arXiv subjects
Publications and source records attributed to Alexander Grishkov.
We classify Bol loops of order $27$, using a combination of theoretical results and computer search. There are $15$ Bol loops of order $27$, including five groups. New constructions for the ten nonassociative Bol loops of order $27$ are given.
Motivated by the recent progress towards classification of simple finite-dimensional Lie algebras over an algebraically closed field of characteristic $2$, we investigate such $15$-dimensional algebras.
We prove that for positive integers $m \geq 1, n \geq 1$ and a prime number $p \neq 2,3$ there are finitely many finite $m$-generated Moufang loops of exponent $p^n$.
In this note we describe the structure of finite dimensional Malcev algebras over a field of real numbers $\mathbb{R}$, which are nilpotent module its Lie center. It is proved that the corresponding analytic global Moufang loops are nilpotent module their nucleus.
We compute low-degree cohomology of current Lie algebras extended over the 3-dimensional simple algebra, compute deformations of related semisimple Lie algebras, and apply these results to classification of simple Lie algebras of absolute toral rank 2 and having a Cartan subalgebra of toral rank one. Everything is in characteristic 2.
A loop is said to be automorphic if its inner mappings are automorphisms. For a prime $p$, denote by $\mathcal A_p$ the class of all $2$-generated commutative automorphic loops $Q$ possessing a central subloop $Z\cong \mathbb Z_p$ such that $Q/Z\cong\mathbb Z_p\times\mathbb Z_p$. Upon describing the free $2$-generated nilpotent class two commutative automorphic loop and the free $2$-generated nilpotent class two commutative automorphic $p$-loop $F_p$ in the variety of loops whose elements have order dividing $p^2$ and whose associators have order dividing $p$, we show that every loop of $\mathcal A_p$ is a quotient of $F_p$ by a central subloop of order $p^3$. The automorphism group of $F_p$ induces an action of $GL_2(p)$ on the three-dimensional subspaces of $Z(F_p)\cong (\mathbb Z_p)^4$. The orbits of this action are in one-to-one correspondence with the isomorphism classes of loops from $\mathcal A_p$. We describe the orbits, and hence we classify the loops of $\mathcal A_p$ up to isomorphism. It is known that every commutative automorphic $p$-loop is nilpotent when $p$ is odd, and that there is a unique commutative automorphic loop of order $8$ with trivial center. Knowing $\mathcal A_p$ up to isomorphism, we easily obtain a classification of commutative automorphic loops of order $p^3$. There are precisely $7$ commutative automorphic loops of order $p^3$ for every prime $p$, including the $3$ abelian groups of order $p^3$.
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.
Let $F^λ_σ [G]$ be a crossed product of a group $G$ and the field $F$. We study the Lie properties of $F^λ_σ [G]$ in order to obtain a characterization of those crossed products which are upper (lower) Lie nilpotent and Lie $(n,m)$-Engel.