# Probabilistic Debiasing of Scene Graphs

Bashirul Azam Biswas and Qiang Ji
Rensselaer Polytechnic Institute, Troy, NY-12180
{biswab<EMAIL_ADDRESS>

###### Abstract

The quality of scene graphs generated by the state-of-the-art (SOTA) models is
compromised due to the long-tail nature of the relationships and their parent
object pairs. Training of the scene graphs is dominated by the majority
relationships of the majority pairs and, therefore, the object-conditional
distributions of relationship in the minority pairs are not preserved after
the training is converged. Consequently, the biased model performs well on
more frequent relationships in the marginal distribution of relationships such
as ‘on’ and ‘wearing’, and performs poorly on the less frequent relationships
such as ‘eating’ or ‘hanging from’. In this work, we propose virtual evidence
incorporated within-triplet Bayesian Network (BN) to preserve the object-
conditional distribution of the relationship label and to eradicate the bias
created by the marginal probability of the relationships. The insufficient
number of relationships in the minority classes poses a significant problem in
learning the within-triplet Bayesian network. We address this insufficiency by
embedding-based augmentation of triplets where we borrow samples of the
minority triplet classes from its neighborhood triplets in the semantic space.
We perform experiments on two different datasets and achieve a significant
improvement in the mean recall of the relationships. We also achieve better
balance between recall and mean recall performance compared to the SOTA de-
biasing techniques of scene graph models. Code is publicly available at
https://github.com/bashirulazam/within-triplet-debias.

##  1 Introduction

Figure 1: (a) Within-triplet dependency of relationship on its parent object
pair; (b) long-tail nature of the pair statistics where 33% pair samples
originated from top $1\%$ pairs; (c) long-tail nature of the relationships
showing the dominance of ‘on’ and ‘wearing’. The skewness in (c) is an effect
of skewness in (b). Since ‘on’, ‘near’ or ‘wearing’ dominates in these top
$1\%$ pairs, they become the majority relationships in (c) and many other
relationships, such as ‘eating’ or ‘flying in’, which dominate in the tail
pairs of (b), are suppressed in the training process.

Any visual relationship can be expressed as a triplet subject-relationship-
object and all triplets in an image can be represented as a concise graph
called Scene Graph (SG) [19] where the nodes represent the objects and the
edges represent relationships. This representation has been proven useful for
many downstream tasks such as image captioning [37], visual reasoning [24],
and image generation [10]. Scene Graph Generation (SGG) has become one of the
major computer vision research arenas after the introduction of Visual Genome
(VG) dataset [11]. The distribution of triplets in VG images has two distinct
characteristics: (1) the presence of strong within-triplet prior, and (2) the
long-tail distribution of the relationship. As shown in Figure 1 (a), the
within-triplet prior dictates that ‘window’ will most likely be ‘on’ the
‘building’ rather than ‘eating’ it. Zeller et al. [43] has utilized this
within-triplet prior as the conditional probability of relationships given
subject and object by proposing a frequency baseline in the SGG task. On the
other hand, the distribution of relationship labels suffers from a long-tailed
nature and Tang et al. [27] addressed this long-tailed issue by considering a
causal interpretation of the biased prediction. We argue that these two
seemingly different characteristics of the relationship distribution are
interrelated. The abundance of the head classes of the relationship
distribution in Figure 1 (c), such as ‘on’ and ‘wearing’, arises from the
abundance of their parent subject and object lying in the head region of
Figure 1 (b).

As indicated by [5], the long-tailed distribution exists both in relationship
and object label. Since relationship labels are dependent on their object pair
because of the within-triplet prior, the long-tail distribution of the
relationship worsens due to the long-tail nature of the object pairs. Crowd-
collection of VG images creates selection bias and crowd-annotation of these
images create label-bias [29] and co-occurring-bias [25]. To analyze such
biases, we investigate the distribution of the object pair of the triplets in
VG database. As shown in Figure 1 (b), ‘window-building’ and ‘man-shirt’ are
the most frequently annotated pairs and top $1\%$ object pair covers $33\%$ of
all triplets. As a result, the dominant relationships in these head pairs,
such as ‘on’ and ‘wearing’, dominate the marginal distribution of Figure 1
(c).

In training a deep-learning-based SGG model, samplers will sample more
relationships from the head pairs. As a result, the Maximum Likelihood
Estimation (MLE) of the parameters is biased to predict the relationship
classes in the head pairs [27] and the object-conditional representation of
the relationship in the tail pairs will be lost in the training process.
Therefore, various deep learning-based models, which attempt to implicitly
capture such object-conditional representation [45, 4], fail to preserve the
representation in the trained model and perform poorly on the tail region of
the relationships.

Previous works attempt to retrieve the tail regions through re-sampling/re-
weighting the minority classes in training [5, 7, 3] or through causal
intervention in testing [27]. Their success is well-demonstrated by the
significant increase of minority-driven evaluation metric mean recall.
However, these approaches do not consider the strong within-triplet prior of
triplets and hurt the performance of majority-driven evaluation metric recall.
Keeping this gap in mind, we propose an inference-time post-processing
methodology that bolsters the minority tail classes as well as hurts the
majority head classes less brutally. We propose a within-triplet Bayesian
Network (BN) that combines the within-triplet prior with uncertain biased
evidence from SOTA models. Posterior inference with this BN simultaneously
eradicates the long-tailed bias in the marginal distribution of the
relationship and restores the object-conditional within-triplet prior.

Learning such a small within-triplet BN from the training data is a seemingly
trivial task where we can perform simple MLE of parameters by counting.
However, because of restricting our training samples only belonging to some
top-$N_{r}$ classes based on the marginal probability of relationship, we
sacrifice many information revealing triplets in the minority pairs. For
example, in the ‘man-pizza’ pair, we see there exist many interesting
relationships such as ‘man-biting-pizza’ or ‘man consuming pizza’ which are
semantically similar to one of the top-$N_{r}$ valid triplets ‘man-eating-
pizza’. This phenomenon is also a result of label bias [29] where the
annotator chooses some labels over another for the same category of objects or
relationships. We propose a novel method of borrowing samples from such
invalid triplets into learning the distribution of the valid triplets using
embedding-based augmentation.

The posterior inference is the most efficient probabilistic tool to combine
domain-dependent prior with instance-dependent evidence and, to the best of
our knowledge, no prior work in SGG literature formulates the problem of
triplet generation as a posterior inference problem. The overview of our
approach is illustrated in Figure 2. In summary, our contribution is proposing
a posterior inference-based post-processing method where we

  * •

integrate the within-triplet priors with the evidence uncertainties generated
by the measurement model and,

  * •

introduce a simple yet novel learning scheme of the within-triplet network
where we borrow samples from the semantically similar yet invalid triplet
categories.

Figure 2: Overview of our proposed approach. For each testing image $I$, SGG
baseline model $\theta_{m}$ generates physically connected triplets with
associated uncertainties for the subject, object, and relationship. Our
within-triplet inference framework takes all the uncertainties as uncertain
evidence and performs posterior inference to infer each triplet separately.
Afterward, a constrained optimization procedure is performed to resolve the
conflict between object entities.

##  2 Related Works

Our proposed SGG model combines prior knowledge of triplets with the uncertain
evidence of measurement models to address the long-tailed issue of
relationships. Moreover, we learn the prior model using the similarity of
triplets in the language embedding space. Hence, we divide the related works
into four major categories as following

Implicit prior and context incorporation: Global context of an image has been
captured either by BiLSTM, Graph Neural Network (GNN), attentional Graph
Convolution Network (aGCN), or Conditional Random Field (CRF) in [43, 32, 26,
34, 36]. Statistical information of triplets is encoded in [33, 2, 4]. Tang et
al. [28] composed a dynamic tree structure to capture triplet context.
Transformer-based context capturing is adopted in [17, 42, 23, 13]. Within-
triplet relations are incorporated in learnable modules as embeddings in [44,
30, 22]. All of these above methods extract the relational information from
the training data whereas several other approaches [6, 40, 19, 16, 41] rely on
external knowledge-base such as ConceptNet [18].

Language models and ambiguity: Language and vision modules are combined
together to guide the training process in [19, 40, 9, 21]. Embeddings from
phrasal context [39], word embedding-based external knowledge incorporation
[6, 41], and caption database [38] are used for supervision. The ambiguity of
scene graph triplets is addressed in [35, 46, 12].

Long-tail distribution: An unbiased metric (mean recall) is proposed by [2,
28] to measure the performance of the SGG models in tail classes. Several
works addressed this long-tailedness in training through modified loss
functions [17, 41] and re-weighting/re-sampling training samples [5, 7, 14,
15, 12, 3]. The most relevant to our work is the inference-time causal
intervention proposed by Tang et al. [27] where they identified the ‘bad bias’
from counterfactual causality and attained significantly higher mean recall
than other SOTA models. We, instead of applying causal intervention, resort to
a graphical model-based approach that can remove the ‘bad bias’ through
uncertain evidence insertion into a Bayesian network while maintaining the
‘good bias’ by within-triplet prior incorporation.

Uncertain evidence: Incorporation of evidence uncertainty into belief networks
has been discussed by Judea Pearl in [20]. A thorough discussion for
interested readers can be found in [1].

##  3 Problem formulation

In scene graph generation database, every image $I$ has an annotation of a
scene graph $\mathcal{G}_{I}=(\mathcal{E},\mathcal{R})$ where
$\mathcal{E}=\\{E_{i},B_{i}\\}_{i=1}^{N_{E}}$ contains the object classes
$E_{i}$ and their bounding boxes $B_{i}$ whereas
$\mathcal{R}=\\{R_{j}(S_{j},O_{j})\\}_{j=1}^{N_{T}}$ contains the
relationships of a scene graph. Each relationship $R_{j}$ exists between its
subject $S_{j}$ and object $O_{j}$ where $S_{j},O_{j}\in\\{E_{i}\\}$. Now, in
training any SGG model parameterized by $\theta$, we can write the cost
function $\mathcal{J}(\theta)$ as

$\mathcal{J}(\theta)=\mathbb{E}_{\sim
p(I,\mathcal{G})}L(I,\mathcal{G},\theta)\approx\frac{1}{M}\sum_{i=1}^{M}L(I_{i},\mathcal{G}_{i},\theta)$
(1)

where $\mathbb{E}(\cdot)$ is the expectation operator, $L$ is the loss
function with parameter $\theta$ for a sample image $I_{i}$ with associated
scene graph $\mathcal{G}_{i}$ and $M$ is total number of images. Now, as shown
in Figure 1, the distribution of scene graphs is skewed towards certain few
categories of object pairs and their relationships. Therefore, while training,
the cost function $\mathcal{J}(\theta)$ is driven by the head pairs, and the
relationships dominating in the tail pairs are ignored. As a result, many
object-conditional distributions of relationships are not preserved after the
training is converged. We propose a test-time post-processing method where the
object-conditional distribution of relationships is restored by a within-
triplet prior Bayesian network.

##  4 Posterior inference of Scene Graphs

###  4.1 Within-triplet Bayesian network

A scene graph is a collection of connected triplets and each triplet has three
semantic components - subject ($S$), object ($O$), and their relationship
($R$) and two spatial components - subject bounding box $B_{s}$ and object
bounding box $B_{o}$. We consider the semantic components as random
categorical variables and we aim to model their joint distribution $P(S,R,O)$.
We assume there exists no latent con-founder between the subject and object
node whereas the relationship node depends on its parent subject and object.
Formally, we assume the following statements hold true for any triplet -

  1. 1.

Relationship label of a triplet depends on its subject and object $\implies
S\rightarrow R\leftarrow O$;

  2. 2.

Subject and object are independent, not given the relationship $\implies
S\perp\\!\\!\\!\perp O\not|R$;

  3. 3.

Subject and object becomes dependent, given the relationship $\implies
S\not\perp\\!\\!\\!\perp O\mid R$

Based on these assumptions, we can build a Bayesian network of triplet, as
shown in Figure 3, which encodes the joint distribution using the chain rule
as follows

$P(S,R,O)=P(S)P(O)P(R|S,O)$ (2)

where $P(S)$ and $P(O)$ represent the marginal distribution of the parents $S$
and $O$, and $P(R|S,O)$ represents the conditional distribution of the
relationship $R$ given its parent subject and object. This Bayesian network
encodes the prior joint distribution which resides within a triplet and hence
we term it as within-triplet Bayesian network. In the next subsection, we
discuss how we can debias the measurement probability of a relationship by
incorporating them as uncertain evidence into this Bayesian network to perform
posterior inference.

Figure 3: Within-triplet Bayesian network where prior probabilities of subject
and object are combined with conditional probability of relationship.

###  4.2 Uncertain evidence

Figure 4: Uncertain evidence of each entity of triplet is incorporated into
the Bayesian network as the conditional probability of virtual evidence node
denoted as $Z_{s},Z_{o},$ and $Z_{r}$.

We denote any trained SGG model with parameter $\theta_{m}$ which generates
measurement probabilities of subject, object, and relationship of every
triplet for an image $I$ as $P_{\theta_{m},I}(S)$, $P_{\theta_{m},I}(O)$, and
$P_{\theta_{m},I}(R)$. We consider these measurements as uncertain evidence of
the nodes in the within-triplet BN in Figure 3. Since this evidence is
uncertain, we incorporate it as virtual evidence nodes into the BN as shown in
Figure 4. Following the virtual evidence method proposed by Judea Pearl in
[20], we introduce binary virtual evidence nodes $Z_{s},Z_{o},$ and $Z_{r}$ as
children of their respective parent evidence nodes $S,R,$ and $O$ and
instantiate them as True. According to Theorem 5 in [1], the conditional
distributions of these virtual nodes maintain the following likelihood ratios

$\small\begin{split}&P(Z_{s}=1|s_{1}):..:P(Z_{s}=1|s_{n})=\frac{P_{I,\theta_{m}}(s_{1})}{P(s_{1})}:..:\frac{P_{I,\theta_{m}}(s_{n})}{P(s_{n})}\\\
&P(Z_{o}=1|o_{1}):..:P(Z_{o}=1|o_{n})=\frac{P_{I,\theta_{m}}(o_{1})}{P(o_{1})}:..:\frac{P_{I,\theta_{m}}(o_{n})}{P(o_{n})}\\\
&P(Z_{r}=1|r_{1}):..:P(Z_{r}=1|r_{n})=\frac{P_{I,\theta_{m}}(r_{1})}{P(r_{1})}:..:\frac{P_{I,\theta_{m}}(r_{n})}{P(r_{n})}\end{split}$
(3)

where $P(S=s),P(O=o),$ and $P(R=r)$ are the marginal probabilities of subject,
object, and relationship node and $P_{\theta_{m},I}(S=s)$,
$P_{\theta_{m},I}(O=o)$, and $P_{\theta_{m},I}(R=r)$ are their observed
measurement probabilities from image $I$ with model $\theta_{m}$. Now, we have
a complete Bayesian network in Figure 4 with well-defined marginal and
conditional probabilities. A brief discussion on uncertain evidence and its
incorporation into the Bayesian network is discussed in Appendix A of the
supplementary material.

###  4.3 Within-Triplet Inference (WTI) of triplets

After the evidence incorporation as virtual evidence nodes, the posterior
joint distribution of triplet nodes becomes

$\small\begin{split}&P(S,R,O|Z_{s}=1,Z_{o}=1,Z_{r}=1)\\\ &\propto
P(Z_{s}=1|S)P(S)P(Z_{o}=1|O)P(O)P(Z_{r}=1|R)P(R|S,O)\\\ &\propto
P_{I,\theta_{m}}(S)P_{I,\theta_{m}}(O)\frac{P_{I,\theta_{m}}(R)}{P(R)}P(R|S,O)\end{split}$
(4)

The Maximum a-Posterior (MAP) of this posterior joint distribution becomes

$\small\begin{split}&S^{*},R^{*},O^{*}=\arg\max_{S,R,O}P(S,R,O|Z_{s}=1,Z_{o}=1,Z_{r}=1)\\\
&=\arg\max_{S,R,O}P_{I,\theta_{m}}(S)P_{I,\theta_{m}}(O)\frac{P_{I,\theta_{m}}(R)}{P(R)}P(R|S,O)\end{split}$
(5)

In Eqn. (5), the within-triplet dependency of relationship is encoded in
$P(R|S,O)$ and the measurement probability of relationship
$P_{I,\theta_{m}}(R)$ is debiased by its marginal probability $P(R)$. The
subject and object uncertainties are encoded in $P_{I,\theta_{m}}(S)$ and
$P_{I,\theta_{m}}(O)$. We include some special cases of the MAP Eqn. (5) in
Appendix B of supplementary material.

###  4.4 Constraint optimization in inferred triplets

Any object entity $E_{i}$ can reside in multiple triplets as subject $S$ or
object $O$ and after individual triplet inference, their inferred values can
be different in different triplets. However, to form a valid scene graph,
their values should be the same after the inference. Formally, if object
entity $E_{i}$ resides in $J$ triplets, their inferred values $E^{*}_{ij}$
must satisfy the following constraint

$E^{*}_{i1}=E^{*}_{i2}=...=E^{*}_{ij}=...=E^{*}_{iJ}=E^{I}_{i}$ (6)

One of the most straightforward ways to satisfy such constraint would be to
take the mode of these inferred values as the final value for $E_{i}$.
However, any object entity $E^{*}_{i}$ should be consistent with respect to
all of its connected triplets, and hence we formulate a two-step optimization
algorithm to infer $E^{*}_{i}$ and $R^{*}_{j}$ from their connections.

##### Object Updating:

In the first step, we infer each object label $E_{i}^{*}$ combining its
measurement probability $P_{I,\theta_{m}}(E_{i})$ with the within-triplet
probabilities of its connected triplets. We denote $T_{i}^{S}$ and $T_{i}^{O}$
as the sets of triplets where $E_{i}$ acts as subject and object respectively

$\begin{split}T_{i}^{S}=\\{t_{p}:t_{p}(S)=E_{i}\\}\\\
T_{i}^{O}=\\{t_{q}:t_{q}(O)=E_{i}\\}\end{split}$ (7)

The updated object probability for object $E_{i}$ and inference of $E_{i}$ is
derived as

$\small\begin{split}f(E_{i})=P_{I,\theta_{m}}(E_{i})\bigg{(}&\sum_{t_{p}\in
T_{i}^{S}}P(R=r_{t_{p}}^{I}|S=E_{i},O=o_{t_{p}}^{I})\\\ +&\sum_{t_{q}\in
T_{i}^{O}}P(R=r_{t_{q}}^{I}|S=s_{t_{q}}^{I},O=E_{i})\bigg{)}\\\
E^{*}_{i}&=\arg\max f(E_{i})\end{split}$ (8)

Intuitively speaking, the updated object probability $f(E_{i})$ derived in
Eqn. (8) combines the uncertain evidence of an object entity
$P_{I,\theta_{m}}(E_{i})$ with the prior probabilities of the within-triplet
Bayesian networks of all of its connected triplet.

##### Relationship Updating:

After the first step is completed for each object entity, the conflicts of
object entities are resolved with updated entity values. In the second step,
we update the relationship label of each triplet based on the updated subject
and object values

$\begin{split}R_{j}^{*}&=\arg\max_{R}P(R_{j}|Z_{r}=1,S=s_{j},O=O_{j})\\\
&=\arg\max_{R}\frac{P_{I,\theta_{m}}(R_{j})}{P(R_{j})}P(R_{j}|S=s_{j},O=o_{j})\end{split}$
(9)

Detailed derivation and pseudo-code are provided in Appendix C of the
supplementary material. We denote this as constraint optimization in our
overview in Figure 2.

##  5 Learning BN with embedding similarity

The within-triplet priors $P(S)\in\mathbb{R}^{N_{s}}$,
$P(O)\in\mathbb{R}^{N_{o}}$, and conditional distribution
$P(R|S,O)\in\mathbb{R}^{N_{s}\times N_{o}\times N_{r}}$ are learned from
annotations of training data where $N_{s}$, $N_{o},$ and $N_{r}$ denote the
number of categories of subject, object, and relationship. A training dataset
of $\mathbb{Z}^{3\times N}$ is created by collecting total $N$ ground truth
(GT) triplets from the training images. Afterward, we apply MLE to estimate
$P(R|S,O)$ and $P(R)$ as follows

$\small\begin{split}P(R=r|S=s,O=o)&=\frac{N^{c}_{s,r,o}}{\sum_{r^{\prime}}N^{c}_{s,r^{\prime},o}}\\\
P(R)&=\sum_{S,O}P(R|S,O)P(S)P(O)\end{split}$ (10)

where $N^{c}_{s,r,o}$ is the count of triplet with $S=s,O=o$ and $R=r$.
However, because of selecting only top $N_{r}$ relationships from the training
data, many semantically similar triplets whose relationships lie outside these
top $N_{r}$, are ignored in this count (e.g. ‘man-consuming-pizza’ is ignored
whereas ‘man-eating-pizza’ is considered as a valid triplet). Hence, we
propose a novel sample augmentation method using off-the-shelf sentence
embedding models [31] where all the ignored invalid triplets, lying within a
$\epsilon$-neighbourhood of a valid triplet in the embedding space, are
counted as augmented samples of that valid triplet. For any subject and object
pair with $S=s,O=o$, we denote any valid triplet as $T=\\{s,r,o\\}$ and
invalid triplet as $T_{i}=\\{s,r_{i},o\\}$ where $r\in N_{r}$ and $r_{i}\notin
N_{r}$. Now, if we denote the original count of $T$ as $N^{c}_{s,r,o}$ and
that of $T_{i}$ as $N_{s,r_{i},o}$, we can augment the original count as
following

$N^{a}_{s,r,o}=\begin{cases}N^{c}_{s,r,o}+\sum_{T_{i}\in\mathcal{N}_{\epsilon}(T)}{N_{s,r_{i},o}}\\\
N^{c}_{s,r,o}\ \ \text{if $\mathcal{N}_{\epsilon}(T)=\emptyset$}\end{cases}$
(11)

Here, $\epsilon$ is a hyper-parameter and $\mathcal{N}_{\epsilon}(T)$ is
defined as the $\epsilon$-neighbourhood of a valid triplet $T$ in the
embedding space using the following criteria

$\mathcal{N}_{\epsilon}(T)=\\{T_{i}:\phi(f(T),f(T_{i}))<\epsilon\\}$ (12)

where $\phi(f(T),f(T_{i}))$ represents the distance between two embedding
vectors of two triplet $f(T)$ and $f(T_{i})$. Since the embeddings lie on a
unit-sphere, we employ cosine similarity to measure the angular distance
between two embedding vectors of two triplets. We visualize the
$\epsilon$-neighbourhood of a valid triplet $T$ in Figure 5.

Figure 5: $\mathcal{N}_{\epsilon}(T)$ in the embedding space where
$f(T_{1}),f(T_{2})\in\mathcal{N}_{\epsilon}(T)$ and
$f(T_{3})\notin\mathcal{N}_{\epsilon}(T)$.

DS | Method | Recall and Mean Recall @K
---|---|---
PredCls | SGCls | SGDet
R@50/100 | mR@50/100 | R@50/100 | mR@50/100 | R@50/100 | mR@50/100
VG |  IMP⋄ [33] | 61.63/ 63.63 | 11.53/ 12.38 | 36.14/ 37.09 | 5.69/ 5.98 | 28.04/ 31.30 | 4.89/ 5.84
Inf-IMP |  59.93/ 62.02 $\downarrow$ |  25.14/ 28.34 $\uparrow$ |  36.02/ 37.09 $\downarrow$ |  12.57/ 14.06 $\uparrow$ |  26.50/ 29.51 $\downarrow$ |  8.58/ 10.67 $\uparrow$
MOTIF⋄ [43] | 59.57/ 63.95 | 12.88/ 15.47 | 36.45/ 38.47 | 7.66/ 8.83 | 26.85/ 30.50 | 5.61/ 6.73
Inf-MOTIF |  51.49/ 55.07 $\downarrow$ |  24.67/ 30.71 $\uparrow$ |  32.15/ 33.79 $\downarrow$ |  14.50/ 17.40 $\uparrow$ |  23.94/ 27.09 $\downarrow$ |  9.36/ 11.71 $\uparrow$
VCTree⋄ [28] | 65.46/ 67.18 | 15.36/ 16.61 | 44.15/ 45.11 | 9.17/ 9.83 | 29.94/ 32.57 | 6.21/ 6.96
Inf-VCTree |  59.50/ 60.97 $\downarrow$ |  28.14/ 30.72 $\uparrow$ |  40.69/ 41.55 $\downarrow$ |  17.31/ 19.40 $\uparrow$ |  27.74/ 30.10 $\downarrow$ |  10.40/ 11.86 $\uparrow$
Unb-MOTIF⋄ [27] | 45.87/ 51.24 | 24.75/ 28.69 | 26.30/ 28.78 | 13.21/ 15.06 | 16.25/ 19.53 | 8.65/ 10.47
Inf-Unb-MOTIF |  42.40/ 46.82 $\downarrow$ |  28.64/ 35.65 $\uparrow$ |  24.13/ 26.28 $\downarrow$ |  15.85/ 18.88 $\uparrow$ |  15.06/ 18.03 $\downarrow$ |  9.60/ 11.94 $\uparrow$
DLFE-MOTIF◆ [3] | 51.63/ 53.28 | 26.87/ 28.75 | 28.79/ 29.66 | 15.61 /16.38 | 24.22/ 27.95 | 10.62/ 12.61
Inf-DLFE-MOTIF |  43.27/ 44.82 $\downarrow$ |  35.25/ 38.20 $\uparrow$ |  24.34/ 25.14 $\downarrow$ |  19.74/ 20.66 $\uparrow$ |  20.61/ 23.80 $\downarrow$ |  14.07/ 16.76 $\uparrow$
BGNN◆ [14] | 58.15/ 60.41 | 29.46/ 31.83 | -/ - | \- /- | 30.26/ 34.98 | 10.37/ 12.31
Inf-BGNN |  55.42/ 57.47 $\downarrow$ |  32.18/ 34.27 $\uparrow$ | -/ - | \- /- |  26.16/ 30.11 $\downarrow$ |  13.24/ 16.10 $\uparrow$
GQA |  IMP⋄ [33] | 61.94/ 63.68 | 13.04/ 13.74 | 34.25/ 34.83 | 7.46/ 7.80 | 25.39/ 27.42 | 5.77/ 6.58
Inf-IMP |  61.87/ 63.98 $\uparrow$ |  35.14/ 37.54 $\uparrow$ |  33.19/ 34.01 $\downarrow$ |  19.06/ 20.33 $\uparrow$ |  23.46/ 25.56 $\downarrow$ |  12.17/ 14.13 $\uparrow$
MOTIF⋄ [43] | 68.29/ 69.65 | 20.67/ 21.56 | 34.89/ 35.43 | 10.90/ 11.31 | 27.83/ 29.38 | 7.38/ 8.32
Inf-MOTIF |  62.95/ 64.23 $\downarrow$ |  37.93/ 40.07 $\uparrow$ |  31.82/ 32.35 $\downarrow$ |  19.09/ 20.00 $\uparrow$ |  25.51/ 26.86 $\downarrow$ |  14.34/ 15.84 $\uparrow$
VCTree⋄ [28] | 68.83/ 70.14 | 22.07/ 23.01 | 35.04/ 35.58 | 10.59/ 10.97 | 27.21/ 28.79 | 7.03/ 7.75
Inf-VCTree |  62.80/ 64.05 $\downarrow$ |  39.44/ 41.63 $\uparrow$ |  32.23/ 32.80 $\downarrow$ |  19.18/ 20.03 $\uparrow$ |  25.04/ 26.44 $\downarrow$ |  13.58/ 15.11 $\uparrow$
Unb-MOTIF⋄ [27] | 51.87/ 55.87 | 27.81/ 32.30 | 26.10/ 28.06 | 14.09/ 16.33 | 18.22/ 21.63 | 10.78/ 12.89
Inf-Unb-MOTIF |  49.86/ 53.60 $\downarrow$ |  34.45/ 40.80 $\uparrow$ |  24.76/ 26.74 $\downarrow$ |  17.18/ 20.59 $\uparrow$ |  16.84/ 19.94 $\downarrow$ |  12.35/ 14.76 $\uparrow$

Table 1: R@K and mR@K results of inference with prefix ‘Inf-’. We observe a
significant increase in mR@K with a slight decrease in R@K for all baseline
models in both datasets. Graph constraint is applied in all settings. Baseline
results are generated by codebase released by [27] ($\diamond$) and by
respective authors ($\blacklozenge$)

.

##  6 Experimental settings

###  6.1 Dataset

We evaluate our proposed method on two datasets: (1) Visual Genome (VG), and
(2) GQA.

(1) Visual Genome: For SGG, the most commonly used database is the Visual
Genome (VG) [11] and from the original database, the most frequent $150$
object and $50$ predicate categories are retained [33, 43]. We adopt the
standard train-test split ratio of $70:30$. The number of prior triplets from
the original training dataset is around $323K$ and after augmenting with
$\epsilon=0.05$, the number rises to around $391K$.

(2) GQA: GQA [8] is a refined dataset derived from the VG images. We retain
images only with the most frequent $150$ object and $50$ relationship
categories. We train on around $50k$ valid images and perform the evaluation
on the validation dataset of valid $7k$ images. We collect over $190K$ prior
triplets and after embedding-based augmentation with $\epsilon=0.05$, the
number rises to around $200K$.

###  6.2 Task description

A triplet is considered correct if the subject, relationship, and object label
matches with the ground truth labels and the boundary boxes of subject and
object have an Intersection over Union (IoU) of at least $50\%$ with the
ground truth annotations. We consider three test-time tasks, defined by [33],
- (1) PredCls: known object labels and locations, (2) SGCls: known object
locations, and (3) SGDet: where no information about objects are known. We
apply graph constraints for all the tasks where for each pair of objects, only
one relationship is allowed.

###  6.3 Evaluation metrics

The performance of our debiased SGGs is evaluated through recall (R@K) and
mean recall (mR@K). R@K of an image is computed as the fraction of ground
truth triplets in top@K predicted triplets [19] whereas mR@K computes recall
for each relationship separately and then the average over all relationships
are computed [2, 28].

###  6.4 Implementation details

We perform training and testing of the baseline models released by [27], [3],
and [14]. We collect measurement results and ground truth annotations of the
testing database using Python. For sample augmentation, we employ the sentence
transformer model ‘all-mpnet-base-v2’ released by HuggingFace [31]. We learn
the within-triplet prior from the original and augmented annotations, and
perform posterior inference in MATLAB on a computer with core i5 7th
generation Intel processor running at $2.5$ MHz with $8.00$ GB RAM. The total
training time for the prior probabilities of VG dataset is $968s$ and that of
GQA is $430s$. The inference task per image requires $0.13s$.

##  7 Experimental results

###  7.1 Quantitative results

We generate triplet measurements from four classical SGG models - (1) IMP
[33], (2) MOTIF [43], (3) VCTree [28], and (4) Unb-MOTIF [27] from codebase
[27] ($\diamond$), and two recent-most SOTA SGG models released by (1) DLFE-
MOTIF [3] and (2) BGNN [14] ($\blacklozenge$). Considering these measurements
as uncertain evidence, we perform within-triplet inference for all three
settings in Sect. 6.2 and conduct two-step updating only for SGCls and SGDet
settings. We denote the final results with the prefix ‘Inf-’. The Bayesian
network is learned from the augmented counts of triplet derived by Eqn. (11).
We report R@K and mR@K for all three tasks with VG and GQA in Table 1. We also
include a separate comparison with other bias removal techniques (1) Unb-
[27], (2) DLFE- [3], and (3) NICE-[12] in Table 2. Our method performs better
in balancing the head and tail classes without any retraining of the biased
model.

Method | Re-train | R@K | mR@K
---|---|---|---
@50/100 | @50/100
DT2-ACBS [5] | Yes | 23.3/ 25.6 | 35.9/ 39.7
BGNN [14] | Yes | 59.2/ 61.3 | 30.4/ 32.9
IMP[33] | - | 61.6/ 63.6 | 11.5/ 12.4
Inf-IMP (Ours) | No | 59.9/ 62.0 | 25.1/ 28.3
MOTIF[33] | - | 59.6/ 64.0 | 12.9/ 15.5
Unb-MOTIF[27] | No | 45.9/ 51.2 | 24.8/ 28.7
DLFE-MOTIF[3] | Yes | 51.6/ 53.2 | 26.9/ 28.8
NICE-MOTIF [12] | Yes | 55.1/ 57.2 | 29.9/ 32.3
Inf-MOTIF (Ours) | No | 51.5/ 55.1 | 24.7/ 30.7
VCTree[28] | - | 65.5/ 67.2 | 15.4/ 16.6
Unb-VCTree [27] | No | 47.2/ 51.6 | 25.4/ 28.7
DLFE-VCTree [3] | Yes | 51.8/ 53.5 | 25.3/ 27.1
NICE-VCTree [12] | Yes | 55.0/ 56.9 | 30.7/ 33.0
Inf-VCTree (Ours) | No | 59.5/ 61.0 | 28.1/ 30.7
Table 2: Comparison with other de-biasing methods in PredCls. Without re-
training, our loss in R@K is significantly lower, and gain in mR@K is higher
or competitive than other SOTA debiasing methods.

###  7.2 Analysis

####  7.2.1 Ablation study on prior and uncertain evidence

We perform an ablation study on the measurement results of the PredCls task by
VCTree [28] in Table 3. We visualize the improvement of tail classes with BN
learned from original and augmented samples in Figure 6(a) and 6(b). In the
former case, mid relationships are improving after inference and in the latter
case both mid and tail are improving.

(a) Recall improvement of each relationship with original samples.

(b) Recall improvement of each relationship with augmented samples.

Figure 6: Improvement of mean recalls with VCTree [28] evidence for PredCls task in VG. Relationships are ordered in descending order of their frequencies. In (a), BN learned with original samples improves the mid region whereas in (b), the embedding-based augmentation improves both the mid and the tail ones. In both cases, the head relationships are worsened after debiasing. Method | R@K | Mean R@K
---|---|---
@50/100 | @50/100
Uncertain evidence (VCTree) | 65.5/ 67.2 | 15.4/ 16.6
WT BN only (org) (FREQ) |  64.3/ 65.8 $\downarrow$ |  16.1/ 17.5 $\uparrow$
WT BN only (aug) (Ours) |  62.7/ 64.2 $\downarrow$ |  16.3/ 17.8 $\uparrow$
WT BN (org) + Unc. Evi (Ours) |  62.5 /64.1 $\downarrow$ |  22.7/ 24.8 $\uparrow$
WT BN (aug) + Unc. Evi (Ours) |  59.5 /61.0 $\downarrow$ |  28.1/ 30.7 $\uparrow$
Table 3: Ablation study on PredCls performance for VG. We observe the
consistent improvement of mR@K starting with uncertain evidence and ending in
posterior inference with uncertain evidence with BN learned from augmented
samples.

####  7.2.2 Ablation study on conflict resolution

As discussed in Sect. 4.4, the potential conflicts of object labels after
within-triplet inference can be resolved with naive mode-selection or by our
proposed constraint optimization. We observe the effectiveness of the proposed
optimization method over mode selection in Table 4.

Method | R@K | mR@K
---|---|---
@50/100 | @50/100
VCTree [28] | 44.2/ 45.1 | 9.2/ 9.8
Inf-VCTree (Conflict res. by mode) | 40.3/ 41.2 | 16.9/ 18.7
Inf-VCTree (Conflict res. by opt.) |  40.7/ 41.6 $\uparrow$ |  17.3/ 19.4 $\uparrow$
Table 4: Ablation study on conflict resolution for VCTree SGCls performance
for Visual Genome. The optimization algorithm performs better than the naive
mode-selection version.

####  7.2.3 Effect of $\epsilon$ on training data augmentation

We observe the effect of neighborhood radius $\epsilon$ in the embedding space
on R@K and mR@K in Table 5. Larger $\epsilon$ tends to hurt the majority
classes more. We choose $\epsilon=0.05$ for our final experiments.

Method | $\epsilon$ | Recall@K | Mean Recal@K
---|---|---|---
@50/@100 | @50/@100 |
Inf- VCTree (org) | - | 62.48/ 64.06 | 22.74/ 24.78 |
Inf- VCTree (aug) | 0.03 | 59.44/ 60.88 | 28.01/ 30.55 |
0.05 | 59.50/ 60.97 | 28.14/ 30.72 |
0.07 | 59.27/ 60.73 | 28.28/ 30.92 |
Table 5: Effect of $\epsilon$ on PredCls performance for VG with VCTree
baseline [28]. We find that using larger $\epsilon$ tends to drop R@K more.
Based on this study, we choose $\epsilon=0.05$ for our final experiments.

####  7.2.4 Effect on zero-shot recall

The zero-shot prediction of the measurement model is compromised after
posterior inference due to the MLE-based learning of BN. However, since unseen
triplets will have higher entropy than the seen ones in the prediction phase,
we can filter out the high-entropy triplets so that they do not get refined by
the BN. An ablation study with respect to entropy threshold is shown in Table
6.

Method | Entropy Th. | ZS R@100 | ZS mR@100
---|---|---|---
org/aug | org/aug
VCTree [28] | - | 6.02 | 16.61
Inf-VCTree | 0 | 6.02/ 6.02 | 16.61/ 16.61
1.5 | 5.81/ 6.13 | 20.20/ 22.71
2.5 | 5.47/ 6.03 | 24.42/ 29.28
3.912 | 1.06/ 2.30 | 24.78/ 30.70
Table 6: Zero-shot recall on PredCls for VG for VCTree [28]. Lower thresholds
restore the zero-shot capability of measurement models by hurting the mean
recall. We choose not to use any threshold (last row) to maximize performance
in the mean recall.

##  8 Limitations

While improving mR@K by our proposed method, we lose performance in R@K. This
phenomenon is prevailing in SGG debiasing works since we are perturbing the
‘head’ classes to gain improvement in the ‘tail’. Moreover, the augmentation
hyper-parameter $\epsilon$ may vary from dataset to dataset and need to be
chosen carefully. Another weakness is the compromise in the zero-shot
prediction capability of a measurement model due to MLE-based learning of BN.

##  9 Conclusion

We proposed a debiasing strategy of scene graphs by combining prior and
uncertain evidence of triplets in the Bayesian framework. We performed MAP
inference and optimally solved the conflict between object entities to predict
the debiased triplet from biased evidence. We augmented the count of valid
triplets with semantically similar invalid triplets to alleviate sample
insufficiency. Our method showcased significant improvement in mean recall
with baseline measurements. We also attained a better balance between majority
and minority performances of the relationships. In the future, we will extend
the MAP inference for multiply connected triplets and we will explore well-
defined criteria for zero-shot refinement to restore the zero-shot recall of
the measurement models.

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# Noetherian enveloping algebras of simple graded Lie algebras

Nicolás Andruskiewitsch  Facultad de Matemática, Astronomía y Física,
Universidad Nacional de Córdoba. CIEM – CONICET.
Medina Allende s/n (5000) Ciudad Universitaria, Córdoba, Argentina
<EMAIL_ADDRESS>and  Olivier Mathieu  CNRS and Université
Claude Bernard Lyon
Institut Camille Jordan UMR 5028 du CNRS,
69622 Villeurbanne Cedex, France<EMAIL_ADDRESS>
SUSTech, Shenzhen International Center for Mathematics, Shenzhen, China

###### Abstract.

It is shown that the universal enveloping algebra of an infinite-dimensional
simple ${\mathbb{Z}}^{n}$-graded Lie algebra is not Noetherian.

_2020 MSC._ 16S30, 17B35. N. A. was partially supported by the Secyt (UNC),
CONICET (PIP 11220200102916CO), FONCyT-ANPCyT (PICT-2019-03660). Both authors
were partially supported by the Shenzhen International Center of Mathematics,
SUSTech.

##  1\. Introduction

Let $K$ be a field of characteristic 0. If a Lie algebra is finite
dimensional, then its enveloping algebra is Noetherian. Whether the converse
is true has been asked by many authors, among them R. Amayo and I. Stewart,
see [1, Question 27], K. A. Brown, see [2, Question B], J. Dixmier, and V.
Latyshev. Besides its intrinsic interest, this is an unavoidable question in
the problem of the classification of Noetherian Hopf algebras. S. Sierra and
C. Walton stated this question as a Conjecture.

######  Conjecture 1.1.

[10] The universal enveloping algebra of an infinite-dimensional Lie algebra
is not Noetherian.

Intuitively, since ‘large’ Lie algebras satisfy the Conjecture, e.g. the
enveloping algebra of a free Lie algebra in two generators is not Noetherian,
one expects that a counterexample to the Conjecture, if any, should be in some
sense ’small’. In this direction, a breakthrough result was obtained in 2013
by Sierra and Walton. Recall that the Witt algebra is
$W(1)\coloneqq\operatorname{Der}K[t]$.

######  Theorem 1.2.

[10, 0.5] The enveloping algebra of $W(1)$ is not Noetherian.

This result allows to conclude that the enveloping algebra of an infinite
dimensional simple ${\mathbb{Z}}$-graded Lie algebra of finite growth is not
Noetherian, by going over the classification of such Lie algebras obtained in
[9].

However there are neither classification results for simple
${\mathbb{Z}}$-graded Lie algebras of arbitrary growth, nor for simple
${\mathbb{Z}}^{n}$-graded Lie algebras for $n\geq 2$. Nevertheless, the
following result will be established in the present paper.

######  Theorem 1.3.

The universal enveloping algebra of an infinite-dimensional simple
${\mathbb{Z}}^{n}$-graded Lie algebra is not Noetherian.

See below for our conventions on ${\mathbb{Z}}^{n}$-graded Lie algebras. The
proof uses the Theorem of Sierra and Walton, and some classification results
of the second author, namely the Theorem 4 and the results of Section 8 of
[9].

##  2\. Conventions and Preliminaries

###  2.1. Conventions about graded vector spaces

In the whole paper, we will adopt the following convention. A vector space $M$
endowed with a decomposition $M=\oplus_{{\mathbf{m}}\in{\mathbb{Z}}^{n}}M_{\bf
m}$ will be called a ${\mathbb{Z}}^{n}$-graded vector space only if all
homogeneous components $M_{\bf m}$ are finite dimensional.

A Lie algebra $\mathcal{L}$ endowed with a ${\mathbb{Z}}^{n}$-grading is
called a ${\mathbb{Z}}^{n}$-graded Lie algebra if we have

$\displaystyle[\mathcal{L}_{\bf n},\mathcal{L}_{\bf m}]$
$\displaystyle\subset\mathcal{L}_{\bf n+m}$ $\displaystyle\text{for any }\bf
n,\bf m\in{\mathbb{Z}}^{n}.$

A ${\mathbb{Z}}^{n}$-graded Lie algebra $\mathcal{L}$ of dimension $\geq 2$
without nontrivial proper ${\mathbb{Z}}^{n}$-graded ideals is called a simple
${\mathbb{Z}}^{n}$-graded Lie algebra. For example, $\mathfrak{sl}(2)\otimes
K[t,t^{-1}]$ is a simple ${\mathbb{Z}}$-graded Lie algebra, but it is not
simple as a Lie algebra. The definitions of a _ ${\mathbb{Z}}^{n}$-graded
$\mathcal{L}$-module_ and a _simple ${\mathbb{Z}}^{n}$-graded
$\mathcal{L}$-module_ are similar.

###  2.2. Criteria for noetherianity of enveloping algebras

A section of a Lie algebra $L$ is a Lie algebra $\mathfrak{s}$ isomorphic to
$\mathfrak{q}/\mathfrak{m}$ for some Lie subalgebra $\mathfrak{q}\subset L$
and some ideal $\mathfrak{m}$ of $\mathfrak{q}$.

The following standard observations are useful, see [10, 1.7] and [4, 2.1].

######  Lemma 2.1.

Let $L$ be a Lie algebra such that $U(L)$ is Noetherian.

  1. (a)

$L$ satisfies the ascending chain condition on Lie subalgebras.

  2. (b)

$L$ is finitely presented and $H_{k}(L)$ is finite dimensional for any $k\geq
0$.

  3. (c)

If $\mathfrak{s}$ is a section of $L$, then $U(\mathfrak{s})$ is also
Noetherian.

  4. (d)

If $\mathfrak{s}$ is an abelian section of $L$, then
$\dim\mathfrak{s}<\infty$.

  5. (e)

If $L$ is a Lie subalgebra of finite codimension of some Lie algebra
$L^{\prime}$, then $U(L^{\prime})$ is also Noetherian. ∎

###  2.3. Examples of enveloping algebras that are not Noetherian

Lemma 2.1 allows to deduce that many Lie algebras satisfy Conjecture 1.1 from
Lie algebras that are already known to fulfill it, for instance:

  1. (i)

The free Lie algebra $\operatorname{Free}(Z)$ on a vector space $Z$ of
dimension $\geq 2$. Indeed $U(\operatorname{Free}(Z))\simeq T(Z)$ is not
Noetherian.

  2. (ii)

[10, 0.5] The positive Witt algebra $W_{+}$. By Lemma 2.1(e), this result is
equivalent to the result stated in the introduction.

See [10, 4] for a list of Lie algebras whose enveloping algebras are not
Noetherian by the remarks above. By Lemma 2.1.(c) and (i), another example is
a Kac–Moody algebra of indefinite type, cf. [6, Corollary 9.12].

##  3\. Growth of modules over ${\mathbb{Z}}$-graded Lie algebras

In this section and in the next three, we investigate the noetherianity
condition for ${\mathbb{Z}}$-graded Lie algebras. The present section involves
the questions of finite generation and growth.

Given a ${\mathbb{Z}}$-graded vector space $M$ and an integer
$n\in{\mathbb{Z}}$, we set

$\displaystyle M_{\geq n}\coloneqq\oplus_{k\geq n}\,M_{k}.$

The subspaces $M_{>n}$, $M_{\leq n}$ and $M_{<n}$ are similarly defined.

###  3.1. Finite generation

Let $\mathcal{L}$ be a ${\mathbb{Z}}$-graded Lie algebra. We set

$\displaystyle\mathcal{L}^{+}=\mathcal{L}_{>0}\quad\text{ and
}\quad\mathcal{L}^{-}=\mathcal{L}_{<0}.$

######  Lemma 3.1.

Assume that the Lie algebra $\mathcal{L}$ is finitely generated. Then
$\mathcal{L}^{+}$ and $\mathcal{L}^{-}$ are finitely generated subalgebras.

Moreover let $M$ be a finitely generated ${\mathbb{Z}}$-graded
$\mathcal{L}$-module. Then the $\mathcal{L}^{+}$-module $M_{\geq 0}$ and the
$\mathcal{L}^{-}$-module $M_{\leq 0}$ are finitely generated.

###### Proof.

By hypothesis, there is an integer $d>0$ such that $\oplus_{-d\leq k\leq
d}\,\mathcal{L}_{k}$ generates $\mathcal{L}$. By Lemma 18 of [8],
$\oplus_{1\leq k\leq d}\,\mathcal{L}_{k}$ generates $\mathcal{L}^{+}$ and
$\oplus_{-d\leq k\leq-1}\,\mathcal{L}_{k}$ generates $\mathcal{L}^{-}$, which
proves the first assertion.

Let $S$ be a finite set of generators of $M$. There is an integer $e$ such
that $S$ lies in $M_{\leq e}$. Since $M_{\leq e}$ is a $\mathcal{L}_{\leq
0}$-module, we have $M=U(\mathcal{L}^{+})\cdot M_{\leq e}$. Since in addition
$\mathcal{L}^{+}$ is generated by $\oplus_{1\leq k\leq d}\,\mathcal{L}_{k}$,
we have

$\displaystyle M_{n}=\sum_{1\leq k\leq d}\,\mathcal{L}_{k}\cdot M_{n-k},$

for any $n>e$. It follows easily that $M_{\geq 0}$ is finitely generated. The
proof of the finite generation of the $\mathcal{L}^{-}$-module $M_{\leq 0}$ is
similar. ∎

###  3.2. Finite and intermediate growth

A ${\mathbb{Z}}$-graded vector space $M$ is called of finite growth if the
function

$\displaystyle n\mapsto\dim\,M_{n}$

is bounded by a polynomial. It is called of intermediate growth if both limits

$\displaystyle\limsup\frac{\log^{+}(\dim M_{n})}{n}$ and
$\displaystyle\limsup\frac{\log^{+}(\dim M_{-n})}{n}$

are zero, where the function $\log^{+}$ is defined by $\log^{+}(x)=\log(x)$ if
$x\geq 1$ and $\log^{+}(x)=0$ otherwise. The formal series

$\displaystyle\chi_{M}^{\pm}(z)\coloneqq\sum_{n\geq 0}\,\dim\,M_{\pm
n}\,z^{n}$

are called the two generating series of $M$. Equivalently, $M$ has
intermediate growth iff both series $\chi_{M}^{+}(z)$ and $\chi_{M}^{-}(z)$
are convergent for $|z|<1$.

Assume now that $M=\oplus_{n\geq 1}\,M_{n}$ is a positively graded vector
space. Then the symmetric algebra $S(M)$ is a nonnegatively graded vector
space. The following lemma is well-known.

######  Lemma 3.2.

Assume that the positively graded vector space $M$ has intermediate growth.
Then $S(M)$ also have intermediate growth.

###### Proof.

For any integer $n\geq 1$, set $a_{n}=\dim\,M_{n}$. We have

$\displaystyle\chi^{+}_{M}(z)$ $\displaystyle=\sum_{n\geq 1}\,a_{n}z^{n},$
$\displaystyle\chi^{+}_{S(M)}(z)$ $\displaystyle=\prod_{n\geq
1}\,\frac{1}{(1-z^{n})^{a_{n}}}.$

If $M$ is finite dimensional, $S(M)$ has finite growth. Otherwise, the lemma
follows because these series have the same radius of convergence. ∎

###  3.3. Growth of ${\mathbb{Z}}$-graded $\mathcal{L}$-modules

Let $\mathcal{L}$ be a ${\mathbb{Z}}$-graded algebra and let $M$ be a
${\mathbb{Z}}$-graded $\mathcal{L}$-module. For any $n$, let
$M_{n}^{\operatorname{int}}$ the subspace of all $m\in M_{n}$ such that
$U(\mathcal{L}^{+})\cdot m$ has intermediate growth. Set
$M^{\operatorname{int}}=\oplus_{n\in{\mathbb{Z}}}M_{n}^{\operatorname{int}}$.

######  Lemma 3.3.

The subspace $M^{\operatorname{int}}$ is a $\mathcal{L}$-submodule.

###### Proof.

Since $M^{\operatorname{int}}$ is clearly a $\mathcal{L}^{+}$-module, it is
enough to show that for any homogeneous elements $u\in\mathcal{L}$ of degree
$d\leq 0$ and $v\in M^{\operatorname{int}}$, $u\cdot v$ belongs to
$M^{\operatorname{int}}$. First note that

$\displaystyle U(\mathcal{L}^{+})u\subset U(\mathcal{L}^{+})\mathcal{L}_{\geq
d}=\mathcal{L}_{\geq
d}U(\mathcal{L}^{+})=U(\mathcal{L}^{+})\oplus\oplus_{d\leq k\leq
0}\,\mathcal{L}_{k}U(\mathcal{L}^{+}).$

Therefore we have

$\displaystyle U(\mathcal{L}^{+})u.v\subset U(\mathcal{L}^{+}).v+\sum_{n\leq
k\leq 0}\,\mathcal{L}_{k}U(\mathcal{L}^{+}).v.$

Thus $U(\mathcal{L}^{+})u.v$ has intermediate growth, i.e. $u\cdot v$ belongs
to $M^{\operatorname{int}}$. ∎

######  Lemma 3.4.

Let $\mathcal{L}$ be finitely generated ${\mathbb{Z}}$-graded Lie algebra and
let $M$ be a simple ${\mathbb{Z}}$-graded module. Assume that, for some
homogeneous $v\in M\setminus 0$, the vector space $\mathcal{L}\cdot v$ has
intermediate growth. Then $M$ has intermediate growth.

###### Proof.

Let $\mathcal{K}^{+}=\\{x\in\mathcal{L}^{+}\mid x.v=0\\}$. As a graded space,
the ${\mathcal{L}^{+}}$-module
$\operatorname{Ind}_{\mathcal{K}^{+}}^{\mathcal{L}^{+}}Kv$ is isomorphic to
$S({\mathcal{L}^{+}}/\mathcal{K}^{+})$. By Lemma 3.2,
$\operatorname{Ind}_{\mathcal{K}^{+}}^{\mathcal{L}^{+}}Kv$ has intermediate
growth. Thus $U(\mathcal{L}^{+})\cdot v$, a quotient of
$\operatorname{Ind}_{\mathcal{K}^{+}}^{\mathcal{L}^{+}}$, has intermediate
growth too.

Since $M$ is simple, from Lemma 3.3 we infer that any cyclic
$U(\mathcal{L}^{+})$-submodule of $M$ has intermediate growth. Now the
$\mathcal{L}^{+}$-module $M_{\geq 0}$ is finitely generated by Lemma 3.1,
hence $M_{\geq 0}$ has intermediate growth. Similarly $M_{\leq 0}$ has
intermediate growth; therefore $M$ has intermediate growth. ∎

##  4\. Rank one Lie algebras of class $\mathscr{V}$

We define first the general notions of roots and rank of a
${\mathbb{Z}}$-graded Lie algebra $\mathcal{L}$. Then we split the proof that
$U(\mathcal{L})$ is not Noetherian into three cases: Lie algebras of class
$\mathscr{V}$ are treated in this section; the next section 5 is devoted to
class $\mathscr{S}$. The last section 6 deals with Lie algebras of rank $\geq
2$.

###  4.1. Roots and rank

Let $\mathcal{L}=\oplus_{n\in{\mathbb{Z}}}$ be a ${\mathbb{Z}}$-graded Lie
algebra. We fix, once and for all, a Cartan subalgebra $\mathfrak{h}$ of
$\mathcal{L}_{0}$, i.e., $\mathfrak{h}$ is a nilpotent self-normalizing
subalgebra of $\mathcal{L}_{0}$ [3]. For any $\alpha\in\mathfrak{h}^{*}$ and
any $n\in{\mathbb{Z}}$, we set

$\displaystyle\mathcal{L}_{n}^{\alpha}=\\{x\in\mathcal{L}_{n}\mid\,(\operatorname{ad}(h)-\alpha(h))^{N}(x)=0\quad\forall
h\in\mathfrak{h}\text{ and }N>>0\\}.$

Also, $\mathcal{L}^{\widetilde{\alpha}}\coloneqq\mathcal{L}^{\alpha}_{n}$ for
$\widetilde{\alpha}=(\alpha,n)\in\mathfrak{h}^{*}\times{\mathbb{Z}}$. The set
of roots of $\mathcal{L}$ is the set
$\Delta\coloneqq\\{\widetilde{\alpha}\mid\mathcal{L}^{\widetilde{\alpha}}\neq
0\\}$ (with our nonstandard definition, $0$ is a root). Therefore

$\displaystyle\mathcal{L}=\oplus_{\widetilde{\alpha}\in\Delta}\,\mathcal{L}^{\widetilde{\alpha}}$

is the generalized root space decomposition of $\mathcal{L}$.

A root $\widetilde{\alpha}=(\alpha,n)$ is called real if $\alpha\neq 0$ and
imaginary otherwise. Let $\Delta_{\operatorname{re}}$, respectively
$\Delta_{\operatorname{im}}$, be the set of real, respectively imaginary,
roots.

The root lattice is the subgroup $Q\subset\mathfrak{h}^{*}\times{\mathbb{Z}}$
generated by $\Delta$. By definition the rank of ${\mathcal{L}}$ is the rank
of $Q$.

###  4.2. Rank one Lie algebras

Let $\mathcal{L}=\oplus_{n\in{\mathbb{Z}}}$ be a ${\mathbb{Z}}$-graded Lie
algebra of rank one. Therefore there exists $\widetilde{\alpha}=(\alpha,1)$
such that $\Delta$ lies in ${\mathbb{Z}}.\widetilde{\alpha}$. We keep the
terminology of [9]. When $\alpha=0$ or, equivalently, when the set of real
roots is void, we say that $\mathcal{L}$ belongs to the class $\mathscr{V}$.
Otherwise, we say that $\mathcal{L}$ belongs to the class $\mathscr{S}$. Here
the letter $\mathscr{S}$ stands for string, because, roughly speaking, all
real roots are on a “string”.

###  4.3. Rank one Lie algebras of class $\mathscr{V}$

This case follows easily from the next result.

######  Lemma 4.1.

[7, Lemma 22] Let $\mathcal{L}$ be a ${\mathbb{Z}}$-graded Lie algebra of
class $\mathscr{V}$. If $\mathcal{L}=[\mathcal{L},\mathcal{L}]$, then
$\mathcal{L}$ is not finitely generated. ∎

######  Corollary 4.2.

Let $\mathcal{L}$ be a simple ${\mathbb{Z}}$-graded Lie algebra of class
$\mathscr{V}$. Then $U(\mathcal{L})$ is not Noetherian.

###### Proof.

Immediate from Lemmas 2.1(c) and 4.1. ∎

##  5\. Rank one Lie algebras of class $\mathscr{S}$

The case of Lie algebras of class $\mathscr{S}$ is more difficult than the
previous one. Recall that a ${\mathbb{Z}}$-graded Lie algebra $\mathcal{L}$
belongs to $\mathscr{S}$ if there exists a nonzero $\alpha\in\mathfrak{h}^{*}$
such that $\mathcal{L}_{n}=\mathcal{L}_{n}^{n\alpha}$ for any
$n\in{\mathbb{Z}}$.

The main step is Theorem 5.6, which is implicit in [9]. Navigating through
chapters 7 and 8 of loc. cit. is not easy. Thus for the sake of the reader, we
rewrite parts of those in a convenient way.

We need the following definition. For $n\neq 0$, let $\mathcal{L}\\{n\\}$ be
the Lie algebra $\mathcal{L}$ endowed with a grading rescaled by a factor of
$n$, i.e. we have

$\displaystyle\mathcal{L}\\{n\\}_{nk}$ $\displaystyle=\mathcal{L}_{k},\quad
k\in{\mathbb{Z}},$ $\displaystyle\mathcal{L}\\{n\\}_{m}$
$\displaystyle=0\text{ if }n\not|m.$

The ${\mathbb{Z}}$-graded Lie algebra $\mathcal{L}\\{n\\}$, again in class
$\mathscr{S}$, is called a rescaling of $\mathcal{L}$.

###  5.1. Local Lie algebras

Let $P$ be the set of pairs of integers $(i,j)$ with $i,j,i+j\in\\{-1,0,1\\}$.
Following [5], see [6, Exercise 1.8, p. 13], a local Lie algebra is a graded
vector space

$\displaystyle G=G_{-1}\oplus G_{0}\oplus G_{1}$

endowed with a degree preserving bracket $[\,,\,]$ which is defined only on
$\cup_{(i,j)\in P}\,G_{i}\times G_{j}$ and which satisfies the Jacobi identity
whenever it makes sense. Equivalently, this means that $G_{0}$ is a Lie
algebra, $G_{1}$ and $G_{-1}$ are $G_{0}$-modules and the bracket
$[,]:G_{-1}\times G_{1}\to G_{0}$ is $G_{0}$-equivariant.

The notions of morphisms between local Lie algebras, local Lie subalgebras and
local ideals are defined in an evident way. Analogously a local Lie algebra
$S$ is a called a section of $G$ if $S$ is isomorphic to $H/K$ for some local
subalgebra $H\subset G$ and some local ideal $K$ of $H$.

Given a ${\mathbb{Z}}$-graded Lie algebra $\mathcal{L}$, its local part

$\displaystyle\mathcal{L}_{loc}:=\mathcal{L}_{-1}\oplus\mathcal{L}_{0}\oplus\mathcal{L}_{1}$

is evidently a local Lie algebra. Conversely, given a local Lie algebra $G$
there are Lie algebras whose local part is $G$. One of them, denoted by
$\mathcal{L}_{\max}(G)$, is defined as follows. As a vector space we have

$\displaystyle\mathcal{L}_{\max}(G)=\operatorname{Free}(G_{-1})\oplus
G_{0}\oplus\operatorname{Free}(G_{1})$

where $\operatorname{Free}(G_{\pm 1})$ denotes the free Lie algebra on the
vector space $G_{\pm 1}$. Then the local Lie bracket and the
${\mathbb{Z}}$-grading extend uniquely to $\mathcal{L}_{\max}(G)$ [5]. Now the
functor $G\to\mathcal{L}_{\max}(G)$ is the left adjoint of the functor
$\mathcal{L}\to\mathcal{L}_{loc}$ [8]. Let $\mathcal{I}$ be the largest graded
ideal of $\mathcal{L}_{\max}(G)$ such that $\mathcal{I}\cap G=0$ and set

$\displaystyle\mathcal{L}_{\min}(G)=\mathcal{L}_{\max}(G)/\mathcal{I}.$

Notice that, if $\mathcal{L}$ is a Lie ${\mathbb{Z}}$-graded Lie algebra which
is generated by its local part $G$, then there are natural epimorphisms

$\displaystyle\mathcal{L}_{\max}(G)$
$\displaystyle\twoheadrightarrow\mathcal{L}$ and $\displaystyle\mathcal{L}$
$\displaystyle\twoheadrightarrow\mathcal{L}_{\min}(G),$

so $\mathcal{L}$ is between the Lie algebras $\mathcal{L}_{\max}(G)$ and
$\mathcal{L}_{\min}(G)$. We conclude:

######  Lemma 5.1.

Let $G$ be a local Lie algebra and let $\mathcal{L}$ be a
${\mathbb{Z}}$-graded Lie algebra. If $G$ is a section of $\mathcal{L}_{loc}$,
then $\mathcal{L}_{\min}(G)$ is a section of $\mathcal{L}$. ∎

###  5.2. Examples of simple Lie algebras of class $\mathscr{S}$

We start recalling the definitions of some Lie algebras of class
$\mathscr{S}$.

  * $\circ$

The centerless Virasoro algebra is $W=\operatorname{Der}K[t,t^{-1}]$. It has a
natural grading, relative to which the element
$t^{n+1}\frac{\mathrm{d}}{\mathrm{d}t}$ is homogeneous of degree $n$. We have
$W_{0}=\mathfrak{h}=K.t\frac{\mathrm{d}}{\mathrm{d}t}$; clearly $W$ is in
class $\mathscr{S}$.

  * $\circ$

The Witt algebra is $W(1)=\operatorname{Der}K[t]$; it is a graded subalgebra
of $W$ and also belongs to $\mathscr{S}$.

  * $\circ$

The contragredient Lie algebra $G(^{2\,2}_{2\,2})$. It is generated by five
elements $h,e_{1},e_{2},f_{1},f_{2}$ and defined by the following relations

$\displaystyle[h,e_{i}]$ $\displaystyle=2e_{i},$ $\displaystyle[h,f_{i}]$
$\displaystyle=-2f_{i},$ $\displaystyle[e_{i},f_{j}]$
$\displaystyle=\delta_{i,j}\,h,$ (1)

for any $i,j\in\\{1,2\\}$, where, as usual, $\delta_{i,j}$ is the Kronecker
symbol.

It has a ${\mathbb{Z}}$-grading relative to which the $e_{i}$’s have degree
one, $h$ has degree zero and the $f_{i}$’s have degree $-1$. Let $G$ be the
local part of the Lie algebra $G(^{2\,2}_{2\,2})$. Since $G(^{2\,2}_{2\,2})$
is generated by its local part and is defined by local relations, we have

$\displaystyle G(^{2\,2}_{2\,2})=\mathcal{L}_{\max}(G).$

######  Lemma 5.2.

We have $G(^{2\,2}_{2\,2})=\mathcal{L}_{\min}(G)$.

###### Proof.

This follows because the Lie algebra $G(^{2\,2}_{2\,2})$ is simple [5]. ∎

###  5.3. A simple criterion for a section isomorphic to $G(^{2\,2}_{2\,2})$

Let $\mathcal{L}$ be a ${\mathbb{Z}}$-graded Lie algebra of class
$\mathscr{S}$ with $\alpha\in\mathfrak{h}^{*}$ as above. In this subsection
and in the next one, we do not assume that $\mathcal{L}$ is simple as a
${\mathbb{Z}}$-graded algebra. We will describe criteria for the existence of
a section of $\mathcal{L}$ isomorphic to $G(^{2\,2}_{2\,2})$.

For $n\neq 0$, let $B_{n}:\mathcal{L}_{-n}\times\mathcal{L}_{n}\to K$ be the
bilinear map

$\displaystyle
B_{n}:(x,y)\in\mathcal{L}_{-n}\times\mathcal{L}_{n}\mapsto\alpha([x,y]).$

Let $\mathcal{K}_{n}$ and $\mathcal{K}_{-n}$ be its right kernel and its left
kernel. Also set $\mathcal{K}_{0}=\operatorname{Ker}\alpha$.

######  Lemma 5.3.

Assume that $H_{0}(\mathcal{K}_{0},\mathcal{L}_{n}/\mathcal{K}_{n})$ has
dimension $\geq 2$ for some $n>0$. Then, up to a rescaling,
$G(^{2\,2}_{2\,2})$ is a section of $\mathcal{L}$.

###### Proof.

We can assume that $n=1$. The bilinear form $B_{1}$ provides a nondegenerate
pairing of $\mathcal{L}_{-1}/\mathcal{K}_{-1}$ and
$\mathcal{L}_{1}/\mathcal{K}_{1}$. Thus $\dim
H^{0}(\mathcal{K}_{0},\mathcal{L}_{-1}/\mathcal{K}_{-1})\geq 2$. Hence there
is a $\mathcal{L}_{0}$-module $\mathcal{L}^{\prime}_{-1}$ with
$\mathcal{K}_{-1}\subset\mathcal{L}^{\prime}_{-1}\subset\mathcal{L}_{-1}$ such
that

$\displaystyle\dim\mathcal{L}^{\prime}_{-1}/\mathcal{K}_{-1}$
$\displaystyle=2$ and
$\displaystyle[\mathcal{K}_{0},\mathcal{L}^{\prime}_{-1}]$
$\displaystyle\subset\mathcal{K}_{-1}.$

Let $\mathcal{K}^{\prime}_{1}$ be the orthogonal in $\mathcal{L}_{1}$ of
$\mathcal{L}^{\prime}_{-1}$. Then

$\displaystyle\dim\mathcal{L}_{1}/\mathcal{K}^{\prime}_{1}$ $\displaystyle=2$
and $\displaystyle[\mathcal{K}_{0},\mathcal{L}_{1}]$
$\displaystyle\subset\mathcal{K}^{\prime}_{1}.$

Therefore,
$\mathcal{I}\coloneqq\mathcal{K}_{-1}\oplus\mathcal{K}_{0}\oplus\mathcal{K}^{\prime}_{1}$
is a local ideal of the local Lie algebra
$\mathcal{G}\coloneqq\mathcal{L}^{\prime}_{-1}\oplus\mathcal{L}_{0}\oplus\mathcal{L}_{1}$
and clearly $\mathcal{G}/\mathcal{I}$ is isomorphic to the local part of
$G(^{2\,2}_{2\,2})$.

It follows from Lemmas 5.1 and 5.2 that $G(^{2\,2}_{2\,2})$ is a section of
$\mathcal{L}$. ∎

###  5.4. An improved criterion for a section isomorphic to
$G(^{2\,2}_{2\,2})$

Using the notation of the previous section, we show the same criterion with a
weaker hypothesis.

######  Lemma 5.4.

Assume that $\mathcal{L}_{n}/\mathcal{K}_{n}$ has dimension $\geq 2$ for some
integer $n$. Then, up to a rescaling, $G(^{2\,2}_{2\,2})$ is a section of
$\mathcal{L}$.

###### Proof.

We can assume that $n=1$ and that $\mathcal{L}$ is generated by its local
part. This implies that $\mathcal{L}^{+}$ is generated by $\mathcal{L}_{1}$.

By Lemma 7.9 of [9], the function $n\mapsto\operatorname{rk}\,B_{n}$ has
infinite growth. The maximal dimension of cyclic modules in
$\mathcal{L}_{1}^{\otimes n}$ growths polynomially, so the same property holds
into its quotient $\mathcal{L}_{n}/\mathcal{K}_{n}$. It follows that the
function
$n\mapsto\dim\,H_{0}(\mathcal{K}_{0},\mathcal{L}_{n}/\mathcal{K}_{n})$, which
measures the minimal number of generators of the $\mathcal{K}_{0}$-module
$\mathcal{L}_{n}/\mathcal{K}_{n}$, has an infinite growth. Therefore, for some
$n$, we have

$\displaystyle\dim\,H_{0}(\mathcal{K}_{0},\mathcal{L}_{n}/\mathcal{K}_{n})$
$\displaystyle\geq 2.$

Thus by Lemma 5.3, $G(^{2\,2}_{2\,2})$ is a section of $\mathcal{L}$. ∎

###  5.5. The dichotomy for the class $\mathscr{S}$

Let $\mathcal{L}$ be a simple ${\mathbb{Z}}$-graded Lie algebra of class
$\mathscr{S}$. It is implicitely proved in [9, Chapter 8] that $\mathcal{L}$
is isomorphic to $W$ or $W(1)$, under the hypothesis

all bilinear forms $B_{n}$ have rank $\leq 1$. ($\mathcal{H}_{1}$)

Unfortunately, the explicit hypothesis used in [9, Chapter 8] is

the Lie algebra $\mathcal{L}$ has intermediate growth. ($\mathcal{H}_{2}$)

It would be long to go into the details of _loc. cit._ to explain why
$(\mathcal{H}_{1})$ can be used instead of $(\mathcal{H}_{2})$. Here we can
assume that $\mathcal{L}$ is finitely generated. Under this additional
hypothesis, the next lemma gives an easy explanation.

######  Lemma 5.5.

If $\mathcal{L}$ is finitely generated, then $(\mathcal{H}_{1})$ implies
$(\mathcal{H}_{2})$.

###### Proof.

Let $M\coloneqq\oplus_{n\in{\mathbb{Z}}}\,\mathcal{L}_{n}^{*}$ be the graded
dual of the adjoint module. The hypothesis $(\mathcal{H}_{1})$ means that the
${\mathbb{Z}}$-graded space $\mathcal{L}\cdot\alpha$ has homogenous components
of dimension $\leq 1$. By Lemma 3.4, we see that $M$ has intermediate growth,
i.e. $(\mathcal{H}_{2})$ holds. ∎

The following result is implicitely proved in [9], even without the hypothesis
of finite generation.

######  Theorem 5.6.

Let $\mathcal{L}$ be a simple ${\mathbb{Z}}$-graded Lie algebra of class
$\mathscr{S}$. Assume that $\mathcal{L}$ is finitely generated. Then

  1. (i)

either $\mathcal{L}$ is isomorphic to $\mathfrak{sl}(2)$, $W(1)$ or $W$,

  2. (ii)

or $\mathcal{L}$ contains a nonabelian free Lie algebra.

###### Proof.

(i) First assume that the bilinear $B_{n}$ has rank $\leq 1$ for any $n$. By
Lemma 5.5, $\mathcal{L}$ has intermediate growth. Thus it follows from
Proposition 8.9 of [9] that $\mathcal{L}$ is isomorphic to $\mathfrak{sl}(2)$,
$W(1)$ or $W$.

(ii) Otherwise, the bilinear form has rank $\geq 2$ for some $n$. By Lemma 5.4
the Lie algebra $G(^{2\,2}_{2\,2})$ is a section of $\mathcal{L}$. By Lemma
5.2, the Lie algebra $G(^{2\,2}_{2\,2})$ contains a nonabelian free Lie
algebra, namely the subalgebra generated by $e_{1}$ and $e_{2}$. Hence
$\mathcal{L}$ contains a nonabelian free algebra. ∎

######  Corollary 5.7.

Let $\mathcal{L}$ be a simple ${\mathbb{Z}}$-graded Lie algebra of class
$\mathscr{S}$. Then $U(\mathcal{L})$ is not Noetherian, except if
$\mathcal{L}$ is isomorphic to $\mathfrak{sl}(2)$.

###### Proof.

Assume that $\mathcal{L}$ is infinite dimensional. By Theorem 5.6,
$\mathcal{L}$ contains a subalgebra isomorphic to the Witt algebra $W(1)$ or a
nonabelian free subalgebra. In the first case, Theorem 1.2 implies that
$U(\mathcal{L})$ is not Noetherian. In the second case, we already observed
that the enveloping algebra of a nonabelian free Lie algebra is not
Noetherian, so neither is $U(\mathcal{L})$. ∎

##  6\. ${\mathbb{Z}}$-graded Lie algebras of rank $\geq 2$

In this section we investigate the noetherianity condition for
${\mathbb{Z}}$-graded Lie algebras of rank $\geq 2$.

We will encounter Lie algebras $\mathcal{M}$ with a decomposition
$\mathcal{M}=\oplus\mathcal{M}_{n}$ satisfying
$[\mathcal{M}_{n},\mathcal{M}_{m}]\subset\mathcal{M}_{n+m}$ where the
homogeneous components could be of infinite dimension; we shall call them
weakly ${\mathbb{Z}}$-graded Lie algebras.

###  6.1. The hypothesis $(\mathcal{H})$

Let $\mathcal{L}$ be a ${\mathbb{Z}}$-graded Lie algebra. Consider the
following hypothesis

There exist $\displaystyle\widetilde{\alpha},\widetilde{\beta}\in
Q,\,\widetilde{\beta}\notin{\mathbb{Q}}.\widetilde{\alpha},\,\text{such
that}\,({\widetilde{\beta}}+{\mathbb{Z}}.{\widetilde{\alpha}})\cap\Delta\,\text{is
infinite.}$ ($\mathcal{H}$)

######  Lemma 6.1.

If $\mathcal{L}$ satisfies the hypothesis $(\mathcal{H})$, then
$U(\mathcal{L})$ is not Noetherian.

###### Proof.

For any integer $k\geq 1$, set
$\Delta(k)=(k.{\widetilde{\beta}}+{\mathbb{Z}}.{\widetilde{\alpha}})\cap\Delta$
and

$\displaystyle\mathcal{M}_{k}=\oplus_{\widetilde{\gamma}\in\Delta(k)}\,\mathcal{L}^{\widetilde{\gamma}}.$

Since we have $[\mathcal{M}_{k},\mathcal{M}_{l}]\subset\mathcal{M}_{k+l}$ for
any $k,l\geq 1$, the vector space

$\displaystyle\mathcal{M}\coloneqq\oplus_{k\geq 1}\,\mathcal{M}_{k}$

is a weakly positively graded Lie algebra. The natural map

$\displaystyle\mathcal{M}_{1}\to
H_{1}(\mathcal{M})=\mathcal{M}/[\mathcal{M},\mathcal{M}]$

is one to-to-one, therefore $H_{1}(\mathcal{M})$ is infinite dimensional.
Since $\widetilde{\beta}$ is not in ${\mathbb{Q}}.\widetilde{\alpha}$, the
sets $\Delta(k)$ are pairwise disjoint. Hence $\mathcal{M}$ is a Lie
subalgebra of $\mathcal{L}$. By Lemma 2.1 (b), $U(\mathcal{L})$ is not
Noetherian. ∎

###  6.2. Constructions of ideals in Lie algebras

The next two lemmas show that certain subspaces of a Lie algebra are indeed
ideals. Results of this kind are useful in the study of simple Lie algebras.

Let $\mathcal{L}$ be a Lie algebra.

######  Lemma 6.2.

[7, Lemma 6] Let $\mathcal{A}$ and $\mathcal{B}$ be linear subspaces of
$\mathcal{L}$ such that $\mathcal{L}=\mathcal{A}+\mathcal{B}$ and
$[\mathcal{A},\mathcal{B}]\subset\mathcal{B}$. Then
$\mathcal{B}+[\mathcal{B},\mathcal{B}]$ is an ideal of $\mathcal{L}$. ∎

Let $L$ be a linear subspace of $\mathcal{L}$. We say that $\mathcal{L}$ is
locally $L$-nilpotent if, for any $x\in\mathcal{L}$, we have
$\mathrm{Ad}(L)^{1+n}(x)=0$ for some integer $n=n(x)\geq 0$.

######  Lemma 6.3.

Let $L$ be a linear subspace such that $\mathcal{L}$ is locally $L$-nilpotent.
Then the following subspace is an ideal of $\mathcal{L}$:

$\displaystyle\mathcal{I}\coloneqq\cap_{N\geq
0}\,\mathrm{Ad}(L)^{N}(\mathcal{L}).$

###### Proof.

Let $x\in\mathcal{L}$. For any $y\in\mathcal{L}$ and any $N>0$, we have

$\displaystyle[x,\mathrm{Ad}(L)^{N}(y)]\subset\sum_{0\leq k\leq
N}\mathrm{Ad}(L)^{N-k}([\mathrm{Ad}(L)^{k}(x),y]).$

Assume now that $\mathrm{Ad}(L)^{n+1}(x)=0$. Thus for any $N\geq n$, we have

$\displaystyle[x,\mathrm{Ad}(L)^{N}(\mathcal{L})]\subset\mathrm{Ad}(L)^{N-n}(\mathcal{L})$

and therefore $[x,\mathcal{I}]\subset\mathcal{I}$. ∎

###  6.3. A dichotomy for the ${\mathbb{Z}}$-graded Lie algebras of rank $\geq
2$

Let $\mathcal{L}$ be a simple ${\mathbb{Z}}$-graded Lie algebra of rank $\geq
2$. We now define two hypothetical properties, and show that any such
$\mathcal{L}$ satisfies one of them. By the end of the section it will be
clear that these properties are mutually exclusive.

To start with, we define the notion of a string. Let $\widetilde{\alpha}\in Q$
and $\widetilde{\beta}\in\Delta$. There are
$a,b\in{\mathbb{Z}}\cup\\{\pm\infty\\}$ with $a<0<b$ such that

  1. (i)

$\widetilde{\beta}+k\widetilde{\alpha}$ belongs to $\Delta$ for any
$k\in]a,b[$, but

  2. (ii)

neither $\widetilde{\beta}+a\widetilde{\alpha}$ nor
$\widetilde{\beta}+b\widetilde{\alpha}$ belongs to $\Delta$.

The set $\\{\widetilde{\beta}+k\widetilde{\alpha}\mid k\in]a,b[\\}$ is called
the _$\widetilde{\alpha}$ -string through $\widetilde{\beta}$_.

The first hypothetical property $(\mathcal{H}_{\operatorname{re}})$ is the
following:

$\displaystyle\begin{aligned} &\text{There exist
}\widetilde{\alpha}\in\Delta_{\operatorname{re}},\quad\widetilde{\beta}\in\Delta,\quad\widetilde{\beta}\notin{\mathbb{Q}}.\widetilde{\alpha},\text{
such that}\\\ &\text{the $\widetilde{\alpha}$-string through
$\widetilde{\beta}$ is infinite.}\end{aligned}$
($\mathcal{H}_{\operatorname{re}}$)

The hypothesis $(\mathcal{H}_{\operatorname{re}})$ is obviously stronger than
$(\mathcal{H})$.

The second hypothetical property is the notion of weak integrability.
Following [9], we say that $\mathcal{L}$ is weakly integrable if, for any
$\widetilde{\alpha}\in\Delta_{\operatorname{re}}$, we have

$\displaystyle\bigcap_{n\geq
0}\,\mathrm{Ad}(\mathcal{L}^{\widetilde{\alpha}})^{n}(\mathcal{L})=0.$

######  Lemma 6.4.

Let $\mathcal{L}$ be a simple ${\mathbb{Z}}$-graded algebra of rank $\geq 2$.
Then either

  1. (a)

$\mathcal{L}$ satisfies the hypothesis $(\mathcal{H}_{\operatorname{re}})$, or

  2. (b)

$\mathcal{L}$ is weakly integrable.

###### Proof.

Assuming that $\mathcal{L}$ does not satisfy
$(\mathcal{H}_{\operatorname{re}})$, we will prove that $\mathcal{L}$ is
weakly integrable. Let $\widetilde{\alpha}\in\Delta_{\operatorname{re}}$. Set

$\displaystyle\mathcal{A}$
$\displaystyle=\oplus_{\widetilde{\beta}\in{\mathbb{Q}}\cdot\widetilde{\alpha}}\,\mathcal{L}^{\widetilde{\beta}}$
and
$\displaystyle\mathcal{B}=\bigoplus_{\widetilde{\beta}\notin{\mathbb{Q}}\cdot\widetilde{\alpha}}$
$\displaystyle\mathcal{L}^{\widetilde{\beta}}.$

By hypothesis any $\widetilde{\alpha}$-string through any root
$\widetilde{\beta}\notin{\mathbb{Q}}\widetilde{\alpha}$ is finite. Hence

  1. (i)

The restriction of $\mathrm{Ad}(\mathcal{L}^{\widetilde{\alpha}})$ to
$\mathcal{B}$ is locally nilpotent, and

  2. (ii)

$\cap_{n\geq
0}\,\mathrm{Ad}(\mathcal{L}^{\widetilde{\alpha}})^{n}(\mathcal{L})\subset\mathcal{A}$.

Since $\mathcal{L}$ has rank $\geq 2$, $\mathcal{B}\neq 0$. By Lemma 6.2, the
simplicity of $\mathcal{L}$ implies that
$\mathcal{L}=\mathcal{B}+[\mathcal{B},\mathcal{B}]$. Hence $\mathcal{L}$ is
$\mathrm{Ad}(\mathcal{L}^{\widetilde{\alpha}})$-locally nilpotent.

By Lemma 6.3, we see that $\mathcal{I}\coloneqq\cap_{n\geq
0}\,\mathrm{Ad}(\mathcal{L}^{\widetilde{\alpha}})^{n}(\mathcal{L})$ is an
ideal. Since we have $\mathcal{I}\subset\mathcal{A}$, we conclude that
$\mathcal{I}=0$. In other words, $\mathcal{L}$ is weakly integrable. ∎

###  6.4. Non-noetherianity for ${\mathbb{Z}}$-graded Lie algebras of rank
$\geq 2$

######  Corollary 6.5.

Let $\mathcal{L}$ be a simple ${\mathbb{Z}}$-graded algebra of rank $\geq 2$.
If $U(\mathcal{L})$ is Noetherian, then $\mathcal{L}$ is finite dimensional.

###### Proof.

By Lemma 6.4, $\mathcal{L}$ satisfies the hypothesis
$(\mathcal{H}_{\operatorname{re}})$ or $\mathcal{L}$ is weakly integrable. In
the first case, $U(\mathcal{L})$ is not Noetherian by Lemma 6.1.

In the latter case, $\mathcal{L}$ is isomorphic to an affine Lie algebra or it
has finite dimension by [9, Theorem 4]. But if $\mathcal{L}$ is an affine Lie
algebra, then it has an infinite dimensional abelian subalgebra, hence
$U(\mathcal{L})$ is not Noetherian. ∎

##  7\. Proof of the main result

###  7.1. Simple weakly ${\mathbb{Z}}$-graded Lie algebras

######  Lemma 7.1.

Let $\mathcal{L}=\oplus_{n\in{\mathbb{Z}}}\,\mathcal{L}_{n}$ be a simple
weakly ${\mathbb{Z}}$-graded Lie algebra such that
$\mathcal{L}\neq\mathcal{L}_{0}$. If $U(\mathcal{L})$ is Noetherian, then
$\dim\mathcal{L}_{n}$ is finite for any $n\in{\mathbb{Z}}$.

###### Proof.

Recall that $\mathcal{L}^{+}=\mathcal{L}_{>0}$ and
$\mathcal{L}^{-}=\mathcal{L}_{<0}$.

_Step 1._ We claim that all homogeneous components of $\mathcal{L}^{+}$ are
finite dimensional and there is an ideal $\mathcal{K}^{+}$ of
$\mathcal{L}_{0}$ such that

$\displaystyle[\mathcal{K}^{+},\mathcal{L}^{+}]=0\text{ and
}\dim\,\mathcal{L}_{0}/\mathcal{K}^{+}<\infty.$

By Lemma 2.1, $U(\mathcal{L}^{+})$ is Noetherian, hence $\mathcal{L}^{+}$ is
finitely generated. Thus

  1. (i)

$\dim\mathcal{L}_{n}<\infty$ for all $n\geq 1$, and

  2. (ii)

there exists $d\in{\mathbb{N}}$ such that
$\mathcal{L}_{+}=\langle\mathcal{L}_{1}\oplus\dots\oplus\mathcal{L}_{d}\rangle$.

Set
$\mathcal{K}^{+}=\\{x\in\mathcal{L}_{0}\mid\operatorname{ad}(x)(\mathcal{L}_{k})=0$
for any $k=1,\dots,d\\}$. It is clear that $\mathcal{K}^{+}$ satisfies the
required conditions, hence the claim is proved.

_Step 2._ Similarly, all homogeneous components of $\mathcal{L}^{-}$ are
finite dimensional and there is an ideal $\mathcal{K}^{-}$ of
$\mathcal{L}_{0}$ such that

$\displaystyle[\mathcal{K}^{-},\mathcal{L}^{-}]=0\text{ and
}\dim\,\mathcal{L}_{0}/\mathcal{K}^{-}<\infty$

_Step 3._ It remains to prove that $\mathcal{L}_{0}$ is also finite
dimensional. Set $\mathcal{K}=\mathcal{K}^{+}\cap\mathcal{K}^{-}$. By the
previous points, $\mathcal{K}$ is an ideal of $\mathcal{L}$. By the simplicity
of $\mathcal{L}$ and the fact that $\mathcal{L}\neq\mathcal{L}_{0}$, we
conclude that $\mathcal{K}=0$, hence $\dim\mathcal{L}_{0}<\infty$. ∎

###  7.2. The endomorphisms of simple ${\mathbb{Z}}^{n}$-graded modules

Let $\mathcal{L}$ be a ${\mathbb{Z}}^{n}$-graded Lie algebra and let $M$ be a
simple ${\mathbb{Z}}^{n}$-graded module.

######  Lemma 7.2.

If $M$ is not simple (as a non-graded module), then there exists
$\theta\in\operatorname{End}_{\mathcal{L}}(M)$ invertible, which is
homogeneous of degree ${\bf p}\in{\mathbb{Z}}^{n}\backslash 0$.

###### Proof.

Any $v\in M$ decomposes as $v=\sum_{\bf m}\,v_{\bf m}$ where $v_{\bf m}\in
M_{\bf m}$. By definition the support of $v$ is the set

$\displaystyle\operatorname{supp}(v)\coloneqq\\{{\bf n}\in{\mathbb{Z}}^{n}\mid
v_{\bf n}\neq 0\\}.$

Assume that $M$ is not simple. Let $v\in M\setminus 0$ be the generator of a
proper submodule with a support of minimal cardinal. Since $M$ is graded
simple, $\operatorname{supp}(v)$ contains distinct elements ${\bf m},{\bf n}$;
otherwise $U(\mathcal{L})\cdot v=U(\mathcal{L})\cdot v_{\bf n}=M$.

We claim that there exists $\theta\in\operatorname{End}_{\mathcal{L}}(M)$
mapping $v_{\bf n}\mapsto v_{\bf m}$.

Set $\theta(u\cdot v_{\bf n})=u\cdot v_{\bf m}$, $u\in U(\mathcal{L})$. We
have to show that $\theta$ is well defined. Let $U_{\bf d}\subset
U(\mathcal{L})$ be the subspace of elements of degree ${\bf
d}\in{\mathbb{Z}}^{n}$. Let $u\in U_{\bf d}$ with $u\cdot v_{\bf n}=0$. The
support of $u\cdot v$ lies in $({\bf
d}+\operatorname{supp}(v))\setminus\\{{\bf d}+{\bf n}\\}$, hence $u\cdot v=0$
and a fortiori $u\cdot v_{\bf m}=0$. Thus $\theta$ is well-defined, implying
the claim.

Clearly, $\theta$ is homogeneous of degree ${\bf p}={\bf m}-{\bf
n}\in{\mathbb{Z}}^{n}$. Since $\operatorname{Ker}\theta$ and
$\mathrm{Im}\,\theta$ are graded submodules, $\operatorname{Ker}\theta=0$ and
$\mathrm{Im}\,\theta=M$, hence $\theta$ is invertible. ∎

###  7.3. Simple ${\mathbb{Z}}^{n}$-graded Lie algebras which are not simple

Let $\mathcal{L}$ be a ${\mathbb{Z}}^{n}$-graded Lie algebra. The algebra of
endomorphisms of the adjoint module is called the centroid of $\mathcal{L}$.

######  Lemma 7.3.

If the simple ${\mathbb{Z}}^{n}$-graded Lie algebra $\mathcal{L}$ is not
simple as a Lie algebra, then it contains an infinite dimensional abelian
subalgebra.

###### Proof.

By Lemma 7.2, there is an element $\theta\neq 0$ in the centroid which is
homogeneous of degree ${\bf m}\in{\mathbb{Z}}^{n}\setminus 0$. Let $0\neq
x\in\mathcal{L}$ be an homogeneous element. Let $\mathfrak{m}$ be the linear
span of $\\{\theta^{p}(x):p\in{\mathbb{Z}}\\}$. For $p,q\in{\mathbb{Z}}$, we
have

$\displaystyle[\theta^{p}(x),\theta^{p}(x)]=\theta^{p+q}([x,x])=0,$

hence $\mathfrak{m}$ is a abelian subalgebra. Moreover the elements
$\theta^{p}(x)$ are nonzero elements of different degrees, hence
$\mathfrak{m}$ is infinite dimensional. ∎

######  Corollary 7.4.

Assume that the simple ${\mathbb{Z}}^{n}$-graded Lie algebra $\mathcal{L}$ is
not simple as a Lie algebra. Then $U(\mathcal{L})$ is not Noetherian.

###### Proof.

This is a consequence of Lemmas 7.3 and 2.1 (d). ∎

###  7.4. Proof of the main result

An equivalent formulation of the main Theorem is the following

######  Theorem 1.3.

Let $\mathcal{L}$ be a simple ${\mathbb{Z}}^{n}$-graded Lie algebra of
infinite dimension. Its enveloping algebra $U(\mathcal{L})$ is not Noetherian.

###### Proof.

We can assume that $\mathcal{L}$ is simple as a Lie algebra, otherwise
$U(\mathcal{L})$ is not Noetherian by Corollary 7.4.

There exists ${\bf m}=(m_{1},\dots,m_{n})\neq 0$ such that $\mathcal{L}_{\bf
m}\neq 0$. Without loss of generality, we can assume that $m_{1}\neq 0$.
Define the weakly ${\mathbb{Z}}$-graded Lie algebra $\mathcal{L}^{\prime}$
(which is $\mathcal{L}$ as Lie algebra) by the requirement that

$\displaystyle\mathcal{L}^{\prime}_{m}$
$\displaystyle=\bigoplus_{(m_{2},\dots,m_{n})\in{\mathbb{Z}}^{n-1}}\mathcal{L}_{(m,m_{2},\dots,m_{n})}.$

We can assume that all homogeneous components of $\mathcal{L}^{\prime}$ are
finite dimensional, otherwise $U(\mathcal{L})$ is not Noetherian by Lemma 7.1.

Therefore $\mathcal{L}^{\prime}$ is a simple ${\mathbb{Z}}$-graded Lie
algebra. If $\mathcal{L}^{\prime}$ has rank one, $U(\mathcal{L})$ is not
Noetherian by corollaries 4.2 and 5.7. Otherwise $\mathcal{L}^{\prime}$ has
rank $\geq 2$ and $U(\mathcal{L})$ is not Noetherian by corollary 6.5. ∎

######  Remark 7.5.

Let $\mathcal{L}$ be a ${\mathbb{Z}}^{n}$-graded Lie algebra. If $\mathcal{L}$
has a simple infinite dimensional graded section, then Theorem 1.3 implies
that $U(\mathcal{L})$ is not Noetherian. In other words, if $U(\mathcal{L})$
is Noetherian, then any simple graded section has finite dimension, in
particular any maximal graded ideal has finite codimension.

### Acknowledgements

The authors thank Efim Zelmanov and Slava Futorny for the warm hospitality
during their visit to the Shenzhen International Center for Mathematics.

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