Noncommuting Variables Research Articles

The paramodular Hecke algebra

AbstractWe give a presentation via generators and relations of the local graded paramodular Hecke algebra of prime level. In particular, we prove that the paramodular Hecke algebra is isomorphic to the quotient of the free $${{\mathbb {Z}}}$$ Z -algebra generated by four non-commuting variables by an ideal generated by seven relations. Using this description, we derive rationality results at the level of characters and give a characterization of the center of the Hecke algebra. Underlying our results are explicit formulas for the product of any generator with any double coset.

Computing the Noncommutative Inner Rank by Means of Operator-Valued Free Probability Theory

AbstractWe address the noncommutative version of the Edmonds’ problem, which asks to determine the inner rank of a matrix in noncommuting variables. We provide an algorithm for the calculation of this inner rank by relating the problem with the distribution of a basic object in free probability theory, namely operator-valued semicircular elements. We have to solve a matrix-valued quadratic equation, for which we provide precise analytical and numerical control on the fixed point algorithm for solving the equation. Numerical examples show the efficiency of the algorithm.

The Chromatic Symmetric Function of a Graph Centred at a Vertex

We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of $e$-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function $Y_G$ in noncommuting variables in this quotient algebra, which defines $y_{G : v}$, the chromatic symmetric function of a graph $G$ centred at a vertex $v$. We then apply our methods to $y_{G : v}$ and find new families of unit interval graphs that are $(e)$-positive, a stronger condition than classical $e$-positivity, thus confirming new cases of the $(3+1)$-free conjecture of Stanley and Stembridge. In our study of $y_{G : v}$, we also describe methods of constructing new $e$-positive graphs from given $(e)$-positive graphs and classify the $(e)$-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of $y_{G : v}$ to acyclic orientations of graphs, including a noncommutative refinement of Stanley's sink theorem.

Non-holomorphic modular forms from zeta generators

We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL(2, ℤ) known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the appearance of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non-commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown’s framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes.

Expanding the reach of quantum optimization with fermionic embeddings

Quadratic programming over orthogonal matrices encompasses a broad class of hard optimization problems that do not have an efficient quantum representation. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a generalization of binary quadratic programs to continuous, noncommutative variables. In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian, thereby enabling the study of this classical problem with the tools of quantum information. This embedding is accomplished by a new representation of orthogonal matrices as fermionic quantum states, which we achieve through the well-known double covering of the orthogonal group. Correspondingly, the embedded LNCG Hamiltonian is a two-body fermion model. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, a quantum analogue to classical semidefinite relaxations. In particular, when optimizing over the special orthogonal group our quantum relaxation obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas our quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution back into the feasible space, we propose rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this rounded quantum relaxation can produce high-quality approximations.

Non-Commutative Classical and Quantum Fractionary Cosmology: FRW Case

In this work, we will explore the effects of non-commutativity in fractional classical and quantum schemes using the flat Friedmann–Robertson–Walker (FRW) cosmological model coupled to a scalar field in the K-essence formalism. In previous work, we have obtained the commutative solutions in both regimes in the fractional framework. Here, we introduce non-commutative variables, considering that all minisuperspace variables qnci do not commute, so the symplectic structure was modified. In the quantum regime, the probability density presents a new structure in the scalar field corresponding to the value of the non-commutative parameter, in the sense that this probability density undergoes a shift back to the direction of the scale factor, causing classical evolution to arise earlier than in the commutative world.

Rational power series in several noncommuting variables and the Skolem–Mahler–Lech theorem

We generalize the Skolem–Mahler–Lech theorem for rational power series in several noncommuting variables having a slender support. This generalization gives a connection between the Skolem–Mahler–Lech theorem and the characterization of slender regular languages proved independently by Păun and Salomaa and by Shallit.

The degree one Laguerre–Pólya class and the shuffle-word-embedding conjecture

Abstract We discuss the class of functions, which are well approximated on compacta by the geometric mean of the eigenvalues of a unital (completely) positive map into a matrix algebra or more generally a type $II_1$ factor, using the notion of a Fuglede–Kadison determinant. In two variables, the two classes are the same, but in three or more noncommuting variables, there are generally functions arising from type $II_1$ von Neumann algebras, due to the recently established failure of the Connes embedding conjecture. The question of whether or not approximability holds for scalar inputs is shown to be equivalent to a restricted form of the Connes embedding conjecture, the so-called shuffle-word-embedding conjecture.

Combinatorial aspects of weighted free Poisson random variables

This paper will be devoted to the study of weighted (deformed) free Poisson random variables from the viewpoint of orthogonal polynomials and statistics of non-crossing partitions. A family of weighted (deformed) free Poisson random variables will be defined in a sense by the sum of weighted (deformed) free creation, annihilation, scalar, and intermediate operators with certain parameters on a weighted (deformed) free Fock space together with the vacuum expectation. We shall provide a combinatorial moment formula of non-commutative Poisson random variables. This formula gives us a very nice combinatorial interpretation to two parameters of weights. One can see that the deformation treated in this paper interpolates free and boolean Poisson random variables, their distributions and moments, and yields some conditionally free Poisson distribution by taking limit of the parameter.

A note on the image of polynomials on upper triangular matrix algebras

In the present paper we shall obtain the following result: Let n ≥ 2 and m ≥ 1 be integers. Let p ( x 1 , … , x m ) be a polynomial with zero constant term in non-commutative variables over an algebraically closed field K. Suppose that 1 ≤ r ≤ n − 1 , where r is the order of p. Then p ( T n ( K ) ) is a dense subset of T n ( K ) ( r − 1 ) (with respect to the Zariski topology). This is a solution of a problem presented by Gargate and de Mello and a supplement to a result obtained by Panja and Prasad in 2023.

Generalized Chromatic Functions

Abstract We define vertex-colourings for edge-partitioned digraphs, which unify the theory of $P$-partitions and proper vertex-colourings of graphs. We use our vertex-colourings to define generalized chromatic functions, which merge the chromatic symmetric and quasisymmetric functions of graphs and generating functions of $P$-partitions. Moreover, numerous classical bases of symmetric and quasisymmetric functions, both in commuting and noncommuting variables, can be realized as special cases of our generalized chromatic functions. We also establish product and coproduct formulas for our functions. Additionally, we construct the new Hopf algebra of $r$-quasisymmetric functions in noncommuting variables, and apply our functions to confirm its Hopf structure, and establish natural bases for it.

Powersum Bases in Quasisymmetric Functions and Quasisymmetric Functions in Non-Commuting Variables

We introduce a new powersum basis for the Hopf algebra of quasisymmetric functions that refines the powersum symmetric basis. Unlike the quasisymmetric powersums of types 1 and 2, our basis is defined combinatorially: its expansion in quasisymmetric monomial functions is given by fillings of matrices. This basis has a shuffle product, a deconcatenate coproduct, and has a change of basis rule to the quasisymmetric fundamental basis by using tuples of ribbons. We lift our quasisymmetric powersum P basis to the Hopf algebra of quasisymmetric functions in non-commuting variables by introducing fillings with disjoint sets. This new basis has a shifted shuffle product and a standard deconcatenate coproduct, and certain basis elements agree with the fundamental basis of the Malvenuto-Reutenauer Hopf algebra of permutations. Finally we discuss how to generalize these bases and their properties by using total orders on indices.

Noncommutative branch‐cut quantum gravity with a self‐coupling inflaton scalar field: The wave function of the Universe

AbstractThis article focuses on the implications of a noncommutative formulation of branch‐cut quantum gravity. Based on a mini‐superspace structure that obeys the noncommutative Poisson algebra, combined with the Wheeler–DeWitt equation and Hořava–Lifshitz quantum gravity, we explore the impact of a scalar field of the inflaton‐type in the evolution of the Universe's wave function. Taking as a starting point the Hořava–Lifshitz action, which depends on the scalar curvature of the branched Universe and its derivatives, the corresponding wave equations are derived and solved. The noncommutative quantum gravity approach adopted preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner Formalism. In this work we delve deeper into a mini‐superspace of noncommutative variables, incorporating scalar inflaton fields and exploring inflationary models, particularly chaotic and nonchaotic scenarios. We obtained solutions to the wave equations without resorting to numerical approximations. The results indicate that the noncommutative algebraic space captures low and high spacetime scales, driving the exponential acceleration of the Universe.

Matrix factorizations and pentagon maps

We propose a specific class of matrices that participate in factorization problems that turn out to be equivalent to constant and entwining (non-constant) pentagon, reverse-pentagon or Yang–Baxter maps, expressed in non-commutative variables. In detail, we show that factorizations of order N = 2 matrices of this specific class are equivalent to the homogeneous normalization map . From order N = 3 matrices, we obtain an extension of the homogeneous normalization map, as well as novel entwining pentagon, reverse-pentagon and Yang–Baxter maps.

Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability

We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability.

The image of polynomials and Waring type problems on upper triangular matrix algebras

Let p be a polynomial in non-commutative variables x1,x2,…,xn with constant term zero over an algebraically closed field K. The object of study in this paper is the image of this kind of polynomial over the algebra of upper triangular matrices Tm(K). We introduce a family of polynomials called multi-index p-inductive polynomials for a given polynomial p. Using this family we will show that, if p is a polynomial identity of Tt(K) but not of Tt+1(K), then p(Tm(K))⊆Tm(K)(t−1). Equality is achieved in the case t=1,m−1 and an example has been provided to show that equality does not hold in general. We further prove existence of d such that each element of Tm(K)(t−1) can be written as sum of d many elements of p(Tm(K)). It has also been shown that the image of Tm(K)× under a word map is Zariski dense in Tm(K)×.

Quantum Central Limit Theorems, Emergence of Classicality and Time-Dependent Differential Entropy.

We derive some quantum central limit theorems for the expectation values of macroscopically coarse-grained observables, which are functions of coarse-grained Hermitian operators consisting of non-commuting variables. Thanks to the Hermiticity constraints, we obtain positive-definite distributions for the expectation values of observables. These probability distributions open some pathway for the emergence of classical behaviours in the limit of an infinitely large number of identical and non-interacting quantum constituents. This is in contradistinction to other mechanisms of classicality emergence due to environmental decoherence and consistent histories. The probability distributions thus derived also enable us to evaluate the non-trivial time-dependence of certain differential entropies.

Bilinear matrix inequalities and polynomials in several freely noncommuting variables

Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear Matrix Inequalities.

Influence of simultaneous weak measurements in Heisenberg uncertainty relation

In quantum mechanics the simultaneous measurement of non-commuter variables has attracted widespread attention. Here we propose a scheme through simultaneously weak measurement to achieve the approximated weak values of two non-commuting variables. It can be observed that the measurement error is better than the Heisenberg uncertainty relation. Furthermore, the effects of free Hamiltonian and higher-order expansion of the time evolution operator are considered.

Noncommutative Nullstellensätze and Perfect Games

The foundations of classical algebraic geometry and real algebraic geometry are the Nullstellensatz and Positivstellensatz. Over the last two decades, the basic analogous theorems for matrix and operator theory (noncommutative variables) have emerged. This paper concerns commuting operator strategies for nonlocal games, recalls NC Nullstellensatz which are helpful, extends these, and applies them to a very broad collection of games. In the process, it brings together results spread over different literature studies, hence rather than being terse, our style is fairly expository. The main results of this paper are two characterizations, based on Nullstellensatz, which apply to games with perfect commuting operator strategies. The first applies to all games and reduces the question of whether or not a game has a perfect commuting operator strategy to a question involving left ideals and sums of squares. Previously, Paulsen and others translated the study of perfect synchronous games to problems entirely involving a $$*$$ -algebra. The characterization we present is analogous, but works for all games. The second characterization is based on a new Nullstellensatz we derive in this paper. It applies to a class of games we call torically determined games, special cases of which are XOR and linear system games. For these games, we show the question of whether or not a game has a perfect commuting operator strategy reduces to instances of the subgroup membership problem and, for linear systems games, we further show this subgroup membership characterization is equivalent to the standard characterization of perfect commuting operator strategies in terms of solution groups. Both the general and torically determined games characterizations are amenable to computer algebra techniques, which we also develop. For context, we mention that Positivstellensätze are behind the standard NPA upper bound on the score players can achieve for a game using a commuting operator strategy. This paper develops analogous NC real algebraic geometry which bears on perfect games.