Nonassociative Group Research Articles

FURTHER REMARKS ON ELEMENTARY RADICALS AND ASSOCIATED FILTERS OF IDEALS

AbstractRyabukhin showed that there is a correspondence between elementary radical classes of rings and certain filters of ideals of the free ring on one generator, analogous to the Gabriel correspondence between torsion classes of left unital modules and certain filters of left ideals of the coefficient ring. This correspondence is further explored here. All possibilities for the intersection of the ideals in a filter are catalogued, and the connections between filters and other ways of describing elementary radical classes are investigated. Some generalisations to nonassociative rings and groups are also presented.

Post-Quantum Cryptography: generalized ElGamal cipher over GF(251^8)

Post-Quantum Cryptography (PQC) attempts to find cryptographic protocols resistant to attacks by means of for instance Shor’s polynomial time algorithm for numerical field problems like integer factorization (IFP) or the discrete logarithm (DLP). Other aspects are the backdoors discovered in deterministic random generators or recent advances in solving some instances of DLP. The use of alternative algebraic structures like non-commutative or non-associative partial groupoids, magmas, monoids, semigroups, quasigroups or groups, are valid choices for these new kinds of protocols. In this paper, we focus in an asymmetric cipher based on a generalized ElGamal non-arbitrated protocol using a non-commutative general linear group. The developed protocol forces a hard subgroup membership search problem into a non-commutative structure. The protocol involves at first a generalized Diffie-Hellman key interchange and further on the private and public parameters are recursively updated each time a new cipher session is launched. Security is based on a hard variation of the Generalized Symmetric Decomposition Problem (GSDP). Working with $GF(251^8)$ a 64-bits security is achieved, and if $GF(251^{16})$ is chosen, the security rises to 127-bits. An appealing feature is that there is no need for big number libraries as all arithmetic if performed in $\Z_{251}$ and therefore the new protocol is particularly useful for computational platforms with very limited capabilities like smartphones or smartcards.

On the axioms for Sabinin algebras

Abstract The tangent space of a Lie loop, a non-associative Lie group, carries the structure of a Sabinin algebra, an algebraic concept generalizing Lie algebras. Alternatively a Sabinin algebra can be interpreted as the universal local covariant of a flat affine connection. Combining this interpretation of Sabinin algebras with a version of the parallel transport equation we prove that every Sabinin algebra structure determines and is determined by the one-sided linear terms in the Baker-Campbell-Hausdorff formula for the multiplication in the underlying loop. Based on this resultwe prove that every abstract Sabinin algebra over ℝ is the tangent algebra of a global Lie loop.

Nonassociative algebras and groups with property (T)

We extend the results of Ershov, Jaikin-Zaiprin, and Kassabov on property (T) to certain groups "coordinatized" by nonassociative algebras, thereby giving new examples of groups with property (T).

Unit Balls, Lorentz Boosts, and Hyperbolic Geometry

The bounded symmetric spaces naturally associated with the Poincare and Beltrami-Klein models of hyperbolic geometry on the open unit ball B in $${\mathbb{R}^n}$$ and with the automorphism group of biholomorphic maps on the open ball in $${\mathbb{C}^n}$$ give rise by a standard construction to specialized loop structures (nonassociative groups), which we use to define canonical metrics, called rapidity metrics. We show that this rapidity metric agrees with the classical Poincare metric resp. the Cayley-Klein metric resp. the Bergman metric. We introduce the Lorentz boost of vectors in B, which turns out to be a loop isomorphism. It induces a similarity of metrics between the rapidity metric of the Einstein or Mobius loop and the trace metric on positive definite matrices restricted to the Lorentz boosts.

Nil-algebras and infinite groups

We simplify our construction of nil-algebras by proving, for any integer d ≥ 2 and over any field $$\mathbb{K}$$ , that there exists a residually nilpotent, nonnilpotent nil-algebra over $$\mathbb{K}$$ generated by d elements. As a consequence, we obtain similar results for nonassociative algebras and groups.

Effects of Blockade of NMDA Receptors and NO Synthase on the Expression of Associative and Non-Associative Sensitization to Effects of Cocaine

After repeated administration of psychostimulant drugs, a sensitization rather than a tolerance to the behavioral effects can be observed. In our own previous studies, it was shown that both blockade of NMDA glutamate receptors and inhibition of NO synthase selectively inhibited the expression of associative but not non-associative sensitization to D-amphetamine. The present experiments were performed in order to study whether a similar selective inhibition of expression of associative sensitization to cocaine can be observed after blockade of NMDA receptors by MK-801 or inhibition of NO synthase by L-NAME. MK-801 as well as L-NAME inhibited the locomotor activity in acutely cocaine-treated rats. Both drugs did not prevent the sensitization either in the associative or the non-associative group. The results suggest that the acute locomotor effects of cocaine were inhibited by both drugs whereas both the non-associative and the associative sensitization to locomotor effects were not inhibited by blockade of NMDA receptors or inhibition of NO synthase. Accordingly, the expression of neither type of sensitization to cocaine was inhibited by any of these drugs.

BURNSIDE AND KUROSH PROBLEMS

For any integer d≥2 and over any field [Formula: see text], we construct a residually finite nonnilpotent nil algebra over [Formula: see text] generated by d elements such that any d-1 elements generate a nilpotent subalgebra. This leads to analogous results for nonassociative algebras and groups. In our proof we do not use the well-known Golod-Šafareviš lemma.

Augmentation of the Development of Behavioral Tolerance to Cannabinoid Administration through Pavlovian Conditioning

This investigation examined the effects, in female rats, of a Pavlovian conditioning paradigm on the development of tolerance to hypolocomotion induced by the cannabinoid agonist HU-210. Rats were administered HU-210 and placebo in either an associative or a nonassociative fashion. The results indicated that rats in the associative paradigm developed tolerance significantly faster than those in the nonassociative group (p < 0.03). Subsequently, once tolerance had developed, the associative group of rats was administered HU-210 and placebo in the opposite environments. There were no differences found in locomotion between the CS+ and CS– environments following administration of HU-210. However, when the placebo was administered in the CS+ environment, there was a trend towards increased activity levels (p = 0.06), suggesting withdrawal-like behavior. These findings indicate that the underlying physiological mechanisms of tolerance development in the cannabinoid system are hastened by conditioning, but that these physiological alterations are not contingent upon the associative parameters used for drug administration.

Quasigroups Connected with Clifford Groups

In Clifford groups, a nonassociative product is defined which leads to the definition of nonassociative groups. These nonassociative groups have matrix representations on the condition that the “row by column” product of two matrices is replaced by the “column by column” product. A nonassociative group of transformations connected with the Lorentz group is determined, together with its irreducible, double-valued matrix representation, whose matrices undergo the “column by column” product.

The Relativistic Noncommutative Nonassociative Group of Velocities and the Thomas Rotation

The bizarre and counterintuitive noncommutativity and nonassociativity of the relativistic composition of nonparallel admissible velocities is sometimes interpreted as a peculiarity of special theory of relativity. It is related to the fact that Lorentz acceleration transformations in three space dimensions do not form a group due to the presence of the so called Thomas rotation. The Thomas rotation turns out to be the effect that provides a means to the presentation of the set of relativistically admissible velocities as an interesting noncommutative, nonassociative group with a group operation given by the relativistic velocity composition. Interestingly, the algebraic structure induced by the Thomas rotation is not an isolated result in special relativity. It was earlier discovered by Karzel, and studied by Kerby and by Wefelscheid in a totally different context.

Axiomatic approach to the nonassociative group of relativistic velocities

The bizarre and counterintuitive noncommutativity and nonassociativity of the relativistic composition of noncollinear velocities is attributed to the presence of the Thomas rotation. The Thomas rotation, in turn, gives rise to anonassociative group structure for the set of relativistically admissible velocities. This nonstandard group structure has been observed in other contexts and, hence, merits axiomatization.

The Thomas rotation formalism underlying a nonassociative group structure for relativistic velocities

Physics is a major external source of mathematical theories. It is shown here that the Thomas rotation effect of special theory of relativity (STR) gives rise to a “group” structure for the set of relativistically admissible velocities. This group structure turns out to be noncommutative and nonassociative. The term nonassociative group is justified by two illustrative examples demonstrating that the new concept of the nonassociative group is forced on us by the study of the laws of relativistic velocities. Thomas rotation is studied in STR as an isolated notion; and the bizarre and counterintuitive noncommutativity and nonassociativity of the relativistic composition of nonparallel admissible velocities is sometimes interpreted as a peculiarity of STR. However, it turns out that the Thomas rotation plays a central role in STR, giving rise to an elegant formalism underlying the noncommutative, nonassociative “group” of relativistically admissible velocities. To demonstrate the Thomas rotation formalism and the group structure to which it gives rise let E?R9,

Associative and non-associative automorphisms of Newtonian space-time

In this article, an algebraic study of the transformations $$(x,t) \mapsto \left( {Rx + \sum\limits_{i = 0}^m {A_i t^i /i!,t + t'} } \right)$$ of Newtonian space-time onto itself is presented. The study is carried out within the framework of Eilenberg and Maclane's co-homology theory of group prolongations, a generalisation of the theory of group extensions (Eilenberg & Maclane, 1947a-d). The ‘loops’ or non-associative groups involving ‘m’ from 0 to 5 are placed in classes of decreasing associativity described in the text. We also discuss the physical applications of the cup-product of co-chains and the vector bundle structure of Newtonian space-time.