delta_{H_{f}}^{(0)}\,H^{(\kappa)}\,+\,\mathcal{O}(\kappa^{2})\,.$
(2.10)

At order $\kappa^{0}$, we have

$\displaystyle\delta_{H_{f}}^{(0)}\,H^{(0)}$ $\displaystyle=$
$\displaystyle-\,(\partial_{-}f)\
\frac{\partial\bar{\partial}}{\partial_{-}}\,\bar{h}\
\frac{\partial\bar{\partial}}{\partial_{-}}\,h\,,$ (2.11)

which vanishes if $\partial_{-}f=0$. Thus, the invariance of the free
Hamiltonian constrains the parameter $f$ to be $f(x,\bar{x},x^{+})$. At the
cubic order, we find

$\displaystyle\delta_{H_{f}}^{(0)}\,H^{(\kappa)}$ $\displaystyle=$
$\displaystyle
2\,\kappa\,\frac{1}{\partial_{-}^{2}}(h\,\partial_{-}^{2}\bar{h})\,\left(\bar{\partial}^{2}f\,\frac{\partial\bar{\partial}}{\partial_{-}}h\,+\,2\,\bar{\partial}f\,\frac{\partial\bar{\partial}^{2}}{\partial_{-}}h\right)\,-\,4\,\kappa\,\partial_{-}^{2}\bar{h}\,\bar{\partial}f\,\frac{\partial\bar{\partial}}{\partial_{-}^{2}}h\,\frac{\bar{\partial}}{\partial_{-}}h\,$
(2.12)
$\displaystyle+\,2\,\kappa\,f\,\partial_{-}\bar{\partial}\partial\bar{h}\,\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\,-\,\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)\,+2\,\kappa\,f\,\partial_{-}^{2}\bar{h}\left(\frac{\partial\bar{\partial}}{\partial_{-}}h\right)\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h$
$\displaystyle+\,2\,\kappa\,f\,\frac{1}{\partial_{-}^{2}}(h\,\partial_{-}^{2}\bar{h})\,\frac{\partial\bar{\partial}^{3}}{\partial_{-}}h\,-\,4\,\kappa\,f\,\partial_{-}^{2}\bar{h}\,\frac{\partial\bar{\partial}^{2}}{\partial_{-}^{2}}h\,\frac{\bar{\partial}}{\partial_{-}}h\,.$

The second and third line, which involve terms with a free $f$, can be
simplified further to cancel exactly against
$\delta_{H_{f}}^{(\kappa)}\,H^{(0)}$. In so doing, we obtain terms involving
$\bar{\partial}f$ and $\bar{\partial}^{2}f$ which cancel against the first
line in (2.12). Therefore, the transformation in (2.9) represents a symmetry
of the light-cone Hamiltonian with the local parameter

$\displaystyle f$ $\displaystyle=$ $\displaystyle f(x,\bar{x},x^{+})\,.$
(2.13)

It is straightforward to check the commutator of $H_{f}$ with the other
generators of the Poincaré algebra. For instance, with the momentum $P$, one
finds

$\displaystyle[\,P\,,\,H_{f}\,]$ $\displaystyle=$ $\displaystyle
H_{\hat{f}}\,,\quad\text{with}\ \hat{f}~{}=~{}\partial f\,,$ (2.14)

which reduces to the familiar case

$\displaystyle[\,P\,,\,H\,]$ $\displaystyle=$ $\displaystyle 0\,,$ (2.15)

when $f$ is a constant.

### Extension of Poincare symmetry - step 2

Having made an extension to the Poincare symmetry focusing on the time
direction, we can now look for similar local extensions in the $x$ and
$\bar{x}$ coordinates

$\displaystyle x~{}\rightarrow~{}x\ +\ Y(x)\,,$
$\displaystyle\partial_{-}Y~{}=~{}\partial Y~{}=~{}0\,,$ (2.16)
$\displaystyle\bar{x}~{}\rightarrow~{}\bar{x}\ +\ \overline{Y}(\bar{x})\,,$
$\displaystyle\partial_{-}\overline{Y}~{}=~{}\bar{\partial}\overline{Y}~{}=~{}0\,,$
(2.17)

under which the fields transform as

$\displaystyle\delta_{Y,\overline{Y}}\,h$ $\displaystyle=$ $\displaystyle
Y(x)\,\bar{\partial}h\,+\,\overline{Y}(\bar{x})\,\partial
h\,+\,(\partial\overline{Y}-\bar{\partial}Y)\,h\,,$
$\displaystyle\delta_{Y,\overline{Y}}\,\bar{h}$ $\displaystyle=$
$\displaystyle
Y(x)\,\bar{\partial}\bar{h}\,+\,\overline{Y}(\bar{x})\,\partial\bar{h}\,-\,(\partial\overline{Y}-\bar{\partial}Y)\,\bar{h}\,.$
(2.18)

The invariance of the Hamiltonian, however, further restricts the parameters
to be

$\displaystyle{\bar{\partial}}^{2}Y=\partial^{2}\overline{Y}=0.$ (2.19)

reducing them to Lorentz rotations. The above condition constrains $Y$ and
$\overline{Y}$ to be linear in the coordinates, ruling out superrotations as
an extension of the Poincaré algebra in the bulk (unlike the case at null
infinity [14]). In terms of the light-cone Poincaré algebra, the
transformations in (2) account for the transformations generated by
$P,\bar{P}$ and $J$ in the appendix.

Nevertheless, it is more convenient to use the $Y,\overline{Y}$ notation for
the Poincaré transformations in the transverse directions $x$ and $\bar{x}$ in
order to demonstrate how the local transformation in (2.9) satisfies non-
trivial commutation relations with the ones given in (2). This allows us to
establish connections with the BMS symmetry.

##  3 Comparison with BMS symmetry

A particularly interesting choice of $f$ involving the rotations $Y$ and
$\overline{Y}$, which obey (2.19), is

$\displaystyle f(x^{+},x,\bar{x})$ $\displaystyle=$ $\displaystyle
T(x,\bar{x})\ +\ \frac{1}{2}\,x^{+}\,(\partial\overline{Y}\ +\
\bar{\partial}Y)\,,$ (3.1)

which coincides with the BMS symmetry [14]. This choice of $f$ follows from
the light-cone gauge conditions imposed on the metric.222The transformations
(2.9) and (2) must respect the light-cone gauge conditions on the metric
$g_{\mu\nu}$ in order to ensure that no new degrees of freedom are introduced
beyond the $h$ and $\bar{h}$. In the light-cone gauge, three metric components
are set equal to zero [15] $g_{--}\ =\ g_{-i}\ =\ 0\,,\quad(i=1,2)\,,$ while
the remaining components are parameterized as follows: $\ g_{+-}\ =\
-e^{\phi},\ \ g_{ij}\ =\ e^{\psi}\gamma_{ij}$, with $\phi,\psi$ real and
$\gamma_{ij}$ real, symmetric and unimodular. The fourth gauge choice is
$\phi=\frac{\psi}{2}\,,$ fixing the time dependence of $f$ in terms of
$Y,\overline{Y}$.  Since the light-cone system is a Carroll hypersurface, the
above relation can also be obtained from the conformal Carroll transformations
[16] in the null time direction $x^{+}$.

The field $h$ transforms as

$\displaystyle\delta_{Y,\overline{Y},f}\,h$ $\displaystyle=$
$\displaystyle\delta_{Y,\overline{Y}}h\ +\ \delta_{f}h\,,$ (3.2)

where the explicit form can be read off from (2.9) and (2). One can choose the
initial surface to be $x^{+}=0$ and evolve the algebra at later times using
the dynamical generators, in which case the function $f=T$ in (3.1).
Therefore, the BMS transformations in light-cone gravity to order $\kappa$
read

$\displaystyle\delta_{Y,\overline{Y},T}\,h$ $\displaystyle=$ $\displaystyle
Y(x)\,\bar{\partial}h+\overline{Y}(\bar{x})\,\partial h\
+(\partial\overline{Y}-\bar{\partial}Y)\,h+T\,\frac{\partial\bar{\partial}}{\partial_{-}}\,h$
(3.3)
$\displaystyle-\,2\,\kappa\,T\,\partial_{-}\,\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\,-\,\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)\
-\
2\,\kappa\,T\,\frac{1}{\partial_{-}}\left(\frac{{\partial}^{2}}{\partial_{-}^{2}}\bar{h}\,\partial_{-}^{2}h\right)\,$
$\displaystyle-\,2\,\kappa\,T\,\frac{{\partial}^{2}}{\partial_{-}^{3}}\,(\bar{h}\,\partial_{-}^{2}h)\,+\
4\,\kappa\,T\,\frac{\partial}{\partial_{-}^{2}}\left(\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}^{2}h\right)\,,$

with the parameters $Y$, $\overline{Y}$ and $T$ satisfying

$\partial Y=\partial_{-}Y=0\
,\quad\bar{\partial}\overline{Y}=\partial_{-}\overline{Y}=0\
,\quad\partial_{-}T=0\ ,$ (3.4)

and

$\bar{\partial}^{2}Y~{}=~{}\partial^{2}\overline{Y}~{}=~{}0\,.$ (3.5)

The commutator of these transformations is

$\displaystyle\left[\,\delta(Y_{1},\overline{Y}_{1},T_{1})\,,\
\delta(Y_{2},\overline{Y}_{2},T_{2})\,\right]\,h$ $\displaystyle=$
$\displaystyle\ \delta(Y_{12},\overline{Y}_{12},T_{12})\,h\ ,$ (3.6)

where the new parameters are

$\displaystyle Y_{12}$ $\displaystyle\equiv$ $\displaystyle
Y_{2}\,\bar{\partial}\,Y_{1}\ -\ Y_{1}\,\bar{\partial}\,Y_{2}\,,$ (3.7)
$\displaystyle\overline{Y}_{12}$ $\displaystyle\equiv$
$\displaystyle\overline{Y}_{2}\,\partial\,\overline{Y}_{1}\ -\
\overline{Y}_{1}\,\partial\,\overline{Y}_{2}\,,$ (3.8) $\displaystyle T_{12}$
$\displaystyle\equiv$ $\displaystyle[Y_{2}\,\bar{\partial}\,T_{1}\ +\
\overline{Y_{2}}\,\partial\,T_{1}\ +\frac{1}{2}\,T_{2}(\bar{\partial}Y_{1}\
+\,\partial\overline{Y}_{1})]\ -\ (1\leftrightarrow 2)\ .$ (3.9)

Thus, two such transformations close on themselves with the parameters
satisfying (3.4) and (3.5). This is the light-cone realization of the BMS
algebra in four dimensions [14].

Following the BMS nomenclature, these “supertranslations” labeled by
$T(x,\bar{x})$ enhance the dynamical part of the Poincaré algebra into an
infinite-dimensional set, while the kinematical part of the algebra remains
unaltered

$\displaystyle\mathbb{K}~{}\rightarrow~{}\mathbb{K}\,,$
$\displaystyle\mathbb{D}~{}\rightarrow~{}\mathbb{D}(T)\,.$ (3.10)

The BMS algebra in (3.6) can then also be written as

$\displaystyle[\,\mathbb{K},\mathbb{K}\,]~{}=~{}\mathbb{K}\,,$
$\displaystyle[\,\mathbb{K},\mathbb{D}(T)\,]~{}=~{}\mathbb{D}(T)\,,$
$\displaystyle[\mathbb{D}(T),\mathbb{D}(T)\,]~{}=~{}0\,.$ (3.11)

By restricting $T$ to be linear in $x$ or $\bar{x}$ with the condition
$\partial^{2}T=\bar{\partial}^{2}T=0$, the dynamical part $\mathbb{D}(T)$
reduces to $\mathbb{D}$, which corresponds to the Poincaré subgroup of the BMS
algebra. Unlike in covariant formulations, where the four spacetime
translations get enhanced to angle-dependent supertranslations, in the light-
cone formalism, the dynamical part of the Poincaré algebra is enhanced to
accommodate supertranslations, thereby, yielding the BMS algebra in four
dimensions. In both cases, however, the enhancement of the Poincaré algebra to
the infinite-dimensional BMS algebra involves only a single local parameter as
a consequence of Lorentz invariance. We must emphasize though that the
invariance of the Hamiltonian under the BMS transformations is only strictly
proven to order $\kappa$. However, we do not anticipate formal difficulties in
extending these results to higher orders, although the explicit calculations
can become extremely cumbersome.

* * *

Even though this analysis has been performed only up to first order in the
coupling constant, one can already draw parallels with existing BMS results
for the full Einstein theory at null and spatial infinity. The BMS analysis at
spatial infinity involves some subtleties that can be better understood in the
linearized theory of gravity, for instance, the canonical realization of boost
generators [17]. In the same spirit, our perturbative approach in the light-
cone formalism offers an unconventional insight into the structure of the BMS
symmetry. Another similarity with the spatial infinity analysis is that the
invariance of the Hamiltonian reduces the parameters, $Y$ and $\overline{Y}$,
to Lorentz rotations, thereby eliminating superrotations [14] from the theory.
There are several ways to interpret the BMS symmetry in gravity apart from the
original work of [4, 5, 6] as the enhancement of the Poincaré algebra with
angle-dependent supertranslations. In [16], for instance, the BMS algebra was
argued to be the conformal extension of the Carroll algebra on null
hypersurfaces. In light-cone gravity, we present a new interpretation of the
BMS symmetry as the local extension of the dynamical part of the Poincaré
algebra in Dirac’s front form analysis [11]. Light-cone gravity offers a
straightforward approach to study the BMS symmetry in a physical gauge, which
might help us better understand its connection with on-shell amplitudes and
soft theorems [7, 18].

A unique result of our light-cone analysis is that the BMS algebra can be
represented on the two physical fields of graviton in the bulk, without any
need for asymptotic limits, suggesting that the BMS symmetry could be more
than just an asymptotic symmetry. Our scope is different from the BMS analyses
relevant for black hole physics as we are mainly interested in the
implications the extended symmetry might have for the quantum theory in a
pertubative regime since the formalism is not suitable for studies with strong
gravity fields. We wish, for example, to utilize the knowledge of this new
symmetry to better understand the ultraviolet behaviour of the theory,
classify counterterms, etc. in the regime of linearized gravity. Light-cone
gravity (formulated in the helicity basis) is closely related to on-shell
physics - MHV Lagrangians, double-copy relations, etc. [19, 20]. Accordingly,
the light-cone formalism is well suited for examining the connections between
these infinite-dimensional symmetries and the perturbative S-matrix and
scattering amplitudes. It will also be interesting to look at the
corresponding symmetry issues in maximal supergravity.

## Acknowledgments

We thank Glenn Barnich and Marc Henneaux for many helpful discussions. The
work of SM is partially supported by the ERC Advanced Grant “High-Spin-Grav”,
by FNRS-Belgium (conventions FRFC PDRT.1025.14 and IISN 4.4503.15), as well as
by funds from the Solvay Family.

##  Appendix A Light-cone Poincaré algebra in four dimensions

We define

$J^{+}\ =\ \frac{J^{+1}+iJ^{+2}}{\sqrt{2}}\ ,\quad\bar{J}^{+}\ =\
\frac{J^{+1}-iJ^{+2}}{\sqrt{2}}\ ,\quad J\ =\ J^{12}\ ,\quad H\ =\ P^{-}\ .$
(A.1)

The non-vanishing commutators of the Poincaré algebra are listed below.

$\displaystyle[H,J^{+-}]\ =\ -iH\ ,\quad$ $\displaystyle[H,J^{+}]=-iP\
,\qquad\qquad[H,\bar{J}^{+}]\ =\ -i\bar{P}$ (A.2)
$\displaystyle[P^{+},J^{+-}]\ =\ iP^{+}\ ,\quad$ $\displaystyle[P^{+},J^{-}]\
=\ -iP\ ,\qquad\quad[P^{+},\bar{J}^{-}]\ =\ -i\bar{P}$
$\displaystyle[P,\bar{J}^{-}]\ =\ -iH\ ,\quad$ $\displaystyle[P,\bar{J}^{+}]\
=\ -iP^{+}\ ,\quad\qquad[P,J]\ =\ P$ $\displaystyle[\bar{P},J^{-}]\ =\ -iH\
,\quad$ $\displaystyle[\bar{P},J^{+}]\ =\ -iP^{+}\ ,\qquad\quad[\bar{P},J]\ =\
-\bar{P}$ $\displaystyle[J^{-},J^{+-}]\ =\ -iJ^{-}\ ,\quad$
$\displaystyle[J^{-},\bar{J}^{+}]\ =\ iJ^{+-}+J\ ,\quad[J^{-},J]\ =\ J^{-}$
$\displaystyle[\bar{J}^{-},J^{+-}]\ =\ -i\bar{J}^{-}\ ,\quad$
$\displaystyle[\bar{J}^{-},J^{+}]\ =\ iJ^{+-}-J\ ,\quad[\bar{J}^{-},J]\ =\
-\bar{J}^{-}$ $\displaystyle[J^{+-},J^{+}]\ =\ -iJ^{+}\ ,\quad$
$\displaystyle[J^{+-},\bar{J}^{+}]\ =\ -i\bar{J}^{+}\ ,$
$\displaystyle[J^{+},J]\ =\ J^{+}\ ,\quad$ $\displaystyle[\bar{J}^{+},J]\ =\
-\bar{J}^{+}\ .$

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# On the dimension of non-abelian tensor square of Lie superalgebras

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 Applied Mathematics and Computing, Institute
of Technical Education and Research
Siksha ‘O’ Anusandhan (A Deemed to be University)
Bhubaneswar-751030
Odisha, India<EMAIL_ADDRESS>and  Sushree Sangeeta Pradhan
Department of Mathematics
C. V. Raman Global University
Bhubaneswar-752054
Odisha, India<EMAIL_ADDRESS>
###### Abstract.

In this paper, we determine upper bound for the non-abelian tensor product of
finite dimensional Lie superalgebra. More precisely, if $L$ is a non-abelian
nilpotent Lie superalgebra of dimension $(k\mid l)$ and its derived subalgebra
has dimension $(r\mid s)$, then $\dim(L\otimes L)\leq(k+l-(r+s))(k+l-1)+2$. We
discuss the conditions when the equality holds for $r=1,s=0$ explicitly.

###### Key words and phrases:

Heisenberg Lie superalgebra; Schur multiplier ; non-abelian tensor and
exterior product

###### 2020 Mathematics Subject Classification:

Primary 17B10; Secondary 17B01.

##  1\. Introduction

In 1991, Rocco [15] proved that if $G$ is a finite $p$-group of order $P^{n}$
with derived subgroup of order $p^{m}$, then $G\otimes G\leq p^{n(n-m)}$.
Further this bound was imporved by Niroomand [9], i.e., $G\otimes G\leq
p^{n(n-m)}$. Ellis [1, 2] developed the theory of tensor product for Lie
algebra. If $L$ and $K$ are two finite dimensional nilpotent Lie algebras,
then the upper bound and lower bound of the dimension of the non-abelian
tensor product $L\otimes K$ have been stuided by salemkar et. al [18] and the
results are generalization of Rocco’s results for finite $p$-group. Recently,
an improved upper bound on the dimension of $L\otimes L$ has been discussed in
[10], explicitly, if $L$ is a non-abelian nilpotent Lie algebra of dimension
$n$ and its derived subalgebra has dimension $m$, then $\dim(L\otimes
L)\leq(n-m)(n-1)+2$.

Lie superalgebras have applications in many areas of Mathematics and
Theoretical Physics as they can be used to describe supersymmetry. Kac [5]
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 [16, 6, 3].
Recently, García-Martínez [3] introduced the notions of non-abelian tensor
product of Lie superalgebras and exterior product of Lie superalgebras over a
commutative ring. In this paper we determine an upper bound on the dimension
of non-abelian tensor product of nilpotent Lie superalgebra.

##  2\. Preliminaries and Auxiliary Results

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 [5, 6]) 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 which 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
$[I,L]\subseteq I$. The ideal

$Z(L)=\\{z\in L:[z,x]=0\;\mbox{for all}\;x\in L\\}$

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 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
the obvious meaning.

Throughout this article, for superdimension of Lie superalgebra $L$ we simply
write $\dim L=(m\mid n)$, where $\dim L_{\bar{0}}=m$ and $\dim L_{\bar{1}}=n$.
Also $A(m\mid n)$ denotes an abelian Lie superalgebra where $\dim A=(m\mid
n)$. A Lie superalgebra $L$ is said to be Heisenberg Lie superalgebra if
$Z(L)=L^{\prime}$ 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 [16]. By Heisenberg Lie superalgebra we mean
special Heisenberg Lie superalgebra in this article, for more details on
Heisenberg Lie superalgebras and its multiplier see [7, 8, 13, 12, 11, 4, 17].
Now we list some useful results from [7], for further use.

######  Theorem 2.1.

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

$H_{\overline{0}}=<x_{1},\ldots,x_{2m},z\mid[x_{i},x_{m+i}]=z,\ i=1,\ldots,m>$

and

$H_{\overline{1}}=<y_{1},\ldots,y_{n}\mid[y_{j},y_{j}]=z,\ j=1,\ldots,n>.$

Further,

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

The following is the established result for the multiplier and cover of
Heisenberg Lie superalgebra of odd center.

######  Theorem 2.2.

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

$H_{m}=<x_{1},\ldots,x_{m},y_{1},\ldots,y_{m},z\mid[x_{j},y_{j}]=z,j=1,\ldots,m>.$

Further,

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

##  3\.  Non-abelian tensor products

Here we recall some of the known notation and results from [3]. Let $P$ and
$M$ be two Lie superalgebras, then by an action of $P$ on $M$ we mean a
$\mathbb{K}$-bilinear map of even degree $P\times M\longrightarrow M,$

$(p,m)\mapsto{}^{p}m,$

such that

  1. (1)

${}^{[p,p^{\prime}]}m={}^{p}({}^{p^{\prime}}m)-(-1)^{|p||p^{\prime}|}~{}{}^{p^{\prime}}({}^{p}m),$

  2. (2)

${}^{p}{[m,m^{\prime}]}=[{}^{p}m,m^{\prime}]+(-1)^{|p||m|}[m,{}^{p}m^{\prime}],$

for all $p,~{}p^{\prime}\in P$ and $m,~{}m^{\prime}\in M$. For any Lie
superalgebra $M$, the Lie multiplication induces an action on itself via
${}^{m}m^{\prime}=[m,m^{\prime}]$. The action of $P$ on $M$ is called trivial
if ${}^{p}m=0$ for all $p\in P$ and $m\in M$.
Given two Lie superalgebras $M$ and $P$ with action of $P$ on $M$, we define
the semidirect product $M\rtimes P$ with underlying supermodule $M\oplus P$
endowed with the bracket given by
$[(m,p),(m^{\prime},p^{\prime})]=([m,m^{\prime}]+{}^{p}m^{\prime}-(-1)^{|m||p^{\prime}|}({}^{p^{\prime}}m),[p,p^{\prime}])$.
A crossed module of Lie superalgebras is a homomorphism of Lie superalgebras
$\partial:M\longrightarrow P$ with an action of $P$ on $M$ satisfying

  1. (1)

$\partial({}^{p}{m})=[p,\partial(m)],$

  2. (2)

${}^{\partial(m)}{m^{\prime}}=[m,m^{\prime}],$ for all $p\in P$ and
$m,m^{\prime}\in M$.

A bilinear function $f:M\times N\longrightarrow T$ is called Lie superpairing
if the following relations are satidfied:

  1. (1)

$f([m,m^{\prime}],n)=f(m,{}^{m^{\prime}}{n})-(-1)^{|m||m^{\prime}|}f(m^{\prime},{}^{m}{n})$,

  2. (2)

$f(m,[n,n^{\prime}])=(-1)^{|n^{\prime}|(|m|+|n|)}f({}^{n^{\prime}}{m},n)-(-1)^{|m||n|}f({}^{n}{m},n^{\prime})$,

  3. (3)

$f({}^{n}{m},{}^{m^{\prime}}{n^{\prime}})=-(-1)^{|m||n|}[f(m,n),f(m^{\prime},n^{\prime})]),$

for every $m,m^{\prime}\in M_{\overline{0}}\cup M_{\overline{1}}$ and
$n,n^{\prime}\in N_{\overline{0}}\cup N_{\overline{1}}$.

Let $M$ and $N$ be two Lie superalgebras with actions on each other. Let
$X_{M,N}$ be the $\mathbb{Z}_{2}$-graded set of all symbols $m\otimes n$,
where $m\in M_{\overline{0}}\cup M_{\overline{1}}$, $n\in N_{\overline{0}}\cup
N_{\overline{1}}$ and the $\mathbb{Z}_{2}$-gradation is given by $|m\otimes
n|=|m|+|n|$. The non-abelian tensor product of $M$ and $N$, denoted by
$M\otimes N$, as the Lie superalgebra generated by $X_{M,N}$ and subject to
the relations:

  1. (1)

$\lambda(m\otimes n)=\lambda m\otimes n=m\otimes\lambda n$,

  2. (2)

$(m+m^{\prime})\otimes n=m\otimes n+m^{\prime}\otimes n$, where $m,m^{\prime}$
have the same degree,
$m\otimes(n+n^{\prime})=m\otimes n+m\otimes n^{\prime}$, where $n,n^{\prime}$
have the same degree,

  3. (3)

$[m,m^{\prime}]\otimes
n=(m\otimes{}^{m^{\prime}}{n})-(-1)^{|m||m^{\prime}|}(m^{\prime}\otimes{}^{m}{n})$,
$m\otimes[n,n^{\prime}]=(-1)^{|n^{\prime}|(|m|+|n|)}({}^{n^{\prime}}{m}\otimes
n)-(-1)^{|m||n|}({}^{n}{m}\otimes n^{\prime})$,

  4. (4)

$[m\otimes n,m^{\prime}\otimes
n^{\prime}]=-(-1)^{|m||n|}({}^{n}{m}\otimes{}^{m^{\prime}}{n^{\prime}}),$

for every $\lambda\in\mathbb{K},m,m^{\prime}\in M_{\overline{0}}\cup
M_{\overline{1}}$ and $n,n^{\prime}\in N_{\overline{0}}\cup N_{\overline{1}}$.
The tensor product $M\otimes N$ has $\mathbb{Z}_{2}$-grading given by
$(M\otimes N)_{\alpha}=\oplus_{\beta+\gamma=\alpha}(M_{\beta}+N_{\gamma})$ for
$\alpha,\beta,\gamma\in{\mathbb{Z}}_{2}$. If $M=M_{\overline{0}}$ and
$N=N_{\overline{0}}$ then $M\otimes N$ is the non-abelian tensor product of
Lie algebras introduced and studied in [2].

Actions of Lie superalgebras $M$ and $N$ on each other are said to be
compatible if

  1. (1)

${}^{({}^{n}{m})}{n^{\prime}}=-(-1)^{|m||n|}[{}^{m}{n},n^{\prime}]$,

  2. (2)

${}^{({}^{m}{n})}{m^{\prime}}=-(-1)^{|m||n|}[{}^{n}{m},m^{\prime}]$,

for all $m,m^{\prime}\in M_{\overline{0}}\cup M_{\overline{1}}$ and
$n,n^{\prime}\in N_{\overline{0}}\cup N_{\overline{1}}$. For instance if $M$,
$N$ are two graded ideals of a Lie superalgebra then the actions induced by
the bracket are compatible.

We will denote the $\mathbb{K}$-module tensor product of $M$ and $N$ as
$M\otimes_{mod}N$. The following result comes immediately from [3, Proposition
3.8].

######  Proposition 3.1.

If $N\subseteq L^{2}\cap Z(G)$, then the sequence $N\otimes L\longrightarrow
L\otimes L\longrightarrow L/N\otimes L/N\longrightarrow 0$ is exact.

######  Proposition 3.2.

[3, Proposition 3.5] If $M$ and $N$ act trivially on each other, then there is
an isomorphism

$M\otimes N\cong M^{ab}\otimes_{mod}N^{ab}.$

where $M^{ab}=M/[M,M]$ and $N^{ab}=N/[N,N].$

######  Proposition 3.3.

[13, Proposition 3.6] Suppose the Lie superalgebras $L$ and $M$ are acting
trivially on each other, then

$(L\oplus M)\otimes(L\oplus M)\cong(L\otimes L)\oplus(L\otimes
M)\oplus(M\otimes L)\oplus(M\otimes M).$

Consider a Lie superalgebra $M=M_{\bar{0}}\oplus M_{\bar{1}}$ and the identity
map on $M$, i.e., the map $id:M\longrightarrow M$. The map $id$ is a crossed
module, and hence the exterior square $M\wedge M$ is obtained from $M\otimes
M$ by imposing the additional relations:

  1. (1)

$m\otimes m^{\prime}+(-1)^{|m||m^{\prime}|}(m^{\prime}\otimes m)=0$

  2. (2)

$m_{\overline{0}}\otimes m_{\overline{0}}=0$

with $m,m^{\prime}\in M_{\overline{0}}\cup M_{\overline{1}}$, and
$m^{\prime}\wedge m$ is the image of $m^{\prime}\otimes m$.

######  Lemma 3.4.

[3, Lemma 6.1] Let $M\square M$ be the submodule of $M\otimes M$ generated by
elements

  1. (1)

$m\otimes m^{\prime}+(-1)^{|m||m^{\prime}|}(m^{\prime}\otimes m)$,

  2. (2)

$m_{\overline{0}}\otimes m_{\overline{0}}$,

with $m,m^{\prime}\in M_{\overline{0}}\cup M_{\overline{1}},~{}m_{0}\in
M_{\overline{0}}$. Then $M\square M$ is a central graded ideal of $M\otimes
M$.

######  Definition 3.5.

The exterior product of $M$ and $M$ is denoted as $M\wedge M$ and is defined
as the quotient Lie superalgebra

$M\wedge M=\frac{M\otimes M}{M\square M}.$

For any $m\otimes m^{\prime}\in M\otimes M$ we denote the coset $m\otimes
m^{\prime}+M\square M$ by $m\wedge m^{\prime}$.

The universal quadratic functor of Lie superalgebra was introduced in [14].
The quadratic functor helps to establish the relations between the Lie
exterior product and the Lie tensor product of Lie superalgebras. Here we
recall the definition of universal quadratic functor of Lie superalgebra and
some results from [14] which will be useful in the next section.

######  Definition 3.6.

Let $L$ be a supermodule. We define the supermodule $\Gamma(M)$ as the direct
sum

$\Gamma(M)=R^{M_{\bar{0}}}\oplus(M\otimes
M)=\\{\gamma(m_{\bar{0})}+m^{\prime}\otimes m^{\prime\prime}\lvert
m_{\bar{0}}\in M_{\bar{0}},\,m^{\prime},m^{\prime\prime}\in M\\},$

and subject to the homogeneous relations

(3.1) $\gamma(\lambda m_{\bar{0}})=\lambda^{2}\gamma(m_{\bar{0}})$

(3.2)
$\gamma(m_{\bar{0}}+m^{\prime}_{\bar{0}})-\gamma(m_{\bar{0}})-\gamma(m^{\prime}_{\bar{0}})=m_{\bar{0}}\otimes
m^{\prime}_{\bar{0}}$ (3.3) $m\otimes
m^{\prime}=(-1)^{|m||m^{\prime}|}m^{\prime}\otimes m$

(3.4) $m_{\bar{1}}\otimes m_{\bar{1}}=0$

where $\lambda\in\mathbb{K}$, $m_{\bar{0}},m^{\prime}_{\bar{0}}\in
M_{\bar{0}}$, $m_{\bar{1}}\in M_{\bar{1}}$ and $m,m^{\prime}\in M$ with the
induced grading.

######  Proposition 3.7.

For any Lie superalgebra $M$, there exists an exact sequence

(3.5) $\Gamma(M^{ab})\overset{\psi}{\longrightarrow}M\otimes
M\overset{\pi}{\longrightarrow}M\wedge M\longrightarrow 0.$

Also, given two graded ideals $I$ and $J$ of $M$, the following sequence is
exact:

(3.6) $\Gamma(\frac{I\cap J}{[I,J]})\overset{\psi}{\longrightarrow}I\otimes
J\overset{\pi}{\longrightarrow}I\wedge J\longrightarrow 0.$

######  Proposition 3.8.

[14, Proposition 7.2.4] Let $M$ and $N$ be two supermodules. Then

$\Gamma(M\oplus N)\cong\Gamma(M)\oplus\Gamma(N)\oplus(M\otimes_{mod}N).$

######  Definition 3.9.

A quadratic map between two supermodules $M$ and $N$ is a map
$\varphi:M\longrightarrow N$ satisfying:

  1. (1)

$\varphi(\lambda m)=\lambda^{2}\varphi(m)$.

  2. (2)

The associated symmetric function $b_{\varphi}:M\times M\longrightarrow N$
defined by
$b_{\varphi}(m,m^{\prime})=\varphi(m+m^{\prime})-\varphi(m)-\varphi(m^{\prime})$
is bilinear.

######  Proposition 3.10.

[14, Proposition 7.2.2] The pair
$\gamma=\gamma_{M}=(\gamma_{\bar{0}},b_{\gamma})$, with
$\gamma_{\bar{0}}:M_{\bar{0}}\longrightarrow\Gamma(M_{\bar{0}})$ defined by
$\gamma_{\bar{0}}(x_{\bar{0}})=\gamma(x_{\bar{0}})$ and $b_{\gamma}:M\times
M\longrightarrow\Gamma(M)$ defined by $b_{\gamma}(x,y)=x\otimes y$ is a
quadratic map.

######  Proposition 3.11.

[14, Proposition 7.2.6] Let $M$ be a free supermodule, and let
$\\{\bar{x_{i}}\\}_{i\in I_{\bar{0}}}\bigcup\\{\bar{x_{i}}\\}_{i\in
I_{\bar{1}}}$ be an ordered basis of $M$ composed by homogeneous elements and
such that the elements $\\{\bar{x_{i}}\\}_{i\in I_{\bar{0}}}$ of $M_{\bar{0}}$
are less or equal than those of $M_{\bar{1}}$, $\\{\bar{x_{i}}\\}_{i\in
I_{\bar{1}}}$. Then $\Gamma(M)$ is free with basis

$\\{\gamma_{\bar{0}}(x_{i})\\}_{i\in
I_{\bar{0}}}\bigcup\\{b_{\gamma}(x_{i},x_{j})\\}_{i,j\in I_{\bar{0}}\bigcup
I_{\bar{1}},i<j}.$

######  Proposition 3.12.

[14, Proposition 7.2.9] Let $M$ be a Lie superalgebra such that $M^{ab}$ is a
free supermodule. Then the homomorphism $\psi$ in the sequence 3.5 is
injective.

######  Corollary 3.13.

Let $M$ be a Lie superalgebra such that $M^{ab}$ is a free supermodule. Then
the following sequence is exact

$0\longrightarrow\Gamma(M^{ab})\overset{\psi}{\longrightarrow}M\otimes
M\overset{\pi}{\longrightarrow}M\wedge M\longrightarrow 0.$

##  4\. Main results

In this section, we show that any $(k\mid l)$-dimensional non-abelian
nilpotent Lie superalgebra $L$ with derived supersubalgebra of dimension
$(r\mid s)$ satisfies $\dim(L\otimes L)\leq(k+l-(r+s))(k+l-1)+2.$ In
particular, for $r=1,s=0$, we determine the structure of $L$ when the equality
holds.

######  Lemma 4.1.

For any two abelian Lie superalgebras $M$ and $N$ there is an isomorphism

$(M\oplus N)\square(M\oplus N)\cong(M\square M)\oplus(N\square
N)\oplus(M\otimes N).$

###### Proof.

The proof follows from Proposition 3.8 and Lemma 4.2. ∎

######  Lemma 4.2.

Let $M$ be be a finite dimension Lie superalgebra. Then

$\Gamma(M/M^{2})\cong M/M^{2}\square M/M^{2}.$

###### Proof.

From the exact Sequence 3.5 we have Im$\psi=$Ker $\pi=M\square M$. Now we
shall prove that

(4.1) $\Gamma(A(m\mid n))\cong A(m\mid n)\square A(m\mid n)$

Let $m+n=1$, then by Corollary 3.13 we have

$\Gamma(A(1\mid 0))\cong A(1\mid 0)\square A(1\mid 0)$ $\Gamma(A(0\mid
1))\cong A(0\mid 1)\square A(0\mid 1)$

Now assume that $m+n\geq 2$. Then from Proposition 3.2, Lemma 4.2 and using
induction hypothesis we have

$\displaystyle\Gamma(A(m+1\mid n))$ $\displaystyle\cong\Gamma(A(m\mid n)\oplus
A(1\mid 0))$ $\displaystyle\cong\Gamma(A(m\mid n))\oplus\Gamma(A(1\mid
0))\oplus A(m\mid n)\otimes A(1\mid 0)$ $\displaystyle\cong A(m\mid n)\square
A(m\mid n)\oplus A(1\mid 0)\square A(1\mid 0)\oplus A(m\mid n)\otimes A(1\mid
0)$ $\displaystyle\cong(A(m\mid n)\oplus A(1\mid 0))\square(A(m\mid n)\oplus
A(1\mid 0))$ $\displaystyle\cong A(m+1\mid n)\square A(m+1\mid n).$

Similarly, we can see that $\Gamma(A(m\mid n+1))\cong A(m\mid n+1)\square
A(m\mid n+1).$ Thus the Relation 4.1 is true for all $m+n\geq 1.$ ∎

######  Corollary 4.3.

Let $M$ be a finite dimension Lie superalgebra, then

$M\square M\cong M/M^{2}\square M/M^{2}\cong\Gamma(M/M^{2}).$

###### Proof.

From Lemma 4.2 and the exact Sequence 3.5, we have the following epimorphism
$M/M^{2}\square M/M^{2}\longrightarrow M\square M$. Now by using Lemma 4.2 and
the natural epimorphism $M\square M\longrightarrow M/M^{2}\square M/M^{2}$ we
get the desire result. ∎

######  Proposition 4.4.

For $m+n\geq 2$, $H(m,n)\otimes H(m,n)\cong H(m,n)/H^{2}(m,n)\otimes
H(m,n)/H^{2}(m,n).$ Moreover, $H(1,0)\otimes H(1,0)\cong A(6\mid 0)$ and
$H(0,1)\otimes H(0,1)\cong A(1\mid 0)$.

###### Proof.

Suppose that $m+n\geq 2$. From Theorem 3.11 and Lemma 4.2, we have

$\dim(H(m,n)\square
H(m,n))=\dim\Gamma(H(m,n)/H^{2}(m,n))=(2m^{2}+m+\frac{n(n-1)}{2}\mid 2mn),$

and by Theorem 2.1 we have $\dim\mathcal{M}(H(m,n))=(2m^{2}-m+n(n+1)/2-1\mid
2mn)$. Then

$\displaystyle\dim(H(m,n)\otimes H(m,n))$ $\displaystyle=\dim(H(m,n)\square
H(m,n))+\dim\mathcal{M}(H(m,n))+\dim H^{2}(m,n)$
$\displaystyle=(2m^{2}+m+\frac{n(n-1)}{2}\mid 2mn)+(2m^{2}-m+n(n+1)/2-1\mid
2mn)+(1\mid 0)$ $\displaystyle=(4m^{2}+n^{2}\mid 4mn)$
$\displaystyle=4m^{2}+4mn+n^{2}.$

On the other hand, $\dim H(m,n)/H^{2}(m,n)\otimes
H(m,n)/H^{2}(m,n)=(2m+n)^{2}=4m^{2}+4mn+n^{2}$. Now the result can be obtained
by the natural epimorphism $H(m,n)\otimes H(m,n)\longrightarrow
H(m,n)H^{2}(m,n)\otimes H(m,n)H^{2}(m,n)$.

Similarly one can see that $H(1,0)\otimes H(1,0)$, (and $H(0,1)\otimes H(0,1)$
) is Lie superalgebra with dimension $(6\mid 0)$, (resp. of dimension $(1\mid
0)$). One can easily check that $H(1,0)\otimes H(1,0)$ and $H(0,1)\otimes
H(0,1)$ are abelian too. ∎

Now we state the same result for a Heisenberg Lie superalgebra with odd
center, as the proof is quite similar to the Proposition 4.4, so we omit it.

######  Proposition 4.5.

When $m\geq 2,~{}H_{m}\otimes H_{m}\cong H_{m}/H^{2}_{m}\otimes
H_{m}/H^{2}_{m}$. Moreover, $H_{1}\otimes H_{1}\cong A(2\mid 3)$.

######  Theorem 4.6.

Let $L$ be an $(k\mid l)$-dimensional non-abelian nilpotent Lie superalgebra
with derived supersubalgebra of dimension $(r\mid s)$. Then

$\dim(L\otimes L)\leq(k+l-(r+s))(k+l-1)+2.$

In particular, for $r=1,s=0$ the equality holds if and only if $L\cong
H(1,0)\oplus A(k-3\mid l)$.

###### Proof.

First, for $r+s=1$, we have the following cases

  1. (1)

$r=1,~{}s=0;$

  2. (2)

$r=0,~{}s=1.$

(1) Let $r=1,~{}s=0$, then from [8, Proposition 3.4], we have $L\cong
H(m,n)\oplus A(k-2m-1\mid l-n)$ for $m+n\geq 1$. Thus by Proposition 3.3,

$\dim(H(m,n)\oplus A(k-2m-1\mid l-n))\otimes(H(m,n)\oplus A(k-2m-1\mid
l-n))=\dim(H(m,n)\otimes H(m,n))$ $+2\dim(H(m,n)\otimes_{mod}A(k-2m-1\mid
l-n))+\dim(A(k-2m-1\mid l-n)\otimes A(k-2m-1\mid l-n)).$

From Proposition 3.2,

$(H(m,n)\otimes A(k-2m-1\mid
l-n))\cong(H(m,n)/H^{2}(m,n)\otimes_{mod}A(k-2m-1\mid l-n)).$

Now Consider the following cases:

(i) When $m=1,n=0$, then by using Proposition 4.4,

$\displaystyle\dim(H(1,0)\oplus A(k-3\mid l))\otimes(H(1,0)\oplus A(k-3\mid
l))$ $\displaystyle=(6\mid 0)+2(2\mid 0)(k-3\mid l)+(k-3\mid l)^{2}$
$\displaystyle=(k+l-1)^{2}+2.$

(ii) When $m=0,n=1$, then again from Proposition 4.4,

$\displaystyle\dim(H(0,1)\oplus A(k-1\mid l-1))\otimes(H(0,1)\oplus A(k-1\mid
l-1))$ $\displaystyle=(1\mid 0)+2(0\mid 1)(k-1\mid l-1)+(k-1\mid l-1)^{2}$
$\displaystyle=(k+l-1)^{2}.$

(iii) When $m+n\geq 2$, then from Proposition 4.4,

$\displaystyle\dim(H(m,n)\oplus$ $\displaystyle A(k-2m-1\mid
l-n))\otimes(H(m,n)\oplus A(k-2m-1\mid l-n))$ $\displaystyle=(4m^{2}+n^{2}\mid
4mn)+2(2m\mid n)(k-2m-1\mid l-n)+(k-2m-1\mid l-n)^{2}$
$\displaystyle=(k+l-1)^{2}.$

(2) Similarly for $r=0,s=1$, we can find that $L\cong H_{m}\oplus A(k-m\mid
l-m-1)$ and $\dim(L\otimes L)\leq(k+l-1)^{2}+2.$

Let $r+s\geq 2$ and assume that the result is true for $r+s-1$. Then for a
graded ideal $N$ contained in $Z(L)\cap L^{2}$ with $\dim N=(1\mid 0)$ (or
$\dim N=(0\mid 1)$), we have by induction hypothesis and Proposition 3.1,

$\dim(L\otimes L)\leq\dim(L/N\otimes L/N)+\dim(L\otimes N),$

and

$\dim(L/N\otimes L/N)\leq(k+l-1-(r+s-1))(k+l-2)+2.$

Now from Proposition 3.2

$\dim(L\otimes N)=\dim(L/L^{2}\otimes_{mod}N).$

Therefore,

$\dim(L\otimes L)\leq(k+l-(r+s))+(k+l-(r+s))(k+l-2)+2$
$=(k+l-(r+s))(k+l-1)+2.$

∎

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