Real Algebra Research Articles

Four-Dimensional Real Third-Power Associative Division Algebras

We study third-power associative division algebras A over a field 𝕂 of characteristic different from 2. Those algebras having dimension ≤2 are commutative. When 𝕂 is the field ℝ of real numbers, those algebras having dimension 4 are power-commutative in each of the following two cases:A contains a central element;A satisfies the additional identity (x, x3, x) = 0.

The QRD and SVD of matrices over a real algebra

Recent work in the field of signal processing has shown that the singular value decomposition of a matrix with entries in certain real algebras can be a powerful tool. In this article we show how to generalise the QR decomposition and SVD to a wide class of real algebras, including all finite-dimensional semi-simple algebras, (twisted) group algebras and Clifford algebras. Two approaches are described for computing the QRD/SVD: one Jacobi method with a generalised Givens rotation, and one based on the Artin–Wedderburn theorem.

On realization of effect algebras

Abstract A well known fact is that there is a finite orthomodular lattice with an order determining set of states which is not order embeddable into the standard quantum logic, the lattice L(𝓗) of all closed subspaces of a separable complex Hilbert space. We show that a finite generalized effect algebra is order embeddable into the standard effect algebra E(𝓗) of effects of a separable complex Hilbert space iff it has an order determining set of generalized states iff it is order embeddable into the power of a finite MV-chain. As an application we obtain an algorithm, which is based on the simplex algorithm, deciding whether such an order embedding exists and, if the answer is positive, constructing it.

On the Zeros of the Slice Bergman Kernels for the Upper Half Space and the Unit Ball on Quaternions and Clifford Algebras

In the complex analysis, it is a well-known problem to determine whether the Bergman kernel for a given domain has zeros or not. The Bergman kernels for the upper half plane and the unit ballare zero-free. We prove that this phenomenon also holds in the setting of quaternions and the real Clifford algebras.

Integrability and conformal data of the dimer model

The central charge of the dimer model on the square lattice is still being debated in the literature. In this paper, we provide evidence supporting the consistency of a description. Using Lieb’s transfer matrix and its description in terms of the Temperley–Lieb algebra at , we provide a new solution of the dimer model in terms of the model of critical dense polymers on a tilted lattice and offer an understanding of the lattice integrability of the dimer model. The dimer transfer matrix is analyzed in the scaling limit, and the result for is expressed in terms of fermions. Higher Virasoro modes are likewise constructed as limits of elements of and are found to yield a realization of the Virasoro algebra, familiar from fermionic bc ghost systems. In this realization, the dimer Fock spaces are shown to decompose, as Virasoro modules, into direct sums of Feigin–Fuchs modules, themselves exhibiting reducible yet indecomposable structures. In the scaling limit, the eigenvalues of the lattice integrals of motion are found to agree exactly with those of the conformal integrals of motion. Consistent with the expression for obtained from the transfer matrix, we also construct higher Virasoro modes with c = 1 and find that the dimer Fock space is completely reducible under their action. However, the transfer matrix is found not to be a generating function for the c = 1 integrals of motion. Although this indicates that Lieb’s transfer matrix description is incompatible with the c = 1 interpretation, it does not rule out the existence of an alternative, c = 1 compatible, transfer matrix description of the dimer model.

Riesz and beta-Riesz distributions

This article derives several properties of the Riesz distributions, such as their corresponding Bartlett decompositions, the inverse Riesz distributions and the distribution of the generalised variance for real normed division algebras. In addition, introduce a kind of generalised beta distribution termed beta-Riesz distribution for real normed division algebras. Two versions of this distributions are proposed and some properties are studied.

Metric and Involution Scores of Clifford Algebras

The notion of a metric score is introduced as a separate tally of the total number of standard (Grassmann) basis elements spanning a Clifford algebra \({C\ell_{p,q}}\) that square to + 1 and −1. A closed-form expression is derived for any given vector space dimension n = p+q. This is then generalized to reversion and Clifford-conjugation. A central application is that two real Clifford algebras are isomorphic if and only if their metric scores are identical.

Extension of algebras and its applications to the real Clifford algebras

Extension of algebras can be constructed through various manners. In this paper, we introduce a new way of construction to extend a given algebra of dimension , which is isomorphic to the real Clifford algebra , to another algebra of dimension over the real numbers. And then, we generalize the construction over a -grading algebra. Moreover, we will show the relation between the new constructed algebras and Clifford algebras.

Two-sided coupled generalized Sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices

In this paper, we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices: Ai∈Cpi×ti, Bi∈Csi×qi, Ci∈Cpi×ti+1, Di∈Csi+1×qi, and Ei∈Cpi×qi (i=1,2,3). We show that from this simultaneous decomposition we can derive some necessary and sufficient conditions for the existence of a solution to the system of two-sided coupled generalized Sylvester matrix equations with four unknowns AiXiBi+CiXi+1Di=Ei (i=1,2,3). Apart from proving an expression for the general solutions to this system, we derive the range of ranks of these solutions using the ranks of the given matrices Ai, Bi, Ci, Di, and Ei. We provide some numerical examples to illustrate our results. Moreover, we present a similar approach to consider the simultaneous decomposition for 5k matrices and the system of k two-sided coupled generalized Sylvester matrix equations with k+1 unknowns AiXiBi+CiXi+1Di=Ei (i=1,…,k, k≥4). The main results are also valid over the real number field and the real quaternion algebra.

Singular invariant structures for Lie algebras admitted by a system of second-order ODEs

Systems of second-order ordinary differential equations (ODEs) arise in mechanics and have several applications. Differential invariants play a key role in the construction of invariant differential equations as well as the classification of a system of ODEs. Like regular invariants, singular invariant structures also possess an important role in the algebraic analysis of a system of ODEs. In this work, singular invariant equations for a system of two second-order ODEs admitting four-dimensional Lie algebras are investigated. Moreover, by using these singular invariant equations, canonical forms for systems of two second-order ODEs are constructed. Furthermore, it is observed that the same Lie algebra admitted by a system of second-order ODEs has different type of realizations some of which are related to regular invariants and some lead to singular invariant equations. Thus realizations of four-dimensional Lie algebras are associated with a regular invariant manifold as well as to a singular invariant manifold defined by a system of second-order ODEs. In addition, a case of a Lie algebra with realization resulting in a conditional singular invariant structure is presented and those cases of singular invariant equations are discussed which do not form a system of second-order ODEs. An integration procedure for the invariant representation of these canonical forms are presented. Illustrative examples are presented with physical applications to mechanics.

Codes Over Subsets of Algebras Obtained by the Cayley–Dickson Process

In this paper, we define binary block codes over subsets of real algebras obtained by the Cayley–Dickson process and we provide an algorithm to obtain codes with a better rate. This algorithm offers more flexibility than other methods known until now.

Fermionic Quantum Orthogonalizations I

We generalize classical orthogonalization procedures from real linear algebra to the setting of fermionic quantum (FQ) operations. In the case of the Gram-Schmidt orthogonalization procedure, the generalization is easy. This, however, helps to obtain general information regarding FQ operations, and to generalize the symmetric orthogonalization procedure.

Realizations of Galilei algebras

All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations.

Recurrence approach and higher rank cubic algebras for the N-dimensional superintegrable systems

By applying the recurrence approach and coupling constant metamorphosis, we construct higher order integrals of motion for the Stackel equivalents of the N-dimensional superintegrable Kepler–Coulomb model with non-central terms and the double singular oscillators of type (). We show how the integrals of motion generate higher rank cubic algebra with structure constants involving Casimir operators of the Lie algebras L1 and L2. The realizations of the cubic algebras in terms of deformed oscillators enable us to construct finite dimensional unitary representations and derive the degenerate energy spectra of the corresponding superintegrable systems.

Wonder of sine-Gordon $Y$-systems

The sine-Gordon Y-systems and the reduced sine-Gordon Y-systems were introduced by Tateo in the 90's in the study of the integrable deformation of conformal field theory by the thermodynamic Bethe ansatz method. The periodicity property and the dilogarithm identities concerning these Y-systems were conjectured by Tateo, and only a part of them have been proved so far. In this paper we formulate these Y-systems by the polygon realization of cluster algebras of types A and D, and prove the conjectured periodicity and dilogarithm identities in full generality. As it turns out, there is a wonderful interplay among continued fractions, triangulations of polygons, cluster algebras, and Y-systems.

Symmetric semi-algebraic sets and non-negativity of symmetric polynomials

The question of how to certify the non-negativity of a polynomial function lies at the heart of Real Algebra and has important applications to optimization. Building on work by Choi, Lam, and Reznick [4], as well as Harris [5], Timofte [9] provided a remarkable method to efficiently certify non-negativity of symmetric polynomials. In this note we slightly generalize Timofte's statements and investigate families of polynomials that allow special representations in terms of power-sum polynomials. We also recover the consequences of Timofte's original statements as a corollary.

Higher Sugawara Operators for the Quantum Affine Algebras of Type A

We give explicit formulas for the elements of the center of the completed quantum affine algebra in type $A$ at the critical level which are associated with the fundamental representations. We calculate the images of these elements under a Harish-Chandra-type homomorphism. These images coincide with those in the free field realization of the quantum affine algebra and reproduce generators of the $q$-deformed classical $W$-algebra of Frenkel and Reshetikhin.

Bott type periodicity for the higher octonions

We study the series of complex nonassociative algebras On and real nonassociative algebras $O_{p,q}$ introduced in [10]. These algebras generalize the classical algebras of octonions and Clifford algebras. The algebras $O_{n}$ and $O_{p,q}$ with $p + q = n$ have a natural $Z_n^2$-grading, and they are characterized by cubic forms over the field $Z_2$. We establish a periodicity for the algebras $O_{n}$ and $O_{p,q}$ similar to that of the Clifford algebras Cln and $Cl_{p,q}$.

A note on the real division algebras with non-trivial derivations

We show that every 4-dimensional real division algebra having non-zero derivations is obtained from the real algebra C by a special kind of duplication process accompanied by an appropriate isotopy. Also this process produces new examples in dimension 8. Mathematics Subject Classification: 17A35, 17A36

Drinfeld Realization of Quantum Twisted Affine Algebras via Braid Group

The Drinfeld realization of quantum affine algebras has been tremendously useful since its discovery. Combining techniques of Beck and Nakajima with our previous approach, we give a complete and conceptual proof of the Drinfeld realization for the twisted quantum affine algebras using Lusztig’s braid group action.