Related papers: Commutative Moufang loops and alternative algebras

It is proved that any free Moufang loop can be embedded in a loop of invertible elements of some alternative algebra.

We construct two infinite series of Moufang loops of exponent $3$ whose commutative center (i.e. the set of elements that commute with all elements of the loop) is not a normal subloop. In particular, we obtain examples of such loops of…

For most (and possibly all) non-associative finite simple Moufang loops, three generators of order 3 can be chosen so that each two of them generate a group isomorphic to $(3, 3 | 3, p)$. The subgroup structure of $(3, 3 | 3, p)$ depends on…

In this work we construct free Moufang loop in the variety generated by code loops. We apply this construction for study the code loops. Moreover, we define and determine all basic representations of code loops of rank 3 and 4.

Let a Moufang loop Q contain a non-unitary subloop, which is a simple loop. Then Q is not embedded into a loop of invertible elements of any alternative algebra.

The paper defines the notion of alternative loop algebra F[Q] for any nonassociative Moufang loop Q as being any non-zero homomorphic image of the loop algebra FQ of a loop Q over a field F. For the class M of all nonassociative alternative…

Moufang loops are one of the best-known generalizations of groups. There is only one countable family of nonassociative finite simple Moufang loops, arising from the split octonion algebras. We prove that every member of this family is…

A loop is automorphic if all its inner mappings are automorphisms. We construct the free commutative automorphic $2$-generated loop of nilpotency class $3$. It has dimension $8$ over the integers.

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 two. $6$-divisible Moufang loops with commuting inner…

It is known that with precision till isomorphism that only and only loops $M(F) = M_0(F)/<-1>$, where $M_0(F)$ denotes the loop, consisting from elements of all matrix Cayley-Dickson algebra $C(F)$ with norm 1, and $F$ be a subfield of…

It is explicitly shown how the Lie algebras can be associated with the analytic Moufang loops. The resulting Lie algebra commutation relations are well known from the theory of alternative algebras and can be seen as a preliminary step to…

We use groups with triality to construct a series of nonassociative Moufang loops. Certain members of this series contain an abelian normal subloop with the corresponding quotient being a cyclic group. In particular, we give a new series of…

A base of a relatively free associative algebra with the identity x^3=0 over a field of arbitrary characteristic is found. As an application a minimal generating system of the 3x3 matrix invariant algebra is determined.

We consider the 2-generated free metabelian associative and Lie algebras over the complex field and the invariants of the dihedral groups of finite order acting on these algebras. In the associative case we find a finite set of generators…

We construct free cubic implication algebras with finitely many generators, and determine the size of these algebras.

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 $V$…

We give some general theorems on free algebras of varieties of Boolean algebras with operators; a hitherto new result is obtained for Pinter's substitution algebras. For n\geq 3, and m>1, there is a generating set of the free algebra freely…

The decomposition theorem for torsion abelian groups holds analogously for torsion commutative diassociative loops. With this theorem in mind, we investigate commutative diassociative loops satisfying the additional condition (trivially…

We generalize the global construction of code loops introduced by Nagy, which is based on the connection between Moufang loops and groups with triality. This follows from the construction of a nilpotent group $G_n$ of class 3 with triality…

We construct a Moufang loop $M$ of order $3^{19}$ and a pair $a,b$ of its elements such that the set of all elements of $M$ that associate with $a$ and $b$ does not form a subloop. This is also an example of a nonassociative Moufang loop…