Related papers: A remark on Ado's Theorem for principal ideal doma…

We prove a refinement of Ado's theorem for Lie algebras over an algebraically-closed field of characteristic zero. We first define what it means for a Lie algebra $L$ to be approximated with a nilpotent ideal, and we then use such an…

We give a natural proof of the Ado theorem.

Let $\mathbb K$ be a field of characteristic zero and $A$ an integral domain over $\mathbb K.$ The Lie algebra $\Der_{\mathbb K} A$ of all $\mathbb K$-derivations of $A$ carries very important information about the algebra $A.$ This Lie…

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of…

In this paper, we construct a faithful representation with the lowest dimension for every complex Lie algebra in dimension $\leq 4$. In particular, in our construction, in the case that the faithful representation has the same dimension of…

Let $p$ be a prime. Given a split semisimple group scheme $G$ over a normal integral domain $R$ which is a faithfully flat $\mathbb Z_{(p)}$-algebra, we classify all finite dimensional representations $V$ of the fiber $G_K$ of $G$ over…

We consider the following question, concerning associative algebras R over an algebraically closed field k: When can the space of (equivalence classes of) finite dimensional irreducible representations of R be topologically embedded into a…

We give a simple proof of the Birkhoff theorem about existence of a faithful representation for any finite-dimensional nilpotent Lie algebra of characteristic zero.

In this paper we construct a graded universal enveloping algebra of a $G$-graded Lie algebra, where $G$ is not necessarily an abelian group. If the grading group is abelian, then it coincides with the classical construction. We prove the…

Toroidal Lie algebras are $n$ variable generalizations of affine Kac-Moody Lie algebras. Full toroidal Lie algebra is the semidirect product of derived Lie algebra of toroidal Lie algebra and Witt algebra, also it can be thought of…

We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We…

We discover a large class of simple affine vertex algebras $V_{k} (\mathfrak g)$, associated to basic Lie superalgebras $\mathfrak g$ at non-admissible collapsing levels $k$, having exactly one irreducible $\mathfrak g$-locally finite…

We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the $\mathrm{AC}_{\mathbb{R}}$ and $\mathrm{NC}_{\mathbb{R}}$-classes for this…

We prove an analog of the Ado theorem - the existence of a finite-dimensional faithful representation - for a certain kind of finite-dimensional nilpotent Hom-Lie algebras.

We show that for any finite-dimensional algebra $\Lambda$ of infinite representation type, over a perfect field, there is a bounded principal ideal domain $\Gamma$ and a representation embedding from $\Gamma -$mod into $\Lambda -$mod. As an…

Let $L$ be a restricted Lie algebra over a field of positive characteristic. We prove that the restricted enveloping algebra of $L$ is a principal ideal ring if and only if $L$ is an extension of a finite-dimensional torus by a cyclic…

We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the `Irreducibility theorem' and `Duflo theorem', to much wider classes of algebras. Our general version of Irreducibility theorem says…

Basic modules of McLain groups $M=M(\Lambda,\leq, R)$ are defined and investigated. These are (possibly infinite dimensional) analogues of Andr\'e's supercharacters of $U_n(q)$. The ring $R$ need not be finite or commutative and the field…

We introduce the concept of a triangular representation of a Lie algebra, give a counterpart of Ado's theorem, and discuss $2$-irreducible triangular modules over a nonreductive Lie algebra.

Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra $\mathfrak{g}_{\infty}$ defined over $\mathbb{C}_{\infty}$-the reduction of $\mathbb{C}$ mod infinitely large prime, and show that…