Related papers: Quantum Matrices by Paths
The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step…
We take a graph theoretic approach to the problem of finding generators for those prime ideals of $\mathcal{O}_q(\mathcal{M}_{m,n}(\mathbb{K}))$ which are invariant under the torus action ($\mathbb{K}^*)^{m+n}$. Launois \cite{launois3} has…
We develop a new approach to the representation theory of quantum algebras supporting a torus action via methods from the theory of finite-state automata and algebraic combinatorics. We show that for a fixed number $m$, the torus-invariant…
In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of…
The notion of quantum matrix pairs is defined. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of…
The ideal I generated by the 2x2 quantum minors in the algebra A = O_q(M_{m,n}(k)) (the quantized coordinate algebra of mxn matrices) is investigated. Analogues of the First and Second Fundamental Theorems of Invariant Theory are proved. In…
Let q be a nonzero complex number that is not a root of unity. We give a criterion for (0) to be a primitive ideal of the algebra O_q(M_{m,n}) of quantum matrices. Next, we describe all height one primes of O_q(M_{m,n}); these two problems…
We study the topology of the prime spectrum of an algebra supporting a rational torus action. More precisely, we study inclusions between prime ideals that are torus-invariant using the $H$-stratification theory of Goodearl and Letzter on…
First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized $n\times r$ matrices as well as quantized factor algebras of $M_q(n)$ are analyzed. The latter are the quantized function…
This paper is an introduction to diagrammatic methods for representing quantum processes and quantum computing. We review basic notions for quantum information and quantum computing. We discuss topological diagrams and some issues about…
A complete determination of the prime ideals invariant under winding automorphisms in the generic 3 by 3 quantum matrix algebra is obtained. Explicit generating sets consisting of quantum minors are given for all of these primes, thus…
The main goal of the paper is to establish the existence of tensor product decompositions for those prime ideals P of the generic algebra A of quantum n by n matrices which are invariant under winding automorphisms of A. More specifically,…
In ArXiv:1604.00338[math.QA] we gave a complete combinatorial characterization of homogeneous quadratic identities for minors of quantum matrices. It was obtained as a consequence of results on minors of matrices of a special sort, the…
The essence of the path integral method in quantum physics can be expressed in terms of two relations between unitary propagators, describing perturbations of the underlying system. They inherit the causal structure of the theory and its…
Building a quantum analog of classical deep neural networks represents a fundamental challenge in quantum computing. A key issue is how to address the inherent non-linearity of classical deep learning, a problem in the quantum domain due to…
Quantum matrix geometry is the underlying geometry of M(atrix) theory. Expanding upon the idea of level projection, we propose a quantum-oriented non-commutative scheme for generating the matrix geometry of the coset space $G/H$. We employ…
We describe a new regularization of quantum field theory on the noncommutative torus by means of one-dimensional matrix models. The construction is based on the Elliott-Evans inductive limit decomposition of the noncommutative torus…
We give a complete combinatorial characterization of homogeneous quadratic relations of "universal character" valid for minors of quantum matrices (more precisely, for minors in the quantized coordinate ring $O_q(M_{m,n}(K))$ of $m\times n$…
In quantum field theory the path integral is usually formulated in the wave picture, i.e., as a sum over field evolutions. This path integral is difficult to define rigorously because of analytic problems whose resolution may ultimately…
Matrix representations of quantum operators are computationally complete but often obscure the structural topology of information flow within a quantum circuit \cite{nielsen2000}. In this paper, we introduce a generalized graph-theoretic…