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Let G be a simple algebraic group over an algebraically closed field k. We classify the spherical conjugacy classes of G.

Let $K$ be a $p$-adically closed field and $G$ a group interpretable in $K$. We show that if $G$ is definably semisimple (i.e. $G$ has no definable infinite normal abelian subgroups) then there exists a finite normal subgroup $H$ such that…

This paper deals with $n$-dimensional algebras, over any field, which have only trivial derivation (automorphism) and simple algebras. It is shown that the corresponding sets of algebras are not empty and, in algebraically closed field…

Let \alpha_0 be an affine action of a discrete group \Gamma on a compact homogeneous space X and \alpha_1 a smooth action of \Gamma on X which is C^1-close to \alpha_0. We show that under some conditions, every topological conjugacy between…

In this article, we establish results concerning the cohomology of Zariski dense subgroups of solvable linear algebraic groups. We show that for an irreducible solvable $\mathbb{Q}$-defined linear algebraic group $\mathbf{G}$, there exists…

In a previous work, the third named author found a combinatorics of line arrangements whose realizations live in the cyclotomic group of the fifth roots of unity and such that their non-complex-conjugate embedding are not topologically…

Using suitable convex functions, we construct a new family of flat Minkowski planes whose automorphism groups are at least $3$-dimensional. These planes admit groups of automorphisms isomorphic to the direct product of $\mathbb{R}$ and the…

If $L$ is a semisimple Lie algebra of vector fields on R^N with a split Cartan subalgebra C, then it is proved that the dimension of the generic orbit of C coincides with the dimension of C. As a consequence one obtains a local canonical…

Let G be a finite group and let p be a prime. A module for G over a field of characteristic p is called algebraic if it satisfies a polynomial, with addition and multiplication given by direct sum and tensor product. In some sense, having…

We first show that any connected algebraic group over a perfect field is the neutral component of the automorphism group scheme of some normal projective variety. Then we show that very few connected algebraic semigroups can be realized as…

Let $\mathfrak{g}$ be a vector space and $[,],[,]'$ be a pair of Lie brackets on $\mathfrak{g}$. By definition they are compatible if $[,]+[,]'$ is again a Lie bracket. Such pairs play important role in bihamiltonian and $r$-matrix…

We show that the number of conjugacy classes of maximal finite subgroups of a lattice in a semisimple Lie group is linearly bounded by the covolume of the lattice. Moreover, for higher rank groups, we show that this number grows sublinearly…

We consider rational projective homogeneous varieties over an algebraically closed field of positive characteristic, namely quotients of a semi-simple group by a possibly non-reduced parabolic subgroup. We determine the group scheme…

We completely determine the autotopism group of the (as of now) largest family of commutative semifields found by G\"olo\u{g}lu and K\"olsch. Since this family of semifields generally does not have large nuclei, this process is considerably…

We give some properties of the subgroup $G_n(\mathbb{C})$ of the group of birational self-maps of $\mathbb{P}^n_\mathbb{C}$ generated by the standard involution and the group of automorphisms of $\mathbb{P}^n_\mathbb{C}$. We prove that…

Let $\mathbf G$ be a connected reductive algebraic group over an algebraically closed field, and let $s\in\mathbf G$ be a semisimple element. We show that the centraliser of $s$ is the semi-direct product of its identity component by its…

For a field $R$ of characteristic $p\ge 0$ and a matrix $c$ in the full $n\times n$ matrix algebra $M_n(R)$ over $R$, let $S_n(c,R)$ be the centralizer algebra of $c$ in $M_n(R)$. We show that $S_n(c,R)$ is a Frobenius-finite,…

We study twisted conjugacy classes of the unit element in different groups. Fel'shtyn and Troitsky showed that the twisted conjugacy class of the unit element of an abelian group is a subgroup for every automorphism. The structure is…

A Lie polynomial is an element of a free Lie algebra $\mathcal F_k$ on $k$-generators, which defines a Lie map on a given Lie algebra $L$, by substituting $k$-elements of $L$. Similar to word maps on groups and polynomial maps on algebras,…

For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on…