Related papers: Linear Algebra In A Jiffy

In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform Zinbiel algebras. It is proven that every non-split central extension of an $n$-dimensional null-filiform Zinbiel algebra is…

With a nilpotent element in a semisimple Lie algebra g one associates a finitely generated associative algebra W called a W-algebra of finite type. This algebra is obtained from the universal enveloping algebra U(g) by a certain Hamiltonian…

Brief proofs of classical results of Lie on finite dimensional subalgebras of vector fields in two and three variables are outlined. The results for algebras of maximal rank for vector fields in $\mathbb{C}^N$ -- $N$ arbitrary -- are also…

We consider the (finite) $W$-algebra $W_{m|n}$ attached to the principal nilpotent orbit in the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb C)$. Our main result gives an explicit description of $W_{m|n}$ as a certain…

Diversities are an extension of the concept of a metric space which assign a non-negative value to every finite set of points, rather than just pairs. A general theory of diversities has been developed which exhibits many deep analogies to…

Using an analogy with the rank theorem in differential geometry, it is shown that for a finite $n$-tuple $X$ in a tracial von Neumann algebra and any finite $m$-tuple $F$ of $*$-polynomials in $n$ noncommuting indeterminates,…

We say that a function $f \in L^1(\mathbb{R})$ tiles at level $w$ by a discrete translation set $\Lambda \subset \mathbb{R}$, if we have $\sum_{\lambda \in \Lambda} f(x-\lambda)=w$ a.e. In this paper we survey the main results, and prove…

For a topological space $X$ we study continuous maps $f : X\to \mathbb R^m$ such that images of every pairwise distinct $k$ points are affinely (linearly) independent. Such maps are called affinely (linearly) $k$-regular embeddings. We…

In this paper it is stressed that there is no {\em physical} reason for symmetries to be linear and that Lie group theory is therefore too restrictive. We illustrate this with some simple examples. Then we give a readable review on the…

An $L$-matrix is a matrix whose off-diagonal entries belong to a set $L$, and whose diagonal is zero. Let $N(r,L)$ be the maximum size of a square $L$-matrix of rank at most $r$. Many applications of linear algebra in extremal combinatorics…

We describe subalgebras of the Lie algebra $\mf{gl}(n^2)$ that contain all inner derivations of $A=M_n(F)$ (where $n\ge 5$ and $F$ is an algebraically closed field of characteristic 0). In a more general context where $A$ is a prime algebra…

This is a survey on the combinatorics and geometry of integrable representations of quantum affine Lie algebras with a particular focus on level 0. Pictures and examples are included to illustrate the affine Weyl group orbits, crystal…

First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies…

We show that there exists $k \in \bbn$ and $0 < \e \in\bbr$ such that for every field $F$ of characteristic zero and for every $n \in \bbn$, there exists explicitly given linear transformations $T_1,..., T_k: F^n \to F^n$ satisfying the…

Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the `` structure sheaf algebras'' in many--in point of fact, in all--the situations of a geometrical character that…

We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and…

We introduce the notion of a non--linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping…

We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding…

Given a graph $F$ and a positive integer $n$, the weak $F$-saturation number $\mathrm{wsat}(K_n,F)$ is the minimum number of edges in a graph $H$ on $n$ vertices such that the edges missing in $H$ can be added, one at a time, so that every…

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. In a recent work, we have determined the maximal dimension for a linear subspace of $n$ by $n$ symmetric matrices with rank less than or equal to…