arXiv subjects
Publications and source records attributed to Péter T. Nagy.
In this paper, we provide a complete classification of real three-dimensional simple Bol algebras $(B, [.,.], \langle .,.,. \rangle )$. The main part of the classification concerns the case $[B,B]=B$. In addition, we identify two isomorphism classes with a nonzero binary product, both satisfying $\dim[B,B]=2$, as well as four simple classes of Lie triple systems in which the binary product vanishes.
We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field $\mathbb K$ of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie algebras, and hence can be considered as the closest relatives of binary Lie algebras. These algebras are extensions of $\mathbb K$ by the 3-dimensional nilpotent Lie algebra and at the same time extensions of a two-dimensional Lie algebra by a two-dimensional abelian algebra. We describe their groups of automorphisms as extensions of a subgroup of the group of automorphisms of the three-dimensional nilpotent Lie algebra by $\mathbb K$.
The aim of our paper is to generalize the tangent prolongation of Lie groups to non-associative multiplications and to examine how the weak associative and weak inverse properties are transferred to the multiplication defined on the tangent bundle. We obtain that the tangent prolongation of a $\mathcal{C}^r$-differentiable loop ($r\geq 1$) is a $\mathcal{C}^{r-1}$-differentiable loop that acquires the classical weak inverse and weak associative properties of the initial loop.
Our paper deals with the investigation of extensions of commutative groups by loops so that the quasigroups that result in the multiplication between cosets of the kernel subgroup are T-quasigroups. We limit our study to extensions in which the quasigroups determining the multiplication are linear functions without constant term, called linear abelian extensions. We characterize constructively such extensions with left-, right-, or inverse properties using a general construction according to an equivariant group action principle. We show that the obtained constructions can be simplified for ordered loops. Finally, we apply our characterization to determine the possible cardinalities of the component loop of finite linear abelian extensions.
The objectives of this paper is to give a systematic investigation of extension theory of loops. A loop extension is (left, right or middle) nuclear, if the kernel of the extension consists of elements associating (from left, right or middle) with all elements of the loop. It turns out that the natural non-associative generalizations of the Schreier's theory of group extensions can be characterized by different types of nuclear properties. Our loop constructions are illustrated by rich families of examples in important loop classes.