A second grammar of associators

It is proved that the commutator ideal of the multiplication algebra of a free commutative alternative algebra of rank n is nilpotent of index n − 1. As a corollary to this fact, the Bruck theorem for special commutative Moufang loops is derived.

A commutative Moufang loop which is not associative has the property that all proper subloops are associative if and only if the loop can be generated by three elements.

Free commutative moufang loops and anticommutative graded rings

An infinite irredundant system of identities in the variety of commutative Moufang loops (and in the variety of distributive Steiner quasigroups) is constructed with the property that no infinite subset is equivalent to a finite subsystem. It follows that the set of subvarieties of the variety of commutative Moufang loops (and of the variety of distributive Steiner quasigroups) has the cardinal of the continuum; it also follows that there is a commutative Moufang loop (a distributive Steiner quasigroup) given by an enumerable set of identical relations, in which the word problem is unsolvable. Finally, the three arguments conjecture is refuted for commutative Moufang loops. In fact, an associator is constructed that is not the identity but has three of its arguments the same.

Commutative Moufang Loops

In this paper it is proved that the semigroup of endomorphisms of a commutative Moufang loop (CML) which admits a decomposition into the direct product of its subloops is isomorphic to the semigroup of M, M-matrices. We introduce the notion of the holomorph of a CML which admits a direct decomposition and give the matrix representation of the holomorph.

We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotence of the free loop proved by Higman, it is established, essentially, by the same method.

Groupes d'Automorphismes des Boucles de Moufang Commutatives

Let G be a discrete group generated by a finite number of generators \( {\gamma _1},...,{\gamma _k} \in G. \) . One defines then a distance d(·,·) on G by requiring that d(gx,gy) = d(x, y) (x,y,g € G) and that d(e) x), the distance of x € G from the neutral point e € G is, by definition, the smallest n ≥ 0 for which we can write \( x = \gamma _{{i_1}}^{{ \in _1}}...\gamma _{in}^{{ \in _n}},({i_1},...,{i_n} = 1,...,k;{ \in _j} = 0, \pm 1) \) .

Marche aléatoire sur un groupe libre : théorèmes limite conditionnellement à la sortie

dx.doi.org/10.7437/NF 2318-4957 /2013.01.003 Inequivalent N = 2 supersymmetrizations of the -conformal Galilei algebra ind-spatial dimensions are constructed from the chiral (2, 2) and the real (1, 2, 1)basic supermultiplets of the N = 2 supersymmetry. For non-negative integer andhalf-integer both superalgebras admit a consistent truncation with a (different)finite number of generators. The real N = 2 case coincides with the superalgebraintroduced by Masterov, while the chiral N = 2 case is a new superalgebra.We present D-module representations of both superalgebras. Then we investi-gate the new superalgebra derived from the chiral supermultiplet. It is shown that itadmits two types of central extensions, one is found for any d and half-integer andthe other only for d = 2 and integer . For each central extension the centrally ex-tended -superconformal Galilei algebra is realized in terms of its super-Heisenbergsubalgebra generators.

Algebraic commutative Moufang loops

Cubic Forms - Algebra, Geometry, Arithmetic

Let V be a cubic surface defined by the equation T_0^3+T_1^3+T_2^3+theta T_3^3=0 over a quadratic extension of 3-adic numbers k=mathbb {Q}_3(theta ), where theta ^3=1. We show that a relation on a set of geometric k-points on V modulo (1-theta )^3 (in a ring of integers of k) defines an admissible relation and a commutative Moufang loop associated with classes of this admissible equivalence is non-associative. This answers a problem that was formulated by Yu. I. Manin more than 50 years ago about existence of a cubic surface with a non-associative Moufang loop of point classes.

In this work, we discovered a dozen of new loop identities we called identities of 'second Bol-Moufang type'. This was achieved by using a generalized and modified nuclear identification model originally introduced by Dr\'{a}pal and Jedli\u{c}ka. Among these twelve identities, eight of them were found to be distinct (from well known loop identities), among which two pairs axiomatize the weak inverse property power associative conjugacy closed (WIP PACC) loop. The four other new loop identities individually characterize the Moufang identities in loops. Thus, now we have eight loop identities that characterize Moufang loops. We also discovered two (equivalent) identities that describe two varieties of Buchsteiner loops. In all, only the extra identities which the Dr\'{a}pal and Jedli\u{c}ka nuclear identification model tracked down could not be tracked down by our own nuclear identification model. The dozen laws $\{Q_i\}_{i=1}^{12}$ induced by our nuclear identification form four cycles in the following sequential format: $\big(Q_{4i-j}\big)_{i=1}^3,~j=0,1,2,3,$ and also form six pairs of dual identities. With the help of twisted nuclear identification, we discovered six identities of lengths five that describe the abelian group variety and commutative Moufang loop variety (in each case). The second dozen identities $\{Q_i^*\}_{i=1}^{12}$ induced by our twisted nuclear identification were also found to form six pairs of dual identities. Some examples of loops of smallest order that obey non-Moufang laws (which do not necessarily imply the other) among the dozen laws $\{Q_i\}_{i=1}^{12}$ were found.