Related papers: The algebra of integro-differential operators on a…

Let $S = \mathbb{C}[x_{i,j}]$ be the ring of polynomial functions on the space of $m \times n$ matrices, and consider the action of the group $\mathbf{GL} = \mathbf{GL}_m \times \mathbf{GL}_n$ via row and column operations on the matrix…

Let $k[X] = k[x_{i,j}: i = 1,..., m; j = 1,..., n]$ be the polynomial ring in $m n$ variables $x_{i,j}$ over a field $k$ of arbitrary characteristic. Denote by $I_2(X)$ the ideal generated by the $2 \times 2$ minors of the generic $m \times…

An interpretation of the multiple Meixner polynomials of the first kind is provided through an infinite Lie algebra realized in terms of the creation and annihilation operators of a set of independent oscillators. The model is used to…

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an $L_2$-form of H\"ormander's kernel condition suffices for the weak…

We study singular integral operators with kernels that are more singular than standard Calder\'on-Zygmund kernels, but less singular than bi-parameter product Calder\'on-Zygmund kernels. These kernels arise as restrictions to two dimensions…

The integral of the Wigner function of a quantum mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval…

We present the most general polynomial Lie algebra generated by a second order integral of motion and one of order M, construct the Casimir operator, and show how the Jacobi identity provides the existence of a realization in terms of…

In this paper our main theorem states the following, Main Theorem : Let B denote the polynomial ring D[x1,.... ,xn] , in the commuting indeterminates x i over a division ring D . Let M be a finitely generated B-module . Let B m denote the…

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…

In this paper we construct a large class of multiplication operators on reproducing kernel Hilbert spaces which are {\em homogeneous} with respect to the action of the M\"{o}bius group consisting of bi-holomorphic automorphisms of the unit…

We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {\rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({\rm YD} in short )modules over…

Let $K<X> =K<X_1,...,X_n>$ be the free $K$-algebra on $X={X_1,...,X_n}$ over a field $K$, which is equipped with a weight $\mathbb{N}$-gradation (i.e., each $X_i$ is assigned a positive degree), and let ${\cal G}$ be a finite homogeneous…

In this paper, we study a class of down-up algebras $\A$ defined over a polynomial base ring $\K[t_{1}, \cdots, t_{n}]$ and establish several analogous results. We first construct a $\K-$basis for the algebra $\A$. As a result, we prove…

Denote by $SL_3(\mathbb R)$ the special linear group of degree 3 over the real numbers, $A$ the subgroup consisting of the diagonal matrices with positive entries. In this paper, we study the algebraic and analytic properties of the…

Directly infinite algebras, those algebras, $E$ which have a pair of elements $x$ and $y$ where $1 = xy \neq yx$, are well known to have a sub-algebra isomorphic to $M_\infty(K)$, the set of infinite $\zplus \times \zplus$-indexed matrices…

The paper is devoted to the so-called complete Leibniz algebras. It is known that a Lie algebra with a complete ideal is split. We will prove that this result is valid for Leibniz algebras whose complete ideal is a solvable algebra such…

The symmetric coinvariant algebra $C[x_1, dots, x_n]_{S_n}$ is the quotient algebra of the polynomial ring by the ideal generated by symmetric polynomials vanishing at the origin. It is known that the algebra is isomorphic to the regular…

Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L x^n= \lambda_n x^n + \nu_n x^{n-2}$ for $\lambda_n$ and $\nu_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L P_n(x) = \lambda_n…

We study operator algebras arising from monomial ideals in the ring of polynomials in noncommuting variables, through the apparatus of subproduct systems and C*-correspondences. We provide a full comparison amongst the related operator…

Let X_0 be a compact connected Riemann surface of genus g with D_0\subset X_0 an ordered subset of cardinality n, and let E_G be a holomorphic principal G-bundle on X_0, where G is a complex reductive affine algebraic group, that admits a…