Real Algebra Research Articles

On the Hausdorff Analyticity for Quaternion-Valued Functions

We consider the concept of the Hausdorff analyticity for functions ranged in real algebras and the corresponding notion of the Hausdorff derivative. Both apply to the real algebra $$\mathbb {H}$$ of Hamilton’s quaternions. The main aim of the work is to compare them with the well-known classes of $$\mathbb {H}$$ -valued functions which have their own definitions of the derivative.

Factorization results for left polynomials in some associative real algebras: State of the art, applications, and open questions

We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put particular emphasis on factorization results for quaternion, dual quaternion and split quaternion polynomials. A general algorithm ensures existence of a factorization for generic polynomials over division rings but we also consider factorizations for non-division rings. We explain the current state of the art, present some new results and provide examples and counter examples.

Unitary group approach for effective potentials in 2D systems: Application to carbon suboxide C3O2

A U(3) algebraic approach is proposed to describe 2D systems for effective potentials. Our approach is based on the 2D vibron model where the addition of a scalar boson is introduced into the 2D harmonic oscillator space. As a crucial ingredient of our approach an algebraic realization of the coordinates and momenta is obtained. This feature provides the tools to obtain the algebraic representation of a 2D Hamiltonian in terms of similitude transformation of a diagonal matrix. As an application of our approach the rotation-bending energy levels of carbon suboxide C3O2 are described in good agreement with experimental data.

Structure of the low-lying positive-parity states in Sm154

The proton-neutron symplectic model with Sp(12,$R$) dynamical algebra is applied to the simultaneous description of the microscopic structure of the low-lying states of the lowest ground, $\ensuremath{\beta}$, and $\ensuremath{\gamma}$ bands in $^{154}\mathrm{Sm}$. For this purpose, the model Hamiltonian is diagonalized in a U(6)-coupled basis, restricted to the state space spanned by the fully symmetric U(6) irreps. A good description of the energy levels of the three bands under consideration as well as the intraband $B(E2)$ transition strengths between the states of the ground band is obtained without the use of an effective charge. The microscopic structure of low-lying collective states in $^{154}\mathrm{Sm}$ shows that there are no admixtures from the higher shells and hence shows the presence of a very good U(6) dynamical symmetry. It is also shown that, in contrast to the Sp(6,$R$) case, the lowest excited bands, e.g., the $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ bands, naturally appear together with the ground band within a single Sp(12,$R$) irreducible representation. The obtained results is given a simple geometrical multiphonon interpretation, based on the algebraic realization of the coupled two-rotor picture, which in turn suggests an interpretation of the low-lying excited bands as relative proton-neutron excitations of the two-component nuclear system, governed by the ${Q}_{p}\ifmmode\cdot\else\textperiodcentered\fi{}{Q}_{n}$ interaction.

Idempotent Geometry in Generic Algebras

Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, Honor of Wolfgang SprßigTrends Math, Birkhäuser/Springer Basel AG, Basel, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.

Invariant functions of vector field realizations of Lie algebras and some applications to representation theory and dynamical systems

Realizations of Lie algebras by means vector fields associated to a linear representation and their corresponding invariant functions are inspected from the perspective of the embedding problem of Lie algebras and the branching rules for the subduction of representations. Some applications concerning the construction of dynamical systems with prescribed Lie algebras of point symmetries and the symmetry breaking with respect to embeddings of algebras into (N) are discussed.

SU(1,1) solution for the Dunkl–Coulomb problem in two dimensions and its coherent states

We study the radial part of the Dunkl–Coulomb problem in two dimensions and show that this problem possesses the SU(1,[Formula: see text]1) symmetry. We introduce two different realizations of the su(1,[Formula: see text]1) Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and eigenfunctions. For the first algebra realization, we apply the Schrödinger factorization to the radial part of the Dunkl–Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, one of them proportional to the Hamiltonian of the radial Schrödinger equation. Finally, we use the SU(1,[Formula: see text]1) Sturmian basis to construct the radial coherent states in a closed form.

Generalized discrete q-Hermite I polynomials and q-deformed oscillator

In this paper, we present an explicit realization of q-deformed Calogero-Vasiliev algebra whose generators are first-order q-difference operators related to the generalized discrete q-Hermite I polynomials recently introduced in [14]. Furthermore, we construct the wave functions and we determine the q-coherent states.

Space and time dimensions of algebras with application to Lorentzian noncommutative geometry and quantum electrodynamics

An analogy with real Clifford algebras on even-dimensional vector spaces suggests assigning an ordered pair (s, t) of space and time dimensions (or equivalently an ordered pair (m, n) of metric and KO dimensions) modulo 8 to any algebraic structure (that we call CPT corepresentation) represented over a Hilbert space by two self-adjoint involutions and an anti-unitary operator having specific commutation relations. It is shown that this assignment is compatible with the tensor product: the space and time dimensions of the tensor product of two CPT corepresentations are the sums of the space and time dimensions of its factors, and the same holds for the metric and KO dimensions. This could provide an interpretation of the presence of such algebras in PT-symmetric Hamiltonians or the description of topological matter. This construction is used to build an indefinite (i.e., pseudo-Riemannian) version of the spectral triple of noncommutative geometry, defined over a Krein space and classified by the pair (m, n) instead of the KO dimension only. Within this framework, we can express the Lagrangian (both bosonic and fermionic) of a Lorentzian almost-commutative spectral triple. We exhibit a space of physical states that solves the fermion-doubling problem. The example of quantum electrodynamics is described.

Gleason’s Problem Associated to a Real Ternary Algebra and Applications

This paper is an introduction of a new class of analytic functions defined on a ternary algebra, a three dimensional structure different from $${{\mathbb {C}}}\times {{\mathbb {R}}}$$ , i.e. a real commutative algebra given by the linear span of $$\{1,\mathbf{e},\mathbf{e}^2\}$$ , where $$\mathbf{e}\not \in {{\mathbb {C}}}$$ is a generating unit. We define a single ternary conjugate and we build a new analytic function theory, different from previous approaches, on the basis of this single conjugation (akin to the quaternionic case). We give the solution to the Gleason problem which gives rise to Fueter-type variables and study the space of rational functions in this case.

Clifford Algebras as Twisted Group Algebras and the Arf Invariant

Some connections between quadratic forms over the field of two elements, Clifford algebras of quadratic forms over the real numbers, real graded division algebras, and twisted group algebras will be highlighted. This allows to revisit real Clifford algebras in terms of the Arf invariant of the associated quadratic forms over the field of two elements, and give new proofs of some classical results.

On a Theorem by Hopf

We prove that the commutative nearly absolute valued division algebras have dimension ≤2. This extends a well-known theorem by H. Hopf asserting the same property for each commutative real division algebra of finite dimension. As an application we prove that R, C and C⋆ are the unique commutative real nearly absolute valued division algebras with norm satisfying ‖x2‖=‖x‖2.

Dynamical structure of Carrollian Electrodynamics

We present an action of ultra-relativistic electrodynamics on a flat Carroll manifold. The model exhibits a couple of physical degrees of freedom per space-point. We observe that the action of the conformal Carroll algebra on the phase space is Hamiltonian in 4 space-time dimensions. Moreover the elements of the algebra give rise to an infinite number of conserved charges and the charge algebra is an exact realization of the kinematical algebra.

A correction of the decomposability result in a paper by Meyer–Neutsch

In their paper of 1993, Meyer and Neutsch established the existence of a 48-dimensional associative subalgebra in the Griess algebra G. By exhibiting an explicit counter example, the present paper shows a gap in the proof one of the key results in Meyer and Neutsch's paper, which states that an idempotent a in the Griess algebra is indecomposable if and only its Peirce 1-eigenspace (i.e. the 1-eigenspace of the linear transformation La:x↦ax) is one-dimensional. The present paper fixes this gap, and shows a more general result: let V be a real commutative nonassociative algebra with an associative inner product, and let c be a nonzero idempotent of V such that its Peirce 1-eigenspace is a subalgebra; then, c is indecomposable if and only if its Peirce 1-eigenspace is one-dimensional. The proof of this result is based on a general variational argument for real commutative metrised algebras with inner product.

Reduction by invariants and projection of linear representations of Lie algebras applied to the construction of nonlinear realizations

A procedure for the construction of nonlinear realizations of Lie algebras in the context of Vessiot–Guldberg–Lie algebras of first-order systems of ordinary differential equations (ODEs) is proposed. The method is based on the reduction of invariants and projection of lowest-dimensional (irreducible) representations of Lie algebras. Applications to the description of parameterized first-order systems of ODEs related by contraction of Lie algebras are given. In particular, the kinematical Lie algebras in (2 + 1)- and (3 + 1)-dimensions are realized simultaneously as Vessiot–Guldberg–Lie algebras of parameterized nonlinear systems in R3 and R4, respectively.

Some results on the projective cone normed tensor product spaces over banach algebras

For two real Banach algebras $\mathbb{A}_1$ and $\mathbb{A}_2$, let $K_p$ be the projective cone in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$. Using this we define a cone norm on the algebraic tensor product of two vector spaces over the Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ and discuss some properties. We derive some fixed point theorems in this projective cone normed tensor product space over Banach algebra with a suitable example. For two self mappings $S$ and $T$ on a cone Banach space over Banach algebra, the stability of the iteration scheme $x_{2n+1}=Sx_{2n}$, $x_{2n+2}=Tx_{2n+1},\;n=0,1,2,...$ converging to the common fixed point of $S$ and $T$ is also discussed here.

The Development of Two High School Students’ Discourses on Geometric Translation in Relation to the Teacher’s Discourse in the Classroom

Research exploring student development on geometric translation in relation to classroom teaching is scarce. We used a qualitative case study design and examined two students’ development of thinking on translation in relation to the teacher’s discourse in a Turkish high school classroom through a discursive framework. We examined the teacher’s discourse through the video-taped classroom session in which he talked about translation. We examined the students’ discursive development through three video-taped task-based interviews. Finally, we compared the students’ discourses on translation with the teacher’s discourse. The teacher’s discourse on translation in the classroom was formal and based on an algebraic realization of the concept as an addition and a geometric transformation. The students adopted particular discursive elements used by the teacher right after instruction but later abandoned some of those and used their more established realizations of translation. The findings revealed the complexity of discursive development as the students continually and actively adjusted their discourses by taking into account the teacher’s discourse while also making it compatible with their own realizations of translation. We conclude that socio-cultural and discursive approaches have the potential to shed additional light on issues regarding learning and teaching of geometric translation.

Classification of involutions on graded-division simple real algebras

We classify, up to isomorphism and up to equivalence, involutions on graded-division finite-dimensional simple real (associative) algebras, when the grading group is abelian.

Comparison of realizations of Lie algebras

The notion of equivalence of Lie algebra realizations is revisited and the quantities stable under the equivalence transformations are proposed. As a result we formulate a practical algorithm that allows to establish the existence of equivalence between any two realizations of a Lie algebra. Several illustrative examples are considered.

Orthogonal-gradings on $$H^*$$H∗-algebras

We study set-gradings on proper $$H^*$$ -algebras A, which are compatible with the involution and the inner product of A, that will be called orthogonal-gradings. If A is an arbitrary $$H^*$$ -algebra with a fine grading, we obtain a (fine) orthogonal-graded version of the main structure theorem for proper arbitrary $$H^*$$ -algebras. If A is associative, we show that any fine orthogonal-grading is either a group-grading or a (non-group grading) Peirce decomposition of A respect to a family of orthogonal projections. If A is alternative, we prove that any fine orthogonal-grading is either a fine orthogonal-grading of a (proper) associative $$H^*$$ -algebra, or a $${\mathbb Z}_2^3$$ -grading of the complex octonions $${\mathbb O}$$ or a non-group grading which is a refinement of the Peirce decomposition of $${\mathbb O}$$ respect to its family of orthogonal projections. Finally, we also show that any orthogonal-grading on the real octonion division algebra is necessarily a group-grading.