Related papers: On two letter identities in Lie rings

We show that a ring $\,R\,$ has two idempotents $\,e,e'\,$ with an invertible commutator $\,ee'-e'e\,$ if and only if $\,R \cong {\mathbb M}_2(S)\,$ for a ring $\,S\,$ in which $\,1\,$ is a sum of two units. In this case, the…

Let $R$ be a ring (not necessarily a commutative ring) with identity. The clean graph $Cl(R)$ of a ring $R$ is a graph with vertices in the form of an ordered pair $(e,u)$, where $e$ is an idempotent and $u$ is a unit of ring $R$,…

Let E and G be free modules of rank e and g, respectively, over a commutative noetherian ring R. The identity map on E^* tensor G induces the Koszul complex ... -> S_mE^* tensor S_nG tensor Wedge^p(E^* tensor G) -> S_{m+1}E^* tensor…

Consider the quotient $G/B$ of a simple matrix Lie group $G$ by a subgroup $B$ isomorphic to a direct product of some of $S^1$s and $S^3$s such that its adjoint representation can be extended over $G$. Then it naturally inherits a stable…

Let $G$ be a group. A ring $R$ is called a graded ring (or $G$-graded ring) if there exist additive subgroups $R_{\alpha }$ of $R$ indexed by the elements $\alpha \in G$ such that $R=\bigoplus_{\alpha \in G}R_{\alpha }$ and $R_{\alpha…

A cover by left ideals of an associative (not necessarily commutative or unital) ring $R$ is a collection of proper left ideals whose set-theoretic union equals $R$. If such a cover exists, then $\eta_\ell(R)$ is the cardinality of a…

Nonparametric rank tests for homogeneity and component independence are proposed, which are based on data compressors. For homogeneity testing the idea is to compress the binary string obtained by ordering the two joint samples and writing…

Let $n\geq m$ be two positive integers, $S_{n,m}=K[x_1,\ldots,x_n,y_1,\ldots,y_m]$ and $I_{n,m}=(x_iy_j\;:\;1\leq i\leq n,1\leq j\leq m)\subset S_{n,m}$ the edge ideal of a complete bipartite graph. Denote…

Let $m,n>1$ be integers and $\mathbb{P}_{n,m}$ be the point set of the projective $(n-1)$-space (defined by [2]) over the ring $\mathbb{Z}_m$of integers modulo $m$. Let $A_{n,m}=(a_{uv})$ be the matrix with rows and columns being labeled by…

Let $B_{n}$ denote the Bernoulli numbers, and $S(n,k)$ denote the Stirling numbers of the second kind. We prove the following identity $$ B_{m+n}=\sum_{\substack{0\leq k \leq n \\ 0\leq l \leq m}}\frac{(-1)^{k+l}\,k!\, l!\,…

In our preceding paper we have introduced the notion of an $s$-homogeneous triple. In this paper we use this technique to study connected $s$-homogeneous algebras with two relations. For such algebras, we describe all possible pairs…

Let $\mathcal K$ be a flag kernel on a homogeneous nilpotent Lie group $G$. We prove that operators $T$ of the form $T(f)= f*\mathcal K$ form an algebra under composition, and that such operators are bounded on $L^{p}(G)$ for $1<p<\infty$.

Let $Q_m$ be the HNN extension of $\Z/m \times \Z/m$ where the stable letter conjugates the first factor to the second. We explore small presentations of the groups $\Gamma_{m,n}=Q_m \ast Q_n$. We show that for certain choices of (m,n), for…

We study the structure of a graded $3$-Lie-Rinehart algebra $\mathcal{L}$ over an associative and commutative graded algebra $A.$ For $G$ an abelian group, we show that if $(L, A)$ is a tight $G$-graded 3-Lie-Rinehart algebra, then…

We do two things. 1. As a corollary to a stronger linearisation result (Theorem A), we prove the finite Morley rank version of the Lie-Kolchin-Malcev theorem on Lie algebras (Corollary A2). 2. We classify Lie ring actions on modules of…

We study the so--called ring $(p,Q)$--mappings which are the natural generalization of quasiregular and quasi isometric mappings. It is proved that open discrete ring $(p,Q)$--mappings are differentiable a.e. and belong to the class $ACL$…

Let $G \to P \to M$ be a flat principal bundle over a closed and oriented manifold $M$ of dimension $m=2d$. We construct a map of Lie algebras $\Psi: \H_{2\ast} (L M) \to {\o}(\Mc)$, where $\H_{2\ast} (LM)$ is the even dimensional part of…

Let $\mathcal{G}^{(\lambda)}$ be a group scheme which deforms $\mathbb{G}_a$ to $\mathbb{G}_m$. We explicitly describe the Cartier dual of the $l$-th Frobenius type kernel $N_l$ of the group scheme $\mathcal{E}^{(\lambda,\mu;D)}$ which is…

We introduce completely bounded kernels taking values in L(A,B) where A and B are C*-algebras. We show that if B is injective such kernels have a Kolmogorov decomposition precisely when they can be scaled to be completely contractive, and…

For appropriate domains $\Omega_{1}, \Omega_{2}$ we consider mappings $\Phi_{\mathbf A}:\Omega_{1}\to\Omega_{2}$ of monomial type. We obtain an orthogonal decomposition of the Bergman space $\mathcal A^{2}(\Omega_{1})$ into finitely many…