Related papers: The Drinfel'd polynomial of a tridiagonal pair

Let $K$ be a field, $R=K[x, y]$ the polynomial ring and $\mathcal{M}(K)$ the set of all pairs of square matrices of the same size over $K.$ Pairs $P_1=(A_1,B_1)$ and $P_2=(A_2,B_2)$ from $\mathcal{M}(K)$ are called similar if…

We consider dissipative operators $A$ of the form $A=S+iV$, where both $S$ and $V\geq 0$ are assumed to be symmetric but neither of them needs to be (essentially) selfadjoint. After a brief discussion of the relation of the operators $S\pm…

Let $\F$ denote a field, and let $V$ denote a vector space over $\F$ with finite positive dimension. A Leonard pair on $V$ is an ordered pair of diagonalizable $\F$-linear maps $A: V \to V$ and $A^* : V \to V$ that each act on an eigenbasis…

Let $\mathbb{K}$ denote an algebraically closed field and let $V$ denote a vector space over $\mathbb{K}$ with finite positive dimension. Let $A,A^*$ denote a tridiagonal pair on $V$. We assume that $A,A^*$ belongs to a family of…

Fix an algebraically closed field $\mathbb{F}$ and an integer $d \geq 3$. Let $\text{Mat}_{d+1}(\mathbb{F})$ denote the $\mathbb{F}$-algebra consisting of the $(d+1) \times (d+1)$ matrices that have all entries in $\mathbb{F}$. We consider…

A square matrix is called {\it Hessenberg} whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $V$ denote a nonzero finite-dimensional vector space over a field $\fld$. We consider an ordered…

Let K be an arbitrary (commutative) field and L be an algebraic closure of it. Let V be a linear subspace of M_n(K), with n>2. We show that if every matrix of V has at most one eigenvalue in K, then dim V<=1+n(n-1)/2. If every matrix of V…

In this paper we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let $d$ denote a nonnegative integer. Let…

Fix an integer $d \geq 0$, a field $\mathbb{F}$, and a vector space $V$ over $\mathbb{F}$ with dimension $d+1$. By a decomposition of $V$ we mean a sequence $\{V_i\}_{i=0}^d$ of $1$-dimensional $\mathbb{F}$-subspaces of $V$ such that $V =…

A detailed version of preprint "Self-linking number of a real algebraic link" by the same author, alg-geom/9410030. For a nonsingular real algebraic curve in 3-dimensional projective space or 3-sphere, a new integer-valued characteristic is…

We extend some of the results of Agler, Knese, and McCarthy [1] to $n$-tuples of commuting isometries for $n>2$. Let $\mathbb{V}=(V_1,\dots,V_n)$ be an $n$-tuple of a commuting isometries on a Hilbert space and let Ann$(\mathbb{V})$ denote…

We consider the transformation $\ev$ which associates to any element in a K-algebra A a function on the the set of its K-points. This is the analogue of the fundamental Gelfand transform. Both $\ev$ and its dual $\ev^*$ are the maps from a…

An arc-weighted digraph is a pair $(D,\omega)$ where $D$ is a digraph and $\omega$ is an \emph{arc-weight function} that assigns\ to each arc $uv$ of $D$ a nonzero real number $\omega(uv)$. Given an arc-weighted digraph $(D,\omega)$ with…

Fix an algebraically closed field $\F$ and an integer $d \geq 3$. Let $V$ be a vector space over $\F$ with dimension $d+1$. A Leonard pair on $V$ is a pair of diagonalizable linear transformations $A: V \to V$ and $A^* : V \to V$, each…

Using adjoint representation we firstly classify two and three dimensional Lie super-bialgebras obtain from decomposable Lie superalgebras. In this way we complete the classification obtained by Eghbali et al., [J. Math. Phys. 51, 073503…

A notion of Drinfeld polynomials is introduced for modules of two-parameter quantum affine algebras. Finite dimensional representations are then characterized by sets of $l$-tuples of pairs of Drinfeld polynomials with certain conditions.

We further define two-parameter quantum affine algebra $U_{r,s}(\widehat{\frak {sl}_n})$ $(n>2)$ after the work on the finite cases (see [BW1], [BGH1], [HS] & [BH]), which turns out to be a Drinfel'd double. Of importance for the quantum…

Define a $\mathcal V^{(d)}$-algebra as an associative algebra with a symmetric and invariant co-inner product of degree $d$. Here, we consider $\mathcal V^{(d)}$ as a dioperad which includes operations with zero inputs. We show that the…

Veldkamp polygons are certain graphs $\Gamma=(V,E)$ such that for each $v\in V$, $\Gamma_v$ is endowed with a symmetric anti-reflexive relation $\equiv_v$. These relations are all trivial if and only if $\Gamma$ is a thick generalized…

For a digraph D and three parameters x, y, z in {0,1,+,-} we define the digraph D^(x,y,z) and call it the (x,y,z)-transformation of D. We show that for every r-regular digraph D the adjacency characteristic polynomial A(t, D^(x,y,z)) of…