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Some 15 years ago M. Kontsevich and A. Rosenberg [KR] proposed a heuristic principle according to which the family of schemes ${Rep_n(A)}$ parametrizing the finite-dimensional represen- tations of a noncommutative algebra A should be…

We show that for any infinite set $A$ in ${\mathbb R}$, there exists a compact set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that does not contain any non-trivial affine copy of $A$. This proves the Erd\"os similarity…

Let F be a p-adic field and n a positive integer. The local Langlands conjecture asserts the existence of a bijection between irreducible admissible representations of GL(n,F) and n-dimensional admissible representations of the Weil-Deligne…

A nonzero locally nilpotent linear derivation of the polynomial algebra K[X] in d variables over a field K of characteristic 0 is called a Weitzenboeck derivation. The classical theorem of Weitzenboeck states that the algebra of constants…

We give a full description of the Lie algebra generated by locally nilpotent derivations (short LNDs) on smooth Danielewski surfaces $D_p$ given by $xy=p(z)$. In case $\mathrm{deg}(p)\geq 3$ it turns out to be not the whole Lie algebra…

In this paper is devoted to nilpotent finite-dimensional evolution algebras E with $dimE^2 = dimE-1$. We described Lie algebras associated with evolution algebras whose nilindex is maximal. Moreover, in terms of this Lie algebra we fully…

A commutative residuated lattice A is said to be subidempotent if the lower bounds of its neutral element e are idempotent (in which case they naturally constitute a Brouwerian algebra A*). It is proved here that epimorphisms are surjective…

We study algebra endomorphisms and derivations of some localized down-up algebras $\A$. First, we determine all the algebra endomorphisms of $\A$ under some conditions on $r$ and $s$. We show that each algebra endomorphism of $\A$ is an…

We prove that every $2$-local automorphism on a finite-dimensional semi-simple Lie algebra $\mathcal{L}$ over an algebraically closed field of characteristic zero is an automorphism. We also show that each finite-dimensional nilpotent Lie…

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $\mathfrak g$ of left-invariant vector fields on a Lie group $\mathbb G$ and…

Let $\varphi: \mathbb{P}^{n}_{F} \to \mathbb{P}^{n}_{F}$ where $F$ is a complete valued field. If $x$ is a fixed point, such that the action of $\varphi$ on $T_{x}$ has eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$, with $\lambda_{1},…

Given a sequence $\vec d=(d_1,\dots,d_k)$ of natural numbers, we consider the Lie subalgebra $\mathfrak{h}$ of $\mathfrak{gl}(d,\mathbb{F})$, where $d=d_1+\cdots +d_k$ and $\mathbb{F}$ is a field of characteristic 0, generated by two block…

Toroidal Lie algebras are universal central extentions of the finite dimensional simple Lie algbera tensored with Laurent Polynomials in several commuteing variables. In this paper we classify irreducible integrable modules for Toroidal Lie…

Let ${\mathbb C}[x_1,\dots,x_n]_{d+1}$ be the vector space of homogeneous forms of degree $d+1$ on ${\mathbb C}^n$, with $n,d\ge 2$. In earlier articles by J. Alper, M. Eastwood and the author, we introduced a morphism, called $A$, that…

Let $K$ be an infinite field of characteristic different from two and let $U_1$ be the Lie algebra of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$. The algebra $U_1$ admits a natural $\mathbb{Z}$-grading. We provide a…

In this paper, we study the analogous Erd\H{o}s similarity conjecture in higher dimensions and generalize the Eigen-Falconer theorem. We show that if $A=\{\boldsymbol{x}_n\}_{n=1}^\infty \subseteq \mathbb{R}^d$ is a sequence of non-zero…

Let $A$ be a unital locally matrix algebra. Among the examples of such algebras are: (1) an infinite tensor product $\otimes M_{n_i}(\mathbb{F})$ of matrix algebras over a field $\mathbb{F}$, and (2) the Clifford algebra of a nondegenerate…

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse…

We investigate the structure of commutative integral domains $B$ of characteristic zero by studying the kernels of locally nilpotent derivations $D : B \to B$.

If $R$ is an integral domain and $A$ is an $R$-algebra, then $A$ has the {\it Laurent cancellation property over $R$} if $A^{[\pm n]}\cong_RB^{[\pm n]}$ implies $A\cong_RB$ ($n\ge 0$ and $B$ an $R$-algebra). Here, $A^{[\pm n]}$ denotes the…