Schedule

In this talk we will present Octonions as group graded algebra obtained from quaternions using Cayley - Dickson Process. Following this aproach some properties of this algebra are revisited.

Inspired by Hamilton's suggestion that quaternions play a fundamental role in the description of reality, possible analogies to physical quantities are explored. Visualization in three dimensions is an important tool for understanding. Thus the stereographic projection is used for an intuitive picture of SO(4), an important transformation when discussing quaternions.

We consider generalized Appell polynomials, a special class of monogenic polynomials that has been introduced in 2006 by using several monogenic hypercomplex variables, cf. [1]. In the past, hypercomplex Appell polynomials successfully have been used, for instance, in 3D-quasi-conformal mappings, in the development of new analytic methods in elasticity or the construction of generalized Hermite, Laguerre, Chebyshev as well as Bernoulli, Euler and other polynomials, to mention only few of their applications.
Starting with a general form of the differential in $(n+1)$-real variables (cf. [2]) we clarify the non-formal reasons why also non-monogenic variables can be used in representations of those monogenic generalized Appell polynomials. The advantage of our approach lies in the fact that immediately follow multivariate generating functions (see [3]) of their corresponding coefficient sets.
The underlying research carried out together with Isabel Cação, Irene Falcão and Graça Tomaz was inspired by our results on intrinsic fundamental relations of continuous (function theoretic) and discrete (enumerative combinatorial) hypercomplex analysis in $\mathbb{R}^{n+1}$.
[1] H. R. Malonek, I. Cação, M. I. Falcão, and G. Tomaz, Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One. In: A. Karapetyants, V. Kravchenko, E. Liflyand (eds) Modern Methods in Operator Theory and Harmonic Analysis. OTHA 2018. Springer, Cham. Springer Proceedings in Mathematics & Statistics. 291, (2019), 93-115
[2] B. Goldschmidt, A theorem about the representation of linear combinations in Clifford algebras. Beiträge Algebra Geom. 13 (1982), 21-24.
[3] E.D. Rainville, Special Functions, Macmillan, New York, 1960.

In this talk we discuss a new class of quaternions namely, balancing quaternions and some quaternions related to them. We establish several identities concerning those quaternions.

Let O be the ring of integers of an algebraic number field K; let E(n,O) be the subgroup of SL(n,O) generated by elementary matrices. Unless n = 2, K is imaginary quadratic, and O is non-Euclidean, E(n,O) = SL(n,O); otherwise, it is an infinite-index, non-normal subgroup. This bizarre behavior was termed "unreasonable slightness" by Bogdan Nica in 2011. In this talk, we'll talk about analogs of this result where we replace the algebraic number field K with a quaternion algebra H, which will allow us to show that unreasonable slightness occurs in at least two other settings.

A convenient method of visualization of a binary algebraic relation $R$ is to define a corresponding graph. Its vertices represent elements or their equivalence classes in an algebraic structure under consideration, and there is an edge from $x$ to $y$ if and only if $xRy$. The most popular relation graphs of various algebras are commutativity, orthogonality, and zero divisor graphs.
The talk is devoted to orthogonality and commutativity graphs of the real sedenion algebra $\mathbb{S}$. We show that any pair of zero divisors in $\mathbb{S}$ produces a double hexagon in its orthogonality graph $\Gamma_O(\mathbb{S})$. The set of vertices of a double hexagon can be extended to a basis of $\mathbb{S}$ which has a convenient multiplication table. We also study the connected components of $\Gamma_O(\mathbb{S})$ and show that the diameter of each connected component equals $3$. We then establish the bijection between the connected components of $\Gamma_O(\mathbb{S})$ and lines in the imaginary part of the octonions. Finally, we consider the commutativity graph $\Gamma_C(\mathbb{S})$ and discover that all elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies between $3$ and $4$.

In this paper we intend to describe generalized Lie-type derivations using, among other things, a generalization for alternative algebras of the result: “If $F:A\to A$ is a generalized Lie n-derivation associated with a Lie n-derivation $D$, then a linear map $H=F-D$ satisfies $H(p_n(x_1,x_2,\ldots ,x_n)) =p_n(H(x_1),x_2,\ldots ,x_n)$ for all $x_1,x_2,\ldots ,x_n\in A$". Thus, if $A$ is a unital alternative algebra with a nontrivial idempotent $e_1$ satisfying certain conditions, then a generalized Lie-type derivation $F : A \rightarrow A$ is of the form $F(x) = \lambda x + \Xi(x)$ for all $x \in A$ , where $\lambda \in Z(A)$ and $\Xi : A \rightarrow A$ is a Lie-type derivation.

In this talk we present some new fundamental aspects in octonionic function theory that are essentially different from quaternionic and Clifford analysis. While any function theoretical tools like for example the Cauchy integral formula or the particular argument principle that we developed in [3] can still be transferred from Clifford analysis to octonionic analysis, the lack of associativity destroys the modular structure of the set of octonionic monogenic functions which however is an absolute key ingredient in the study of analogues for reproducing kernel Hilbert modules. Here, one has to be very careful in the consideration of inner products, since fundamental theorems like the Riesz representation theorem or the existence of an adjoint operator relies on a Cauchy-Schwarz inequality. The latter inequality however does not hold for granted in the context of octonionvalued inner products. To introduce generalizations of octonionic monogenic Bergman and Hardy modules one has to be very careful. One possibility is to work with real-valued inner products on the sets of L2-integrable monogenic functions instead and use corresponding functional analytic results in a context where they are applicable and see how they can be lifted to the octonionic setting. We present some explicit formulas for the Bergman and Szegö kernels of some domains (see also [4]) and an adjoint of the Cauchy transform together with some octonionic Kerzman-Stein operators, where orthogonality is understood in the sense of a properly chosen real-valued inner product, see [5]. We also discuss as a simple application a link to the Hilbert Riesz transform in R8. These results show very nicely how much different the theory octonionic monogenic functions is from Clifford analysis.
In the second part of the talk we present an application of octonionic monogenic functions to class eld theory that cannot be constructed with Clifford monogenic functions, either - giving another essential example for octonionic function theory for being essentially di erent from Clifford analysis. The closed multiplicative structure of octonions namely admit the construction of lattices in R8 that are closed under multiplication in R8 and one can 1 consider corresponding left and right ideals. They provide us with the natural analogue of lattices with complex multiplications to the eight dimensional setting. Now the division values of the normalized classical complex analytic Weierstrass-function are elements of algebraic Galois eld extensions of imaginary quadratic number elds when taking as period lattice a lattice with complex multiplication. Now the octonionic monogenic analogues of the Weierstrass-function turns out to play a similar role for tri-quadratic number elds instead, cf. [1]. This is another essentially di erent feature between octonionic monogenic functions and Cli ord monogenic functions in R8. Finally we round o with some remarks on conformality and Mobius transformations, see [2].
The work partially contains joint work with Prof. Dr. Denis Constales from Ghent University.
[1] R.S. Kraußhar. Function Theories in Cayley-Dickson algebras and Number Theory, Milan Journal of Mathematics 89 No. 1 (2021), DOI: 10.1007/s00032-021-00325-y, 26pp.
[2] R.S. Kraußhar. Conformal mappings revisited in the octonions and Clifford algebras of arbitrary dimension, Advances in Applied Cli ord Algebras 30:36 (2020), 14pp.
[3] R.S. Kraußhar. Differential topological aspects in octonionic monogenic function theory, Advances in Applied Cli ord Algebras 30:51 (2020), 25pp.
[4] R.S. Kraußhar. Recent and new results on octonionic Bergman and Szegö kernels, to appear in: Mathematical Methods in the Applied Sciences (2021), https://doi.org/10.1002/mma.7316, 14pp.
[5] D. Constales and R.S. Kraußhar. Octonionic Kerzman-Stein Operators, submitted for publication. Preprint available at: http://arxiv.org/abs/2012.11925, 20 pp.

I considered definition and properties of polynomial in non\Hyph commutative algebra. I considered division of polynomials with remainder. There exists polynomial which has finite, infinite or empty set of roots.

Using the standard solutions in terms of Mathieu functions for the Helmholtz equation $(\Delta+\lambda^2)F=0$ for a function $F(x,y)$ converted to elliptical coordinates $f(\xi,\eta)$, we construct $\mathbb{R}^3$-valued functions of two real variables which are solutions of $\mbox{Ker}(D+\lambda)f=0$. These functions form a complete orthogonal set in the appropriate $L^2$ space on an elliptical domain.

A contragenic function in a domain $\Omega \subseteq \mathbb{R}^3$ is a reduced-quaternion-valued (i.e., the last coordinate function is zero) harmonic function, which is orthogonal in $L_2(\Omega)$ to all monogenic functions and their conjugates. Contragenicity is not a local property. We investigate the relationships between standard orthogonal bases of harmonic and contragenic functions for one domain to another via computational formulas for spheroids of different eccentricities, showing that there are common contragenic functions to all spheroids of all eccentricities.

We discuss how to use the association of quaternions with rotations in $\mathbb{R}^3$ to prove various theorems in spherical geometry.