# One-shot recognition of any material anywhere using contrastive learning
with physics-based rendering

Manuel S. Drehwald3,*,#, Sagi Eppel 1,2,4,*,#, Jolina Li 2,4, Han Hao2, Alan
Aspuru-Guzik1,2,#

###### Abstract

We present MatSim: a synthetic dataset, a benchmark, and a method for computer
vision based recognition of similarities and transitions between materials and
textures, focusing on identifying any material under any conditions using one
or a few examples (one-shot learning). The visual recognition of materials is
essential to everything from examining food while cooking to inspecting
agriculture, chemistry, and industrial products. In this work, we utilize
giant repositories used by computer graphics artists to generate a new CGI
dataset for material similarity. We use physics-based rendering (PBR)
repositories for visual material simulation, assign these materials random 3D
objects, and render images with a vast range of backgrounds and illumination
conditions (HDRI). We add a gradual transition between materials to support
applications with a smooth transition between states (like gradually cooked
food). We also render materials inside transparent containers to support
beverage and chemistry lab use cases. We then train a contrastive learning
network to generate a descriptor that identifies unfamiliar materials using a
single image. We also present a new benchmark for a few-shot material
recognition that contains a wide range of real-world examples, including the
state of a chemical reaction, rotten/fresh fruits, states of food, different
types of construction materials, types of ground, and many other use cases
involving material states, transitions and subclasses. We show that a network
trained on the MatSim synthetic dataset outperforms state-of-the-art models
like Clip on the benchmark, despite being tested on material classes that were
not seen during training. The dataset, benchmark, code and trained models are
available at these URLs: 1,2

††footnotetext:  1Vector institute, 2University of Toronto,
3Karlsruhe Institute of Technology, 4Innoviz
* Equal Contributions, # Corresponding authors
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Figure 1: The
MatSim benchmark for identifying materials from every aspect of the world
using one or a few real images (one shot). Top-1 results of ConvNeXt trained
on the synthetic MatSim dataset and Clip H14. Neither of the networks was
trained with these classes. Only selected samples are shown. Figure 2:
Selected examples from the MatSim synthetic dataset. The top line contains
random materials on random objects. The bottom line contains random materials
inside transparent vessels.

##  1 Introduction

The visual recognition of materials is essential to understanding nearly every
aspect of the world (Figure 1). Cooking, chemical research, inspecting
materials in construction, agriculture, and industry, and everyday object
handling all depend on the ability to identify materials and their state [33,
38, 25, 15, 17]. This can involve anything from evaluating whether the ground
is wet, a fruit is rotten, water is clean, or metal is rusted to
distinguishing between different types of rocks or fabrics. There are several
major challenges that make this problem difficult for computer vision methods.
First, there is an almost infinite amount of material and textures, and each
can look very different under different illumination conditions (unlike object
shapes, which are invariant to illumination). The second problem is that
transitions between materials and textures tend to be gradual and have a
continuous intermediate state, as in the case of cooking food, which makes it
hard to use discrete categories to describe the material. Another challenge is
that liquids and materials in environments like laboratories, hospitals, and
kitchens are usually handled inside transparent vessels that distort the view
of the materials. Previous studies on image recognition for materials have
mostly focused on distinguishing between a limited set of material classes,
like metal, plastic, and wood, or physical phases, like liquids and solids.
The limitation of these approaches is that they can only recognize the classes
they were trained on [14, 9, 18, 32, 36, 15, 8]. Adding new classes or
distinguishing between subclasses requires one to collect a new dataset and
retrain the neural networks. Another problem with these approaches is that
since they are based on pictures collected in a specific setting, they are
usually limited to scenarios in which the given materials appear on specific
objects. For example, windows will always be made out of glass, while trees
will always be made out of wood. Hence, they are likely to identify materials
by associating them with specific objects and are less likely to recognize,
for example, a duck made of wood or a tree made of glass[6, 19]. In recent
years, contrastive learning methods such as Clip, which are based on learning
the similarity between images and texts, have been proven to be very powerful
one-shot learners that can identify new examples and classes using a single
example or a few images [28, 30]. The main advantage of these approaches is
that when they are trained on enough data, they are not limited to a specific
set of classes and conditions and can work with classes not seen during
training. In addition, such models have proven to be very powerful backbones
for solving other problems, like image generation[29]. The main limitation of
these models is that they require a massive amount of data and computing
resources to be trained properly. In this work, we solve this problem by
generating a huge material similarity synthetic dataset that utilizes large-
scale CGI artist repositories[1, 37] used for animation and computer games
(Figure 2). We use highly diverse physics-based rendering (PBR) materials
repositories to simulate realistic materials[27]. We overlay the PBR materials
on 3D objects taken from the ShapeNet dataset with tens of thousands of
different objects [12](Figure 3); these objects are then put into scenes with
random illumination conditions and backgrounds by utilizing PolyHaven
repository for HDRI images[2]. The HDRI image is wrapped around the scene and
provides a realistic 360-degree background and illumination to the scene
(Figure 2).

Figure 3: Dataset creation. Random materials are created using the Blender
BSDF node or PBR materials downloaded from large-scale artist repositories
(AmbientCG, CGBookCases, 3D Textures). The material is UV mapped on the
surface of a random object loaded from the ShapeNet dataset. Random background
and illumination conditions are loaded from HDRI Haven repositories, ground
plane and background objects are added, and the scene is rendered.

Combining these repositories allows us to generate a large-scale, highly
diverse dataset (Figures 2 and 3). The large number of materials forces the
network to generalize instead of identifying only specific classes. The fact
that every material can be used on any object in any environment means that
the net has to identify the material everywhere and prevents the net from
associating the material with specific objects or environments. Gradual
transformations between materials in the dataset allow the net to detect
smooth transitions between materials. Additionally, rendering some of the
materials inside transparent containers allows the network to learn to
recognize materials stored inside glass vessels. In addition to the synthetic
dataset, we also supply a new real-world image benchmark for few-shot material
recognition (Figure 1). The benchmark contains a large number of examples of
material states and transitions, ranging from food in different states of
cooking, various powders, chemical reactions, metals and rust, wet and dry
sidewalks, construction materials, water and beverages in various states
(frozen/turbid), fruits in rotten and fresh states, and even different types
of candy, pasta, and cereals. The goal of the net is to identify each of these
states using one or a few examples. A second benchmark focus on covering
random objects with random materials, making the object and material
completely uncorrelated. We use the benchmark to test the network trained on
the MatSim dataset, as well as to test a pretrained Clip network. We show that
a net trained on the MatSim synthetic data significantly outperformed the Clip
model and gave good results despite never being trained on any of the material
classes in the benchmark (or real world images in general).

###  1.1 Contribution

To conclude, the main contributions of our work are as follows:

  * •

Demonstrating the first method, benchmark, and dataset for one-shot material
recognition that enable the recognition of unfamiliar materials using one or a
few examples, as well as identifying material gradual transitions and
subclasses.

  * •

Showing how large-scale CGI artist repositories can be combined to create
highly diverse synthetic training data in a way that can be used for many
other image recognition problems.

  * •

Proving that a net trained only on simulated data with a limited amount of
resources (a single GPU) can outperform large-scale models like Clip on
general real-world one-shot learning tasks (Clip H14 was trained on billions
of real images and hundreds of GPUs).

##  2 Related work

###  2.1 One-shot contrastive learning

One-shot contrastive learning methods like Clip have been proven to be very
powerful for image recognition tasks and have achieved state-of-the-art
results on major benchmarks without training on them [28, 30]. In addition,
these models are far more robust than nets trained on specific datasets[28,
19]. These networks work by predicting a descriptor vector for an image and
using the distance between vectors of different images as a similarity metric.
This approach allows such nets to recognize new classes using only a single
example (one shot). The main limitation of this approach is the need for a
very large amount of data. For example, Laion Clip H14 [30] was trained on two
billion images and their text labels. An alternative approach like SimClr [13]
used the self-similarity between images and their augmented versions and
utilized the same type of descriptor-similarity approach, but so far, it has
proven less effective than fully supervised networks or Clip.

###  2.2 Computer vision for material recognition

Computer vision for material recognition has mainly focused on three aspects:
1) classifying material classes, such as wood and metal or liquids and solids;
2) finding visual similarities between materials; and 3) predicting material
properties such as the BRDF, reflectance, glossiness, and turbidity [11, 35,
23, 34, 35, 20, 31].

###  2.3 Material classification datasets

There are a large number of material classification tasks and related
datasets. Datasets for general material classes (wood, metal, glass) include
CURet[14], KTH-TIPS [18], FMD[32], MINC[9], and OpenSurfaces[9, 8]; all of
these datasets use real-world images. Other datasets focus on more specific
properties, such as material phases (liquids, solids, powders, foam)[15], and
more specific topics, such as construction materials, soil type,
crystallization, turbidity, etc[24, 33, 38, 25, 17].

###  2.4 Material similarity prediction

Work on identifying the visual similarity between materials has been done
using both synthetic datasets and unsupervised learning[31, 26, 4]. Schwartz
and Nishino, [31] used 9000 generated images with uniform materials to train a
network that measures the similarity of the materials. The visual similarity
between the simulated materials was determined by human annotators. Perroni et
al.[26] used transfer learning to determine the visual similarity by using the
inner layer of a network trained on classification tasks with eight classes of
materials. Both works are limited to either a small set of classes or uniform
materials, and neither work tried to solve the general problem of being able
to recognize unseen materials and textures in a different environment or
identifying gradual transitions between materials.

###  2.5 Synthetic Datasets

Simulated data have a significant advantage for material datasets because of
the relative accuracy and simplicity with which materials can be simulated and
rendered using software like Blender[10]. Simulated materials datasets mostly
focus on homogenous materials, which enable the prediction of numeric
properties like surface reflectance (BRDF, BSDF)[20, 7, 5, 23, 34, 16].
Textured PBR materials have also been used mainly to generate datasets for
texture map extraction and classification [27, 21, 26]. One of the main
limitations of the current datasets is the use of a limited number of objects,
material classes, and environments, which makes networks trained on these
datasets limited in their scope. While large CGI repositories have existed for
a while, they remained almost unutilized by the computer vision community.

##  3 The MatSim Synthetic dataset generation

The goal of the MatSim dataset is to train a computer vision system capable of
recognizing any material and texture on any surface in any reasonable
environment and illumination conditions. The visual appearance of a material
depends on its physical properties but also on the object’s shape,
environment, and illumination. If, during training, any of these properties
are restricted in some way (for example, only indoor scenes are used) or
correlated with other properties (for example, wood materials only appear on
trees), the net is unlikely to achieve true generalization for material
recognition [19, 6].

Figure 4: Dataset structure. The dataset is composed of sets. Each set
involves two materials and six scenes with a gradual transition between the
two materials. Each image column corresponds to a different mixing ratio (R)
of the two materials. The ratio of the mixture of the two materials is given
in the top column. All images in the same column involve the same material
mixture on different objects and in different environments. Each of the six
scenes involves one main object. The material on this object gradually
transitions from one material to another. The mask of the object is given in
the right column. For scenes 1–2, the background remains exactly the same for
all images in the scene. For scenes 3–4, the background HDRI is randomly
rotated between images, leading to small changes in illumination. For scenes
5–6, the background HDRI is completely replaced between images, leading to
large changes in illumination.

###  3.1 General dataset structure

The dataset is divided into sets; each set contains two random materials and
six scenes involving a gradual transition between these two materials (Figure
4). The objects, backgrounds, and environment are randomly selected for each
scene separately (Figure 3). Each scene involves one static main object and a
static camera, which are both positioned randomly, and both remain unchanged
for the scene (Appendix 10.1). The object’s material is gradually changed
between images in the scene (Figure 4). For each scene, we render five images
with different mixtures of the two materials. The mixture ratios (R) of the
two materials are 0%, 25%, 50%, 75%, and 100%. 0% means that the object is
only made of material A, while 100% means the object is made completely of
material B, and 50% indicates an equal mixture of the two materials (Section
3.3). For scenes 1–2, all aspects of the scene remain the same. Only the
object material is changed gradually between images. In addition, the
translation and rotation of the texture UV mapping to the object were randomly
changed between each render (Appendix 10.2). For scenes 3–4, the procedure is
the same, except that the HDRI background is randomly rotated between images
(Appendix 10.3), leading to a small variation in illumination. For scenes 5–6,
the HDRI background for each image is randomly replaced, leading to a large
variation in illumination. This provides almost any possible variation of each
material’s appearance. Since all the scenes in a set are composed of the same
two materials, it is possible to compare the appearance of the materials
between two scenes with different objects and backgrounds, and light. The
gradual transition allows the network to learn to distinguish between highly
similar materials. Some scenes contain several random objects and a ground
plane for variability (Appendix 10.4). Since a scene can contain many
background objects, for each scene, we provide the mask (region) of the object
on which the material is used (Figure 4, right). If the material is uniform
(BSDF), the values of the material properties (color, transparency, etc.) are
given in the dataset. Altogether, 30k sets, with about 1 million images, were
rendered. For more details see the appendix.

Figure 5: A material’s visual appearance is controlled by several main
properties (color, transparency, etc.). For uniform materials (BSDF), each
property has a single value across the surface. Textured materials are
represented by texture maps for each property. These maps are wrapped around
the object (UV mapping) and provide properties for each point on the object’s
surface.

###  3.2 BSDF and PBR materials

The visual appearance of materials is controlled by several physical
properties, including color, transparency, roughness, etc. All of these
properties can be assigned values and controlled in Blender[10] by a single
system (BSDF node, Figure 5a)[7, 5]. However, in most real materials, these
properties are non-uniformly distributed and cover the object’s surface in
some unique patterns (textures). Materials are often assigned in the CGI
community using physics-based rendering or PBR (Figure 5b). PBR materials use
2D texture maps to control the distribution of each visual property. When a
material is assigned to an object, the texture maps are wrapped around the
object and assign the material properties to each point of the object’s
surface (Figure 5b)[27]. We downloaded over 4400 textures from open
repositories. To further increase the diversity of the materials, we randomly
combined texture maps from different materials (by averaging and mixing maps).
To generate uniform BSDF materials (Figure 5a), we simply select a random
value for each material property and set it uniformly on all of the object
surfaces using the Blender 3.1 BSDF node. Transparent materials are unique in
the sense that their appearance depends on the object’s volume (and is not
limited to the surface). For these, we use the Blender Volume Glossy node.

###  3.3 Representing mixtures and gradual material transformation

All material properties are continuous values, and two or more texture maps
can be combined through the pixel-wise addition (weighted sum). This makes a
gradual transition between materials simple to generate. First, the texture
maps of the two materials are combined using pixel-wise weighted addition to
create a new set of maps. The new maps are then wrapped around the object to
create the transition state. The ratios used are 0/1, 0.25/0.75, 0.5/0.5,
0.75/0.25, and 1/0, where 0.25/0.75 indicates the weighted sum of 25% material
A and 75% material B (Figure 4).

Figure 6: Procedurally generating material inside transparent containers. A
random 2D curve is generated by combining random polynomial and trigonometric
functions. The curve is used as a profile for the symmetric 3D object. The
object is assigned random transparent materials and content objects. The
content object is assigned a material.

###  3.4 Materials in transparent vessels

Liquids and other materials in kitchens, hospitals, and labs are usually
handled inside transparent containers (glassware, flasks, tubes, etc.). To
support these applications, we generated scenes in which we put the material
inside a transparent container. The vessel object was procedurally generated.
The curvature of the transparent vessel was generated by creating a random 2D
curve (by randomly combining linear, polynomial, and trigonometric functions;
Figure 6). This curve was used as the profile of a symmetric vessel by
creating a cylindrical (or other symmetrical) shape with the curve as the
vertical profile (Figure 6). In some cases, the shape was also randomly
stretched to create more variability. The vessel object was assigned a random
transparent material and a random thickness. The content of the vessel was
either a random object loaded from ShapeNet or a mesh that filled the bottom
part of the vessel (similar to static liquid or powder). As before, the
content was assigned a random material. Otherwise, the dataset creation was
the same as in Section 3.1. Examples can be seen in Figure 8 (Appendix).

Figure 7: Training and loss function. The loss function is based on the cosine
similarity with cross-entropy loss. For every three images in the batch, one
image is defined as an anchor ($A$). A second image, specifically the image
that has material more similar to the anchor (section 4.2), is defined as
positive ($P$), and the third image is defined as negative ($N$). All images
and the material masks are passed through the neural network to produce
descriptors ($D_{a}$, $D_{n}$, $D_{p}$). The cosine similarity is calculated
between the descriptor of the anchor and the positive and negative examples.
These cosine similarities are the input for the softmax function, which
returns the probability of a match between the anchor and the positive image
($P_{p}$). If this probability is below the threshold defined as
$0.5+(sim(A,P)-sim(A,N))*0.25$, we calculate the cross-entropy loss
($Loss=(-\log(P_{p})$); otherwise, the loss is set to zero. $sim(A,P$) is the
material similarity between the anchor ($A$) material and the positive
material $P$ (Section 4.2).

##  4 Training and architecture

We examined several standard image encoder architectures and found that
ConvNeXt[22] gave the best results. Two modifications were made in ConvNeXt:
First, the input of the network was changed from a standard three-channel
(RGB) image to an image plus the mask of the region containing the material.
This was done by simply adding the ROI mask as another layer to the RGB image
and changing the first layer of the network so that it could receive a four-
layer (R, G, B, mask) image instead of a three-layer (RGB) image. In addition,
the final linear layer of ConvNeXt was changed to produce a 512-value vector
(instead of 1000 ImageNet classes). This vector was then normalized using L2
normalization and used as the output descriptor. The training itself was done
using the AdamW optimizer for ten epochs (200000 steps with a batch size of
12) using a single RTX3090. This took around three days per training session.
We used the ConvNeXt-base model pretrained on ImageNet.

###  4.1 General loss function

The loss function is described in Figure 7. Similar to recent works, we use
the cosine similarity combined with the cross-entropy loss for training [13].
An important aspect of our dataset is that the levels of similarity between
materials are continuous and can have any value between zero and one
(depending on the mixture; Figure 4). This was accounted for by using a semi-
hard loss with a threshold depending on the similarity level. We found that a
semi-hard loss, in which only loss terms that are above some threshold are
used, works better than a hard loss (in which all loss terms are used). The
threshold, in this case, was controlled by the difference in the similarities
between the anchor material and the positive and negative examples.

###  4.2 Material similarity

We define the real similarity of the two materials as the difference between
their mixture ratios (R, Figure 4). As described in Section 3.3, each material
in a set is a linear mixture between two materials $A$ and $B$. We define this
ratio as $R(0<R<1)$. The similarity between the materials in the two images
($i_{1}$ and $i_{2}$) is defined as:

$sim(i_{1},i_{2})=1-|R(i_{1})-R(i_{2})|$

Where $R(i_{1})$ is the mixture ratio in image $i_{1}$ (Figure 4). Assuming
both images are from the same set.

###  4.3 Predicted material similarity

The predicted material similarity between two images, $A$ and $N$, is
calculated by first passing the images and their corresponding masks through
the neural network and predicting the descriptors $D_{a}$ and $D_{n}$, which
are then normalized using L2 normalization. The predicted similarity is the
cosine similarity between the two descriptors ($S_{an}$, Figure 7).

###  4.4 Loss calculation

The training was performed as shown in Figure 7: For each batch, we randomly
selected a single set (Section 3.1) and sampled 12 random images from this
set. The loss for every three images in the batch was calculated separately
using the procedure shown in Figure 7. One of the three images was chosen as
an anchor (A). The similarity between the material in the anchor image and the
other two images was calculated as described in Section 4.2. The image that
was more similar to the anchor was defined as positive (P), and the other
image was defined as negative (N); hence, $sim(A,P)>sim(A,N)$. If the
similarities of the anchor to the two images were equal $(sim(A,P)=sim(A,N))$,
the loss for this triplet was set to zero. Next, each of the three images and
their material masks were passed through the neural net to predict the
material descriptors (Section 4, Figure 7). The cosine similarities between
the descriptors of the anchor and the negative and positive images were
calculated as described in Section 4.3. These similarities were then passed to
a softmax function to calculate the probability of a match between the anchor
and the positive image:

$P_{p}=\frac{e^{S_{ap}/t}}{e^{S_{ap}/t}+e^{S_{an}/t}}.$ (1)

$S_{ap}$ and $S_{an}$ are the cosine similarity between the anchor ($A$) and
the negative($N$) and positive($P$) descriptors, respectively, $P_{p}$ is the
probability for a match between the anchor and the positive image, and $t$ is
a constant (0.2 in our case). We then calculate the semi-hard loss using the
following condition: If $P_{p}>0.5+(sim(A,P)-sim(A,N))*0.25$, the loss is set
to zero; otherwise, we calculate the standard cross-entropy loss
$(Loss=-log(P_{p}))$.

##  5 Testing Clip models

The Clip neural network has achieved state-of-the-art (SOTA) results on a
large number of image recognition benchmarks and is considered the top general
one-shot net to date [30, 28]. Similar to our model, Clip predicts image
descriptors as a vector, which can then be used to find similarities between
the image and other images or texts. Unlike our model, Clip H14 was trained by
comparing 2 billion real photos to their corresponding text captions. There
are two limitations to using Clip for this task. The first is that the Clip
descriptor is not focused on materials and might contain information regarding
the object and environment. Second, Clip receives just an image without an
attention mask. Hence, it is harder to point to a specific region of the image
where the target material is. To solve this issue, we tried a few methods: The
first involves cropping the region of the image around the target material and
using it as the input. The second involves masking the image region outside
the material (by setting it to black), and the third involves combining
masking and cropping. All of these approaches gave a significant improvement
compared to using the image as is. Cropping without masking gave the best
results. We tested all the pretrained Clip versions available and found that
ViT-B32 gave the best results for open A.I models (Table 1), but the recently
published Open Clip H/14 significantly outperforms all other Clip Models.
Nonetheless, MatSim net still outperformed all Clip models on all benchmarks.

##  6 Benchmark creation

Perhaps the most important part of dealing with the task of general material
recognition is defining a proper benchmark that will cover the main aspects of
this challenge. We define three main capabilities that we want to evaluate: 1)
The ability of the network to visually identify unfamiliar materials in new
environments using one or a few examples with no restrictions on the material
or environment setting. 2) The ability to identify transitions between
material states (e.g., wet/dry, rotten/fresh) and fine-grained subcategories
of materials (e.g., types of pasta). 3) The ability to identify materials
regardless of the object they appear on (e.g., a cow made out of wood)[6, 19].
We are not aware of a benchmark that covers any of these three tasks.
Therefore, we created two benchmarks by manually taking pictures of materials
in a variety of states.

###  6.1 Set 1: Material transition states and subclasses

The first test set is designed to test the hypothesis that the network is able
to recognize the similarity between images of the same material, even when
they are presented in transparent containers and different surroundings, while
distinguishing between subcategories or different states of the same material.
To do this, we collected a large number of images of materials from a wide
range of fields (Figure 1); for example, we collected images of eggs in raw,
cooked, and burned states and sidewalks in wet and dry states (Figure 1). For
each state, we collected at least two images. We divided this test set into
superclasses (e.g., eggs) and divided each superclass into subclasses (e.g.,
cooked/raw/burned; Figure 1). We tried to make different examples of the same
material subclass appear in different environments as much as possible, making
the recognition of the material based on the environment type less likely. In
addition, for each image, we created a mask of the material region in the
image. Note, however, that in this dataset, there will always be some
correlation between materials and objects, i.e., liquids appear in glassware,
and rotten textures appear on fruit. This set contained 416 images divided
into 116 material types.

###  6.2 Set 2: Uncorrelated materials and objects

The second test set is designed to test the hypothesis that the network is
able to recognize the material regardless of the object on which it appears.
Materials in the real world are strongly correlated to objects (trees made out
of wood). Creating this set meant that we needed to generate our own objects.
To achieve this, we collected a number of random objects and materials. The
materials were ribbons and sheets that could be wrapped around the object or
granular and fiber materials that could be scattered on the object’s surface.
These materials were used to cover the surfaces of objects using glue for
attachment. When creating this test set, we made sure that each material did
not appear on the same object or in the same environment in more than one
image, forcing the net to use only the material for recognition. This set
contains 86 images in 16 material types.

##  7 Evaluation methods

For the evaluation of the network, we consider the standard Top-1 accuracy
metric: Given an image of a material, we test the ability of the network to
identify another image of the same material from a group of images. An example
is correct if the highest cosine similarity calculated by the network is
between the given image to another image of the same material subclass. We
calculate two accuracy metrics. The first evaluation method measures the
ability of the network to find another example of the same material from the
set of all images of material the superclass (Figure 1). For example, the task
could be to identify an image with spotted bananas from a set of images of
bananas in various states (Figure 1, Table 1 (subclass)). In the second test
case, we still expect the network to find another example of the same
material. However, we now present it with all the other images in the test
set, including those of different superclasses. For example, the image of a
spotted banana is compared to all images, including those of rust, rocks,
cheese, etc. Note that in this case, it is much easier to use the object type
to distinguish between materials (banana vs. not a banana), which gives the
Clip network an advantage. For the second test set we used only the second
approach. The results given in Table 1, Figure 1, are the average accuracy per
class (All classes given equal weights).

##  8 Results

Method | Set1 Subclass | Set1 All | Set2
---|---|---|---
Random | 0.30 | 0.006 | 0.07
CLIP B32 | 0.51 | 0.32 | 0.44
CLIP B32+C | 0.56 | 0.38 | 0.49
CLIP B32+M | 0.56 | 0.28 | 0.37
CLIP B32+C+M | 0.56 | 0.35 | 0.56
Open CLIP H14 | 0.55 | 0.44 | 0.47
Open CLIP H14+C | 0.67 | 0.52 | 0.77
Open CLIP H14+M | 0.59 | 0.40 | 0.53
Open CLIP H14+C+M | 0.66 | 0.52 | 0.67
MatSim | 0.71 | 0.56 | 0.73
MatSim+C | 0.77 | 0.56 | 0.85
MatSim+M | 0.78 | 0.56 | 0.91
MatSim+C+M | 0.72 | 0.61 | 0.85
Table 1: Results. $+C$ indicates cropping, $+M$ indicates masking. Random
stands for random matching.

As can be seen in Figure 1 and Table 1, the MatSim network performed well
across almost all types of materials and environments. The MatSim network
significantly outperforms the best Clip model on all test sets (Table 1). This
suggests that despite the fact that the MatSim network was trained only on
simulated data, it learned to generalize to unfamiliar real-world materials.
Both Clip and MatSim significantly outperformed random matching (Table 1),
suggesting both approaches achieved a good understanding of materials. One
unexpected aspect that made the benchmark very challenging is the fact that
the camera itself often adjusts the color and smooths the image. This image-
processing step can make even the same object in the same setting look very
different. This variation is not taken to account by the synthetic dataset. To
account for this, significant augmentations were used during training and
resulted in 10% improvement (Appendix 10.6).

###  8.1 Comparing masking and cropping for focusing attention

Focusing the network on the material region of the image by cropping the
material region and using it as the input image seems to improve the accuracy
of both Clip and the MatSim network (Table 1). Masking the background by
blacking out the background pixels seems to improve the results for the MatSim
network but proves to be less effective than cropping for Clip. The fact
MatSim net is affected by cropping and masking is surprising because the
MatSim network already receives the region of the material as input (Section
4), so ideally, it should be able to focus on this region without either
cropping or masking. It seems that the MatSim network is not yet able to make
full use of having the mask image as an extra input in addition to the pure
image. Not surprisingly, Clip with no cropping or masking has the worst
performance since it has no inherent way to focus on the material region.

##  9 Conclusion

This work has demonstrated a new benchmark and a synthetic dataset for the
problem of one-shot material similarity recognition. A solution to this
problem can be beneficial to almost any field in which materials are
manipulated; it can be applied in autonomous chemistry labs, cooking and food
inspection, or the handling of materials in construction and agriculture. The
MatSim network managed to outperform Clip on this general challenge despite
being trained only on synthetic data and using 1/1000 of the images and
computer resources of Clip. The key to this success is the use of large-scale
CGI artist repositories, which allowed us to reach a much higher level of
diversity compared to most existing synthetic datasets. Generative AI methods
are now capable of generating both 3D models and PBR textures, and they can
therefore be used to generate an almost infinite variety of assets for the
next generation of synthetic datasets. This is likely to make this approach
even more powerful in the future.

##  10 Appendix: Building the MatSim Dataset

Figure 8: Set of MatSim dataset with materials inside transparent vessels. The
set structure is described in Section 3.1 and Figure 3. The vessel generation
is described in section 3.4.

###  10.1 Setting objects in the scene

Objects for the dataset were taken from the ShapeNet core2 dataset, which
includes 56000 3D model objects with over 200 categories. During scene
creation, a random object was loaded, randomly scaled, and rotated, and its
original materials were removed. In addition, overlapping faces on the surface
were removed.

###  10.2 UV mapping

Since the PBR materials are given as 2D texture maps, assigning a material to
a 3D object involves wrapping these 2D maps around the object’s surface.
Wrapping 2D maps around complex 3D objects (UV mapping) is a rather complex
problem. Blender 3.1 contains some automatic wrapping techniques, which we
use. A major problem that we encounter is that many objects contain
overlapping faces (surfaces), which causes the textures to be wrapped around
the same areas several times, causing strong unrealistic visual artifacts. We
managed to solve this by removing vertexes and faces that were too close to
each other using the remove overlapping vertexes tool of Blender. This led to
the slight deformation of some objects’ shapes. The wrapping itself depends on
various parameters, such as the relative scale, orientation, and translation
of the texture map relative to the object, as well as the coordinate system.
We randomize all of these parameters between each image to achieve maximum
variability.

###  10.3 Background illumination and HDRIs

The environment and illumination greatly affect a material’s appearance. The
main method used by the CGI community to control the background and
illumination involves high dynamic range images (HDRIs). These are panoramic
images that wrap around the scene and provide 360 illuminations. The range of
intensities of light is from 0–6550, compared to 0–255 in standard images.
This allows HDRIs to represent a much broader range of illumination. We
downloaded over 500 HDRIs from HDRI Haven repositories. These HDRIs are highly
diverse scenes covering both daytime and nighttime conditions in a large
number of indoor and outdoor environments. These HDRIs were used to add both
illumination and background to the scenes. To increase diversity, the HDRIs
were randomly rotated and scaled, and their intensities were randomly
increased or decreased for each scene.

###  10.4 Adding ground plane and background objects

Random objects were scattered in the scene to make the environment more
diverse and add shadows and back reflections. This was done by loading random
objects from the ShapeNet dataset and randomly scaling, rotating, and
positioning them in the scenes. In addition, a ground plane was generated by
adding a flat plane below the object and assigning a random PBR material to
it.

###  10.5 Limitation of the dataset

A major limitation of the dataset is that phase transitions between materials
are often not the average between the two states; instead, they are either a
completely different state (the transition between wood and coal is fire), or
they are not uniformly distributed, like the nucleation of ice in the water.
Both cases are not completely addressed by the dataset and require further
work.

###  10.6 Data augmentation

A major problem with using only simulated data for learning is the fact that
modern cameras significantly modify an image’s appearance by smoothing and
adjusting colors. To account for this, we use extensive augmentation for the
image during training. This includes random Gaussian smoothing, the darkening
or brightening of the image, partial or complete decoloring (to greyscale),
and noise addition. Each of these augmentations was applied for about 10

###  10.7 Data sampling

Sampling is done by choosing one set for each batch and, from this set,
choosing 12 random images (with no repetition). Sets with materials inside and
outside transparent vessels were selected with equal probability.

###  10.8 CGI assets from artist repositories

One of the main challenges in creating the MatSim dataset was collecting large
and diverse enough examples of backgrounds, objects, and materials. Large-
scale repositories of assets that serve the CGI artist community have been
available for a while but remain almost unutilized by the machine-learning
community. Utilizing these repositories made it possible for us to create a
wide range of scenes on a large scale and with sufficient diversity; this
would have been very difficult if we had created these data ourselves. The
main three repositories used for this work were AmbientCG[1], CGBookCase[37]
and TextureBox[3] for materials, HDRI Haven for HDRI environments [2], and
ShapeNet for objects[12].

###  10.9 Blender render setting and hardware.

The dataset was rendered using Blender 3.1, with CYCLEs rendering engine, 120
rendering cycles per frame. The rendering was done using Nvidia RTX 3090 GPU.

###  10.10 Realism vs. generality

There are two ways to go about generating a synthetic dataset. One approach is
to make the dataset as realistic as possible by maximizing the similarity of
the generated data to the real world. The other approach is to maximize
variability and make the scenes as diverse as possible, even at the cost of
realism. In this work, we prefer variability over realism, which means we
assign random materials to random objects and set them in random environments.
For example, an image in the dataset may contain a car made out of wood on the
ground made of metal in a forest surrounded by random objects. This makes the
dataset easier to generate and helps the network achieve a much higher level
of generalization and identify materials regardless of the environment or the
object they appear on.

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# Realizations and star-product of doubly $\kappa$-deformed Yang models

T<EMAIL_ADDRESS>Faculty of Science, University of Split,
Ruđera Boškovića 33, 21000 Split, Croatia  S<EMAIL_ADDRESS>Division of Theoretical Physics, Ruđer Bošković Institute,
Bijenička cesta 54, 10002 Zagreb, Croatia  S<EMAIL_ADDRESS>Dipartimento di Matematica e Informatica, Università di Cagliari,
via Ospedale 72, 09124 Cagliari, Italy  INFN, Sezione di Cagliari, Cittadella
Universitaria, 09042 Monserrato, Italy

The Yang algebra was proposed a long time ago as a generalization of the
Snyder algebra to the case of curved background spacetime. It includes as
subalgebras both the Snyder and the de Sitter algebras and can therefore be
viewed as a model of noncommutative curved spacetime. A peculiarity with
respect to standard models of noncommutative geometry is that it includes
translation and Lorentz generators, so that the definition of a Hopf algebra
and the physical interpretation of the variables conjugated to the primary
ones is not trivial. In this paper we consider the realizations of the Yang
algebra and its $\kappa$-deformed generalization on an extended phase space
and in this way we are able to define a Hopf structure and a twist.

##  1 Introduction

The Yang model [1] is an extension of the Snyder model [2] to the full phase
space, obtained assuming that both position and momentum operators do not
commute among themselves. It can therefore be interpreted as a noncommutative
geometry defined on a spacetime of constant curvature. Algebraically, it is
based on an $so(1,5)$ algebra which includes the Lorentz generators and the
position and momentum operators, together with a further generator, necessary
to close the algebra. Contrary to most models of noncommutative geometry, the
action of the Lorentz algebra on phase space is not deformed. It also enjoys
an invariance under a (generalized) Born duality [3].

The relevance of noncommutative geometries for Planck-scale physics has been
originally highlighted in [4] and since then the subject has become
fashionable. In particular the formalism of Hopf algebras has shown to be
useful for the description of the physical implications of the theory [5].
Among these, particular attention have received the deformation of the
symmetries of spacetime [6, 7, 8] and the applications to phenomenology and to
the search for observable effects, especially in the context of Doubly Special
Relativity [7, 9]. In particular, the introduction of a curved background can
be useful in cosmological contexts, especially when considering
phenomenological effects on the propagation of photons from distant sources
[10].

Thus, after a long oblivion, the Yang model was resumed in recent years. Some
generalizations were presented by Khruschev and Leznov (KL) [11] and in ref.
[12], where an extension of the model that includes also the related Triply
Special Relativity (TSR) theory [13] was introduced. Further investigations
concerning in particular its realizations on a canonical phase space have been
recently performed in [14, 15], using the methods introduced in [16, 17] for
the study of Snyder space. Other contributions to the study of Yang model are
given in [18], while different models of noncommutative geometry in curved
spaces can be found in [19].

Finally, in [20] the Yang model has been further generalized by deforming the
$so(1,5)$ algebra to an $so(1,5;g)$ algebra, so that it also includes
$\kappa$-Poincaré deformations of the kind introduced in [6] for the standard
Poincaré algebra, both in position and momentum spaces. Although from a
mathematical point of view the two algebras are isomorphic, their physical
interpretation is of course different, since, for example, in $so(1,5;g)$ the
Poincaré algebra is deformed.

In this paper, we study the explicit transformations leading from the
generators of $so(1,5)$ to those of $so(1,5;g)$. This will allow us to obtain
a Hopf algebra structure for the Yang model, by defining a coproduct and a
twist and then the star product. This will be done by using the methods that
have been proved successful in the case of Snyder [16] and $\kappa$-deformed
Snyder [21] spaces. In particular, exploiting the isomorphism of the Yang
model to an orthogonal algebra, one can use the results of [22], where the
Hopf structure associated to orthogonal groups has been discussed in detail.

As shown in [15] a realization of the Yang model can be obtained on an
extended phase space, that includes momenta canonically conjugated to both the
original position and momentum variables. Hence, in contrast with standard
models of noncommutative geometry, in our case a coproduct must be introduced
also for the momentum variables. The physical interpretation of this fact is
presently under investigation.

##  2 Generalized Yang models

The Yang model [1] is based on the Yang algebra, which is a Lie algebra
generated by $\hat{x}_{i}$, $\hat{p}_{i}$, ${\hat{M}}_{ij}$ and $\hat{h}$ and
is isomorphic to the orthogonal algebra $so(1,5)$, $so(2,4)$ or $so(3,3)$,
depending on the signature chosen for the metric tensor [23]. It is defined by
the commutation relations444Latin indices run from 0 to 3, Greek indices from
0 to 5. For definiteness in the following we consider the $so(1,5)$ case with
metric $\eta_{ij}={\rm diag}(-1,1,1,1,1,1)$, but our considerations trivially
extend to the other cases. We use natural units, in particular we set
$\hbar=1$.

$[\hat{x}_{i},\hat{x}_{j}]=i\beta^{2}{\hat{M}}_{ij},\qquad[\hat{p}_{i},\hat{p}_{j}]=i\alpha^{2}{\hat{M}}_{ij},\qquad[\hat{x}_{i},\hat{p}_{j}]=i\eta_{ij}\hat{h},$
$[\hat{h},\hat{x}_{i}]=i\beta^{2}\hat{p}_{i},\qquad[\hat{h},\hat{p}_{i}]=-i\alpha^{2}\hat{x}_{i},$
(1)

where the ${\hat{M}}_{ij}$ generate the Lorentz algebra $so(3,1)$ and
$\hat{x}_{i}$, $\hat{p}_{i}$ transform as vectors under the Lorentz algebra,
while $\hat{h}$ is a scalar. The parameters $\alpha$ and $\beta$ are constant
with dimensions $[L]^{-1}$ and $[M]^{-1}$ respectively and are usually
identified with the square root of the cosmological constant and with the
inverse of the Planck mass. In spite of the notation, $\alpha^{2}$ and
$\beta^{2}$ can be taken negative leading to the $so(2,4)$ and $so(3,3)$
cases, which we shall not consider in detail here.

The operators $\hat{x}_{i}$ and $\hat{p}_{i}$ can be interpreted as the
position and the momentum operators in a quantum phase space. The operator
$\hat{h}$ is necessary to close the algebra and generates rotations in the
$x$-$p$ hyperplane. The Yang algebra satisfies the Born duality [3], being
invariant for $\alpha\leftrightarrow\beta$, $\hat{x}_{i}\to-\hat{p}_{i}$,
$\hat{p}_{i}\to\hat{x}_{i}$, ${\hat{M}}_{ij}\leftrightarrow{\hat{M}}_{ij}$,
$\hat{h}\leftrightarrow\hat{h}$.

Defining

${\hat{M}}_{i4}={\hat{x}_{i}\over\beta},\qquad{\hat{M}}_{i5}={\hat{p}_{i}\over\alpha},\qquad{\hat{M}}_{45}={\hat{h}\over\alpha\beta}.$
(2)

then the algebra (1) can be put in the explicit $so(1,5)$ form

$[{\hat{M}}_{\mu\nu},{\hat{M}}_{\rho\sigma}]=i\big{(}\eta_{\mu\rho}{\hat{M}}_{\nu\sigma}-\eta_{\mu\sigma}{\hat{M}}_{\nu\rho}-\eta_{\nu\rho}{\hat{M}}_{\mu\sigma}+\eta_{\nu\sigma}{\hat{M}}_{\mu\rho}\big{)}.$
(3)

One can find alternative realizations of this algebra by defining linear
combinations of the generators. The most general new generators linear in
$\hat{x}_{i}$, $\hat{p}_{i}$, ${\hat{M}}_{ij}$ are

$\displaystyle{\tilde{X}}_{i}=A(\cos\varphi\,\hat{x}_{i}+{\beta\over\alpha}\sin\varphi\,\hat{p}_{i})+\beta
a_{k}{\hat{M}}_{ik},$ (4) (5)
$\displaystyle{\tilde{P}}_{i}=B(\cos\psi\,\hat{p}_{i}+{\alpha\over\beta}\sin\psi\,\hat{x}_{i})+\alpha
b_{k}{\hat{M}}_{ik},$ (6)

with ${\tilde{M}}_{ij}={\hat{M}}_{ij}$. The parameters $A$, $B$, $\varphi$,
$\psi$, $a_{i}$, $b_{i}$ are dimensionless with $AB\neq 0$. The
transformations inverse to (4) are

$\displaystyle\hat{x}_{i}$ $\displaystyle=$
$\displaystyle{A^{-1}\alpha\cos\psi({\tilde{X}}_{i}-\beta
a_{j}{\tilde{M}}_{ij})-B^{-1}\beta\sin\varphi({\tilde{P}}_{i}-\alpha
b_{j}{\tilde{M}}_{ij})\over\alpha\cos(\varphi+\psi)}$ (7)
$\displaystyle\hat{p}_{i}$ $\displaystyle=$
$\displaystyle{B^{-1}\beta\cos\varphi({\tilde{P}}_{i}-\alpha
b_{j}{\tilde{M}}_{ij})-A^{-1}\alpha\sin\psi({\tilde{X}}_{i}-\beta
a_{j}{\tilde{M}}_{ij})\over\beta\cos(\varphi+\psi)}$ (8)

The new generators ${\tilde{X}}_{i}$ and ${\tilde{P}}_{i}$ generate a new
class of Lie algebras isomorphic to the initial Yang algebra. The new algebra
generated by ${\tilde{X}}_{i}$, ${\tilde{P}}_{i}$, ${\tilde{M}}_{ij}$ and
${\tilde{H}}$ is given by the following commutation relations

$\displaystyle[{\tilde{X}}_{i},{\tilde{X}}_{j}]$ $\displaystyle=$
$\displaystyle
i\left(\beta^{2}\tilde{A}{\tilde{M}}_{ij}+\beta(a_{i}{\tilde{X}}_{j}-a_{j}{\tilde{X}}_{i})\right),$
(9) $\displaystyle[{\tilde{P}}_{i},{\tilde{P}}_{j}]$ $\displaystyle=$
$\displaystyle\left(\alpha^{2}\tilde{B}{\tilde{M}}_{ij}+\alpha(b_{i}{\tilde{P}}_{j}-b_{j}{\tilde{P}}_{i})\right),$
(11) $\displaystyle[{\tilde{X}}_{i},{\tilde{P}}_{j}]$ $\displaystyle=$
$\displaystyle i\left(\eta_{ij}{\tilde{H}}+\alpha b_{i}{\tilde{X}}_{j}-\beta
a_{j}{\tilde{P}}_{i}+\alpha\beta{\tilde{\rho}}{\tilde{M}}_{ij}\right),$ (13)
$\displaystyle[{\tilde{M}}_{ij},{\tilde{X}}_{k}]$ $\displaystyle=$
$\displaystyle
i\Big{(}\eta_{ik}{\tilde{X}}_{j}-\eta_{jk}{\tilde{X}}_{i}+\beta(a_{i}{\tilde{M}}_{kj}-a_{j}{\tilde{M}}_{ki})\Big{)},$
(15) $\displaystyle[{\tilde{M}}_{ij},{\tilde{P}}_{k}]$ $\displaystyle=$
$\displaystyle
i\Big{(}\eta_{ik}{\tilde{P}}_{j}-\eta_{jk}{\tilde{P}}_{i}+\alpha(b_{i}{\tilde{M}}_{kj}-b_{j}{\tilde{M}}_{ki})\Big{)},$
(17) $\displaystyle[{\tilde{M}}_{ij},{\tilde{H}}]$ $\displaystyle=$
$\displaystyle
i\Big{(}\alpha(b_{j}{\tilde{X}}_{i}-b_{i}{\tilde{X}}_{j})-\beta(a_{j}{\tilde{P}}_{i}-a_{i}{\tilde{P}}_{j})\Big{)},$
(19) $\displaystyle[{\tilde{H}},{\tilde{X}}_{i}]$ $\displaystyle=$
$\displaystyle
i\left(\beta^{2}\tilde{A}{\tilde{P}}_{i}-\alpha\beta{\tilde{\rho}}\,{\tilde{X}}_{i}-\beta
a_{i}{\tilde{H}}\right),$ (21) $\displaystyle[{\tilde{H}},{\tilde{P}}_{i}]$
$\displaystyle=$ $\displaystyle
i\left(-\alpha^{2}\tilde{B}{\tilde{X}}_{i}+\alpha\beta{\tilde{\rho}}\,{\tilde{P}}_{i}+\alpha
b_{i}{\tilde{H}}\right),$ (23)

where we have defined

${\tilde{H}}=AB\cos(\varphi+\psi)\hat{h}+\beta a\\!\cdot\\!{\tilde{P}}-\alpha
b\\!\cdot\\!{\tilde{X}}-\alpha\beta a_{h}b_{k}{\hat{M}}_{hk},$ (24)
${\tilde{\rho}}=AB\rho+a\\!\cdot\\!b,\qquad\tilde{A}=A^{2}+a^{2},\qquad\tilde{B}=B^{2}+b^{2},$
(25)

with $\rho=\sin(\varphi+\psi)$. A generalized Born duality still holds for
$\alpha\leftrightarrow\beta$, $a_{i}\to-b_{i}$, $b_{i}\to a_{i}$,
$\tilde{A}\leftrightarrow\tilde{B}$,
${\tilde{\rho}}\leftrightarrow-{\tilde{\rho}}$,
${\tilde{X}}_{i}\to-{\tilde{P}}_{i}$, ${\tilde{P}}_{i}\to{\tilde{X}}_{i}$,
${\tilde{M}}_{ij}\leftrightarrow{\tilde{M}}_{ij}$,
${\tilde{H}}\leftrightarrow{\tilde{H}}$.

These commutation relations are of the kind introduced in [15]. They can be
put in the form of as an $so(1,5;g)$ algebra if one defines the generators
${\tilde{M}}_{\mu\nu}$ as

${\tilde{M}}_{i4}={{\tilde{X}}_{i}\over\beta},\qquad{\tilde{M}}_{i5}={{\tilde{P}}_{i}\over\alpha},\qquad{\tilde{M}}_{45}={{\tilde{H}}\over\alpha\beta}.$
(26)

They satisfy the algebra

$[{\tilde{M}}_{\mu\nu},{\tilde{M}}_{\rho\sigma}]=i\big{(}g_{\mu\rho}{\tilde{M}}_{\nu\sigma}-g_{\mu\sigma}{\tilde{M}}_{\nu\rho}-g_{\nu\rho}{\tilde{M}}_{\mu\sigma}+g_{\nu\sigma}{\tilde{M}}_{\mu\rho}\big{)},$
(27)

with $g_{\mu\nu}$ a symmetric matrix of the form

$g_{\mu\nu}=\left(\begin{array}[]{cccccc}-1&0&0&0&a_{0}&b_{0}\\\
0&1&0&0&a_{1}&b_{1}\\\ 0&0&1&0&a_{2}&b_{2}\\\ 0&0&0&1&a_{3}&b_{3}\\\
a_{0}&a_{1}&a_{2}&a_{3}&\tilde{A}&{\tilde{\rho}}\\\
b_{0}&b_{1}&b_{2}&b_{3}&{\tilde{\rho}}&\tilde{B}\end{array}\right).$ (28)

Notice that $\det g=A^{2}B^{2}\cos^{2}(\varphi+\psi)$, hence one must require
that $\cos(\varphi+\psi)\neq 0$, otherwise the matrix $g$ becomes singular.

The matrix $g_{\mu\nu}$ can be reduced to a diagonal form by a transformation
with a matrix $S$ such that $g=S\,\eta\,S^{T}$. This matrix is defined up to
multiplication by an orthogonal matrix $O$ such that $O\,\eta\,O^{T}=\eta$.
One can choose $S$ in a lower triangular form, as

$S=\left(\begin{array}[]{cccccc}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\ 0&0&1&0&0&0\\\
0&0&0&1&0&0\\\ -a_{0}&a_{1}&a_{2}&a_{3}&\sigma&0\\\
-b_{0}&b_{1}&b_{2}&b_{3}&\upsilon&\tau\end{array}\right),$ (29)

with

$\sigma=A,\qquad\upsilon=B\sin(\varphi+\psi),\qquad\tau=B\cos(\varphi+\psi),$
(30)

and $\det S=\sigma\tau=AB\cos(\varphi+\psi)$.

Clearly, the generators ${\hat{M}}_{\mu\nu}$ such that
${\tilde{M}}_{\mu\nu}=(S\,{\hat{M}}\,S^{T})_{\mu\nu}$, satisfy the commutation
relations (3) and can then be identified with those defined in (2). The
explicit relation between their components are then
${\tilde{M}}_{ij}={\hat{M}}_{ij}$, and

$\displaystyle{\tilde{X}}_{i}={1\over\sigma}(\hat{x}_{i}-a^{k}{\hat{M}}_{ik}),$
(31) (32)
$\displaystyle{\tilde{P}}_{i}={1\over\tau}(\hat{p}_{i}-b^{k}{\hat{M}}_{ik})-{\upsilon\over\sigma\tau}(\hat{x}_{i}-a^{k}{\hat{M}}_{ik}),$
(33) (34)
$\displaystyle{\tilde{H}}={1\over\sigma\tau}(\hat{h}+b^{k}\hat{x}_{k}-a^{k}\hat{p}_{k}+a^{h}b^{k}{\hat{M}}_{hk}),$
(35)

with inverse transformations

$\displaystyle\hat{x}_{i}=\sigma{\tilde{X}}_{i}+a^{k}{\tilde{M}}_{ik},$ (36)
(37)
$\displaystyle\hat{p}_{i}=\tau{\tilde{P}}_{i}+\tau{\tilde{X}}_{i}+b^{k}{\tilde{M}}_{ik},$
(38) (39) $\displaystyle\hat{h}=\sigma\tau\,{\tilde{H}}+(\sigma b^{k}+\upsilon
a^{k}){\tilde{X}}_{k}+\tau a^{k}{\tilde{P}}_{k}+a^{h}b^{k}{\tilde{M}}_{hk}.$
(40)

Note that if $a_{k}=b_{k}=0$, then the KL model with $\tilde{\alpha}$,
$\tilde{\beta}$, ${\tilde{\rho}}$, ${\tilde{H}}$ is isomorphic to the original
Yang model with $A$, $B$, $\rho$, $h$ if
$\alpha^{2}=\tilde{\alpha}^{2}/B^{2}$, $\beta^{2}=\tilde{\beta}^{2}/A^{2}$ and
$h={\tilde{H}}/(AB\cos(\varphi+\psi))$ where $A$, $B$ are real numbers and
$A^{2}B^{2}>{\tilde{\rho}}^{2}$.

The same construction as (4)-(9) holds also for Yang models isomorphic to
$so(3,3)$. For models isomorphic to $so(2,4)$, in eq. (4) the trigonometric
functions should be replaced by the corresponding hyperbolic functions and the
parameter $\rho$ can be arbitrary; in particular, when $\varphi=\psi$ it
follows that $\rho=0$.

##  3 Weyl representation of $so(1,5;g)$

We shall now obtain representations for the $\kappa$-deformed Yang model by
using the previous relations between $so(1,5)$ and $so(1,5;g)$. Notice that
from a four-dimensional point of view our primary fields are the generators
${\tilde{M}}_{ij}$, ${\tilde{X}}_{i}$, ${\tilde{P}}_{i}$ and ${\tilde{H}}$, to
each of which we shall assign a conjugate momentum. The formalism is therefore
analog to that introduced for the extended Snyder model [21].

Consider then the generalized Heisenberg algebra

$[x_{\mu\nu},x_{\rho\sigma}]=[k^{\mu\nu},k^{\rho\sigma}]=0,\qquad[x_{\mu\nu},k^{\rho\sigma}]=i(\delta_{\mu}^{\
\rho}\delta_{\nu}^{\ \sigma}-\delta_{\mu}^{\ \sigma}\delta_{\nu}^{\ \rho}),$
(41)

where $k_{\mu\nu}$ are momenta conjugated to the $x_{\mu\nu}$, and define
$X_{\mu\nu}=(S\,x\,S^{T})_{\mu\nu}$ and
$K^{\mu\nu}=(S^{\ddagger}\,k\,S^{-1})^{\mu\nu}$, where
$S^{\ddagger}=(S^{-1})^{T}$.

These variables satisfy commutation relations analogous to (41),

$[X_{\mu\nu},X_{\rho\sigma}]=[K^{\mu\nu},K^{\rho\sigma}]=0,\qquad[X_{\mu\nu},K^{\rho\sigma}]=i(\delta_{\mu}^{\
\rho}\delta_{\nu}^{\ \sigma}-\delta_{\mu}^{\ \sigma}\delta_{\nu}^{\ \rho}),$
(42)

but their indices are raised and lowered by the metric $g_{\mu\nu}$, for
example $K_{\mu\nu}=g_{\mu\rho}g_{\nu\sigma}K^{\rho\sigma}$.

The variables $X_{\mu\nu}$ can be decomposed as

$X_{i4}={X_{i}\over\beta},\quad X_{i5}={P_{i}\over\alpha},\quad
X_{45}={H\over\alpha\beta}.$ (43)

Moreover, the explicit relations between the four-dimensional components of
$K_{\mu\nu}$ and $k_{\mu\nu}$ can be written as

$\displaystyle
K_{ij}=k_{ij}+{\beta\over\sigma}(a_{i}q_{j}-a_{j}q_{i})+{\alpha\over\tau}(b_{i}y_{j}-b_{j}y_{i})-{\alpha\upsilon\over\sigma\tau}(a_{i}y_{j}-a_{j}y_{i})+\alpha\beta(a_{i}b_{j}-a_{j}b_{i})w,$
(44) (45) $\displaystyle Q_{i}={1\over\sigma\tau}\left(\upsilon
q_{i}-{\alpha\tau\over\beta}y_{i}+\alpha b_{i}w\right),$ (46) (47)
$\displaystyle Y_{i}={1\over\sigma\tau}(\sigma y_{i}-\alpha a_{i}w),$ (48)
(49) $\displaystyle W={w\over\sigma\tau},$ (50)

with inverse

$\displaystyle
k_{ij}=K_{ij}-\beta(a_{i}Q_{j}-a_{j}Q_{i})-\alpha(b_{i}Y_{j}-b_{j}Y_{i})+{\alpha\beta}(a_{i}b_{j}-a_{j}b_{i})W,$
(51) (52) $\displaystyle q_{i}=\sigma(Q_{i}-\alpha
b_{i}W)+\upsilon\left({\alpha\over\beta}Y_{i}+\alpha a_{i}W\right),$ (53) (54)
$\displaystyle y_{i}=\tau(Y_{i}+\beta a_{i}W),$ (55) (56) $\displaystyle
w=\sigma\tau W,$ (57)

where we have defined $k^{i4}=\beta q^{i}$, $k^{i5}=\alpha y^{i}$,
$k^{45}=\alpha\beta w$ and

$K^{i4}=\beta Q^{i},\quad K^{i5}=\alpha Y^{i},\quad K^{45}=\alpha\beta W,$
(58)

so that

$[X_{i},Q_{j}]=i\delta_{ij},\qquad[P_{i},Y_{j}]=i\delta_{ij},\qquad[H,W]=i.$
(59)

We now want to find a realisation of the ${\tilde{M}}_{\mu\nu}$ defined in
sect. 1 in terms of the Heisenberg algebra generated by $X_{\mu\nu}$ and
$K^{\mu\nu}$. The ${\tilde{M}}_{\mu\nu}$ satisfy the $so(1,N;g)$ algebra (27),
that we write as

$[{\tilde{M}}_{\mu\nu},{\tilde{M}}_{\rho\sigma}]=i\,C_{{\mu\nu},\,\rho\sigma}^{\qquad\alpha\beta}\,{\tilde{M}}_{\alpha\beta}.$
(60)

The structure constants are given by

$C_{{\mu\nu},\,\rho\sigma}^{\qquad{\alpha\beta}}={1\over
2}\Big{[}-g_{\nu\rho}(\delta_{\mu}^{\ \alpha}\delta_{\sigma}^{\
\beta}-\delta_{\mu}^{\ \beta}\delta_{\sigma}^{\
\alpha})+g_{\mu\sigma}(\delta_{\rho}^{\ \alpha}\delta_{\nu}^{\
\beta}-\delta_{\rho}^{\ \beta}\delta_{\nu}^{\
\alpha})-(\mu\leftrightarrow\nu)\Big{]},$ (61)

and obey the symmetry properties
$C_{{\mu\nu},\rho\sigma}^{\qquad{\alpha\beta}}=-C_{\nu\mu,\rho\sigma}^{\qquad{\alpha\beta}}=-C_{{\mu\nu},\sigma\rho}^{\qquad{\alpha\beta}}=-C_{{\mu\nu},\rho\sigma}^{\qquad\beta\alpha}$
$=-C_{\rho\sigma,{\mu\nu}}^{\qquad{\alpha\beta}}$. Note that structure
constants in (60) are multiplied by $\hbar$, which is set to 1 in our
conventions, and in the classical limit $\hbar=0$ all generators commute.

In general, if the operators $M_{\mu\nu}$ generate a Lie algebra with
structure constants $C_{{\mu\nu},\rho\sigma}^{\qquad{\alpha\beta}}$, the
universal realization of $M_{\mu\nu}$ in terms of the Heisenberg algebra (42),
corresponding to Weyl-symmetric ordering, is given by [22, 24]

$M_{\mu\nu}=X_{\alpha\beta}\left[{{\cal C}\over 1-e^{-\cal
C}}\right]_{{\mu\nu}}^{\quad{\alpha\beta}},$ (62)

where ${\cal C}_{{\mu\nu}}^{\quad{\alpha\beta}}=-{1\over
2}\,C_{{\mu\nu},\,\rho\sigma}^{\qquad{\alpha\beta}}K^{\rho\sigma}$.

This realization enjoys the property

$e^{{i\over 2}t^{\mu\nu}{\tilde{M}}_{\mu\nu}}\triangleright 1=e^{{i\over
2}t^{\mu\nu}X_{\mu\nu}},$ (63)

where the $t_{\mu\nu}$ are real numbers transforming as tensors under
$so(1,N;g)$ and the action $\triangleright$ is defined as

$X_{\mu\nu}\triangleright
f(X_{\alpha\beta})=X_{\mu\nu}f(X_{\alpha\beta}),\qquad
K^{\mu\nu}\triangleright f(X_{\alpha\beta})=-i{\partial
f(X_{\alpha\beta})\over\partial X_{\mu\nu}}=[K^{\mu\nu},f(X_{\alpha\beta})].$
(64)

In particular,

$\displaystyle X_{\mu\nu}\triangleright 1=X_{\mu\nu},\qquad
K_{\mu\nu}\triangleright 1=0,$ (65) (66) $\displaystyle
K^{\mu\nu}\triangleright e^{{i\over
2}t^{\alpha\beta}X_{\alpha\beta}}=t^{\mu\nu}e^{{i\over
2}t^{\alpha\beta}X_{\alpha\beta}}$ (67)

We can now expand (62) in powers of the structure constants
$C_{{\mu\nu},\rho\sigma}^{\qquad{\alpha\beta}}$ (i.e. in terms of $\hbar$),
Then the Weyl realization of ${\tilde{M}}_{\mu\nu}$ in terms of the
generalized Heisenberg algebra generated by $X_{\mu\nu}$ and $K^{\mu\nu}$
reads up to second order,

${\tilde{M}}_{\mu\nu}=X_{\mu\nu}+{1\over 2}X_{\alpha\beta}\,{\cal
C}_{{\mu\nu}}^{\quad{\alpha\beta}}+{1\over 12}X_{\alpha\beta}\left({\cal
C}^{2}\right)_{{\mu\nu}}^{\quad{\alpha\beta}}.$ (68)

where

$\displaystyle{\cal C}_{{\mu\nu}}^{\quad{\alpha\beta}}={1\over
2}\Big{(}\delta_{\mu}^{\ \alpha}K_{\nu}^{\ \beta}+\delta_{\nu}^{\
\beta}K_{\mu}^{\ \alpha}-(\alpha\leftrightarrow\beta)\Big{)},$ (69)
$\displaystyle\left({\cal C}^{2}\right)_{\mu\nu}^{\quad{\alpha\beta}}={1\over
2}\Big{(}2K_{\mu}^{\ \alpha}K_{\nu}^{\ \beta}+\delta_{\nu}^{\
\beta}K_{\mu\rho}K^{\rho\alpha}+\delta_{\mu}^{\
\alpha}K_{\nu\rho}K^{\rho\beta}-(\alpha\leftrightarrow\beta)\Big{)},$ (70)

and the indices are lowered by means of the metric $g_{\mu\nu}$.

Inserting $\cal C$ in (68), we find up to first order,

${\tilde{M}}_{\mu\nu}=\ X_{\mu\nu}+{1\over 2}\left(X_{\mu\alpha}K_{\nu}^{\
\alpha}-X_{\nu\alpha}K_{\mu}^{\ \alpha}\right),$ (71)

and

$[{\tilde{M}}_{\mu\nu},K^{\rho\sigma}]=i(\delta_{\mu}^{\ \rho}\delta_{\nu}^{\
\sigma}-\delta_{\mu}^{\ \sigma}\delta_{\nu}^{\ \rho})+{i\over
2}(\delta_{\mu}^{\ \rho}K_{\nu}^{\ \sigma}-\delta_{\nu}^{\ \rho}K_{\mu}^{\
\sigma}+\delta_{\nu}^{\ \sigma}K_{\mu}^{\ \rho}-\delta_{\mu}^{\
\sigma}K_{\nu}^{\ \rho}).$ (72)

We can write (71) in terms of four-dimensional variables, defined by (26),
(43) and (58), as

$\displaystyle{\tilde{M}}_{ij}=X_{ij}+{1\over 2}\Big{(}X_{ik}(K_{j}^{\
k}-\beta a_{j}Q^{k}-\alpha b_{j}Y^{k})+X_{i}(Q_{j}-\alpha
b_{j}W)+P_{i}(Y_{j}+\beta a_{j}W)$ (73) $\displaystyle\qquad\
-(i\leftrightarrow j)\Big{)},$ (74) (75)
$\displaystyle{\tilde{X}}_{i}=X_{i}+{1\over 2}\Big{(}-\beta
X_{ij}(a_{k}K^{jk}+\beta\tilde{A}Q^{j}+\alpha{\tilde{\rho}}Y^{j})+\beta
X_{i}(a_{j}Q^{j}-\alpha{\tilde{\rho}}W)$ (76)
$\displaystyle\qquad+X_{j}(K_{i}^{\ j}-\beta a_{i}Q^{j}-\alpha
b_{i}Y^{j})+\beta P_{i}(a_{j}Y^{j}+\beta\tilde{A}\,W)-H(Y_{i}+\beta
a_{i}W)\Big{)},$ (77) (78) $\displaystyle{\tilde{P}}_{i}=P_{i}+{1\over
2}\Big{(}-\alpha
X_{ij}(b_{k}K^{jk}+\alpha\tilde{B}Y^{j}+\beta{\tilde{\rho}}Q^{j})+\alpha
P_{i}(b_{j}Y^{j}+\beta{\tilde{\rho}}W)$ (79)
$\displaystyle\qquad+P_{j}(K_{i}^{\ j}-\alpha b_{i}Y^{j}-\beta
a_{i}Q^{j})+\alpha X_{i}(\beta b_{j}Q^{j}-\alpha\tilde{B}\,W)+H(Q_{i}-\alpha
b_{i}W)\Big{)},$ (80) (81) $\displaystyle{\tilde{H}}=H+{1\over 2}\Big{(}\alpha
X_{i}(\alpha\tilde{B}Y^{i}-\beta{\tilde{\rho}}Q^{i}+b_{j}K^{ij})-\beta
P_{i}(\beta\tilde{A}Q^{i}+\alpha{\tilde{\rho}}Y^{i}+a_{j}K^{ij})$ (82)
$\displaystyle\qquad\ +H(\beta a_{i}Q^{i}+\alpha b_{i}Y^{i})\Big{)},$ (83)

where Latin indices are lowered using the flat metric.

Other realizations can be obtained from the Weyl realization
${\tilde{M}}_{\mu\nu}^{W}$ using similarity transformations of the type

${\tilde{M}}_{\mu\nu}=S\,{\tilde{M}}^{W}_{\mu\nu}\,S^{-1}$ (84)

where $S=\exp(G)$ with $G$ of the form $G=XF(K)$. Then the corresponding
realizations will be linear in $X$ and can be written as series in $K$. The
corresponding coproducts, star products and twists can be obtained using the
same similarity transformations [12].

##  4 Coproduct and star product in Weyl realization

Formulae for coproduct and deformed addition of momenta can be deduced using
the results of [21, 24], with the difference that now the sums are performed
with the curved metric $g_{\mu\nu}$ instead of the flat metric. This formalism
allows us to construct a coproduct for the ($\kappa$-deformed ) Yang model, in
particular, both the momenta conjugated to ${\tilde{X}}_{i}$ and to
${\tilde{P}}_{i}$ will admit a coproduct.

Defining

$e^{{i\over 2}s^{{\mu\nu}}{\tilde{M}}_{{\mu\nu}}}e^{{i\over
2}t^{\rho\sigma}{\tilde{M}}_{\rho\sigma}}=e^{{i\over 2}(s^{{\mu\nu}}\oplus
t^{\mu\nu}){\tilde{M}}_{{\mu\nu}}}\equiv e^{{i\over 2}{\cal
D}^{\mu\nu}(s,t){\tilde{M}}_{\mu\nu}},$ (85)

where $s^{\mu\nu}$ and $t^{\mu\nu}$ transform as $so(1,5;g)$ tensors, one has
at first order

${\cal D}^{\mu\nu}(s^{\alpha\beta},t^{\alpha\beta})=\
s^{\mu\nu}+t^{\mu\nu}-{1\over 2}\Big{(}s^{\mu\alpha}t^{\nu}_{\
\alpha}-s^{\nu\alpha}t^{\mu}_{\ \alpha}\Big{)}.$ (86)

In the following, we shall write all the formulas up to first order, without
explicitly mentioning it.

The coproduct $\Delta K^{\mu\nu}$ is then

$\Delta K^{\mu\nu}={\cal D}^{\mu\nu}(K^{\mu\nu}\otimes 1,1\otimes
K^{\mu\nu})=\ \Delta_{0}K^{\mu\nu}-{1\over 2}\Big{(}K^{\mu\alpha}\otimes
K^{\nu}_{\ \alpha}-K^{\nu\alpha}\otimes K^{\mu}_{\ \alpha}\Big{)},$ (87)

where $\Delta_{0}K^{\mu\nu}=K^{\mu\nu}\otimes 1+1\otimes K^{\mu\nu}$. The
coproduct (87) is coassociative.

In components, it reads

$\displaystyle\Delta K^{ij}$ $\displaystyle=$
$\displaystyle\Delta_{0}K^{ij}-{1\over 2}\Big{(}K^{ik}\otimes K^{j}_{\
k}+\beta^{2}\tilde{A}Q^{i}\otimes Q^{j}+\alpha^{2}\tilde{B}Y^{i}\otimes Y^{j}$
(90) $\displaystyle+\alpha\beta{\tilde{\rho}}(Q^{i}\otimes Y^{j}+Y^{i}\otimes
Q^{j})+\beta a_{k}(K^{ik}\otimes Q^{j}+Q^{i}\otimes K^{jk})$
$\displaystyle+\alpha b_{k}(K^{ik}\otimes Y^{j}+Y^{i}\otimes
K^{jk})-(i\leftrightarrow j)\Big{)},$ $\displaystyle\Delta Q^{i}$
$\displaystyle=$ $\displaystyle\Delta_{0}Q^{i}-{1\over 2}\Big{(}K^{ik}\otimes
Q_{k}-Q_{k}\otimes K^{ik}+\alpha\beta{\tilde{\rho}}(Q^{i}\otimes W-W\otimes
Q^{i})$ (93) $\displaystyle+\alpha^{2}\tilde{B}(Y^{i}\otimes W-W\otimes
Y^{i})+\alpha b_{k}(K^{ik}\otimes W-W\otimes K^{ik})\Big{)},$
$\displaystyle\Delta Y^{i}$ $\displaystyle=$
$\displaystyle\Delta_{0}Y^{i}-{1\over 2}\Big{(}K^{ik}\otimes
Y_{k}-Y_{k}\otimes K^{ik}-\alpha\beta{\tilde{\rho}}(Y^{i}\otimes W-W\otimes
Y^{i})$ (96) $\displaystyle+\beta^{2}\tilde{A}(Q^{i}\otimes W-W\otimes
Q^{i})-\beta a_{k}(K^{ik}\otimes W-W\otimes K^{ik})\Big{)},$
$\displaystyle\Delta W$ $\displaystyle=$ $\displaystyle\Delta_{0}W-{1\over
2}\Big{(}Q^{k}\otimes Y_{k}-Y^{k}\otimes Q_{k}\Big{)}.$ (98)

Using the relations (44), the coproduct can also be written in terms of
tensors transforming under $so(1,5)$. It is also easy to see that the
antipodes are trivial.

The star product can easily be deduced from the previous relations, since it
is defined as

$e^{{i\over 2}s^{\mu\nu}X_{\mu\nu}}\star e^{{i\over
2}t^{\rho\sigma}X_{\rho\sigma}}=e^{{i\over 2}{\cal
D}^{\mu\nu}(s,t)X_{\mu\nu}},$ (99)

with ${\cal D}^{\mu\nu}(s,t)$ given in (86). This star product is associative.

It may be useful to explicitly write down the four-dimensional expression of
${\cal D}^{\mu\nu}(s,t)$: setting ${\cal D}^{i}={\cal D}^{i4}$, $\bar{\cal
D}^{i}={\cal D}^{i5}$, ${\cal D}={\cal D}^{45}$, one has

$\displaystyle{\cal D}^{ij}(s,t)$ $\displaystyle=$ $\displaystyle\
s^{ij}+t^{ij}-{1\over 2}\Big{(}s^{ik}t^{j}_{\
k}+\beta^{2}\tilde{A}s^{i}t^{j}+\alpha^{2}\tilde{B}{\bar{s}}^{i}{\bar{t}}^{j}+\alpha\beta{\tilde{\rho}}(s^{i}{\bar{t}}^{j}+{\bar{s}}^{i}t^{j})$
(101) $\displaystyle+\ \beta a_{k}(s^{ik}t^{j}+s^{i}t^{jk})+\alpha
b_{k}(s^{ik}{\bar{t}}^{j}+{\bar{s}}^{i}t^{jk})-(i\leftrightarrow j)\Big{)},$
$\displaystyle{\cal D}^{i}(s,t)$ $\displaystyle=$ $\displaystyle\
s^{i}+t^{i}-{1\over
2}\Big{(}s^{ik}t_{k}-t^{ik}s_{k}+\alpha\beta{\tilde{\rho}}(s^{i}t-st^{i})+\alpha^{2}\tilde{B}({\bar{s}}^{i}t-s{\bar{t}}^{i})$
(104) $\displaystyle+\ \alpha b_{k}(s^{ik}t-st^{ik})\Big{)}$
$\displaystyle\bar{\cal D}^{i}(s,t)$ $\displaystyle=$ $\displaystyle\
{\bar{s}}^{i}+{\bar{t}}^{i}-{1\over
2}\Big{(}s^{ik}{\bar{t}}_{k}-{\bar{s}}_{k}t^{ik}-\alpha\beta{\tilde{\rho}}({\bar{s}}^{i}t-s{\bar{t}}^{i})+\beta^{2}\tilde{A}(s^{i}t-st^{i})$
(107) $\displaystyle-\ \beta a_{k}(s^{ik}t-st^{ik})\Big{)}$
$\displaystyle{\cal D}(s,t)$ $\displaystyle=$ $\displaystyle\ s+t-{1\over
2}\Big{(}s^{k}{\bar{t}}_{k}-{\bar{s}}^{k}t_{k}\Big{)},$ (109)

where we have defined the components of the $so(1,5;g)$ tensors $t^{\mu\nu}$
as $t^{i}=t^{i4}$, $\bar{t}^{i}=t^{i5}$, $t=t^{45}$ and analogously for
$s^{ij}$.

##  5 The twist for the Weyl realization

In this section, we construct the twist operator at first order, again using
the results of [21, 24]. The twist is defined as a bilinear operator such that
$\Delta m={\cal F}\Delta_{0}m{\cal F}^{-1}$ for each $m$ belonging to
$so(1,5;g)$.

The twist in a Hopf algebroid sense can be computed by means of the formula
[25]

${\cal F}^{-1}\equiv e^{F}=e^{-{i\over 2}K^{\mu\nu}\otimes
X_{\mu\nu}}e^{{i\over 2}K^{\rho\sigma}\otimes{\tilde{M}}_{\rho\sigma}}.$ (110)

Using the CBH formula one gets

$F={i\over 2}\,K^{\mu\nu}\otimes({\tilde{M}}_{\mu\nu}-X_{\mu\nu}).$ (111)

and substituting (71) in (111), one obtains

$F={i\over 2}K^{\alpha\gamma}\otimes X_{\alpha\beta}K_{\gamma}^{\ \beta}.$
(112)

It is easy to check that

${\cal F}\Delta_{0}K^{\mu\nu}{\cal F}^{-1}=\Delta K^{\mu\nu},$ (113)

with $\Delta K^{\mu\nu}$ given in (87).

In terms of components, one can write

$\displaystyle F$ $\displaystyle=$ $\displaystyle
K^{ij}\otimes\Big{[}X_{ik}(K_{j}^{\ k}-\beta a_{j}Q^{k}-\alpha
b_{j}Y^{k})+X_{i}(Q_{j}-\alpha b_{j}W)+P_{i}(Y_{j}+\beta a_{j}W)\Big{]}$ (120)
$\displaystyle+\ Q^{i}\otimes\Big{[}-\beta
X_{ij}(\beta\tilde{A}Q^{j}+\alpha{\tilde{\rho}}Y^{j}+a_{k}K^{jk})+\beta
X_{i}(a_{j}Q^{j}-\alpha{\tilde{\rho}}W)$ $\displaystyle+\ X_{j}(K_{i}^{\
j}-\beta a_{i}Q^{j}-\alpha b_{i}Y^{j})+\beta
P_{i}(a_{j}Y^{j}+\beta\tilde{A}\,W)-H(Y_{i}+\beta a_{i}W)\Big{]}$
$\displaystyle+\ Y^{i}\otimes\Big{[}-\alpha
X_{ij}(\alpha\tilde{B}Y^{j}+\beta{\tilde{\rho}}Q^{j}+b_{k}K^{jk})+\alpha
P_{i}(b_{j}Y^{j}+\beta{\tilde{\rho}}W)$ $\displaystyle+\ P_{j}(K_{i}^{\
j}-\alpha b_{i}Y^{j}-\beta a_{i}Q^{j})+\alpha X_{i}(\beta
b_{j}Q^{j}-\alpha\tilde{B}\,W)+H(Q_{i}-\alpha b_{i}W)\Big{]}$ $\displaystyle+\
W\otimes\Big{[}\alpha
X_{i}(b_{j}K^{ij}+\alpha\tilde{B}Y^{i}-\beta{\tilde{\rho}}Q^{i})-\beta
P_{i}(\beta\tilde{A}Q^{i}+\alpha{\tilde{\rho}}Y^{i}+a_{j}K^{ij})$
$\displaystyle+\ H(\beta a_{i}Q^{i}+\alpha b_{i}Y^{i})\Big{]}.$

##  6 Coproduct and twist for the original Yang model

Of course the Hopf structure for the original Yang model can be derived from
the previous results simply setting $A=B=1$, $\varphi=\psi=0$ and $a=b=0$.
Since these results are not discussed in the literature we briefly report them
here.

The coproduct can be written in terms of the four-dimensional variables
defined in section 3 as

$\displaystyle\Delta K^{ij}$ $\displaystyle=$
$\displaystyle\Delta_{0}K^{ij}-{1\over 2}\Big{(}K^{ik}\otimes K^{j}_{\
k}+\beta^{2}Q^{i}\otimes Q^{j}+\alpha^{2}Y^{i}\otimes Y^{j}$ (122)
$\displaystyle+\alpha\beta(Q^{i}\otimes Y^{j}+Y^{i}\otimes
Q^{j})-(i\leftrightarrow j)\Big{)},$ $\displaystyle\Delta Q^{i}$
$\displaystyle=$ $\displaystyle\Delta_{0}Q^{i}-{1\over 2}\Big{(}K^{ik}\otimes
Q_{k}-Q_{k}\otimes K^{ik}+\alpha\beta(Q^{i}\otimes W-W\otimes Q^{i})$ (125)
$\displaystyle+\alpha^{2}(Y^{i}\otimes W-W\otimes Y^{i})\Big{)},$
$\displaystyle\Delta Y^{i}$ $\displaystyle=$
$\displaystyle\Delta_{0}Y^{i}-{1\over 2}\Big{(}K^{ik}\otimes
Y_{k}-Y_{k}\otimes K^{ik}-\alpha\beta(Y^{i}\otimes W-W\otimes Y^{i})$ (128)
$\displaystyle+\beta^{2}(Q^{i}\otimes W-W\otimes Q^{i})\Big{)},$
$\displaystyle\Delta W$ $\displaystyle=$ $\displaystyle\Delta_{0}W-{1\over
2}\Big{(}Q^{k}\otimes Y_{k}-Y^{k}\otimes Q_{k}\Big{)}.$ (130)

It is evident that one cannot disentangle the various components of the
conjugated momenta.

Analogously, the twist takes the form

$\displaystyle F$ $\displaystyle=$ $\displaystyle
K^{ij}\otimes\Big{[}X_{ik}K_{j}^{\ k}+X_{i}Q_{j}+P_{i}Y_{j}\Big{]}$ (134)
$\displaystyle+\ Q_{i}\otimes\Big{[}-\beta X_{ik}(\beta Q^{k}+\alpha
Y^{k})+X_{k}K_{i}^{\ k}+\beta(\beta P_{i}-\alpha X_{i})W-HY^{i}\Big{]}$
$\displaystyle+\ Y_{i}\otimes\Big{[}-\alpha X_{ik}(\alpha Y^{k}+\beta
Q^{k})+P_{k}K_{i}^{\ k}+\alpha(\beta P_{i}-\alpha X_{i})W+HQ^{i}\Big{]}$
$\displaystyle+\ W\otimes\Big{[}\alpha X_{k}(\alpha Y^{k}-\beta Q^{k})-\beta
P_{k}(\beta Q^{k}+\alpha Y^{k})\Big{]}.$

##  7 Conclusions

The Yang model represents a noncommutative geometry defined on a curved
background, which is interesting because of possible applications to quantum
cosmology and for its dual nature for the interchange of positions and
momenta.

In this paper we have discussed a generalization of that model, called doubly
$\kappa$-deformed Yang model, originally proposed in [20] performing a
deformation of the flat metric appearing in the definition of the Yang
algebra, so that both the de Sitter and Snyder subalgebras are deformed.

The formalism introduced in this paper, inspired to the one used in ref. [16]
for the Snyder model, permits to define an associative star product and a
coassociative coproduct, together with a twist. To our knowledge, a Hopf
algebra structure for the Yang model has never been discussed before in the
literature. Using this formalism, we have been able to calculate in a
straightforward way several properties of the associated Hopf algebra. The
achievement of these results necessitates the introduction as primary fields
of extended tensorial coordinates and a scalar coordinate, besides position
and momentum. In addition also the momenta conjugated to these variables must
be considered.

The problem of the interpretation of all these extended coordinates is
crucial. One possibility is that they derive from the symmetry breaking of an
$so(1,5,g)$ algebra to an $so(1,3)$ algebra as proposed in [20]. Also the
possibility to relate Yang models to Kaluza-Klein theories, interpreting the
extra degrees of freedom as higher dimensions is under investigation.

At this point, the question of what are the physical consequences of such
models and the possible physical predictions of new observable effects that
could be measured arises. A first step in this direction would be to define a
dynamics for the theory, writing down a suitable Hamiltonian for particles
leaving in Yang spacetime. Further developments may include the definition of
a quantum field theory compatible with this structure.

## Acknowledgements

We thank Jerzy Lukierski for useful discussions. S. Mignemi acknowledges
support of Gruppo Nazionale di Fisica Matematica.

## References

  * [1] C.N. Yang, Phys. Rev. 72, 874 (1947).
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