# Multi-channel Sensor Network Construction, Data Fusion and Challenges for
Smart Home

He Zhang  0000-0002-8169-1653<EMAIL_ADDRESS>The Future Laboratory, Tsinghua
UniversityNo. 160 Chengfu RoadBeijingP.R.C.100084 Pennsylvania State
UniversityWestgate BuildingUniversity ParkPennsylvaniaUSA16801 ,  Robin Ananda
0000-0001-7126-1302<EMAIL_ADDRESS>Academy of Arts & Design,
Tsinghua UniversityTsinghua UniversityBeijingP.R.C.100084 ,  Xinyi Fu
0000-0001-6927-0111<EMAIL_ADDRESS>The Future Laboratory, Tsinghua
UniversityNo. 160 Chengfu RoadBeijingP.R.C.100084 ,  Zhe Sun
0000-0002-4821-2366<EMAIL_ADDRESS>Academy of Arts & Design, Tsinghua
UniversityTsinghua UniversityBeijingP.R.C.100084 ,  Xiaoyu Wang
0009-0007-6812-7031<EMAIL_ADDRESS>Yuanpei College, Peking
UniversityNO.5 Yiheyuan RoadBeijingP.R.C100871 ,  Keqi Chen
0000-0001-9383-9910<EMAIL_ADDRESS>Nanyang Technological
UniversitySingapore and  John M. Carroll  0000-0001-5189-337X<EMAIL_ADDRESS>Pennsylvania State UniversityWestgate BuildingUniversity
ParkPennsylvaniaUSA16801

(2023)

###### Abstract.

Both sensor networks and data fusion are essential foundations for developing
the smart home Internet of Things (IoT) and related fields. We proposed a
multi-channel sensor network construction method involving hardware,
acquisition, and synchronization in the smart home environment and a smart
home data fusion method (SHDFM) for multi-modal data (position, gait, voice,
pose, facial expression, temperature, and humidity) generated in the smart
home environment to address the configuration of a multi-channel sensor
network, improve the quality and efficiency of various human activities and
environmental data collection, and reduce the difficulty of multi-modal data
fusion in the smart home. SHDFM contains 5 levels, with inputs and outputs as
criteria to provide recommendations for multi-modal data fusion strategies in
the smart home. We built a real experimental environment using the proposed
method in this paper. To validate our method, we created a real experimental
environment — a physical setup in a home-like scenario where the multi-channel
sensor network and data fusion techniques were deployed and evaluated. The
acceptance and testing results show that the proposed construction and data
fusion methods can be applied to the examples with high robustness,
replicability, and scalability. Besides, we discuss how smart homes with
multi-channel sensor networks can support digital twins.

smart home, internet of things, data fusion, sensor network, system

††journalyear: 2023††copyright: rightsretained††conference: Chinese CHI 2023;
November 13–16, 2023; Denpasar, Bali, Indonesia††booktitle: Chinese CHI 2023
(CHCHI 2023), November 13–16, 2023, Denpasar, Bali, Indonesia††doi:
10.1145/3629606.3629638††isbn: 979-8-4007-1645-4/23/11††ccs: Human-centered
computing HCI design and evaluation methods††ccs: Human-centered computing
Empirical studies in HCI††ccs: Information systems Information systems
applications††ccs: Hardware Communication hardware, interfaces and
storage††ccs: Hardware Sensor devices and platforms

##  1\. Introduction

With the improvement of sensor technology, the development of artificial
intelligence, the popularization of high-speed Internet, and the application
of the concept of smart home Internet of Things, the concept of the smart home
is being accepted by more and more people. At the same time, with the
popularity of consumer-grade smart home products and the increase in human
society’s demand, the smart home market has a certain scale and is growing
rapidly. Statista states that the global smart home market could reach $182.45
billion by 2025 (Lasquety-Reyes, 2019). According to the scenario-based needs
of users, a smart home mainly involves several key areas such as device
control and linkage (Zheng et al., 2017), scene atmosphere (Eggen et al.,
2003), entertainment and leisure (Kim et al., 2018), intelligent security
(Touqeer et al., 2021), energy management (Badar and Anvari-Moghaddam, 2022),
intelligent applications (Kornyshova et al., 2022), and design for specific
people or needs (Jamwal et al., 2022; Schak et al., 2022; Radha, 2022). On the
one hand, there is a growing demand for smarter and more efficient smart homes
and related products, and the various needs, motivations, goals, and
implementation methods of using smart homes in smart home scenarios are
repeatedly emphasized (Marikyan et al., 2019; Hargreaves et al., 2018). Among
them, data and sensor networks are one of the most important foundations for
studying smart home problems. On the other hand, traditional interaction
methods have been found to have a number of interactivity problems that still
affect the interaction experience, such as intrusiveness (Benlian et al.,
2020), fatigue (Liu et al., 2020), and interaction with specific groups of
people (Kim and Choudhury, 2021). The sensor network construction and data
fusion application are the preconditions to help solve the above problems. In
addition, humans have moved into the information society, where more and more
interactions are moving from physical to virtual. Highly informative and
virtualized interactions are gradually accepted and taken for granted. The
diversity of data collected through multi-channel sensor networks
significantly impacts future interaction methods, especially for further smart
home digital twins.

There is currently no comprehensive summary of sensor network construction
methods in the smart home. The main problems are the following two: (1) There
are few comprehensive, integrated application and sensor network construction
methods covering the whole smart home. Most of the existing studies are sensor
networks built for specific studies, which are characterized by using only a
small number of channels, a relatively single type, and a single purpose of
sensors. Although similar sensor networks are still somewhat usable for
specific research problems, with the further diversification of scenarios and
requirements in smart homes, a series of unpredictable cross-influences on the
research problems in real smart home environments cannot be effectively
addressed or planned; (2) Existing studies rarely mention the specific
implementation problems encountered in the construction of sensor networks,
especially the configuration of hardware, acquisition, and synchronization in
multi-channel sensor networks. The smart home is a significant research sub-
field in human-computer interaction (HCI) and even in the interdisciplinary
intersection. There are a lot of research problems related to people’s
livelihoods that can be explored (FU et al., 2023). At the same time, the a
priori skills of researchers and technologies are likely to be limited in the
interdisciplinary field, which limits further research.

With the development of sensor technologies and the growing demand for HCI,
ubiquitous computing (Mehrotra et al., 2021) and data-driven (Sarker, 2021)
approaches have received significant attention from researchers. The
processing and fusion methods of the large amount of multimodal data generated
in the smart home environment directly affect the effectiveness of the
relevant models. The data fusion mentioned in this paper refers to the process
of fusing real data from multiple sources, where disparate data are fused into
usable integrated data. Traditional data fusion has certain challenges, mainly
regarding uncertainty, integrity, and consistency after fusion [9]. In
addition to the conventional challenges, data fusion solutions for smart homes
also face the following new challenges:

  1. (1)

Existing data fusion solutions are not applicable to smart home problems.

  2. (2)

Multimodal data fusion has limitations in terms of the types and numbers of
modalities involved.

  3. (3)

Only specific problems are addressed without global considerations, and
portability is insufficient.

Based on the above background and challenges, this paper firstly introduces
the current situation, development trend, problems, and difficulties of multi-
channel sensor network construction and data fusion and related research, and
then proposes a multi-sensor network construction method applied to the smart
home environment in conjunction with the development trend of smart home,
future research-ability, sensor availability and related environment
construction problems, and thus proposes a data fusion scheme in the smart
home environment. Concurrently, the proposed methods for constructing the
sensor network and the data fusion scheme were preliminarily tested through
real scenario assessments on the comprehensive experimental platform we’ve
built.

##  2\. Related Work

###  2.1. The Construction of Multi-channel Sensor Networks

The sensor network consists of multiple sensor nodes deployed in the smart
home, forming a multi-hop self-organizing network using wireless
communication. Liang et al. (Liang et al., 2021) built a wireless sensor
network for the smart home security system based on ZigBee protocol and other
IoT technologies. Their proposed sensor network mainly consists of a digital
access control system(hub) and sensor nodes. The sensor nodes mainly focus on
detecting intrusion and indoor environmental information, including PIR, water
level, smoke, and reed switch sensors. Ramson and Moni (Jino Ramson and Moni,
2017) reviewed various applications of wireless sensor networks, and the
examples cited in their study regarding environmental monitoring, healthcare,
and smart building categories are highly relevant to smart homes. Liu et al.
(Liu et al., 2023) described the types of multi-channel sensors that may be
used in smart environments and their uses.

Multi-device synchronization in multi-channel sensor networks poses
significant challenges. While the widely used Network Time Protocol (NTP)
(Mills et al., 2010) offers universal time coordination, it lacks precision
(its time accuracy is low, providing 1-50ms time accuracy and relying on real-
time network load), which may result in substantial errors in systems
requiring high-frequency data, such as multi-camera video sensors.
Additionally, to ensure the smooth functioning of the sensor network, a smart
home system often requires a robust control base station, data acquisition,
data processing, wireless communication, and power supply systems (Shaikh and
Zeadally, 2016). While various studies (Intille et al., 2005; Seo et al.,
2021) have made valuable contributions to multi-channel sensor networks in
smart homes, we aim to extend this research further. We aim to develop a
system that integrates a broader array of channels with a more diverse range
of sensors and smart home devices. Our proposed construction route is designed
to be comprehensive, enhancing the system’s overall quality. In our ongoing
work, we strive for a system that excels in terms of wholeness, accessibility,
robustness, reproducibility, and intelligence.

###  2.2. Multi-channel Data Fusion

Researchers have widely discussed data fusion and related techniques. The
underlying logic is to combine different types, dimensions, and qualities of
data from multiple sources and use them for any parameter estimation
(Castanedo, 2013). In related studies (Ming-Hao and Jian-Hua, 2019; Sun and
Deng, 2009), the conceptual terms of multimodal data fusion, multi-channel
data fusion, and multisensor data fusion are often substitutable. Hall and
Linas (Hall and Llinas, 1997) proposed a definition of data fusion in which
data from multiple sources are processed and combined based on relevant
information to obtain higher accuracy and greater interpretability than
single-source data. On the one hand, researchers have conducted a large number
of studies related to multimodal data fusion, involving fusion strategies
(Chen et al., 2021), data processing methods (Azcarate et al., 2021),
multisensor data fusion (Hall and Llinas, 1997), information fusion (Leung et
al., 2019), and data fusion applications (King et al., 2017), etc. Meanwhile,
some technical engineering and methodological guidelines on data fusion are
provided in related studies (Liggins II et al., 2017; Esteban et al., 2005).
On the other hand, researchers have continuously reviewed and organized
techniques and fusion methods related to data fusion (Castanedo, 2013; Esteban
et al., 2005; Durrant-Whyte and Henderson, 2016; Khaleghi et al., 2013).

The data fusion architecture system proposed by Dasarathy (Dasarathy, 1997) is
one of the most famous data fusion methods, and its system consists of five
categories based on input and output conditions, which are (1) Data In-Data
Out (DAI-DAO): this method is the most basic data fusion method, where the raw
data is fused, and the data itself is output directly after the sensor
acquires the data; (2)Data In-Feature Out (DAI-FEO): Combine the sensors’
original data and output the fused data’s features; (3)Feature In-Feature Out
(FEI-FEO): Both input and output are data features that are suitable for
multi-source data with different data structures; (4) Feature In-Decision Out
(FEI- DEO): Goal-oriented by inputting features and deriving decision labels
based on a priori knowledge or pre-trained models. This category is often
referred to as feature fusion; (5) Decision In-Decision Out (DEI- DEO): Both
input and output are decisions, and this category is often referred to as
decision fusion. All the five fusion strategies proposed in the method have
some limitations. Among them, methods (1) to (4) are the common fusion
methods, while method (5) has higher technical configuration requirements,
especially for the deployment and configuration of full sensor networks.
However, (5) has a good scope for development and is more suitable for the
development trend in the context of artificial intelligence and machine
learning. Based on this data fusion architecture, Fawzy et al. (Fawzy et al.,
2021) proposed a spatiotemporal data fusion (STDF) method for low-level data
input-output fusion for real-time spatial IoT resource aggregation by k-mean
clustering for spatial clustering and tiny AG-gregation (TAG) for temporal
aggregation, which reduced the data size by 95% and saved 80% of the
processing time with 90% accuracy.

Joint Directors of Laboratories (JDL) (White, 1991), although originating from
the military domain, is currently the most popular data fusion model (Khaleghi
et al., 2013), which is based on a data fusion process divided into five
levels and includes an associated information database and an information
channel for connecting the five levels, the database, the management system
and the user interface.

###  2.3. Technology Mediation

The use of information technology to create new ”virtual spaces” is no longer
just an idea but has become a happening today. Creations using virtual reality
(VR) (Eckstein et al., 2019), augmented reality (AR) (Ullah et al., 2012), and
mixed reality (MR) (Speicher et al., 2019) technologies are beginning to blur
the boundaries of the world. Dourish’s (Dourish, 2006) concept of relying on
data and technology and developing new practices, new responses, and new
environmental perceptions from everyday real-world experiences has been
integrated into and become part of human society. The impact of data on the
digital twin and the future of interaction is now widely recognized by
academia. In the Smart Home domain, many possible devices and concept
prototypes are made possible through data support. For example, the ”Smart
Home Virtual Tour” created by Kučera and Haffner et al. (Kučera et al., 2018)
uses multiple sensors connected to an Arduino microcontroller to enable
virtual tours and events to react to each other. Hu and Mao et al. (Hu et al.,
2022) proposed a digital twin and mixed reality based remote collaboration
system for smart homes (RCSSH) in which cameras, environmental sensors (for
temperature, harmful gas, smoke/fire), weight sensors, audio sensors (by
headphones), and infrared sensors are deployed. Remote assistance is provided
to users by collecting data from sensors in a mixed reality environment.
Gopinath and Srija et al. (Gopinath et al., 2019) proposed a solution for
redesigning the smart home using the digital twin, which relies on continuous
connection and data transfer between sensors and virtual models and
applications to enable real-time tracking and visualization of the smart home
in a digital twin environment. However, there are few digital twins for the
entire smart home environment.

##  3\. Multi-channel Sensor Network Construction for Smart Home

This paper proposes the whole sensor network system consists of four
subsystems covering sensor acquisition, data processing, data transmission,
database maintenance management, and automation functions, namely: (1)
hardware system, (2) acquisition system, (3) synchronization system, and (4)
robustness system.

###  3.1. Hardware, Data Collection and Synchronization System

The hardware system of the smart home multi-channel sensor network contains
(1) multi-channel sensors for collecting various activities and environmental
data in the smart home environment; (2) Interprocess communication (IPC)
system components for handling the data acquisition process; (3)
Synchronization control system components for solving multi-channel sensor
synchronization acquisition problems; (4) Network attached storage (NAS).
Common types of sensors include multi-camera systems, microphone arrays,
pressure sensors, electronic nose, gas sensors, sensors attached to furniture
and home appliances, etc. The hardware system determines the upper limit of
the quality of the collected data. It is recommended to focus on the sensor
type, deployment location, synchronization method, and feasibility of the
hardware system.

The multi-channel sensor network, as outlined in our real-time hardware model,
has been deployed in a real-life smart home integrated experiment platform
spread over approximately 60m2. The setup comprises multiple FLIR cameras,
connected to an Industrial PC (IPC) via a 10 GigE network, generating about
6.7TB of data daily, excluding bedrooms and bathrooms to maintain privacy.
Microphones, olfactory, environmental, and specific device sensors, such as
pressure and infrared, are positioned strategically. These sensors relay data
to the IPC system using data cables, Bluetooth, or WiFi. The synchronized data
is pre-processed and stored on a dedicated server for future use.

##  4\. Multi-channel Sensor Networks: Smart Home Data Fusion Model

After the multi-channel sensor network system is built, human activity data
and environmental data will be collected in the smart home environment. We
propose a data fusion model for a multi-channel sensor network in smart home
environments based on the JDL model in Figure 1, called Smart Home Data Fusion
Model (SHDFM).

Figure 1. Smart Home Data Fusion Model (SHDFM): Human activity in the smart
home environment directly influences the Resources of Data, Sensors, Database,
and Prior knowledge on the left side of the diagram, which is processed
through the information channel for smart home data fusion and then acts on
smart home devices through data sources alignment, data cleaning feature
extraction, advanced behavioral feature extraction, complex decision-making to
influence human behavior. In addition, the data will also be used in smart
cities through external resources after complex decisions have been made and
through process optimization using virtual interactions to influence human
behavioral activities. The final result is a Scenario and end-user task-based
design and application.

SHDFM

  1. Level 0

This level implements the most basic data fusion, but unlike the traditional
JDL model, the focus of this level is on multi-source data alignment, where
the input is data from separate timelines and the output is data with the same
timestamp. The synchronization components and timestamp alignment methods
mentioned above are applied to align the multi-modal data in time units. At
this level, the other modal data sizes can be aligned to the primary modal by
specifying a few primary modalities. In the example, the amount of information
contained in the images and audio is taken as a reference. The high-frequency
data, such as the data of the multi-camera and microphone system, are chosen
as the primary modality for alignment, considering the large amount of
semantic information contained in the images and audio, but not in the
sensors, such as environment and usage.

This level addresses the critical limitation of mismatched timestamps often
found in traditional data fusion methods. Here, rather than just fusing data,
we lay emphasis on multi-source data alignment. The input comprises data from
disparate timelines, and the output synchronizes them to have identical
timestamps. By aligning high-frequency data, such as images and audio from
multi-camera and microphone systems, which carry significant semantic
information, to a primary modality, this level establishes a foundational
solution for subsequent levels.

  2. Level 1

At this level, multi-model data sources are used for data filtering and
labeling. For example, in a smart home environment using a multi-camera
system, camera data is aligned at a specific frame rate. Main cameras are
identified, unoccupied frames removed via person detection algorithms, and
relevant frames are calibrated or deleted based on comparison with voice data.
Building on the alignment achieved in Level 0, this level harnesses multi-
modal data sources for data filtering and labeling. In practical scenarios
like a smart home, this helps eliminate redundancy and enhances accuracy.

Data from multi-channel sensors is fused to extract more useful data and
features. Using the multi-camera system again as an example, a spatial
coordinate system is established. Each camera’s relative spatial coordinates
are calibrated, and skeletal point coordinates are extracted and reconstructed
in 3D, improving data accuracy. This, combined with aligned floor data, helps
obtain the subject’s movements and position in the room.

Ideally, the processes of these initial stages can feed into and optimize the
outputs of further stages.

  3. Level 2

This level involves further extraction of data features and high-level states,
such as types of activities, emotions, health conditions, and interactions
between objects. Here, the labeling results are more complex, influenced by
prior knowledge and human factors. Also, the fusion process goes through
cycles for optimization, knowledge updates, and expansion of dynamic variable
influence on the output. Studies at this level focus on advanced state
behavior recognition, multi-modal emotion detection, and specific interaction
studies. For instance, in ongoing research, researchers are trying to identify
thirteen advanced states of occupants based on camera and floor sensor data
(see example in the Figure 2(a)). Furthermore, they performed emotion
recognition based on gait and speech information, categorizing the multi-modal
data into nine emotional states.

  4. Level 3

Complex Decision Making: The advanced tags, such as various human activities,
emotions, events, and behaviors in the smart home environment obtained in the
previous level, are fused and combined with environmental data for complex
decision-making or higher-level activity recognition. Compared to level 2,
level 3 has more uncertainty in the output but more personalization. Also,
from this level onwards, there will be interactions initiated by things to
people, such as active recommendation systems and emotional interventions.
Therefore, from this level, intelligent interactive devices within the smart
home system are invoked.

  5. Level 4.1

Smart City Access: External resources beyond the limits of the smart home
begin to be called upon, such as big data and more shared data beyond the
scope of the smart home. This level can result from quantitative changes, such
as independent smart home networks accessing the broader smart city. At the
core of this is the fact that sensor channels and coverage areas reach a
sizable scale in a given area of human activity, and data sharing is no longer
bound by space and time , and the data can support multiple events and context
awareness.

  6. Level 4.2

Virtual Smart Home: it represents an untapped yet promising avenue parallel to
level 4.1. It encompasses the virtualization of a smart home, achieved through
continuous data feature interaction, extraction, recognition, fusion, and
decision-making. The integral processes of this level and level 4.1 mutually
influence each other without a hierarchical relationship. The multi-channel
sensor network data can be rendered into a virtual space through digital
twins, establishing a connection between actions within the virtual space and
the real world. This virtual smart home transcends traditional notions of
space and time, entirely supported by the underlying data infrastructure.

SHDFM weaves the smart home environment with a multi-channel sensor network in
a dynamic, adaptable architecture. As we climb the SHDFM ladder, requirements
and resources evolve, becoming more intricate. Interestingly, as the
complexity increases, the volume of output data decreases, a testament to the
refined processes at play. While each level has the flexibility for self-
iteration and influences across levels, they all present vast research
potential.

Presently, extensive research efforts are directed toward levels 0 to 3, and
with technological advancements, level 4 is also being explored. Moving
forward, we anticipate continued improvements across levels 0 to 3,
potentially in a non-linear fashion. Moreover, we foresee the exploration of
levels 4.1 and 4.2 as the focus of future work, provided the resource ceiling
is lifted. However, delving into these advanced levels may bring ethical
considerations into sharper focus.

##  5\. Further Discussion

The smart home environment introduces unique challenges that conventional data
fusion methods often need to be equipped to tackle. Traditional solutions need
to improve in the face of smart homes’ dynamic nature, diversity of data, and
the pivotal role of privacy and security.

Smart homes are characterized by the simultaneous evolution of numerous
factors, including residents’ behaviors, varied ambient conditions, and the
status of interconnected smart devices. These homes generate vast data types,
requiring a flexible fusion strategy to manage disparate data scales
efficiently. Additionally, a high emphasis on privacy and security becomes
necessary, which isn’t typically a primary concern in other data fusion
scenarios.

SHDFM is designed to address these specific challenges, utilizing a non-
unidirectional flow architecture that permits dynamic adjustments and
iterative optimization across various layers. This novel approach facilitates
the model’s adept adaptation to the dynamic nature of smart homes.

Conventional data fusion methods, founded on a linear or unidirectional data
processing flow, need help to adapt efficiently to smart homes’ dynamism and
diverse data types. Furthermore, they may need more mechanisms to ensure
adequate privacy and security protections. While these methods could be
modified to a degree, they may not offer robust or optimal solutions in the
smart home context.

Our proposed SHDFM, with its structured yet adaptable approach, confronts
these challenges head-on while vigilantly maintaining user privacy and
security. As we continue to refine this model, we anticipate it is playing a
transformative role in advancing the capabilities and user-friendliness of
future smart home systems.

##  6\. Future Work

One of our ongoing works focuses on advancing the application of Digital Twin
technology in our smart home system. We are striving to optimize the multi-
modal data collection from our network of over 100 IoT devices and enhance the
accuracy of our smart home virtual representation. In parallel, we enrich our
understanding of human-centric data through our multi-channel sensor network,
emphasizing capturing and interpreting behavioral and cognitive insights. The
ultimate goal is to extend our capabilities to virtually map human
interactions within the smart home, providing a comprehensive picture of user
behavior.

The prototype (Figure 2(b)) we’ve created is a detailed replication of a real-
world smart home environment, capturing key elements such as furniture,
devices, and sensor placements. Its objective is to provide an extensive
visual interpretation of the smart home setup. Our next step involves
upgrading the prototype to allow interaction with specific devices within this
virtual space, driven by data from the actual smart home. This enhancement
would be possible through further improvements in the prototype’s
capabilities.

Additionally, we are progressing with a project that leverages the SHDFM to
analyze and synthesize primary behavioral data along with environmental data.
The goal of this project is to extract high-level semantic information like
human emotion, intent, and cognition (Zhang et al., 2023). For a more thorough
digital twin, we gather human data through multi-channel sensors, which aid in
affective analysis.

##  7\. Conclusion

This paper delves into the research landscape of smart homes, scrutinizes
multi-channel sensor networks and data fusion literature specific to this
realm, and pinpoints existing challenges. We propose a robust method for
constructing multi-channel sensor networks and present a comprehensive example
as a reference for researchers, particularly those venturing into
interdisciplinary fields. We further introduce a modified version of the JDL
data fusion model, dubbed the SHDFM, to establish a blueprint for smart home
data fusion studies. To date, we have accomplished Level 0 and 1, along with
aspects of Level 3 and 4.2, with the remaining levels slated for future work.

Key areas of focus for next-generation smart home sensor networks include
amplifying network transmission rates and incorporating snap-in sensors that
are simpler to install. The emphasis on Level 4 of the SHDFM is apparent and
poised to grow. Overcoming technical obstacles while addressing ethical
concerns affiliated with smart homes remains paramount. Furthering our
understanding of user psychology and needs will allow for more targeted and
practical frameworks and platforms.

Our proposed framework and methodology are designed to aid in constructing
smart home research platforms, underpin theoretical research on sensor
networks and data fusion, and guide the development of smart home products. By
providing examples of smart home platforms built on the foundation of this
framework and theory, we aim to streamline subsequent research. Looking
forward, the creation of advanced smart home experimental platforms,
incorporating comprehensive sensor networks and data fusion, will stand as a
crucial focus for our future endeavors in HCI studies.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grant No. 62172252)

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##  Appendix A APPENDICESx

(a) Example of behavior-posture annotation. The pose estimation diagram with
background is shown in the upper left corner. The upper right corner shows the
pose estimation without background. The lower left corner shows the gesture
estimation with background, including hand gestures. The lower right corner
shows the gesture estimation with no background and hand gestures.

(b) Smart Home Digital Twin Prototype. The prototype shows an overview of the
entire smart home environment and the deployment status of the multi-channel
sensor. The diagram includes the house space content, furniture, devices, and
sensors deployment.

Figure 2. Example of Behavior-posture Annotation and Smart Home Digital Twin
Prototype.


# Menichetti’s nonassociative $G$-crossed product algebras

Susanne Pumpluen<EMAIL_ADDRESS>School of Mathematical
Sciences
University of Nottingham
University Park
Nottingham NG7 2RD
United Kingdom

###### Abstract.

We construct unital central nonassociative algebras over a field $F$ which
have either an abelian Galois extensions $K/F$ or a central simple algebra
over a separable extension of $F$ in their nucleus. We give conditions when
these algebras are division algebras. Our constructions generalize algebras
studied by Menichetti over finite fields. The algebras are examples of non-
trivial semiassociative algebras and thus relevant for the semiassociative
Brauer monoid recently defined by Blachar, Haile, Matzri, Rein, and Vishne.
When ${\rm Gal}(K/F)=G$ the algebras of the first type can be viewed as
nonassociative $G$-crossed product algebras.

###### Key words and phrases:

Division algebras, nonassociative algebras.

###### 2010 Mathematics Subject Classification:

Primary: 17A35; Secondary: 17A60, 17A99

##  1\. Introduction

Let $F$ be a field. An associative algebra $A$ over $F$ is called a _$G$
-crossed product algebra_ or _crossed product algebra_ , if it contains a
maximal field extension $K/F$ which is Galois with Galois group $G={\rm
Gal}(K/F)$. Crossed product algebras play an important role in the theory of
central simple algebras, since every element in the Brauer group of $F$ is
similar to a crossed product algebra.

What is a reasonable generalization of a $G$-crossed product algebra to the
nonassociative setting? The first paper addressing this question is by Albert
[1]. The construction defined therein is very general and includes, for
instance, even the iterated Cayley Dickson doubling process. It employs groups
of bijective linear maps and generalizations of factor sets.

We propose instead to think of nonassociative $G$-crossed product algebras in
terms of unital central algebras of dimension $m^{2}$ that have a Galois field
extension $K/F$ of degree $m$ with Galois group $G$ in their nucleus. To avoid
some pathological cases, we assume that a nonassociative $G$-crossed product
algebra $A$ is faithful as a $K^{e}$-module via the action $(e\otimes
e^{\prime})a=eae^{\prime}$ for all $a\in A$, $e,e^{\prime}\in K$, where
$K^{e}=K\otimes_{F}K$. If $A$ is minimally faithful as a $K^{e}$-module, then
such an algebra is a semiassociative algebra [2], and has dimension $m^{2}$.

In 1975, Menichetti introduced a class of semifields of dimension $m^{2}$ over
a finite field $F$ that he constructed out of a cyclic field extension $K/F$
of degree $m$ [4]. We generalize Menichetti’s approach. His construction
yields nonassociative unital central algebras $(K/F,k_{0},\dots,k_{m-1})$ of
dimension $m^{2}$ over $F$ for any base field $F$ that has a Galois field
extension $K/F$ of degree $m$, and can be generalized even more to construct
nonassociative unital algebras $(D,k_{0},\dots,k_{m-1})$ of dimension
$n^{2}m^{2}$ over $F$ employing a central simple algebra of degree $n$ over
some separable field extension of $F$.

We take some steps toward systematically understanding the generalized
Menichetti algebras $(K/F,k_{0},\dots,k_{m-1})$. These algebras are special
cases of Albert’s nonassociative crossed extensions in [1], and can be viewed
as nonassociative crossed products.

After defining the algebras in Section 2, we collect Menichetti’s results, but
generalize them immediately to any abelian Galois field extension $K/F$ in
Section 3. We then look at the cases when field extensions of degree three and
four are used in the construction in Sections 4 and 5.

The field $K$ is a maximal commutative subalgebra of
$(K/F,k_{0},\dots,k_{m-1})$, so Menichetti algebras are examples of
nonassociative $G$-crossed product algebras. They include the opposite
algebras of nonassociative cyclic algebras as special cases.

Semiassociative algebra were recently defined in [2] in order to generalize
the classical setup of central simple algebras, splitting fields, the Brauer
group, etc.: proper Menichetti algebras make up a large class of non-trivial
semiassociative algebras.

A few special cases of some of the results here appeared in the PhD thesis of
A. Steele [6].

##  2\. The construction

###  2.1. Preliminaries

Let $F$ be a field. A _(nonassociative) algebra_ $A$ over $F$ is an $F$-vector
space together with an $F$-bilinear map $A\times A\to A$, $(x,y)\mapsto x\cdot
y$, denoted simply by juxtaposition $xy$, which is the multiplication of $A$.
$A$ is called _unital_ if there is an element in $A$, denoted by 1, such that
$1x=x1=x$ for all $x\in A$. We only consider finite-dimensional unital
algebras. An algebra $A\not=0$ is called a division algebra if for any
$0\not=a\in A$, the left and right multiplication with $a$, $L_{a}(x)=ax$ and
$R_{a}(x)=xa$, are bijective. Since our $A$ is finite-dimensional as an
$F$-vector space, $A$ is a division algebra if and only if $A$ has no zero
divisors. The left, middle and _right nucleus_ of $A$ are defined as ${\rm
Nuc}_{l}(A)=\\{x\in A\,|\,[x,A,A]=0\\}$, ${\rm Nuc}_{m}(A)=\\{x\in
A\,|\,[A,x,A]=0\\}$ ${\rm Nuc}_{r}(A)=\\{x\in A\,|\,[A,A,x]=0\\}$,
respectively, where $[x,y,z]=(xy)z-x(yz)$ is the associator. These nuclei are
associative subalgebras of $A$. Their intersection ${\rm Nuc}(A)=\\{x\in
A\,|\,[x,A,A]=[A,x,A]=[A,A,x]=0\\}$ is called the nucleus of $A$. The nucleus
of $A$ is an associative subalgebra of $A$, and $x(yz)=(xy)z$ whenever one of
the elements $x,y,z$ is in ${\rm Nuc}(A)$. The center of $A$ is ${\rm
C}(A)=\\{x\in\text{Nuc}(A)\,|\,xy=yx\text{ for all }y\in A\\}$. A
nonassociative algebra is called a _proper_ nonassociative algebra if it is
not associative. The algebra $A$ is called _simple_ , if $A\not=\\{0\\}$ and
$A$ has no ideals apart from $\\{0\\}$ and $A$, and _central_ or _$F$
-central_, if $C(A)=F$.

An algebra $A$ over $F$ is called an _étale algebra_ if it is a finite direct
product of finite separable field extensions of $F$. An étale algebra is
semisimple and the class of étale algebras is closed under tensor products.

A finite dimensional nonassociative $F$-central algebra $A$ is called
_semiassociative_ if its nucleus has an étale $F$-subalgebra $E$, such that
$A$ is cyclic and faithful over $E\otimes_{F}E$ via the action $(e\otimes
e^{\prime})a=eae^{\prime}$ for all $a\in A$, $e,e^{\prime}\in E$ [2]. The
dimension of a semiassociative algebra $A$ is a square [2, Corollary 3.4 ] and
the root of this dimension is called the _degree_ of $A$. Every associative
central simple algebra of degree $n$ has a maximal étale subalgebra $E$ of
dimension $n$ and is semiassociative.

If $A$ is a semiassociative algebra with respect to one étale subalgebra of
its nucleus, then it is semiassociative with respect to all étale subalgebras
of its nucleus [2, Proposition 3.6].

If A is semiassociative of degree $n$, then any $n$-dimensional étale
subalgebra $E$ of ${\rm Nuc}(A)$ is a maximal commutative subalgebra of $A$
[2, Corollary 7.3].

A semiassociative algebra $A$ over $F$ is called _semicentral_ , if all of the
simple components of the semisimple quotient $\sigma(A)={\rm Nuc}(A)/J({\rm
Nuc}(A))$ are $F$-central [2, Definition 16.1]. A semiassociative algebra is
_homogeneous_ if it is semicentral, and the atoms are all Brauer equivalent to
each other [2, Definition 17.2].

Two semiassociative algebras $A$ and $B$ over $F$ are _Brauer equivalent_ , if
there exist skew matrix algebras $M_{n}(F;c)$ and $M_{m}(F,c^{\prime})$ such
that $A\otimes_{F}M_{n}(F;c)\cong B\otimes_{F}M_{m}(F;c^{\prime})$. The
_semiassociative Brauer monoid_ $Br^{sa}(F)$ is the set of equivalence classes
with respect to Brauer equivalence, with product
$[A]^{sa}[B]^{sa}=[A\otimes_{F}B]^{sa}$ and unit element $[F]^{sa}$.

###  2.2. Menichetti algebras

Let $K/F$ be a Galois field extension of degree $m$ with ${\rm
Gal}(K/F)=\\{\tau_{0},\dots,\tau_{m-1}\\}$. Let $k_{i}\in K^{\times}$, $i\in
0,\dots,m-1$, and put

$c_{i,j}=k_{0}^{-1}k_{1}^{-1}\cdots k_{j-1}^{-1}k_{i}k_{i+1}\cdots
k_{i+j-1}\text{ for all }i,j\in\mathbb{Z}_{m}.$

Let $z_{0},\dots,z^{m-1}$ be an $F$-basis of the vector space $A=K^{m}$ and
define a multiplication on $A$ via

$(az_{i})\cdot(bz_{j})=\tau_{j}(a)b(z_{i}\cdot z_{j}),$ $z_{i}\cdot
z_{0}=z_{0}\cdot z_{i}=z_{i}\text{ for all }i\in\mathbb{Z}_{m},$ $z_{i}\cdot
z_{j}=c_{ji}z_{i+j}\text{ for all }i\in\mathbb{Z}_{m}\setminus\\{0\\}$

for all $a,b\in K$. Then $A$ is a nonassociative unital algebra over $F$ of
dimension $m^{2}$ we denote by $(K/F,k_{0},\dots,k_{m-1})$. Define

$M(x_{0},\dots,x_{m-1})=\left[\begin{array}[]{cccccc}x_{0}&c_{m-1,1}\tau_{1}(x_{m-1})&...&c_{1,m-1}\tau_{m-1}(x_{1})\\\
x_{1}&\tau_{1}(x_{0})&...&c_{2,m-1}\tau_{m-1}(x_{2})\\\
x_{2}&c_{1,1}\tau_{1}(x_{1})&\tau_{2}(x_{0})&c_{3,m-1}\tau_{m-1}(x_{3})\\\
...&...&...&...\\\
x_{m-2}&c_{m-3,1}\tau_{1}(x_{m-3})&...&c_{m-1,m-1}\tau_{m-1}(x_{m-1})\\\
x_{m-1}&c_{m-2,1}\tau_{1}(x_{m-2})&...&\tau_{m-1}(x_{0})\\\
\end{array}\right].$

Identify $x_{0}z_{0}+\dots+x_{m-1}z_{m-1}$ with $(x_{0},\dots,x_{m-1})\in
K^{m}$, then the multiplication of $A$ can also be expressed as

$(x_{0},\dots,x_{m-1})\cdot(y_{0},\dots,y_{m-1})=M(x_{0},\dots,x_{m-1})(y_{0},\dots,y_{m-1})^{t}.$

It is easy to see that $K$ is a subfield of $(K/F,k_{0},\dots,k_{m-1})$.

Note that if $K/F$ is a cyclic Galois field extension then

$(K/F,1,\dots,1,k)^{op}=(K/F,\sigma,k)$

is an associative or nonassociative cyclic algebra [3, 6], depending on the
choice of $k\in K^{\times}$.

######  Remark 2.1.

Menichetti algebras are special cases of Albert’s nonassociative crossed
extensions in [1]: Here, we “extend” a Galois field extension $K/F$ with
Galois group $G$. In Albert’s notation we construct special cases of crossed
extension of the kind $(K,G,g)$, where $g$ is an extension set. To construct a
crossed extension $(K,G,g)$, Albert only requires every $\sigma:K\to K$ in a
finite group $G$ to be an $F$-linear bijective map satisfying $\sigma(1)=1$,
and $g=\\{g_{i,j}\\}$ to be a set of elements in $K^{\times}$ for all
$i,j\in\mathbb{Z}_{m}$. For a field extension $K/F$ of dimension $n$, Albert
defines a generalization of the Petit algebra
$K[t;\sigma]/K[t;\sigma](t^{m}-d)$ [5] as well (again, Albert does not assume
that $\sigma$ is an automorphism).

######  Proposition 2.2.

The field $K$ is contained in the left, middle and right nucleus of
$(K/F,k_{0},\dots,k_{m-1})$.

###### Proof.

(i) For ease of notation we use the opposite algebras in this proof: Let $l\in
K$, $x=(x_{0},\dots,x_{m-1})$, $y=(y_{0},\dots,y_{m-1})$, then $x\cdot
l=(x_{0}l,\dots,x_{m-1}l)$ and

$l\cdot x=(lx_{0},\tau_{0}(l)\dots,\tau_{m-1}(l)x_{m-1}).$

This implies that

$(x\cdot l)\cdot y=(x_{0}l,\dots,x_{m-1}l)M(y_{0},\dots,y_{m-1}),$
$x\cdot(l\cdot
y)=(x_{0},\dots,x_{m-1})M(ly_{0},\tau_{0}(l),\dots,\tau_{m-1}(l)y_{m-1}).$

Looking at the matrix representing right multiplication, these terms are
identical, thus $K\subset{\rm Nuc}_{r}(A)$. Moreover, we have

$l\cdot(x\cdot
y)=(l,0,\dots,0)\cdot((x_{0},\dots,x_{m-1})M(y_{0},\dots,y_{m-1})),$ $(l\cdot
x)\cdot
y=(lx_{0},\tau_{0}(l)\dots,\tau_{m-1}(l)x_{m-1})M(y_{0},\dots,y_{m-1})$

Looking at the matrix representing right multiplication, these terms are
identical, thus $K\subset{\rm Nuc}_{l}(A)$. Also,

$(x\cdot y)\cdot
l=((x_{0},\dots,x_{m-1})M(y_{0},\dots,y_{m-1}))M(l,0,\dots,0),$ $x\cdot(y\cdot
l)=(x_{0},\dots,x_{m-1})M(y_{0}l,\dots,y_{m-1}l).$

These terms are again identical due to the definition of the matrix
representing right multiplication, thus $K\subset{\rm Nuc}_{m}(A)$. ∎

Since $K\subset{\rm Nuc}((K/F,k_{0},\dots,k_{m-1}))$, Menichetti algebras are
$K$-associative algebras. They are also semiassociative algebras, since
$(K/F,k_{0},\dots,k_{m-1})$ is cyclic and faithful over $K\otimes_{F}K$ via
the action $(e\otimes e^{\prime})a=eae^{\prime}$ for all
$a\in(K/F,k_{0},\dots,k_{m-1})$, $e,e^{\prime}\in K$. Therefore, $K$ is a
maximal commutative subalgebra of $(K/F,k_{0},\dots,k_{m-1})$ [2, Corollary
7.3].

Since $K\subset{\rm Nuc}((K/F,k_{0},\dots,k_{m-1}))$ we know that
$(K/F,k_{0},\dots,k_{m-1})$ is not semicentral. Since
$(K/F,k_{0},\dots,k_{m-1})$ is not semicentral, if $(K/F,k_{0},\dots,k_{m-1})$
is a proper nonassociative algebra then it is a nontrivial example of a
semiassociative algebra: it does not lie in the similarity class of any
$F$-central simple algebra $B$ in $Br^{sa}(F)$ and is thus not homogeneous
[2].

###  2.3. Generalized Menichetti algebras

Let $D$ be a central simple algebra of degree $n$ over its center $C$. Let
$\sigma\in{\rm Aut}(D)$ such that $\sigma|_{C}$ has finite order $m$, and put
$F={\rm Fix}(\sigma)\cap C$. Then $C/F$ is a cyclic Galois field extension of
degree $m$ with $\mathrm{Gal}(C/F)=\langle\sigma|_{C}\rangle$. Let $k_{i}\in
C^{\times}$, $i\in 0,\dots,m-1$, and put

$c_{i,j}=k_{0}^{-1}k_{1}^{-1}\cdots k_{j-1}^{-1}k_{i}k_{i+1}\cdots k_{i+j-1}$

for all $i,j\in\mathbb{Z}_{m}$. For all $x_{i}\in D$, let

$M(x_{0},\dots,x_{m-1})=\left[\begin{array}[]{cccccc}x_{0}&c_{m-1,1}\sigma(x_{m-1})&...&c_{1,m-1}\sigma^{m-1}(x_{1})\\\
x_{1}&\sigma(x_{0})&...&c_{2,m-1}\sigma^{m-1}(x_{2})\\\
x_{2}&c_{1,1}\sigma(x_{1})&\sigma^{2}(x_{0})&c_{3,m-1}\sigma^{m-1}(x_{3})\\\
...&...&...&...\\\
x_{m-2}&c_{m-3,1}\sigma(x_{m-3})&...&c_{m-1,m-1}\sigma^{m-1}(x_{m-1})\\\
x_{m-1}&c_{m-2,1}\sigma(x_{m-2})&...&\sigma^{m-1}(x_{0})\\\
\end{array}\right]$

and then define a multiplication on $D^{m}$ via

$(x_{0},\dots,x_{m-1})\cdot(y_{0},\dots,y_{m-1})=M(x_{0},\dots,x_{m-1})(y_{0},\dots,y_{m-1})^{t}.$

This way we obtain a nonassociative unital algebra over $F$ of dimension
$n^{2}m^{2}$ we denote by $(D,\sigma,k_{0},\dots,k_{m-1})$ and call a
_generalized Menichetti algebra of degree $mn$_.

Let $E$ be a maximal étale subalgebra of $D$ of dimension $n$. Then
$(D,\sigma,k_{0},\dots,k_{m-1})$ has the étale algebra $E/F$ of dimension $mn$
in its nucleus. The algebra $(D,\sigma,k_{0},\dots,k_{m-1})$ is a faithful
$E^{e}$-module, thus is cyclic as a $E^{e}$-module [2, Remark 3.3] and is a
semiassociative algebra [2]. Since $E\subset{\rm Nuc}((D,\sigma,d))$ we know
that $(D,\sigma,)$ is not semicentral. Since $(K/F,\sigma,d)$ is not
semicentral, it does not lie in the similarity class of any $F$-central simple
algebra $B$ in $Br^{sa}(F)$ and is thus not homogeneous [2].

###  2.4. Tensor products of central simple algebras and Menichetti algebras

Let $D_{0}$ be a central simple algebra over $F$ of degree $n$.

Let $K/F$ be a cyclic field extension of degree $m$ with ${\rm
Gal}(K/F)=\langle\sigma\rangle$. Put $D=D_{0}\otimes_{F}K$. Let
$\widetilde{\sigma}$ be the unique extension of $\sigma$ to $D$ such that
$\widetilde{\sigma}|_{D_{0}}=id_{D_{0}}$. Then the tensor product

$A=D_{0}\otimes_{F}(K/F,k_{0},\dots,k_{m-1})\cong(D,\widetilde{\sigma},k_{0},\dots,k_{m-1})$

of the central simple algebra $D_{0}$ and the Menichetti algebra
$(K/F,k_{0},\dots,k_{m-1})$ is a nonassociative generalized Menichetti algebra
over $F$ of degree $mn$ with multiplication defined via

$M(x_{0},\dots,x_{m-1})=\left[\begin{array}[]{cccccc}x_{0}&c_{m-1,1}\widetilde{\sigma}(x_{m-1})&...&c_{1,m-1}\widetilde{\sigma}^{m-1}(x_{1})\\\
x_{1}&\widetilde{\sigma}(x_{0})&...&c_{2,m-1}\sigma^{m-1}(x_{2})\\\
x_{2}&c_{1,1}\widetilde{\sigma}(x_{1})&\widetilde{\sigma}^{2}(x_{0})&c_{3,m-1}\widetilde{\sigma}^{m-1}(x_{3})\\\
...&...&...&...\\\
x_{m-2}&c_{m-3,1}\widetilde{\sigma}(x_{m-3})&...&c_{m-1,m-1}\widetilde{\sigma}^{m-1}(x_{m-1})\\\
x_{m-1}&c_{m-2,1}\widetilde{\sigma}(x_{m-2})&...&\widetilde{\sigma}^{m-1}(x_{0})\\\
\end{array}\right]$

for $x_{0},\dots,x_{m-1}\in D$. In the semiassociative Brauer monoid
$Br(F)^{sa}$, we thus obtain

$[D_{0}]^{sa}[(K/F,k_{0},\dots,k_{m-1})]^{sa}=[(D,\widetilde{\sigma},k_{0},\dots,k_{m-1})]^{sa}.$

Let now $K/F$ be a Galois field extension of degree $m$ with $G={\rm
Gal}(K/F)=\\{\tau_{1},\dots,\tau_{m}\\}$, and again put $D=D_{0}\otimes_{F}K$.
Let $\widetilde{\tau_{i}}$ be the unique extension of $\tau_{i}$ to $D$ such
that $\widetilde{\tau_{i}}|_{D_{0}}=id_{D_{0}}$. The tensor product of two
semiassociative algebras is semiassociative algebra, thus the tensor product

$A=D_{0}\otimes_{F}(K/F,k_{0},\dots,k_{m-1})$

of the central simple algebra $D_{0}$ and the Menichetti algebra
$(K/F,k_{0},\dots,k_{m-1})$ is a semiassociative algebra over $F$ of degree
$mn$ with multiplication defined via the matrix

$M(x_{0},\dots,x_{m-1})=\left[\begin{array}[]{cccccc}x_{0}&c_{m-1,1}\widetilde{\tau}_{1}(x_{m-1})&...&c_{1,m-1}\widetilde{\tau}_{m-1}(x_{1})\\\
x_{1}&\widetilde{\tau}_{1}(x_{0})&...&c_{2,m-1}\widetilde{\tau}_{m-1}(x_{2})\\\
x_{2}&c_{1,1}\widetilde{\tau}_{1}(x_{1})&\tau_{2}(x_{0})&c_{3,m-1}\widetilde{\tau}_{m-1}(x_{3})\\\
...&...&...&...\\\
x_{m-2}&c_{m-3,1}\widetilde{\tau}_{1}(x_{m-3})&...&c_{m-1,m-1}\widetilde{\tau}_{m-1}(x_{m-1})\\\
x_{m-1}&c_{m-2,1}\widetilde{\tau}_{1}(x_{m-2})&...&\widetilde{\tau}_{m-1}(x_{0})\\\
\end{array}\right]$

for all $x_{i}\in D$. We denote this $F$-algebra by
$(D,G,k_{0},\dots,k_{m-1})$ and note that
$\\{\widetilde{\tau}_{1},\dots,\widetilde{\tau}_{m-1}\\}$ is a subgroup of
${\rm Aut}(D)$.

Since for $A=(D,G,k_{0},\dots,k_{m-1})$, we know that ${\rm
Nuc}_{l}(A)=D_{0}\otimes{\rm Nuc}_{l}((K/F,k_{0},\dots,k_{m-1}))$, ${\rm
Nuc}_{m}(A)=D_{0}\otimes{\rm Nuc}_{m}((K/F,k_{0},\dots,k_{m-1}))$ and ${\rm
Nuc}_{r}(A)=D_{0}\otimes{\rm Nuc}_{r}((K/F,k_{0},\dots,k_{m-1}))$, we have
$D\subset{\rm Nuc}_{l}(A)$, $D\subset{\rm Nuc}_{m}(A)$ and $D\subset{\rm
Nuc}_{r}(A)$. Therefore $A$ is not semicentral and thus it does not lie in the
similarity class of any $F$-central simple algebra $B$ in $Br^{sa}(F)$. It is
not homogeneous [2].

In the semiassociative Brauer monoid $Br(F)^{sa}$, we have

$[D_{0}]^{sa}[(K/F,k_{0},\dots,k_{m-1})]^{sa}=[(D,G,k_{0},\dots,k_{m-1})]^{sa}.$

This observation leads us to define an $F$-algebra $(D,G,k_{0},\dots,k_{m-1})$
as follows: Let $D$ be a central simple algebra of degree $n$ over its center
$C$. Let $\tau_{1},\dots,\tau_{m-1}\in{\rm Aut}(D)$ such that
$\\{\tau_{1},\dots,\tau_{m-1}\\}$ is a subgroup of automorphisms of $D$,
${\tau_{i}}|_{C}$ has finite order $m_{i}$, and put $F_{i}={\rm
Fix}(\tau_{i})\cap C$. Then $C/F_{i}$ is a cyclic Galois field extension of
degree $m_{i}$ with $\mathrm{Gal}(C/F_{i})=\langle{\tau_{i}}|_{C}\rangle$.
Define $F=F_{1}\cap\dots\cap F_{m}$. Suppose that
$G=\\{{\tau_{1}}|_{C},\dots,{\tau_{m-1}}|_{C}\\}$ is a group and that $C/F$ is
a Galois field extension with Galois group $G$ (this can happen, see our
previous example).

Let $k_{i}\in C^{\times}$, $i\in 0,\dots,m-1$, and put

$c_{i,j}=k_{0}^{-1}k_{1}^{-1}\cdots k_{j-1}^{-1}k_{i}k_{i+1}\cdots k_{i+j-1}$

for all $i,j\in\mathbb{Z}_{m}$. Define

$M(x_{0},\dots,x_{m-1})=\left[\begin{array}[]{cccccc}x_{0}&c_{m-1,1}\tau_{1}(x_{m-1})&...&c_{1,m-1}\tau_{m-1}(x_{1})\\\
x_{1}&\tau_{1}(x_{0})&...&c_{2,m-1}\tau_{m-1}(x_{2})\\\
x_{2}&c_{1,1}\tau_{1}(x_{1})&\tau_{2}(x_{0})&c_{3,m-1}\tau_{m-1}(x_{3})\\\
...&...&...&...\\\
x_{m-2}&c_{m-3,1}\tau_{1}(x_{m-3})&...&c_{m-1,m-1}\tau_{m-1}(x_{m-1})\\\
x_{m-1}&c_{m-2,1}\tau_{1}(x_{m-2})&...&\tau_{m-1}(x_{0})\\\
\end{array}\right]$

for all $x_{i}\in D$. Again define a multiplication on $D^{m}$ via

$(x_{0},\dots,x_{m-1})\cdot(y_{0},\dots,y_{m-1})=M(x_{0},\dots,x_{m-1})(y_{0},\dots,y_{m-1})^{t}.$

We obtain a nonassociative unital algebra over $F$ of dimension $n^{2}m^{2}$
which has $D$ as a subalgebra. We denote it by $(D,G,k_{0},\dots,k_{m-1})$ We
will not investigate these algebras here.

##  3\. First results

Menichetti proved his results only for finite fields $F$, where every Galois
field extension is cyclic. We put his results into the context of general
abelian Galois field extensions.

Let $K/F$ be an abelian Galois field extension of degree $m$ with ${\rm
Gal}(K/F)=\\{\tau_{0},\dots,\tau_{m-1}\\}$. Let $k_{i}\in K^{\times}$, $i\in
0,\dots,m-1$, and

$c_{i,j}=k_{0}^{-1}k_{1}^{-1}\cdots k_{j-1}^{-1}k_{i}k_{i+1}\cdots
k_{i+j-1}\text{ for all }i,j\in\mathbb{Z}_{m}.$

The analogous arguments as the one used in [4, Proposition 1] show that

$\det(M(x_{0},\dots,x_{m-1}))=\det(M(\tau_{i}(x_{0}),\dots,\tau_{i}(x_{m-1})))$

for all $x_{i}\in K$, $i\in\mathbb{Z}_{m}$. More precisely,
$\det(M(x_{0},\dots,x_{m-1}))$ is a sum of terms of the type

$k_{0}^{n_{0}}k_{1}^{n_{1}}\cdots
k_{m-1}^{n_{m-1}}\rho_{0}(x_{0})\rho_{1}(x_{1})\cdots\rho_{m-1}(x_{m-1})$

with $n_{i}\in\mathbb{Z}$ and the $\rho_{i}$ are either 0 or
$\rho_{i}(x)=\tau_{r_{1}}(x)\cdots\tau_{r_{s}}(x)$, where the
$r_{j}\in\mathbb{Z}_{m}$ are all distinct. Therefore
$\det(M(x_{0},\dots,x_{m-1}))$ is a sum of elements that look like

$k_{0}^{n_{0}}k_{1}^{n_{1}}\cdots
k_{m-1}^{n_{m-1}}f_{n_{0},n_{1},\dots,n_{m-1}}(x_{0},\dots,x_{m-1}),$

where the $f_{n_{0},n_{1},\dots,n_{m-1}}(x_{0},\dots,x_{m-1})$ are polynomial
functions in $x_{0},\dots,x_{m-1}$, $n_{i}\in\mathbb{Z}$, and
$f_{n_{0},n_{1},\dots,n_{m-1}}(x_{0},\dots,x_{m-1})\in F$ for all choices of
$x_{i}\in K$.

This is proved by using clever matrix manipulations completely analogously as
in [4, Corollary 1] and immediately yields the following general result:

######  Theorem 3.1.

If the elements $k_{0}^{n_{0}}k_{1}^{n_{1}}\cdots k_{m-1}^{n_{m-1}}$ that
appear in the sum of the terms in the determinant are linearly independent
over $F$, then $(K/F,k_{0},\dots,k_{m-1})$ is a division algebra over $F$.

Since this general criterium is not tractable, like Menichetti we will focus
on some special cases: Let

$k_{0}=1,k_{1}=k_{2}=\dots=k_{m-2}=k\text{ and }k_{m-1}=kk^{\prime}\text{ with
}k,k^{\prime}\in K^{\times}.$

Then

$c_{i,j}=k\text{ for }i+j<m,c_{i,j}=kk^{\prime}\text{ for }i+j=m,\text{ and
}c_{i,j}=k^{\prime}\text{ for }i+j>m,$

that means for $(K/F,1,k,\dots,k,kk^{\prime})$ we have

$M(x_{0},\dots,x_{m-1})=\left[\begin{array}[]{ccccccc}x_{0}&kk^{\prime}\tau_{0}(x_{m-1})&kk^{\prime}\tau_{2}(x_{m-2})&...&kk^{\prime}\tau_{m-1}(x_{1})\\\
x_{1}&\tau_{0}(x_{0})&k^{\prime}\tau_{2}(x_{m-1})&...&k^{\prime}\tau_{m-1}(x_{2})\\\
x_{2}&k\tau_{1}(x_{1})&\tau_{2}(x_{0})&...&k^{\prime}\tau_{m-1}(x_{3})\\\
...&...&...&...\\\
x_{m-2}&k\tau_{0}(x_{m-3})&...&...&k^{\prime}\tau_{m-1}(x_{m-1})\\\
x_{m-1}&k\tau_{0}(x_{m-2})&k\tau_{2}(x_{m-3})&...&\tau_{m-1}(x_{0})\\\
\end{array}\right].$

If $k=\alpha_{1}+\alpha_{2}k^{\prime}$ with $\alpha_{i}\in F$, then
$\det(M(x_{0},\dots,x_{m-1}))$ is a polynomial of the form

$\det
M(x_{0},\dots,x_{m-1})=N_{K/F}(x_{0})+kk^{\prime}f_{1}(x_{0},\dots,x_{m-1})+\dots+kk^{\prime
m-1}f_{m-1}(x_{0},\dots,x_{m-1})$

and $f_{j}(x_{0},\dots,x_{m-1})\in F$ for all choices of $x_{i}\in K$. In
particular,

$f_{0}(x_{0},\dots,x_{m-1})=x_{0}\tau_{1}(x_{0})\cdots\tau_{m-1}(x_{0})=N_{K/F}(x_{0}),$
$f_{1}(0,\dots,0,x_{i},x_{i+1},\dots,x_{m-1})=(-1)^{(i+2)(m-i)}\alpha_{1}^{m-i-1}x_{i}\tau_{0}(x_{i})\cdots\tau_{m-1}(x_{i}),$

and consequently we find that

$\det M(0,\dots,0,x_{i},x_{i+1},\dots,x_{m-1})=kk^{\prime
i}(f_{1}(0,\dots,0,x_{i},x_{i+1},\dots,x_{m-1})+k^{\prime}(\dots))$

analogously as in [4]. We generalize [4, p. 344]:

######  Theorem 3.2.

If $k,k^{\prime}\in K^{\times}$, $k=\alpha_{1}+\alpha_{2}k^{\prime}$ with
$\alpha_{i}\in F$, $\alpha_{1}\not=0$, are chosen such that
$1,kk^{\prime},kk^{\prime 2},\dots,kk^{\prime m-1}$ are linearly independent
over $F$, then $(K/F,1,k,\dots,k,kk^{\prime})$ is a division algebra over $F$.

###### Proof.

If $1,kk^{\prime},kk^{\prime 2},\dots,kk^{\prime m-1}$ are linearly
independent over $F$, then

$\det
M(x_{0},\dots,x_{m-1})=N_{K/F}(x_{0})+kk^{\prime}f_{1}(x_{0},\dots,x_{m-1})+\dots+kk^{\prime
m-1}f_{m-1}(x_{0},\dots,x_{m-1})=0$

if and only if $f_{0}=\dots=f_{m-1}=0$, which in turn is equivalent to
$x_{0}=\dots=x_{m-1}=0$, employing that

$\det M(0,\dots,0,x_{i},x_{i+1},\dots,x_{m-1})=kk^{\prime
i}(f_{1}(0,\dots,0,x_{i},x_{i+1},\dots,x_{m-1})+k^{\prime}(\dots)).$

∎

######  Corollary 3.3.

(i) Choose $k^{\prime},k=\alpha_{1}+\alpha_{2}k^{\prime}\in K^{\times}$,
$\alpha_{i}\in F$, $\alpha_{1}\not=0$, such that $1,k^{\prime},k^{\prime
2},\dots,k^{\prime m-1}$ are linearly independent over $F$ (for instance, let
$k^{\prime}\in K$ be a primitive element). Then

$k^{\prime m}=\sum_{i+0}^{m-1}\lambda_{i}k^{\prime i}$

for suitable $\lambda_{i}\in F$. If

$\alpha_{1}^{m-1}+\sum_{i=1}^{m-1}(-1)^{i+1}\lambda_{m-i}\alpha_{1}^{m-i-1}\alpha_{2}^{i}\not=0,$

then $(K/F,1,k,\dots,k,kk^{\prime})$ is a division algebra over $F$.
(ii) Let $k,k^{\prime}\in K^{\times}$, Choose

$k_{0}=1,k_{1}=\dots=k_{m-2}=k,k_{m-1}=kk^{\prime},k^{\prime}=\alpha_{1}+\alpha_{2}k\in
K^{\times},\alpha_{i}\in F,\alpha_{1}\not=0,$

such that $1,k^{\prime}k,k^{\prime}k^{2},\dots,k^{\prime}k^{m-1}$ are linearly
independent over $F$. Then $(K/F,1,k,\dots,k,kk^{\prime})$ is a division
algebra over $F$.

This generalizes [4, p. 345].

###### Proof.

(i) The elements $1$, $kk^{\prime}=\alpha_{1}k^{\prime}+\alpha_{2}k^{\prime
2},\dots,kk^{\prime m-2}=\alpha_{1}k^{\prime m-2}+\alpha_{2}k^{\prime m-1}$
and $kk^{\prime
m-1}=\alpha_{2}\lambda_{0}+\alpha_{2}\lambda_{1}k^{\prime}+\dots+(\alpha_{1}+\alpha_{2}\lambda_{m-1})k^{\prime
m-1}$ are linearly independent over $F$ if and only if

$\det(\left[\begin{array}[]{ccccccc}1&0&0&...&0\\\
0&\alpha_{1}&\alpha_{1}&...&0\\\ 0&0&\alpha_{1}&...&0\\\ ...&...&...&...\\\
0&0&...&\alpha_{1}&\alpha_{2}\\\
\alpha_{2}\lambda_{0}&\alpha_{2}\lambda_{1}&...&\alpha_{2}\lambda_{m-2}&\alpha_{1}+\alpha_{2}\lambda_{m-1}\\\
\end{array}\right])\not=0$

for $(x_{0},\dots,x_{m-1})\not=(0,\dots,0)$.
(ii) The proof is analogous to the one for (i):

$\det
M(x_{0},\dots,x_{m-1})=f_{0}(x_{0},\dots,x_{m-1})+k^{\prime}\sum_{j=1}^{m-1}k^{j}f_{j}(x_{0},\dots,x_{m-1})$

and $f_{0}(x_{0},\dots,x_{m-1})=N_{K/F}(x_{0}),$

$f_{j}(0,x_{1},\dots,x_{m-j},0,\dots,0)=(-1)^{(j+2)(m-j)}\alpha_{1}^{m-j-1}N_{K/F}(x_{m-j}).$

∎

Every four-dimensional unital proper nonassociative algebra over $F$ which has
a quadratic separable field extension of $F$ as its nucleus is a
nonassociative quaternion algebra [8] and hence a Menichetti algebra. Thus the
well-known nonassociative quaternion algebras are the only Menichetti division
algebras of dimension four, up to isomorphism.

In the next two sections we therefore look at nine- and 16-dimensional
Menichetti algebras.

##  4\. Cubic Field Extensions

Let $K/F$ be a cubic Galois field extension with norm $N=N_{K/F}$ and trace
$T=T_{K/F}$. Let $a,b,c\in K^{\times}$. The Menichetti algebra $(K/F,a,b,c)$
has the multiplication

$\displaystyle x\cdot y$
$\displaystyle=(x_{0}y_{0}+ca^{-1}\sigma(x_{2})y_{1}+ca^{-1}\sigma^{2}(x_{1})y_{2})$
$\displaystyle+(x_{1}y_{0}+\sigma(x_{0})y_{1}+cb^{-1}\sigma^{2}(x_{2})y_{2})z$
$\displaystyle+(x_{2}y_{0}+ba^{-1}\sigma(x_{1})y_{1}+\sigma^{2}(x_{0})y_{2})z^{2},$

that is

$M(x_{0},x_{1},x_{2})=\left(\begin{array}[]{ccc}x_{0}&ca^{-1}\sigma(x_{2})&ca^{-1}\sigma^{2}(x_{1})\\\
x_{1}&\sigma(x_{0})&cb^{-1}\sigma^{2}(x_{2})\\\
x_{2}&ba^{-1}\sigma(x_{1})&\sigma^{2}(x_{0})\end{array}\right).$

Moreover,

$\det(M(y_{0},y_{1},y_{2}))=N(y_{0})+ca^{-1}N(y_{1})+b^{-1}c^{2}a^{-1}N(y_{2})-ca^{-1}T(y_{0}\sigma(y_{1})\sigma^{2}(y_{2})),$
$=N(y_{0})+b^{-1}c^{2}a^{-1}N(y_{2})+ca^{-1}(N(y_{1})-T(y_{0}\sigma(y_{1})\sigma^{2}(y_{2}))).$

In particular, we have

$z\cdot z=ba^{-1}z^{2},\quad z^{2}\cdot z=z\cdot z^{2}=ca^{-1},\quad
z^{2}\cdot z^{2}=cb^{-1}z\quad(z\cdot z)\cdot z=ca^{-1}\sigma(ba^{-1}),\quad
z\cdot(z\cdot z)=ca^{-1}ba^{-1}.$

Thus if $ba^{-1}\in K\setminus F$, then $(K/F,a,b,c)$ is not third power-
associative.

######  Remark 4.1.

Note that $(K/F,a,a,c)=(K/F,\sigma,ca^{-1})$, and
$(K/F,\sigma^{2},ca^{-1})\cong(K/F,a,c,c)$ are nonassociative cyclic algebras
of degree three, with the isomorphism given by $x_{0}+x_{1}z+x_{2}z^{2}\mapsto
x_{0}+x_{2}z+x_{1}z^{2}.$ These are associative or nonassociative depending on
the choice of $ca^{-1}\in K^{\times}$.

Let $k,k^{\prime}\in K^{\times}$. The multiplication in
$(K/F,1,k,kk^{\prime})$ is given by

$M(x_{0},x_{1},x_{2},x_{3})=\left(\begin{array}[]{ccc}x_{0}&kk^{\prime}\sigma(x_{2})&kk^{\prime}\sigma^{2}(x_{1})\\\
x_{1}&\sigma(x_{0})&k^{\prime}\sigma^{2}(x_{2})\\\
x_{2}&k\sigma(x_{1})&\sigma^{2}(x_{0})\end{array}\right)$

Put $k=a^{-1}b$, $k^{\prime}=b^{-1}c$, then
$(K/F,1,k,kk^{\prime})=(K/F,a,b,c)$. We therefore investigate w.l.o.g.
algebras of the type $(K/F,1,k,kk^{\prime})$.

If $k,k^{\prime}\in K^{\times}$, $k=\alpha_{1}+\alpha_{2}k^{\prime}$ with
$\alpha_{i}\in F$, $\alpha_{1}\not=0$, are chosen such that
$1,kk^{\prime},kk^{\prime 2}$ are linearly independent over $F$, then
$(K/F,1,k,kk^{\prime})$ is a division algebra over $F$ (Theorem 3.2). For
$m=3$ we can prove this also directly:

######  Proposition 4.2.

If 1, $kk^{\prime}=ca^{-1}$, $kk^{\prime 2}=b^{-1}c^{2}a^{-1}$ are linearly
independent over $F$, then $(K/F,a,b,c)$ is a division algebra.

###### Proof.

Suppose that 1, $ca^{-1}\in K\setminus F$, and $b^{-1}c^{2}a^{-1}\in
K\setminus F$ are linearly independent over $F$. Then
$N(x_{0})+b^{-1}c^{2}a^{-1}N(x_{2})+ca^{-1}(N(x_{1})-T(x_{0}\sigma(x_{1})\sigma^{2}(x_{2})))=0$
implies that $N(x_{0})=0$, hence $x_{0}=0$, $N(x_{2})=0$, hence $x_{2}=0$, and
$N(x_{1})-T(x_{0}\sigma(x_{1})\sigma^{2}(x_{2}))=0$, i.e. $N(x_{1})=0$, that
means $(x_{0},x_{1},x_{2})=0$. ∎

We obtain from Corollary 3.3:

######  Corollary 4.3.

(i) Choose $k^{\prime},k=\alpha_{1}+\alpha_{2}k^{\prime}\in K^{\times}$,
$\alpha_{i}\in F$, $\alpha_{1}\not=0$, such that $1,k^{\prime},k^{\prime 2}$
are linearly independent over $F$. Then $k^{\prime
3}=\sum_{i=0}^{2}\lambda_{i}k^{\prime i}$ for suitable $\lambda_{i}\in F$. If

$\alpha_{1}^{2}+\sum_{i=1}^{2}(-1)^{i+1}\lambda_{3-i}\alpha_{1}^{3-i-1}\alpha_{2}^{i}\not=0,$

then $(K/F,1,k,kk^{\prime})$ is a division algebra over $F$.
(ii) Choose $k$, $k^{\prime}=\alpha_{1}+\alpha_{2}k\in K^{\times}$,
$\alpha_{i}\in F$, $\alpha_{1}\not=0$, such that
$1,k^{\prime}k,k^{\prime}k^{2}$ are linearly independent over $F$. Then
$(K/F,1,k,kk^{\prime})$ is a division algebra over $F$.

We now derive additional criteria for when $(K/F,a,b,c)$ is a division algebra
based on choosing linearly independent $ca^{-1}=kk^{\prime}$ and
$b^{-1}c=k^{\prime}$:

######  Theorem 4.4.

Suppose that $ca^{-1}$ and $b^{-1}c$ are two linearly independent elements.
(i) $ca^{-1}\in F^{\times}$ and $b^{-1}c\in K\setminus F$: If $ca^{-1}\not\in
N(K^{\times})$, then $(K/F,a,b,c)$ is a division algebra over $F$.
(ii) $b^{-1}c\in F^{\times}$ and $ca^{-1}\in K\setminus F$: If $b^{-1}c\not\in
N(K^{\times})$, then $(K/F,a,b,c)$ is a division algebra over $F$.
(iii) $b^{-1}c\in K\setminus F$ and $ca^{-1}\in K\setminus F$ are two linearly
independent elements:
If $kk^{\prime 2}=b^{-1}c^{2}a^{-1}\in F$ and $b^{-1}c^{2}a^{-1}\not\in
N(K^{\times})$, then $(K/F,a,b,c)$ is a division algebra over $F$.
If $kk^{\prime 2}=b^{-1}c^{2}a^{-1}\not\in F$ and $b^{-1}c\not\in
N(K^{\times})$, then $(K/F,a,b,c)$ is a division algebra over $F$.

###### Proof.

We assume that

$\det(M(y_{0},y_{1},y_{2}))=N(y_{0})+b^{-1}c^{2}a^{-1}N(y_{2})+ca^{-1}(N(y_{1})-T(y_{0}\sigma(y_{1})\sigma^{2}(y_{2})))=0$

and want to conclude that this implies $y_{0}=y_{1}=y_{2}=0$.
(i) By our assumptions, we have
$N(y_{0})+ca^{-1}(N(y_{1})-T(y_{0}\sigma(y_{1})\sigma^{2}(y_{2})))=0$ and
$b^{-1}c^{2}a^{-1}N(y_{2})=0$, the later implying $N(y_{2})=0$ thus $y_{2}=0$.
This means $N(y_{0})+ca^{-1}N(y_{1})=0$, therefore
$ca^{-1}=N(-\frac{y_{0}}{y_{1}})\in N(K^{\times})$ if $y_{1}\not=0$,
contradicting our assumption. Thus $y_{1}=0$ which means also $y_{0}=0$.
(ii) By our assumptions, we immediately obtain $N(y_{0})=0$, thus $y_{0}=0$,
and so $b^{-1}cN(y_{2})+N(y_{1})=0$. If also $y_{2}=0$ then this means
$y_{1}=0$ and we are done, so assume $y_{2}\not=0$. This yields
$b^{-1}c=N(-\frac{y_{1}}{y_{2}})$. Since we assumed $b^{-1}c\not\in
N(K^{\times})$ this cannot happen.
(iii) We distinguish two subcases.
If $b^{-1}c^{2}a^{-1}\in F$ then we have
$N(y_{0})+b^{-1}c^{2}a^{-1}N(y_{2})=0$ and
$N(y_{1})-T(y_{0}\sigma(y_{1})\sigma^{2}(y_{2}))=0$. If $y_{2}=0$ this implies
$y_{0}=y_{1}=0$. So assume that $y_{2}\not=0$, then
$\alpha\beta=N(-\frac{y_{0}}{y_{2}})\in N(K^{\times})$, contradicting our
assumption.
If $b^{-1}c^{2}a^{-1}\not\in F$ then this means $y_{0}=0$ and thus
$ca^{-1}(b^{-1}cN(y_{2})+N(y_{1}))=0$. It follows that
$b^{-1}cN(y_{2})+N(y_{1})=0$, so that $b^{-1}c\in N(K^{\times})$, a
contradiction. ∎

Since the norm map is surjective for finite fields, the above criteria are not
strong enough to check when $(K/F,a,b,c)$ is a semifield. A stronger result is
obtained when we argue a bit differently:

######  Theorem 4.5.

Suppose that $ca^{-1}$ and $b^{-1}c$ are two linearly independent elements.
(i) $ca^{-1}\in F^{\times}$ and $b^{-1}c\in K\setminus F$: If
$N(a^{-2}bc)\not\in F^{\times 3}$, then $(K/F,a,b,c)$ is a division algebra
over $F$.
(ii) $b^{-1}c\in F^{\times}$ and $ca^{-1}\in K\setminus F$: If
$N(ab^{-2}c)\not\in N(K^{\times})^{3}$, then $(K/F,a,b,c)$ is a division
algebra over $F$.
(iii) $b^{-1}c\in K\setminus F$ and $ca^{-1}\in K\setminus F$ are two linearly
independent elements: If $b^{-1}c^{2}a^{-1}\not\in F$ and $N(ab^{-2}c)\not\in
N(K^{\times})^{3}$, then $(K/F,a,b,c)$ is a division algebra over $F$.

When $kk^{\prime 2}=b^{-1}c^{2}a^{-1}\in F^{\times}$ we cannot seem to derive
a stronger condition for $(K/F,a,b,c)$ to be a division algebra over $F$.

###### Proof.

Let
$\det(M(y_{0},y_{1},y_{2}))=N(y_{0})+b^{-1}c^{2}a^{-1}N(y_{2})+ca^{-1}(N(y_{1})-T(y_{0}\sigma(y_{1})\sigma^{2}(y_{2})))=0.$
(i) By our assumptions $N(y_{2})=0$, therefore we have $y_{2}=0$. Now

$x\cdot
y=(x_{0}y_{0}+ca^{-1}x_{2}\sigma^{2}(y_{1}))+(x_{0}y_{1}+x_{1}\sigma(y_{0}))z+(ba^{-1}x_{1}\sigma(y_{1})+x_{2}\sigma^{2}(y_{0}))z^{2}$

implies $x_{0}y_{0}=-ca^{-1}x_{2}\sigma^{2}(y_{1})=0,$
$x_{0}y_{1}=-x_{1}\sigma(y_{0})$ and
$ba^{-1}x_{1}\sigma(y_{1})=-x_{2}\sigma^{2}(y_{0})$.
If also $y_{1}=0$ then

$x\cdot y=x_{0}y_{0}+x_{1}\sigma(y_{0})z+x_{2}\sigma^{2}(y_{0})z^{2}=0$

implies that $x_{0}y_{0}=0,$ $x_{1}\sigma(y_{0})=0$ and
$x_{2}\sigma^{2}(y_{0})=0$. If $y_{0}=0$ then $y=0$ and we are done, so assume
$y_{0}\not=0$ then $x_{0}=x_{1}=x_{2}=0$ and we are done as well.
So let us assume that $y_{1}\not=0$.
If $x_{0}=0$ then

$x\cdot
y=ca^{-1}x_{2}\sigma^{2}(y_{1})+x_{1}\sigma(y_{0})z+(ba^{-1}x_{1}\sigma(y_{1})+x_{2}\sigma^{2}(y_{0}))z^{2}$

implies $x_{2}=0$, hence $ba^{-1}x_{1}\sigma(y_{1})=0$, so $x_{2}=0$, thus
$x=0$ and we are done.
So assume $x_{0}\not=0$. Then applying $N$ on both sides of the equations
yields $N(x_{0})N(y_{0})=-N(ca^{-1})N(x_{2})N(y_{1}),$
$N(x_{0})N(y_{1})=-N(x_{1})N(y_{0})$ and
$N(ba^{-1})N(x_{1})N(y_{1})=-N(x_{2})N(y_{0})$. Thus
$-\frac{1}{N(ca^{-1})}\frac{1}{N(y_{1})}N(x_{0})N(y_{0})=N(x_{2}),$ and
plugging this into the third equation yields

$N(ba^{-1})N(x_{1})N(y_{1})=\frac{1}{N(ca^{-1})}\frac{1}{N(y_{1})}N(x_{0})N(y_{0})N(y_{0})$

i.e.

$N(ba^{-1})N(x_{1})=\frac{1}{N(ca^{-1})}\frac{1}{N(y_{1})^{2}}N(x_{0})N(y_{0})^{2}.$

Assume $y_{0}=0$. Then by the second equation (since we are assuming
$x_{0}\not=0$), also $y_{1}=0$ and we are done.
So $y_{0}\not=0$. By the second equation, we have
$N(x_{1})=-N(x_{0})N(y_{1})\frac{1}{N(y_{0})}$, so this in turn yields

$N(ba^{-1})N(x_{0})N(y_{1})\frac{1}{N(y_{0})}=-\frac{1}{N(ca^{-1})}\frac{1}{N(y_{1})^{2}}N(x_{0})N(y_{0})^{2}.$

Therefore, since $x_{0}\not=0$, we get

$N(ba^{-1})N(y_{1})\frac{1}{N(y_{0})}=-\frac{1}{N(ca^{-1})}\frac{1}{N(y_{1})^{2}}N(y_{0})^{2}$

and hence $N(bca^{-2})=N(-\frac{y_{0}}{y_{1}})^{3}.$ This implies
$y_{1}\not=0$. If $N(a^{-2}bc)\not\in N(K^{\times})^{3}$ then this is a
contradiction.
(ii) By our assumptions, we immediately obtain $N(y_{0})=0$, thus $y_{0}=0$.
Hence

$\displaystyle x\cdot y$
$\displaystyle=(ca^{-1}x_{1}\sigma(y_{2})+ca^{-1}x_{2}\sigma^{2}(y_{1}))$
$\displaystyle+(x_{0}y_{1}+cb^{-1}x_{2}\sigma^{2}(y_{2}))z$
$\displaystyle+(x_{0}y_{2}+ba^{-1}x_{1}\sigma(y_{1}))z^{2}$

and this implies $ca^{-1}(x_{1}\sigma(y_{2})+x_{2}\sigma^{2}(y_{1}))=0$,
$x_{0}y_{1}+cb^{-1}x_{2}\sigma^{2}(y_{2})=0$ and
$x_{0}y_{2}+ba^{-1}x_{1}\sigma(y_{1})=0$.
If also $y_{1}=0$ we get $ca^{-1}x_{1}\sigma(y_{2})=0$,
$cb^{-1}x_{2}\sigma^{2}(y_{2})=0$ and $x_{0}y_{2}=0$, so either $y_{2}=0$ as
well and we are done, or $y_{2}\not=0$ but then $x_{0}=x_{1}=x_{2}=0$ and we
are done as well. So assume that $y_{1}\not=0$, and apply $N$ to both sides of
the above equations to obtain $N(x_{1})N(y_{2})=-N(x_{2})N(y_{1})$
$N(x_{0})N(y_{1})=-N(cb^{-1})N(x_{2})N(y_{2})$, and
$N(x_{0})N(y_{2})=-N(ba^{-1})N(x_{1})N(y_{1})$. If also $y_{2}=0$ we are done,
so let $y_{2}\not=0$ as well. Then the last two equations imply that $x_{0}=0$
means $x=0$ so we may assume $x_{0}\not=0$. They also imply $x_{1}=0$ means
$x=0$ so we may assume $x_{1}\not=0$. They also imply $x_{2}=0$ means $x=0$ so
we may assume $x_{2}\not=0$. We obtain
$N(x_{1})=-N(x_{2})N(y_{1})N(y_{2}^{-1})$ and plugging this into the third
equation yields

$N(x_{0})N(y_{2})^{2}=N(ba^{-1})N(x_{2})N(y_{1})^{2}.$

Plugging in $N(x_{0})=-N(cb^{-1})N(x_{2})N(y_{2})N(y_{1})^{-1}$ we get

$-N(cb^{-1})N(x_{2})N(y_{2})N(y_{1})^{-1}N(y_{2})^{2}=N(ba^{-1})N(x_{2})N(y_{1})^{2}$

which simplifies to $N(cb^{-1})=N(ba^{-1})N(-\frac{y_{1}}{y_{2}})^{3}.$ This
cannot happen by our assumption.
(iii) If $b^{-1}c^{2}a^{-1}\not\in F$ then $y_{0}=0$ and thus the
multiplication becomes

$\displaystyle x\cdot y$
$\displaystyle=(ca^{-1}x_{1}\sigma(y_{2})+ca^{-1}x_{2}\sigma^{2}(y_{1}))$
$\displaystyle+(x_{0}y_{1}+cb^{-1}x_{2}\sigma^{2}(y_{2}))z$
$\displaystyle+(x_{0}y_{2}+ba^{-1}x_{1}\sigma(y_{1}))z^{2}$

as in (ii). The same argument as in (ii) now shows the assertion. ∎

######  Example 4.6.

([7]) Let $d\in K\setminus F$. For $(K/F,d,1,d)$ we have

$\det(M(x_{0},x_{1},x_{2}))=N(x_{0})+dN(x_{2})+(N(x_{1})-T(x_{0}\sigma(x_{1})\sigma^{2}(x_{2}))).$

Thus if $y_{2}\not=0$ then $\det(M(y_{0},y_{1},y_{2}))\not=0$. This implies
that if $x\cdot y=0$ and $y_{2}\not=0$ then $x=0$. So there are no nontrivial
right zero divisors $y\in(K/F,d,1,d)$ with $y_{2}\not=0$. Similarly, if $xy=0$
and $y_{2}=0$ but $N(x_{0})\not=-N(x_{1})$ then again $x\cdot y=0$ implies
that $x=0$, so there are no nontrivial zero divisors $x$, $y$ of this type.

##  5\. Menichetti algebras of dimension 16

###  5.1. Cyclic Field Extensions of Degree 4

Let $K/F$ be a cyclic field extension of degree $4$ with ${\rm
Gal}(K/F)=\langle\sigma\rangle$, norm $N$ and trace $T$. Choose $a,b,c,d\in
K^{\times}$ and let $A=(K/F,a,b,c,d)$. We already know from Section 3 that

$\det(M(x_{0},x_{1},x_{2},x_{3}))=\sum_{i}a^{n_{i1}}b^{n_{i2}}c^{n_{i3}}d^{n_{i4}}f_{i}(x_{0},x_{1},x_{2},x_{3}),$

where the $n_{ij}$ are integers and the $f_{i}$ are functions in
$x_{0},x_{1},x_{2},x_{3}$ which take values in $F$. We now explicitly compute
the determinant: Let $F_{0}={\rm Fix}(\sigma^{2})$ be the intermediate field
of $K$ and $F$ fixed by $\sigma^{2}$. Then

$\displaystyle{\rm det}(M(x_{0},x_{1},x_{2},x_{3}))$
$\displaystyle=N(x_{0})-\frac{bcd}{a^{3}}N(x_{1})+\frac{c^{2}d^{2}}{a^{2}b^{2}}N(x_{2})$
$\displaystyle-\frac{d^{3}}{abc}N(x_{3})+\frac{cd}{a^{2}}T(x_{0}\sigma(x_{1})\sigma^{2}(x_{1})\sigma^{3}(x_{2}))$
$\displaystyle-\frac{d}{a}T(x_{0}\sigma(x_{0})\sigma^{2}(x_{1})\sigma^{3}(x_{3}))$
(1)
$\displaystyle+\frac{d^{2}}{ab}T(x_{0}\sigma(x_{2})\sigma^{2}(x_{3})\sigma^{3}(x_{3}))$
$\displaystyle+\frac{cd^{2}}{a^{2}b}T(x_{1}\sigma(x_{2})\sigma^{2}(x_{2})\sigma^{3}(x_{3}))$
$\displaystyle-\frac{cd}{ab}T_{F_{0}/F}(x_{0}\sigma^{2}(x_{0})\sigma(x_{2})\sigma^{3}(x_{2}))$
$\displaystyle+\frac{d^{2}}{a^{2}}T_{F_{0}/F}(x_{1}\sigma^{2}(x_{1})\sigma(x_{3})\sigma^{3}(x_{3})).$

Let $k,k^{\prime}\in K^{\times}$. Then $(K/F,1,k,k,kk^{\prime})$ has
multiplication given by

$M=M(x_{0},x_{1},x_{2},x_{3})=\left(\begin{array}[]{cccc}x_{0}&kk^{\prime}\sigma(x_{3})&kk^{\prime}\sigma^{2}(x_{2})&kk^{\prime}\sigma^{3}(x_{1})\\\
x_{1}&\sigma(x_{0})&k^{\prime}\sigma^{2}(x_{3})&k^{\prime}\sigma^{3}(x_{2})\\\
x_{2}&k\sigma(x_{1})&\sigma^{2}(x_{0})&k^{\prime}\sigma^{3}(x_{3})\\\
x_{3}&k\sigma(x_{2})&k\sigma^{2}(x_{1})&\sigma^{3}(x_{0})\\\
\end{array}\right).$

Put $k=a^{-1}b$, $k^{\prime}=b^{-1}d$, then
$(K/F,1,k,k,kk^{\prime})=(K/F,a,b,b,d)$.

If $k,k^{\prime}\in K^{\times}$, $k=\alpha_{1}+\alpha_{2}k^{\prime}$ with
$\alpha_{i}\in F$, $\alpha_{1}\not=0$, are chosen such that
$1,kk^{\prime},kk^{\prime 2},kk^{\prime 3}$ are linearly independent over $F$,
then $(K/F,1,k,k,kk^{\prime})$ is a division algebra over $F$ (Theorem 3.2).
Corollary 3.3 becomes:

######  Corollary 5.1.

Let $k,k^{\prime}\in K^{\times}$.
(i) Let $k=\alpha_{1}+\alpha_{2}k^{\prime}$ with $\alpha_{i}\in F$,
$\alpha_{1}\not=0$, such that $\\{1,k^{\prime},k^{\prime 2},k^{\prime 3}\\}$
is a basis for $K/F$. If $k^{\prime
4}=\lambda_{0}+\lambda_{1}k^{\prime}+\lambda_{2}k^{\prime
2}+\lambda_{3}k^{\prime 3}$, such that
$\alpha_{1}^{3}+\alpha_{1}^{2}\alpha_{2}\lambda_{3}-\alpha_{1}\alpha_{2}^{2}\lambda_{2}+\alpha_{2}\lambda_{1}\neq
0$, then $1,kk^{\prime},kk^{\prime 2},kk^{\prime 3}$ are linearly independent
over $F$, and $(K/F,1,k,k,kk^{\prime})$ is a division algebra over $F$.
(ii) Let $k^{\prime}=\alpha_{1}+\alpha_{2}k\in K^{\times}$, $\alpha_{i}\in F$,
$\alpha_{1}\not=0$, such that $1,k^{\prime}k,k^{\prime}k^{2},k^{\prime}k^{3}$
are linearly independent over $F$. Then $(K/F,1,k,k,kk^{\prime})$ is a
division algebra over $F$.

######  Lemma 5.2.

(Steele [7]) Pick $d\in K\setminus F$ such that $1,d,d^{2},d^{3}$ are linearly
independent over $F$. Then the following are division algebras:
$(K/F,d,1,1,1)$, $(K/F,1,d,1,1)$, $(K/F,1,1,d,1)$, $(K/F,d,d,1,d)$,
$(K/F,d,1,d,d)$, $(K/F,1,d,d,d)$.

###### Proof.

For $(K/F,d,1,1,1)$, the explicit formula for ${\rm
det}(M(x_{0},x_{1},x_{2},x_{3}))$ from (5.1) is a polynomial of degree 3 in
$1/d$ with coefficients in $F$. Since $1,d,d^{2},d^{3}$ are linearly
independent over $F$, so are $1,1/d,1/d^{2},1/d^{3}$. Moreover, the only term
without a factor of $1/d$ is $N_{K/F}(x_{0})$ and the only term with a factor
of $1/d^{3}$ is $-N_{K/F}(x_{3})$. Hence if ${\rm
det}(M(x_{0},x_{1},x_{2},x_{3}))=0$, then $x_{0}=x_{3}=0$. Putting this back
into (5.1), the only remaining coefficient of $1/d$ is $-N_{K/F}(x_{1})$ and
the only remaining coefficient of $1/d^{2}$ is $N_{K/F}(x_{2})$, so these must
be zero also, implying that $x_{1}=x_{2}=0$. The other cases are proved
similarly. ∎

Note that if $c^{-1}d\in K\setminus F$ then $(K/F,c,c,c,d)$ is a
nonassociative cyclic algebra, and if $c^{-1}d\in F$ then $(K/F,c,c,c,d)$ is
an associative cyclic algebra of degree $4$.

###  5.2. Menichetti algebras from biquadratic field extensions

Let now $char(F)\not=2$, $K=F(\sqrt{u},\sqrt{v})$ be a biquadratic field
extension with norm $N$ and trace $T$, ${\rm
Gal}(K/F)=\\{1,\sigma,\tau,\sigma\tau\\},$ where $\sigma(\sqrt{u})=\sqrt{u},$
$\sigma(\sqrt{v})=-\sqrt{v},$ $\tau(\sqrt{u})=-\sqrt{u}$,
$\tau(\sqrt{v})=\sqrt{v}.$ Let $a,b,c,d\in K^{\times}$, $A=(K/F,a,b,c,d)$ and
$F_{0}=F(\sqrt{v})={\rm Fix}(\tau)$. Then

$\displaystyle{\rm det}(M(x_{0},x_{1},x_{2},x_{3}))$
$\displaystyle=N(x_{0})-\frac{bcd}{a^{3}}N(x_{1})+\frac{c^{2}d^{2}}{a^{2}b^{2}}N(x_{2})$
$\displaystyle-\frac{d^{3}}{abc}N(x_{3})+\frac{cd}{a^{2}}T(x_{0}\sigma(x_{1})\tau(x_{1})\sigma\tau(x_{2}))$
$\displaystyle-\frac{d}{a}T(x_{0}\sigma(x_{0})\tau(x_{1})\sigma\tau(x_{3}))$
(2)
$\displaystyle+\frac{d^{2}}{ab}T(x_{0}\sigma(x_{2})\tau(x_{3})\sigma\tau(x_{3}))$
$\displaystyle+\frac{cd^{2}}{a^{2}b}T(x_{1}\sigma(x_{2})\tau(x_{2})\sigma\tau(x_{3}))$
$\displaystyle-\frac{cd}{ab}T_{F_{0}/F}(x_{0}\tau(x_{0})\sigma(x_{2})\sigma\tau(x_{2}))$
$\displaystyle+\frac{d^{2}}{a^{2}}T_{F_{0}/F}(x_{1}\tau(x_{1})\sigma(x_{3})\sigma\tau(x_{3})),.$

Let us consider the special case $(K/F,1,k,k,kk^{\prime})$. Here, the
multiplication is given by

$M(x_{0},x_{1},x_{2},x_{3})=\left(\begin{array}[]{cccc}x_{0}&kk^{\prime}\sigma(x_{3})&kk^{\prime}\tau(x_{2})&kk^{\prime}\sigma\tau(x_{1})\\\
x_{1}&\sigma(x_{0})&k^{\prime}\tau(x_{3})&k^{\prime}\sigma\tau(x_{2})\\\
x_{2}&k\sigma(x_{1})&\tau(x_{0})&k^{\prime}\sigma\tau(x_{3})\\\
x_{3}&k\sigma(x_{2})&k\tau(x_{1})&\sigma\tau(x_{0})\\\ \end{array}\right)$

and if $k,k^{\prime}\in K^{\times}$ are chosen such that
$1,kk^{\prime},kk^{\prime 2},kk^{\prime 3}$ are linearly independent over $F$,
then $(K/F,1,k,k,kk^{\prime})$ is a division algebra over $F$ (Theorem 3.2):

######  Corollary 5.3.

Let $k,k^{\prime}\in K^{\times}$, such that one of the following holds:
(i) $k=\alpha_{1}+\alpha_{2}k^{\prime}$ with $\alpha_{i}\in F$,
$\alpha_{1}\not=0$, such that $1,k^{\prime},k^{\prime 2},k^{\prime 3}$ or
$1,kk^{\prime},kk^{\prime 2},kk^{\prime 3}$ is a basis for $K/F$. E.g., if
$k^{\prime 4}=\lambda_{0}+\lambda_{1}k^{\prime}+\lambda_{2}k^{\prime
2}+\lambda_{3}k^{\prime 3}$, $\lambda_{i}\in F$, and
$\alpha_{1}^{3}+\alpha_{1}^{2}\alpha_{2}\lambda_{3}-\alpha_{1}\alpha_{2}^{2}\lambda_{2}+\alpha_{2}\lambda_{1}\neq
0$, then $1,kk^{\prime},kk^{\prime 2},kk^{\prime 3}$ is a basis for $K/F$.
(ii) $k^{\prime}=\alpha_{1}+\alpha_{2}k\in K^{\times}$ with $\alpha_{i}\in F$,
$\alpha_{1}\not=0$, such that $1,k^{\prime}k,k^{\prime}k^{2},k^{\prime}k^{3}$
are linearly independent over $F$.
Then $(K/F,1,k,k,kk^{\prime})$ is a division algebra over $F$.

######  Example 5.4.

(Steele [7]) Pick $d\in K^{\times}$ such that $1,d,d^{2},d^{3}$ are linearly
independent over $F$, then $(K/F,d,1,1,1)$, $(K/F,1,d,1,1)$, $(K/F,1,1,d,1)$,
$(K/F,1,1,1,d)$, $(K/F,d,d,d,1)$, $(K/F,d,d,1,d)$, $(K/F,d,1,d,d)$,
$(K/F,1,d,d,d)$ are division algebras. The proof is straightforward, looking
at the determinant.

_A_ cknowledgement: This paper was written while the author was a visitor at
the University of Ottawa. She would like to thank the Department of
Mathematics and Statistics for its hospitality.

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