arXiv subjects

Alexey Galt

Publications and source records attributed to Alexey Galt.

9 recordsLinked to original sources

arXiv2026

Elements of finite order in the normalizer of a maximal torus of a semisimple group

We prove that the set of elements of a given finite order in the connected component $N_w$ of the normalizer $N_G(T)$ of a maximal torus $T$ of a semisimple group $G$ is either empty or a disjoint union of finitely many irreducible subvarieties $C_i$. The dimension of each $C_i$ equals the dimension of the subspace of fixed vectors for the action of the element $w$ of the Weyl group $W$ corresponding to the component $N_w$. Moreover, each $C_i$ is an orbit of the action of the torus $T$ on the component $N_w$ by conjugation.

math.GR↗

arXiv2025

On Rota--Baxter operators on finite simple groups of Lie type

Rota--Baxter operators on groups were introduced by L. Guo, H. Lang, Yu. Sheng in 2020. In 2023, V. Bardakov and the second author showed that all Rota--Baxter operators on simple sporadic groups are splitting, i.\,e. they correspond to exact factorizations of groups. In 2024, the authors of the current paper described all non-splitting Rota--Baxter operators on alternating groups. Now we describe Rota--Baxter operators on finite simple exceptional groups of Lie type and projective special linear groups of degree two.

math.GR↗

arXiv2024

Rota-Baxter operators on dihedral and alternating groups

Rota-Baxter operators on algebras, which appeared in 1960, have connections with different versions of the Yang-Baxter equation, pre- and postalgebras, double Poisson algebras, etc. In 2020, the notion of Rota-Baxter operator on a group was defined by L. Guo, H. Lang, Yu. Sheng. In 2023, V. Bardakov and the second author showed that all Rota-Baxter operators on simple sporadic groups are splitting, i. e. they are defined via exact factorizations. In the current work, we clarify for which $n$, there exist non-splitting Rota-Baxter operators on the alternating group $\mathrm{A}_n$. For the corresponding $n$, we describe all non-splitting Rota-Baxter operators on $\mathrm{A}_n$. Moreover, we describe Rota-Baxter operators on dihedral groups $D_{2n}$ providing the general construction which lies behind all non-splitting Rota-Baxter operators on $\mathrm{A}_n$ and $D_{2n}$.

math.GR↗

arXiv2022

On splitting of the normalizer of a maximal torus in finite groups of Lie type

Let $G$ be a finite group of Lie type and $T$ a maximal torus of $G$. In this paper we complete the study of the question of the existence of a complement for the torus $T$ in its algebraic normalizer $N(G,T)$. It is proved that every maximal torus of the group $G\in\{G_2(q), {}^2G_2(q), {}^3D_4(q)\}$ has a complement in its algebraic normalizer. The remaining twisted classical groups ${}^2A_n(q)$ and ${}^2D_n(q)$ are also considered.

math.GR↗

arXiv2021

Double axes and subalgebras of Monster type in Matsuo algebras

Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov \cite{hrs,hrs1} as a broad generalisation of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional simple groups, and Matsuo algebras corresponding to $3$-transposition groups. Thus, axial algebras of Monster type unify several strands in the theory of finite simple groups. It is shown here that double axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion law of Monster type $(2η,η)$. Primitive subalgebras generated by two single or double axes are completely classified and $3$-generated primitive subalgebras are classified in one of the three cases. These classifications further lead to the general flip construction outputting a rich variety of axial algebras of Monster type. An application of the flip construction to the case of Matsuo algebras related to the symmetric groups results in three new explicit infinite series of such algebras.

math.RA↗

arXiv2020

On splitting of the normalizers of maximal tori in $E_7(q)$ and $E_8(q)$

Let $G$ be a finite group of Lie type $E_l$ with $l\in\{6,7,8\}$ over $F_q$ and $W$ be the Weyl group of $G$. We describe all maximal tori $T$ of $G$ such that $T$ has a complement in its algebraic normalizer $N(G,T)$. Let $T$ correspond to an element $w$ of $W$. When $T$ does not have a complement, we show that $w$ has a lift in $N(G,T)$ of order $|w|$ in all considered groups, except the simply-connected group $E_7(q)$. In the latter case we describe the elements $w$ that have a lift in $N(G,T)$ of order $|w|$.

math.GR↗

arXiv2020

Minimal supplements in normalizers of maximal tori of $F_4(q)$

Let $G$ be a finite group of Lie type $F_4$ with the Weyl group $W$. For every maximal torus $T$ of $G$, we find the minimal order of a supplement to $T$ in its algebraic normalizer $N(G,T)$. In particular, we obtain all maximal tori having complements in $N(G,T)$. Assume that $T$ corresponds to an element $w$ of $W$. We find the minimal order of lifts of $w$ to $N(G,T)$.

math.GR↗

arXiv2018

On splitting of the normalizer of a maximal torus in $E_6(q)$

Let $G$ be a finite group of Lie type $E_6$ over $F_q$ (adjoint or simply connected) and $W$ be the Weyl group of $G$. We describe maximal tori $T$ such that $T$ has a complement in its algebraic normalizer $N(G,T)$. It is well known that for each maximal torus $T$ of $G$ there exists an element $w\in W$ such that $N(G,T)/T\simeq C_W(w)$. When $T$ does not have a complement isomorphic to $C_W(w)$, we show that $w$ has a lift in $N(G,T)$ of the same order.

math.GR↗

arXiv2018

On Shalev's conjecture for type $A_n$ and ${}^{2}A_n$

In the paper we consider images of finite simple projective special linear and unitary groups under power words. In particular, we show that if $G\simeq \PSL_n^\varepsilon (q)$, then for every power words of type $x^M$ there exist constant $c$ and $N$ such that $\vert ω(G)\vert >c\frac{\vert G\vert }{n}$ whenether $\vert G\vert >N$.

math.GR↗