Composition Algebras Research Articles

Two lives: Compositions of unimodular rows

The paper lays the foundation for the study of unimodular rows using Spin groups. We show that elementary orbits of unimodular rows (of any length n≥3) are equivalent to elementary Spin orbits on the unit sphere. (This bijection is proved over all commutative rings.) In the special case n=3, this leads to an interpretation of the Vaserstein symbol using Spin groups.In addition, we introduce a new composition law that operates on certain subspaces of the underlying quadratic space (using the multiplication in composition algebras). In particular, the special case of split-quaternions leads to the composition of unimodular rows (discovered by L. Vaserstein and later generalized by W. van der Kallen). Strikingly, with this approach, we now see the possibility of new orbit structures not only for unimodular rows (using octonion multiplication) but also for more general quadratic spaces.

Fine gradings on Kantor systems of Hurwitz type

We give a classification up to equivalence of the fine group gradings by abelian groups on the Kantor pairs and triple systems associated with Hurwitz algebras (i.e., unital composition algebras), under the assumption that the base field is algebraically closed of characteristic different from 2. The universal groups and associated Weyl groups are computed. We also determine, in the case of Kantor pairs, the induced (fine) gradings on the associated Lie algebras given by the Kantor construction.

Vector product and composition algebras in braided monoidal additive categories

This is an account of some work of Markus Rost and his students Dominik Boos and Susanne Maurer. We adapt it to the braided monoidal setting.

The Hopf monoid of orbit polytopes

Many families of combinatorial objects have a Hopf monoid structure. Aguiar and Ardila introduced the Hopf monoid of generalized permutahedra and showed that it contains various other notable combinatorial families as Hopf submonoids, including graphs, posets, and matroids. We introduce the Hopf monoid of orbit polytopes, which is generated by the generalized permutahedra that are invariant under the action of the symmetric group. We show that modulo normal equivalence, these polytopes are in bijection with integer compositions. We interpret the Hopf structure through this lens, and we show that applying the first Fock functor to this Hopf monoid gives a Hopf algebra of compositions. We describe the character group of the Hopf monoid of orbit polytopes in terms of noncommutative symmetric functions, and we give a combinatorial interpretation of the basic character and its polynomial invariant.

The Riemann Curvature Tensor and Higgs Scalar Field within CAM Theory

The composition algebra based methodology (CAM) (Wolk in Pap Phys 9:090002, 2017, Phys Scr 94:025301, 2019, Adv Appl Clifford Algebras 27(4):3225, 2017, J Appl Math Phys 6:1537, 2018, Phys Scr 94:105301, 2019) has previously been shown to generate the pre-Higgs Standard Model Lagrangian. In this paper the symmetry of general covariance is incorporated into CAM. The Riemann curvature tensor thereby arises, from which gravity-field Lagrangians are constructed. A Higgs-like scalar field coupled to the spacetime metric tensor also manifests.

Renominative logics with extended renomination, equality and predicate complement

A new class of program-oriented logical formalisms is investigated – renominative logics with extended renominations, equality predicates, and predicate complement composition. Composition algebras and languages of such logics are described; their semantic properties are investigated. For these logics, a number of logical consequence relations are proposed and investigated, in particular, the logical consequence relations with undefinedness conditions. Properties of these relations form the semantic basis for further construction of sequent-type calculi for the proposed logics.

Split Octonionic Cauchy Integral Formula

Cauchy integral formulas for the quaternions have been known since 1926. This result has previously been extended to the octonions and split quaternions. These techniques are extended to the split octonions. Thus Cauchy integral formulas are now known for all composition algebras over the real numbers.

Gradings on tensor products of composition algebras and on the Smirnov algebra

We give classifications of group gradings, up to equivalence and up to isomorphism, on the tensor product of a Cayley algebra C and a Hurwitz algebra over a field of characteristic different from 2. We also prove that the automorphism group schemes of C⊗n and Cn are isomorphic.On the other hand, we prove that the automorphism group schemes of a Smirnov algebra T(C) (a 35-dimensional simple exceptional structurable algebra constructed from a Cayley algebra C) and C are isomorphic. This is used to obtain classifications, up to equivalence and up to isomorphism, of the group gradings on Smirnov algebras.

The division algebraic constraint on gauge mediated proton decay

The composition algebra based methodology—previously shown to generate the pre-Higgs Standard Model Lagrangian in intrinsically left-chiral and local gauge invariant form—places a strict constraint on gauge mediated proton decay which phenomenologically differentiates it from the Yang–Mills formalism.

Першопорядковi композиційно-номінативні логіки з предикатами слабкої та строгої рівності

Development of the new software-oriented logical formalisms is a topical problem. The paper introduces lo­gics of partial predicates with predicate complement and equality predicates, we denote them LCE. They ex­tend logics of quasiary predicates with equality and logics with predicate complement. The composition of the predicate complement is used in Floyd-Hoare pro­­gram logics’ extensions on the class of partial predi­cates. We define predicates of weak equality and of strong equality. Thus, LCE with predicates of weak equality (denoted by LCEw) and LCE with predicates of strong equality (denoted by LCEs) can be specified. LCE can be studied on the first order and renominative levels. We consider composition algebras of LCE, investigate properties of their compositions and describe first order languages of such logics. We concentrate on the properties related to the equality predicates and the composition of the predicate complement. Various variants of logical consequence relations for the first order LCE are introduced and studied: P |= T , P |= F , R |= T , R |= F , P |= TF , R |= TF , P |= IR . In particular, we obtained that LCEw are somewhat degenerate, as for them all the relations are incorrect except for the irrefutability logical consequence relation under the conditions of undefinedness |= IR ^ . At the same time, all of the listed relations are correct for LCEs. Properties of the logical consequence relations are the semantic basis for con­struction of the respective calculi of sequential type. Further investigation of logical consequence relations for LCE includes adding the conditions of undefined­ness and constructing the corresponding sequent calcu­li; it is planned to be displayed in the forthcoming ar­ticles. Problems in programming 2019; 3: 28-44

Відношення логічного наслідку в логіках часткових предикатів з композицією предикатного доповнення

In this paper we study software-oriented logics of partial predicates with new special non-monotonic operation (composition) of the predicate complement. We denote these logics by LC and composition of the predicate complement by . Such operations are used in various versions of the Floyd-Hoare program logic with partial pre- and post-conditions. We describe first order composition algebras and LC languages. For LC, a number of logical consequence relations ( Pc |= T , Pc |= F , Rc |= T , Rc |= F , Pc |= TF , Rc |= TF , P с |= IR ) and logical consequence relations under the conditions of undefinedness ( P |= T ^ , P |= F ^ , R |= T ^ , R |= F ^ , P |= TF ^ , R |= TF ^ ) are specified. Properties of the defined relations are investigated, differences and the relationship between them are given. For the introduced relations, we describe the conditions for their guaranteed presence, the decomposition conditions for formulas and the properties of quantifier elimination. The theorem of elimination of the conditions of undefinedness for the relations |= T ^ and |= F is proved. Thus, the relations P |= T ^ , P |= F ^ , R |= T ^ and R |= F ^ can be expressed by Pc |= T , Pc |= F , Rc |= T and Rc |= F respectively. However, it is shown that |= IR ^ cannot be expressed by P с |= IR . Moreover, it is impossible to define correctly the decomposition conditions for formulas for P с |= IR . Properties of decomposition conditions for formulas are different for the relations |= T and |= F , therefore properties of decomposition and equivalent transformations must be specified indirectly through the corresponding properties of |= T and |= F . First order sequent calculi for the introduced logical consequence relations for LC and logical consequence relations under the conditions of undefinedness will be constructed in in the forthcoming articles. Problems in programming 2019; 3: 11-27

Associative systems generated by symmetric elements of a system with involution

In this paper we study associative systems (algebras, pairs or triples) with involution, and we extend to them well known results of Herstein (see, for example [5]). In particular, we show that any ⁎-simple associative algebra is either generated by traces or it is a composition algebra for some involution. In the case of ⁎-simple associative pairs or triple systems, they are either generated by traces, or their sets of symmetric elements satisfy a certain Clifford homotope PI. Moreover, we show that the associative subpair generated by any ample subspace inherits ⁎-regularity conditions such as ⁎-primeness, ⁎-simplicity and ⁎-primitivity from an associative pair, and that the Baer, Properly Nilpotent, Locally Nilpotent and Jacobson radicals of the associative subpair generated by any ample subspace are directly inherited from the same radical of the whole pair. We end up with a close description of ample subspaces for ⁎-simple algebras, pairs or triple systems generated by traces.

Exceptional sequences and Drinfeld double Hall algebras

Let A be a finitary hereditary abelian category and D(A) be its reduced Drinfeld double Hall algebra. By giving explicit formulas in D(A) for left and right mutations, we show that the subalgebras of D(A) generated by exceptional sequences are invariant under mutation equivalences. As an application, we obtain that if A is the category of finite dimensional modules over a finite dimensional hereditary algebra, or the category of coherent sheaves on a weighted projective line, the double composition algebra of A is generated by any complete exceptional sequence. Moreover, for the Lie algebra case, we also have paralleled results.

Order 3 Elements in G2 and Idempotents in Symmetric Composition Algebras

AbstractOrder three elements in the exceptional groups of type G2 are classiûed up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras.Over an algebraically closed field, there are two conjugacy classes of order three elements in G2 in characteristic not 3 and four of them in characteristic 3. The centralizers in characteristic 3 fail to be smooth for one of these classes.

Analysis of relative abundances with zeros on environmental gradients: a multinomial regression model.

Ecologists often analyze relative abundances, which are an example of compositional data. However, they have made surprisingly little use of recent advances in the field of compositional data analysis. Compositions form a vector space in which addition and scalar multiplication are replaced by operations known as perturbation and powering. This algebraic structure makes it easy to understand how relative abundances change along environmental gradients. We illustrate this with an analysis of changes in hard-substrate marine communities along a depth gradient. We fit a quadratic multivariate regression model with multinomial observations to point count data obtained from video transects. As well as being an appropriate observation model in this case, the multinomial deals with the problem of zeros, which often makes compositional data analysis difficult. We show how the algebra of compositions can be used to understand patterns in dissimilarity. We use the calculus of simplex-valued functions to estimate rates of change, and to summarize the structure of the community over a vertical slice. We discuss the benefits of the compositional approach in the interpretation and visualization of relative abundance data.

Octonions, Exceptional Jordan Algebra and The Role of The Group $$F_4$$ F 4 in Particle Physics

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra J. The automorphism group F_4 of J and its maximal Borel - de Siebenthal subgroups are studied in detail and applied to the classification of fundamental fermions and gauge bosons. Their intersection in F_4 is demonstrated to coincide with the gauge group of the Standard Model of particle physics. The first generation's fundamental fermions form a basis of primitive idempotents in the euclidean extension of the Jordan subalgebra JSpin_9 of J.

Isotopes of Hurwitz Algebras

We study the class of all algebras that are isotopic to a Hurwitz algebra. Isomorphism classes of such algebras are shown to correspond to orbits of a certain group action. A complete, geometrically intuitive description of the category of isotopes of Hamilton's quaternions is given. As an application, we demonstrate how some results concerning the classification of finite-dimensional composition algebras can be deduced from our general results.

On Orthogonal Tensors and Best Rank-One Approximation Ratio

As is well known, the smallest possible ratio between the spectral norm and the Frobenius norm of an $m \times n$ matrix with $m \le n$ is $1/\sqrt{m}$ and is (up to scalar scaling) attained only by matrices having pairwise orthonormal rows. In the present paper, the smallest possible ratio between spectral and Frobenius norms of $n_1 \times \dots \times n_d$ tensors of order $d$, also called the best rank-one approximation ratio in the literature, is investigated. The exact value is not known for most configurations of $n_1 \le \dots \le n_d$. Using a natural definition of orthogonal tensors over the real field (resp., unitary tensors over the complex field), it is shown that the obvious lower bound $1/\sqrt{n_1 \cdots n_{d-1}}$ is attained if and only if a tensor is orthogonal (resp., unitary) up to scaling. Whether or not orthogonal or unitary tensors exist depends on the dimensions $n_1,\dots,n_d$ and the field. A connection between the (non)existence of real orthogonal tensors of order three and the classical Hurwitz problem on composition algebras can be established: existence of orthogonal tensors of size $\ell \times m \times n$ is equivalent to the admissibility of the triple $[\ell,m,n]$ to the Hurwitz problem. Some implications for higher-order tensors are then given. For instance, real orthogonal $n \times \dots \times n$ tensors of order $d \ge 3$ do exist, but only when $n = 1,2,4,8$. In the complex case, the situation is more drastic: unitary tensors of size $\ell \times m \times n$ with $\ell \le m \le n$ exist only when $\ell m \le n$. Finally, some numerical illustrations for spectral norm computation are presented.

Web Service Interaction Modeling and Verification Using Recursive Composition Algebra

The design principle of composability among Web services is one of the most crucial reasons for the success and popularity of Web services. However, achieving error-free automatic Web service composition is still a challenge. In this paper, we propose a recursive composition based modeling and verification technique for Web service interaction. The application of recursive composition over a Web service with respect to a given set of Web services yields a recursive composition interaction graph (RCIG). In order to capture the requirement specifications of a Web service interaction scenario, we propose recursive composition specification language (RCSL) as a requirement specification language. Further, we employ the proposed RCIG as an interpretation model to interpret the semantics of a RCSL formula. Our verification technique is based on the generation and analysis of all possible interaction patterns. Performance evaluation results, provided in this paper, show that our proposition is implementable for the real world applications. The key advantages of the proposed approach are: (i) it does not require explicit system modeling as in model checking based approaches, (ii) it captures primitive characteristics of Web service interaction patterns, such as recursive composition, sequential and parallel flow, etc, and (iii) it supports automatic composition of services.

Ringel–Hall algebra construction of quantum Borcherds–Bozec algebras

We give the Ringel–Hall algebra construction of the positive half of quantum Borcherds–Bozec algebras as the generic composition algebras of quivers with loops.