Composition algebras and outer automorphisms of algebraic groups via triality

In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form n over a field k of characteristic not two, and a category arising from an action of the projective similarity group of n on certain pairs of automorphisms of the group scheme PGO⁺(n) defined over k. This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known results on composition algebras from our equivalence.     

The Number of Sylow p-Subloops in Finite Moufang Loops

We prove that if L is a finite Moufang loop and p is a Sylow prime for L then the number of Sylow p-subloops of L is congruent to one modulo p.     

Triality and Dual Equivalence Between Dirac Field and Topologically Massive Gauge Field

It is proved that there exists a vector representation of Dirac＇s spinor field and. in one sense it is equivalent to biquaternion （i.e. complexified quaternion） representation. This can be considered as a generalization of Cartan＇s idea of triality to Dirac＇s spinors. In the vector representation the first-order Dirac Lagrangian is dual-equivalent to the two-order Lagrangian of topologically massive gauge field. The potential field which corresponds to the Dirac field is obta/ned by using master （or parent） action approach. The novel gauge field is self-dual and contains both anti-symmetric Lee and symmetric Jordan structure.     

Duality, triality and complementary extremum principles in non-convex parametric variational problems with applications

This paper presents a reasonably complete duality theory anda nonlinear dual transformation method for solving the fullynonlinear, non-convex parametric variational problem inf{W(u- µ) - F(u)}, and associated nonlinear boundary valueproblems, where is a nonlinear operator, W is either convexor concave functional of p = u, and µ is a given parameter.Detailed mathematical proofs are provided for the complementaryextremum principles proposed recently in finite deformationtheory. A method for obtaining truly dual variational principles(without a dual gap and involving the dual variable p* of uonly) in n-dimensional problems is proposed. It is proved thatfor convex W(p), the critical point of the associated LagrangianL_µ(u, p*) is a saddle point if and only if the so-calledcomplementary gap function is positive. In this case, the systemhas only one dual problem. However, if this gap function isnegative, the critical point of the Lagrangian is a so-calledsuper-critical point, which is equivalent to the Auchmuty'sanomalous critical point in geometrically linear systems. Wediscover that, in this case, the system may have more than oneprimal-dual set of problems. The critical point of the Lagrangianeither minimizes or maximizes both primal and dual problems.An interesting triality theorem in non-convex systems is proved,which contains a minimax complementary principle and a pairof minimum and maximum complementary principles. Applicationsin finite deformation theory are illustrated. An open problemleft by Hellinger and Reissner is solved completely and a purecomplementary energy principle is constructed. It is provedthat the dual Euler-Lagrange equation is an algebraic equation,and hence, a general analytic solution for non-convex variational-boundaryvalue problems is obtained. The connection between nonlineardifferential equations and algebraic geometry is revealed.     

Pure Spinor Formalism for Conformal Fermion and Conserved Currents

Pure spinor formalism and non-integrable exponential factors are used for constructing the conformal-invariant wave equation and Lagrangian density for massive fermion. It is proved that canonical Dirac Lagrangian for massive fermion is invariant under induced projective conformal transformations.     

Triality and Dual Equivalence Between Dirac Field and Topologically Massive Gauge Field

It is proved that there exists a vector representation of Dirac‘s spinor field andin one sense it is equivalent to biquaternion (i.e. complexified quaternion) representation. This can be considered as a generalization of Cartan‘s idea of triality to Dirac‘s spinors. In the vector representation the first-order Dirac Lagrangian is dual-equivalent to the two-order Lagrangian of topologically massive gauge field. The potential field which corresponds to the Dirac field is obtained by using master (or parent) action approach. The novel gauge field is self-dual and contains both anti-symmetric Lee and symmetric Jordan structure.     

Direct construction of code loops

Code loops were introduced by R.L. Griess Jr. [Code loops, J. Algebra 100 (1) (1986) 224-234] and T. Hsu [Explicit constructions of code loops as centrally twisted products, Math. Proc. Cambridge Philos. Soc. 128 (2) (2000) 223-232] gave methods to construct the corresponding code loop from any given doubly even binary code; both these methods used some kind of induction. In this paper, we present a global construction of the loop, where we apply the correspondence between the concepts of Moufang loops and groups with triality.     

Finite-temperature gluodynamics with test charges in various representations

We analytically investigate the finite-temperature gluodynamics with test charges, which transform under arbitrary representations of the group SU(3), on an anisotropic lattice in the spherical model approximation. We demonstrate that at below-critical temperatures, the potential for arbitrary charges with trialities differing in sign increases linearly with the distance between the charges, while the potential for sources with zero trialities has the Debye form. The model predicts that the attractive Coulomb potential exists for all the representations, which indicates that bound states exist at all above-critical temperatures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 466–475, September, 2007.     

From wave-particle duality to wave-particle-mixedness triality: an uncertainty approach

The wave-particle duality, as a manifestation of Bohr’s complementarity, is usually quantified in terms of path predictability and interference visibility. Various characterizations of the wave-particle duality have been proposed from an operational perspective, most of them are in forms of inequalities, and some of them are expressed in forms of equalities by incorporating entanglement or coherence. In this work, we shed different insights into the nature of the wave-particle duality by casting it into a form of information conservation in a multi-path interferometer, with uncertainty as a unified theme. More specifically, by employing the simple yet fundamental concept of variance, we establish a resolution of unity, which can be interpreted as a complementarity relation among wave feature, particle feature, and mixedness of a quantum state. This refines or reinterprets some conventional approaches to wave-particle duality, and highlights informational aspects of the issue. The key idea of our approach lies in that a quantum state, as a Hermitian operator, can also be naturally regarded as an observable, with measurement uncertainty (in a state) and state uncertainty (in a measurement) being exploited to quantify particle feature and wave feature of a quantum state, respectively. These two kinds of uncertainties, although both are defined via variance, have fundamentally different properties and capture different features of a state. Together with the mixedness, which is a kind of uncertainty intrinsic to a quantum state, they add up to unity, and thus lead to a characterization of the wave-particle-mixedness complementarity. This triality relation is further illustrated by examples and compared with some popular wave-particle duality or triality relations.