Related papers: Two-generation of traceless matrices over finite f…

Let $F[X]$ be the polynomial ring over a finite field $F$. It is shown that, for $n\geq 3$, the special linear group $SL_n(F[X])$ is boundedly generated by the elementary matrices.

Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the…

Let $F$ be a field and let $E$ be an \'etale algebra over $F$, that is, a finite product of finite separable field extensions $E = F_1 \times \dots \times F_r$. The classical primitive element theorem asserts that if $r = 1$, then $E$ is…

It is shown that the product of two nonscalar conjugacy classes of the special linear group SL$(n,K)$ contains matrices of arbitrary trace if $n \ge 4$ and $K$ is an abitrary field or $n=3$ and $K$ is finite.

In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: $D_2$, the algebra of $2\times 2$ diagonal matrices and $C_2$, the algebra of $2…

Over an algebraically closed fields, an alternative to the method due to Kostrikin and Shafarevich was recently suggested. It produces all known simple finite dimensional Lie algebras in characteristic p>2. For p=2, we investigate one of…

A classification of the semisimple subalgebras of the Lie algebra of traceless $3\times 3$ matrices with complex entries, denoted $A_2$, is well-known. We classify its nonsemisimple subalgebras, thus completing the classification of the…

Let $A$ be an associative algebra over a field of characteristic $\neq 2$ that is generated by a finite collection of nilpotent elements. We prove that all Lie derived powers of $A$ are finitely generated Lie algebras.

Let $R$ be a commutative ring that is free of rank $k$ as an abelian group, $p$ a prime, and $SL(n,R)$ the special linear group. We show that the Lie algebra associated to the filtration of $SL(n,R)$ by $p$-congruence subgroups is…

Let $F$ be a field of characteristic not $2$ . An associative $F$-algebra $R$ gives rise to the commutator Lie algebra $R^{(-)}=(R,[a,b]=ab-ba).$ If the algebra $R$ is equipped with an involution $*:R\rightarrow R$ then the space of the…

Let F_m=F_m(var(sl(2,K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl(2,K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication…

In this note, we derive explicitly the local relative trace formula for the symmetric space F*\SL(2,F) at the level of Lie algebras, where F is a p-adic field of residue characteristic greater than two and F* is the set of invertible…

Let L be a restricted Lie superalgebra with its restricted enveloping algebra u(L) over a field F of characteristic p>2. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2\times 2 matrices over F. We…

Let A be an algebra whose group of units U(A) satisfies a Laurent polynomial identity (LPI). We establish conditions on these polynomials in such a way that nil-generated algebras and group algebras with torsion groups over infinite fields…

We prove that if $L=\mbox{}^2F_4(2^{2n+1})'$ and $x$ is a nonidentity automorphism of $L$ then $G=\langle L,x\rangle$ has four elements conjugate to $x$ that generate $G$. This result is used to study the following conjecture about the…

We give necessary conditions for the existence of degenerations between two complex Lie superalgebras of dimension $(m,n)$. As an application, we study the variety $\mathcal{LS}^{(2,2)}$ of complex Lie superalgebras of dimension $(2,2)$.…

We present two noncommutative algebras over a field of characteristic zero that each posses a family of actions by cyclic groups of order $2n$, represented in $n \times n$ matrices, requiring generators of degree $3n$.

Let $K$ be a non-archimedean local field with residue field of characteristic $p$. We give necessary and sufficient conditions for a two-generator subgroup $G$ of ${\rm PSL_2}(K)$ to be discrete, where either $K=\mathbb{Q}_p$ or $G$…

For the $p$-dimensional filiform Lie algebra ${\mathfrak m}_2(p)$ over a field ${\mathbb F}$ of prime characteristic $p\ge 5$ with nonzero Lie brackets $[e_1,e_i] = e_{i+1}$ for $1<i<p$ and $[e_2,e_i]=e_{i+2}$ for $2<i<p-1$, we show that…

The class of minimal non-elementary Lie algebras over a field F are studied. These are classified when F is algebraically closed and of characteristic different from 2,3. The solvable algebras in this class are also characterised over any…