Involutive automorphisms of $N_\circ^\circ$ groups of finite Morley rank

Simple groups of Morley rank 5 are bad

By exploiting the geometry of involutions in $N_\circ^\circ$-groups of finite Morley rank, we show that any simple group of Morley rank $5$ is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most $2$. The main result is then used to catalog the nonsoluble connected groups of Morley rank $5$.

Involutive latin solutions of the Yang-Baxter equation

Wolfgang Rump showed that there is a one-to-one correspondence between nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation and binary algebras in which all left translations $L_x$ are bijections, the squaring map is a bijection, and the identity $(xy)(xz) = (yx)(yz)$ holds. We call these algebras \emph{rumples} in analogy with quandles, another class of binary algebras giving solutions of the Yang-Baxter equation. We focus on latin rumples, that is, on rumples in which all right translations are bijections as well. We prove that an affine latin rumple of order $n$ exis…

On the realization of the Gelfand Character of a finite group as a twisted trace

We show that the Gelfand character $ χ_G$ of a finite group $G $ (i.e. the sum of all irreducible complex characters of $G$ ) may be realized as a `` twisted trace'' $ g \mapsto Tr( ρ_g \circ T) $ for a suitable involutive linear automorphism $T$ of $L^2(G)$, where $(L^2(G), ρ)$ is the right regular representation of $G$. Moreover, we prove that under certain hypotheses $T(f)= f \circ L \;\; (f \in L^2(G)), $ where $ L $ is an involutive antiautomorphism of $ G.$ The natural representation $τ$ of $G$ associated to the natural $L$-conjugacy action of $G$ in the fixed point set $Fix_G(L)$ of $L$…

李歪 brace 的结构和刚性性质

Structural and rigidity properties of Lie skew braces

本文研究了李歪 brace 的结构和刚性性质，揭示了其在不同李群类别中的存在条件及构造方法。

斜辫括形的表示

Representations of skew braces

本文研究了斜左辫括形的线性表示，证明了同源性在表示理论中的保持，并计算了若干素数幂阶斜左辫括形的不可约表示维度。