Subloops of indecomposable RA loops

Summary An RA loop is a (necessarily Moufang) loop whose loop rings in any characteristic are alternative, but not associative. There are seven classes of finite indecomposable RA loops. In this paper, we find the indecomposable subloops and the indecomposable nonabelian groups which can appear inside the loops in each class.     

A class of loops with right alternative loop rings

The purpose of this paper is to exhibit a class of loops which have strongly right alternative loop rings that are not alternative..     

Alternative loop rings

The right alternative law implies the left alternative law in loop rings of characteristic other than 2. We also exhibit a loop which fails to be a right Bol loop, even though its characteristic 2 loop rings are right alternative.     

When is an L(B, m, n, r, s, t, z, w) loop SRAR?

In [5, 6], the second author and D. A. ROBINSON initiated a study of non-Moufang Bol loops with the property that over a field, necessarily of characteristic 2, their loop rings satisfy the right, but not the left, Bol identity. They called such loops SRAR and showed that the family of SRAR loops includes those Bol loops which have a unique non-identity commutator/associator. In [4, 2], the current authors presented a construction for a new class of Bol loops denoted L(B,m,n,r,s,t,z,w) with initial data a given (possibly associative) Bol loop B, elements, r, s, t, z and w in the centre of B, and integers m and n.     

A Construction of Loops which admit Right Alternative Loop Rings

In this note, the authors offer a specific construction of loops whose loop rings are right, but not left, alternative.     

SRAR loops with more than two commutators

Possession of a unique nonidentity commutator/associator is a property that dominates the theory of loops whose loop rings, while not associative, nevertheless satisfy an “interesting” identity. For instance, until now, all loops with loop rings satisfying the right Bol identity (such loops are called SRAR) have been known to have this property. In this paper, we present various constructions of other kinds of SRAR loops.     

Nilpotent right alternative rings and bol circle loops

We determine the nilpotent right alternative rings of prime power oirder pⁿ n ≥ 4, which are not left alternative. Those which are strongly right alternative become Bol loops under the circle operation. The smallest Bol circle loop has order 16. There are six such loops, all of which appear to be new.     

Jordan decompositions in alternative loop rings

It is observed that the additive as well as multiplicative Jordan decompositions hold in alternative loop algebras of finiteRA loops and theRA loops for which the additive Jordan decomposition holds in the integral loop ring are characterized. Multiplicative Jordan decomposition (MJD) inZL, whereL is a finiteRA loop with cyclic centre is analysed, besides settling MJD for integral loop rings of allRA loops of order ≤32. It is also shown that for any finiteRA loopL,U (ZL) is an almost splittable Moufang loop. Research of the second author is supported by CSIR.     

Isomorphisms of integral alternative loop rings

This paper first settles the “isomorphism problem” for alternative loop rings; namely, it is shown that a Moufang loop whose integral loop ring is alternative is determined up to isomorphism by that loop ring. Secondly, it is shown that every normalized automorphism of an alternative loop ringZ L is the product of an inner automorphism ofQ L and an authomorphism ofL.     

Units in alternative loop rings

In this paper, we show that certain well known theorems concerning units in integral group rings hold more generally for integral loop rings which are alternative. This research was supported in part by the Natural Sciences and Engineering Research Council, Grants No. A9087 and A8775.     

Units in Alternative Integral Loop Rings

We prove the isomorphism problem for integral loop rings of finitely generated RA loops using a decomposition of the loop of units. Also we describe the finitely generated RA loops whose loops of units satisfy a certain property.     

On the rigidity of the coisotropic Maslov index on certain rational symplectic manifolds

We revisit the definition of the Maslov index of loops in coisotropic submanifolds tangent to the characteristic foliation of this submanifold. This Maslov index is given by the mean index of a certain symplectic path which is a lift of the holonomy along the loop. We prove a Maslov index rigidity result for stable coisotropic submanifolds in a broad class of ambient symplectic manifolds. Furthermore, we establish a nearby existence theorem for the same class of ambient manifolds.     

Finite subloops of units in an alternative loop ring

An RA loop is a loop whose loop rings, in characteristic different from , are alternative but not associative. In this paper, we show that every finite subloop of normalized units in the integral loop ring of an RA loop is isomorphic to a subloop of . Moreover, we show that there exist units in the rational loop algebra such that . Thus, a conjecture of Zassenhaus which is open for group rings holds for alternative loop rings (which are not associative).

     

COMMUTATIVE ALTERNATIVE RINGS: A CONSTRUCTION

In this paper, the authors establish a connection between commutative, alternative rings and Engel rings(of type 2), thereby providing a method for constructing commutative, alternative rings which are not associative.     

On rings asymptotically close to associative rings

The subject of this work is an extension of A. R. Kemer’s results to a rather broad class of rings close to associative rings, over a field of characteristic 0 (in particular, this class includes the varieties generated by finite-dimensional alternative and Jordan rings). We prove the finite-basedness of systems of identities (the Specht property), the representability of finitely-generated relatively free algebras, and the rationality of their Hilbert series. For this purpose, we extend the Razymslov-Zubrilin theory to Kemer polynomials. For a rather broad class of varieties, we prove Shirshov’s theorem on height.     

Loop algebras of code loops

In this paper we study loop algebras of code loops in the modular case and, in particular, we show that code loops are determined by their loop algebras over the field with two elements. Actually, many of our results hold for a wider family of loops L which we introduce in the second section.     

Moufang loops with commuting inner mappings

We investigate the relation between the structure of a Moufang loop and its inner mapping group. Moufang loops of odd order with commuting inner mappings have nilpotency class at most 2. The 6-divisible Moufang loops with commuting inner mappings have nilpotency class at most 2. There is a Moufang loop of order 2¹⁴ with commuting inner mappings and of nilpotency class 3.     

Optimal System of Loops on an Orientable Surface

Every compact orientable boundaryless surface M can be cut along simple loops with a common point v₀, pairwise disjoint except at v₀, so that the resulting surface is a topological disk; such a set of loops is called a {\it system of loops} for M. The resulting disk may be viewed as a polygon in which the sides are pairwise identified on the surface; it is called a polygonal schema. Assuming that M is a combinatorial surface, and that each edge has a given length, we are interested in a shortest (or optimal) system of loops homotopic to a given one, drawn on the vertex-edge graph of M. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its homotopy class. We give an algorithm to build such a system, which has polynomial running time if the lengths of the edges are uniform. As a byproduct, we get an algorithm with the same running time to compute a shortest simple loop homotopic to a given simple loop.