Schedule

Since the work of J. C. Maxwell biquaternions are present in electromagnetic theory. The talk is an overview of the most important mathematical results obtained in electromagnetism with the aid of the algebra of biquaternions and hypercomplex analysis. Biquaternionic reformulation of the Maxwell system for time-dependent and time-harmonic fields in homogeneous and inhomogeneous, chiral and achiral media; integral representations and solution of boundary value problems both for sourceless and for models with sources will be considered, including a biquaternionic approach for their numerical treatment.

An algebra $A$ with a strictly nondegenerate quadratic form $n(\cdot)$ is called a composition algebra if $n(ab) = n(a)n(b)$ for all elements $a, b \in A$. By Jacobson's theorem, any unital composition algebra over an arbitrary field $F$ can be obtained by the Cayley-Dickson process either from $F$ ($\mathrm{char} \, F \neq 2$) or from a certain two-dimensional subalgebra over $F$ ($\mathrm{char} \, F = 2$). Such algebras are called Hurwitz algebras, and they always have dimension $2^n$, where $0 \leq n \leq 3$. For each Hurwitz algebra $A$, there are three new non-unital composition algebras which can be constructed from $A$. These algebras, together with Hurwitz algebras, are called standard composition algebras.
The length function is a numerical invariant of finite-dimensional algebras, which is closely related to their identities. It was introduced by Spencer in 1959, and since then the lengths of various algebras, especially, matrix algebras, were studied. The non-associative case proves to be the most complicated and interesting, since the length of an associative algebra cannot exceed its dimension, but this is not true for non-associative algebras. In this talk we present a new method which allows us to compute the lengths of standard composition algebras over an arbitrary field.
The talk is based on a joint work with Alexander Guterman.

This is a joint work with Alexander Zubkov. We discuss the classification of orbits of pairs of octonions with respect to the simultaneous action by $G_2$ over an algebraically closed field of arbitrary characteristic. As an application, we describe separating $G_2$-invariants of octonions over an algebraically closed field of characteristic two.

Group cohomology is intimately connected with algebraic structures. For example, if $G$ is a finite group, then $H^2(G,K^*)$, where $K^*=K\setminus 0$ classifies "twisted" forms of the group ring $K[G]$. This is a very useful construction in Galois theory: if $G=Gal(K/k)$, then the twisted rings are central simple $k$-algebras representing elements of the relative Brauer group $Br(K/k)$. In fact, in that case $Br(K/k) \cong H^2(G,K^*)$. These twisted rings also admit a construction in terms of extensions of $G$ by $K^*$, highlighting another important aspect of (second degree) group cohomology. One is interested in finding whether these facts hold true for higher degrees. In particular, for degree three group cohomology, a close analog of twisted group rings is provided by linear Gr-categories. If $G$ is, say, a finite group, then $C(G,\omega,k)$ is a category of $G$-graded $k$-vector spaces, but where the usual (graded) tensor product is twisted by correcting the usual canonical associativity of three factors by a scalar linear map $\omega$. To satisfy the axioms, $\omega$ must represent a class in $H^3(G,k^*)$, and it is important in applications that $\omega$ be explicitly known. Of interest are the algebras in linear Gr-categories, and in particular the division algebras. Recently, new examples of non-associative division algebras have been produced using this framework (the non associativity is controlled by $\omega$), and one can hope to make progress in the interpretation of higher terms in Galois cohomology by using these results. It is known t hat the octonions can be realized as an ample division algebra in a special linear group category $G$, where $G=Z_2\times Z_2\times Z_2$, and a third co-cycle is a co-boundary. Our work in this field has two immediate goals. The first is to explicitly compute 3-cocycles (representing elements in $H^3(G,k^*)$) for a large class of finite groups, such as products of cyclic groups. The second, which relies on the first, is to study division algebras in linear Gr-categories (as mentioned, this is of interest for higher Brauer groups and Galois cohomology). This has two subgoals: one is to employ techniques from non-abelian cohomology in Algebraic Topology to give a theoretical classification; the other is to employ the results of the first goal to construct new explicit examples of (possibly non-associative) division algebras, especially over fields other than $\mathbb{R}$ or $\mathbb{C}$, in particular local fields $K/\mathbb{Q}_{p}$, where $p$ is prime.
Joint work with Ettore Aldrovandi

Consider vector space overnon-commutative division algebra. Set of automorphisms of this vector space is group $GL$. Group $GL$ acts on the set of bases of vector space (basis manifold) single transitive and generates active representation. Twin representation on basis manifold iscalled passive representation. There is no automorphism associated with passive transformation. However passive transformation generates transformation of coordinates of vector with respect to basis. If we consider homomorphism of vector space $V$ into vector space $W$, then we can learn how passive transformation in vector space $V$ generates transformation of coordinates of vector in vector space $W$. Vector in vector space $W$ is called geometric object in vector space $V$. Covariance principle states that geometric object does not depend on the choice of basis. I considered transformation of coordinates of vector and polylinear map.

Our best understanding of fundamental particles is summarized in the Standard Model of Particle Physics. Loosely speaking, these particles are described by a long list of irreducible representations of the gauge group SU(3)$\times$SU(2)$\times$U(1)/$\mathbb{Z}_6$ and spacetime symmetries of the (universal cover of the) Poincare group. On one hand, this description works remarkably well -to the extent that over thelast few decades, experiment has succeeded little in luring the standard model from its originally proposed particle content. On the other hand, this choice of groups and representations continues to evade logical explanation. Why these groups and representations, while not others? In this talk, we demonstrate that a set of vector spaces, quite close to the standard model's particle content, can be embedded inside the 256-dimensional Euclidean Jordan algebra $\mathcal{H}_{16}(\mathbb{C})$.$\mathcal{H}_{16}(\mathbb{C})$ is an algebra that arises naturally from the left-multiplication algebra of $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$. From its right multiplication algebra, we propose the construction of some familiar building blocks of spacetime. We close by observing how division algebraic substructure within this $\mathcal{H}_{16}(\mathbb{C})$ provides a set of possible measurements on the system. These measurementsomitcolour as an observable (related to the phenomenon of confinement).
This talk will not assume a background in particle physics. Everyone is welcome

A symmetric space is a Riemannian manifold whose group of isometries contains the geodesic reflection about every point. Examples include the $n$-dimensional sphere, Grassmanian manifolds, and Lie groups. By exploiting the algebraic characterization of symmetric spaces, E. Cartan obtained a complete classification of all symmetric spaces. Up to the compact/non-compact duality, there are precisely seven infinite series and twelve exceptional irreducible symmetric spaces. The exceptional symmetric spaces are intimately related with the octonions. In this talk I will discuss the geometry of some of these exceptional symmetric spaces.

In this talk we explain how an adjoint Cauchy transform can be meaningfully introduced in the context of octonionic monogenic functions. Due to the lack of associativity we cannot expect to have an octonionic linear adjoint transform $T$ satisfying a relation of the form $(Tf,g)$ for an octonion valued inner product. Instead, one has to consider octonionic para-linear operators and to work on the level of the real parts of the inner products. In this context we discuss an adjoint of the octonionic monogenic Cauchy transform $C^*$ satisfying $Re(Cf,g) = Re(f,C^*,g)$. As an application we look at an octonionic generalization of the Kerzman-Stein operator and establish relations to the Hilbert Riesz transform in $\mathbb{R}^8$.

We present some vector cross product differential and difference equations expressed by using a matrix approach of the 2-fold vector cross product in $\mathbb{R}^3$ and in $\mathbb{R}^7$. Either the classical theory or convenient Drazin inverses of elements belonging to the class of index 1 matrices are applied to obtain closed formulas for the solution of those equations.

In the last years, we have introduced the quaternionic hyperbolic Fourier transform and studied its basic properties,inversion and Parseval Theorems, and uncertainty principles. Now, we extend the work to the quaternionic windowed Fouriertransform, also called the quaternionic hyperbolic Gabor transform. We show some of its properties and use the transform to study Gabor frames in our context. We prove Janssen's and Walnut's representations of quaternionic hyperbolic Gabor frames, as well as modified versions of the Wexler-Raz biorthogonality. Our work extends results obtained in the Euclidean case to the hyperbolic case.

Quaternions with quantum integer components have been studied recently. In this study, we introduce a new class of dual generalized complex numbers with quantum integer components. We obtain some basic properties of these numbers such as conjugation identities, generating functions, Binet formulas, and sum formulas. Moreover, we derive Vajda's like identity for these numbers which generalizes the well-known Cassini's identity.

Based on studies around the quaternions and octonions of Padovan, an application of these numbers to the Perrin numerical sequence is then performed. Thus, there is the process of complexification of this sequence, investigating mathematical properties of these numbers, from which their matrix form, generating function, and Binet's formula stand out. Therefore, identities inherent to these numbers are obtained, emphasizing the complexification process. For future work, the integration of this mathematical investigation with other contents is encouraged.

The aim of this talk is to obtain all the complex Lie algebras obtained by graded contractions of the fine $\mathbb Z_2^3$-grading of $\mathfrak g_2$, the exceptional Lie algebra of derivations of the (complex) octonion algebra. Also, we study the properties of the obtained algebras, many of them nilpotent or solvable.

In 1840 Rodrigues wrote a paper about the laws of geometry that control the displacement of a solid system in space. This work was a precursor to Hamilton's quaternions. Rodrigues' work went unnoticed until 1846 when Cayley acknowledged Euler's and Rodrigues' priority describing orthogonal transformations in a letter to the Editors of the Philosophical Magazine. In this talk, Dr. Familton will discuss the history of Rodrigues' work, and his 1840 paper. He will analyze its connection with Euler's rotation theorem, and Hamilton's quaternions.

A function $f$ from a domain in $\mathbf{R}^3$ to the quaternions is said to be inframonogenic if $\overline{\partial} f\overline{\partial} =0$. However, there are different possibilities for defining $\overline{\partial}$, and we take $\overline{\partial} = \partial/\partial x_0+ (\partial/\partial x_1)e_1+(\partial/\partial x_2) e_2$ which is a different context from much previous work. In the context of functions $f=f_0+f_1e_1+f_2e_2$ taking values in the reduced quaternions, we show that the inframonogenic homogeneous polynomials of degree $n$ form a subspace of dimension $6n+3$. We use the homogeneous polynomials to construct a orthogonal basis for the Hilbert space of square-integrable inframonogenic functions defined in the ball in $\mathbf{R}^3$.
Joint work with C. Álvarez-Peña and J. Morais.

We recently showed how to interpret the elements of the exceptional Lie algebra $\mathfrak{e}_{8(-24)}$ as objects in the Standard Model, including lepton and quark spinors with the usual properties, the Standard Model Lie algebra $\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$, and the Lorentz Lie algebra $\mathfrak{so}(3,1)$. An overview of this model will be presented here, emphasizing the underlying octonionic structure.

NTQO 2022 is an excellent occasion for celebrating 40 years of the book

Clifford Analysis

by Richard Delanghe and his disciples Fred Brackx and Frank Sommen, [1]. It was published in the famous Red Edition of Pitman Research Notes in Mathematics Series as volume number 76. Today, a comprehensive, 14-page review [2] in the Bulletin of the AMS, by R. D. Carmichael, can be considered an immediate expression of its substantial effect in many areas of analysis and algebra. One could even statistically evaluate the influence of the book, as is customary nowadays. However, and above all, one should remember that it also had a significant influence in other respects. Research groups were inspired in different countries; together analysists, algebraists, physicists, geometers, and others initiated conferences (successfully active until now), founded journals, etc. - the list is long. But, in my opinion, the question still arises: to what extent does Clifford Analysis live up to or can live up to its reputation as a generalization of the theory of holomorphic functions of one complex variable? Several chapters of the 1982 book are devoted to this problem. However, despite progress in recent years that goes beyond the contents of [1], I believe that a relatively self-contained generalization of the theory of holomorphic functions of one complex variable by using Clifford algebras, as once dreamt by R. Fueter [3], is still pending. In our talk, we want to address some related mathematical questions and encourage further investigations.
[1] F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis, Pitman 76, London, 1982.
[2] R. D. Carmichael, Review: F. Brackx, R. Delanghe and F. Sommen, Clifford analysis, Bull. Amer. Math. Soc. (N.S.) 11(1): 227-240 (July 1984).
[3] R. Fueter, Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 8 (1935 - 1936) 371-378.