Related papers: Noncommutative Symmetric Functions and the Inversi…
Let $K$ be any unital commutative $\mathbb Q$-algebra and $z=(z_1, ..., z_n)$ commutative or noncommutative free variables. Let $t$ be a formal parameter which commutes with $z$ and elements of $K$. We denote uniformly by $\kzz$ and…
Let $K$ be any unital commutative $\bQ$-algebra and $W$ any non-empty subset of $\bN^+$. Let $z=(z_1, ..., z_n)$ be commutative or noncommutative free variables and $t$ a formal central parameter. % Denote uniformly by $\kzz$ and $\kttzz$…
This paper is the first of a sequence papers ([Z4]--[Z7]) on the {\it ${\mathcal N}$CS $(\text{noncommutative symmetric})$ systems} over differential operator algebras in commutative or noncommutative variables ([Z4]); the ${\mathcal N}$CS…
This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables.…
Recent work on perturbative quantum field theory has led to much study of the Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf algebra of rooted trees, had already been studied by algebraists. L. Foissy introduced a…
This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric…
In this paper, we construct explicitly a noncommutative symmetric (${\mathcal N}$CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the ${\mathcal N}$CS system formed by the generating…
Let $z=(z_1, z_2, ..., z_n)$ be noncommutative free variables and $t$ a formal parameter which commutes with $z$. Let $k$ be any unital integral domain of any characteristic and $F_t(z)=z-H_t(z)$ with $H_t(z)\in {k[[t]]< < z >>}^{\times n}$…
In the last decennia two generalizations of the Hopf algebra of symmetric functions have appeared and shown themselves important, the Hopf algebra of noncommutative symmetric functions NSymm and the Hopf algebra of quasisymmetric functions…
We argue that Hopf-algebra deformations of symmetries -- as encountered in non-commutative models of quantum spacetime -- carry an intrinsic content of $operator$ $entanglement$ that is enforced by the coproduct-defined notion of composite…
The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the multiplication, while the second group is the…
The goal of this paper is to study the structure of noncommutative weighted shifts, their properties, and to understand their role as models (up to similarity) for $n$-tuples of operators on Hilbert spaces as well as their implications to…
Let us fix a positive integer $\nu>1$. For each positive integer $n>1$, we consider a normal supercharacter theory $\mathcal{S}_n$ of $G_n$, where $G_n$ is the direct-product of $n-1$ copies of the cyclic group of order $\nu$. Then we endow…
We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions which was recently studied as a vector space by Rosas and Sagan. The bases for this algebra are indexed by set partitions. We show that there…
In the Hopf algebra of symmetric functions, Sym, the basis of Schur functions is distinguished since every Schur function is isomorphic to an irreducible character of a symmetric group under the Frobenius characteristic map. In this note we…
The ring of symmetric functions carries the structure of a Hopf algebra. When computing the coproduct of complete symmetric functions $h_\lambda$ one arrives at weighted sums over reverse plane partitions (RPP) involving binomial…
Let $\CRF_S$ denote the category of $S$-colored rooted forests, and $\H_{\CRF_S}$ denote its Ringel-Hall algebra as introduced in \cite{KS}. We construct a homomorphism from a $K^+_0 (\CRF_S)$--graded version of the Hopf algebra of…
Let T:=[T_1,..., T_n] be an n-tuple of operators on a Hilbert space such that T is a completely non-coisometric row contraction. We establish the existence of a "one-to-one" correspondence between the joint invariant subspaces under…
In this paper, using a Hopf-algebraic method, we construct deformed Poincar\'e SUSY algebra in terms of twisted (Hopf) algebra. By adapting this twist deformed super-Poincar\'e algrebra as our fundamental symmetry, we can see the…
The main objects of study in this paper are those functionals that are analytic in the sense that they annihilate the non-commutative disc algebra. In the classical univariate case, a theorem of F. and M. Riesz implies that such functionals…