Infinite-dimensional Representations Research Articles

Positive-Energy Dirac Particles and Dark Matter

The relativistic positive-energy wave equation proposed by P. Dirac in 1971 is an old but largely forgotten subject. The purpose of this note is to speculate that particles described by this equation (called here Dirac particles) are natural candidates for the dark matter. The reasoning is based on a fact that the internal structure of such particles simply prohibits their interaction with electromagnetic fields (at least with the minimal coupling) which is exactly what is required for dark matter. Dirac particles have quite unusual properties. In particular, they are transformed by an infinite-dimensional representation of the homogeneous Lorentz group, which clearly distinguishes them from all known elementary particles described by finite-dimensional representations and hints to a physics beyond the Standard Model. To clarify the topic, a brief review of the main features of the above-mentioned Dirac equation is given.

A distant descendant of the six-vertex model

In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local “spin variables” can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent “dual” formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra Uq(slˆ(2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.

A Q-Operator for Open Spin Chains II: Boundary Factorization

One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine sl2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}_2$$\\end{document} and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.

Some Infinite-Dimensional Representations of Certain Coxeter Groups

Some Infinite-Dimensional Representations of Certain Coxeter Groups

Generating operators of symmetry breaking—From discrete to continuous

Based on the “generating operator” of the Rankin–Cohen brackets introduced in Kobayashi–Pevzner [arXiv:2306.16800], we present a method to construct various fundamental operators with continuous parameters such as invariant trilinear forms on infinite-dimensional representations, the Fourier and the Poisson transforms on the anti-de Sitter space, and integral symmetry breaking operators for the fusion rules, among others, out of a countable set of differential symmetry breaking operators.

A simple and general axiomatization of average utility maximization for infinite streams

This paper provides the most general preference axiomatization of average utility (AU) maximization over infinite sequences presently available, reaching almost complete generality. The only restriction is that all periodic sequences should be contained in the domain. Infinite sequences may designate intertemporal outcomes streams where AU models patience, welfare allocations where AU models fairness, or decisions under ambiguity where AU models complete ignorance. As a methodological contribution, this paper shows that infinite-dimensional representations can be simpler, rather than more complex, than finite-dimensional ones. Infinite dimensions provide a richness that may be convenient rather than cumbersome. In particular, (empirically problematic) continuity assumptions are not needed in our axiomatization. Continuity is optional.

On polynomial symmetry algebras underlying superintegrable systems in Darboux spaces

We review three different approaches to polynomial symmetry algebras underlying superintegrable systems in Darboux spaces. The first method consists of using deformed oscillator algebra to obtain finite-dimensional representations of quadratic algebras. This allow one to gain information on the spectrum of the superintegrable systems. The second method has similarities with the induced module construction approach in the context of Lie algebras and can be used to construct infinite dimensional representations of the symmetry algebras. Explicit construction of these representations is a non-trivial task due to the non-linearity of the polynomial algebras. This method allows the construction of states of the superintegrable systems beyond the reach of separation of variables. As a result, we are able to construct a large number of states in terms of Airy, Bessel and Whittaker functions which would be difficult to obtain in other ways. We also discuss the third approach which is based on the notion of commutants of subalgebras in the enveloping algebra of a Poisson algebra or a Lie algebra. This allows us to discover new superintegrable models in the Darboux spaces and to construct their integrals and symmetry algebras via polynomials in the enveloping algebras.

An infinite-dimensional representation of the Ray-Knight theorems

The classical Ray-Knight theorems for Brownian motion determine the law of its local time process either at the first hitting time of a given value a by the local time at the origin, or at the first hitting time of a given position b by Brownian motion. We extend these results by describing the local time process jointly for all a and all b, by means of stochastic integral with respect to an appropriate white noise. Our result applies to µ-processes, and has an immediate application: a µ-process is the height process of a Feller continuous-state branching process (CSBP) with immigration (Lambert [10]), whereas a Feller CSBP with immigration satisfies a stochastic differential equation driven by a white noise (Dawson and Li [7]); our result gives an explicit relation between these two descriptions and shows that the stochastic differential equation in question is a reformulation of Tanaka's formula.

Infinite dimensional representations of orthogonal groups of quadratic forms with finite index

Infinite dimensional representations of orthogonal groups of quadratic forms with finite index

Infinite-dimensional representations of cubic and quintic algebras and special functions

Infinite-dimensional representations of cubic and quintic algebras and special functions

Banach space representations of Drinfeld–Jimbo algebras and their complex-analytic forms

We prove that every nondegenerate Banach space representation of the Drinfeld–Jimbo algebra Uq(g) of a semisimple complex Lie algebra g is finite dimensional when |q|≠1. As a corollary, we find an explicit form of the Arens–Michael envelope of Uq(g), which is similar to that of U(g) obtained by Joseph Taylor in 1970s. In the case when g=sl2, we also consider the representation theory of the corresponding analytic form, the Arens–Michael algebra U˜(sl2)ℏ (with eℏ=q) and show that it is simpler than for Uq(sl2). For example, all irreducible continuous representations of U˜(sl2)ℏ are finite dimensional for every admissible value of the complex parameter ℏ, while Uq(sl2) has a topologically irreducible infinite-dimensional representation when |q|=1 and q is not a root of unity.

Second cohomology groups of the Hopf * -algebras associated to universal unitary quantum groups

We compute the second (and the first) cohomology groups of $^*$-algebras associated to the universal quantum unitary groups of not neccesarily Kac type, extending our earlier results for the free unitary group $U_d^+$. The extended setup forces us to use infinite-dimensional representations to construct the cocycles.

Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.

Finite Dimensional Koopman Form of Polynomial Nonlinear Systems*

The Koopman framework is a popular approach to transform a finite dimensional nonlinear system into an infinite dimensional, but linear model through a lifting process using so-called observable functions. While there is an extensive theory on infinite dimensional representations in the operator sense, there are few constructive results on how to select the observables to realize them. When it comes to the possibility of finite Koopman representations, which are highly important from a practical point of view, there is no constructive theory. Hence, in practice, often a data-based method and ad-hoc choice of the observable functions is used. When truncating to a finite number of basis, there is also no clear indication of the introduced approximation error. In this paper, we propose a systematic method to compute the finite dimensional Koopman embedding of a specific class of polynomial nonlinear systems in continuous-time, such that the embedding can fully represent the dynamics of the nonlinear system without any approximation.

Infinite-dimensional representation of nD nonlinear state space systems

Infinite-dimensional representation of nD nonlinear state space systems

Searching for gravity without a metric

Recently it has been explicitly shown how a theory with global $GL(d,\mathbb{R})$ coordinate (affine) invariance which is spontaneously broken down to its Lorentz subgroup will have as its Goldstone fields enough degrees of freedom to create a metric and a covariant derivative arXiv:1105.5848. Such a theory would constitute an effective theory of gravity. So far however, no explicit theory has been found which exhibits this symmetry breaking pattern, mainly due to the difficulty of even writing down a $GL(d,\mathbb{R})$ invariant actions in the absence of a metric. In this paper we explicitly construct an affine generalization of the Dirac action employing infinite dimensional spinorial representations of the group. This implies that it is built from an infinite number of spinor Lorentz multiplets. We introduce a systematic procedure for obtaining $GL(d,\mathbb{R})$ invariant interaction terms to obtain quite general interacting models. Such models have order operators whose expectation value can break affine symmetry to Poincar\'{e} symmetry. We discuss possible interactions and mechanisms for this symmetry breaking to occur, which would provide a dynamical explanation of the Lorentzian signature of spacetime.

Error-feedback temperature regulation for a reverse flow reactor driven by a distributed parameter exosystem

This work is devoted to solve the state and error-feedback control problems for a catalytic reverse flow reactor (CFRR), which is modeled by nonlinear partial differential equations (PDEs). These two regulation problems will be solved based on the linearized infinite-dimensional representation. The objective is to track a desired output reference under the presence of disturbances. Both the reference trajectory and the disturbance profiles are generated by a distributed parameter exosystem. First, a state feedback stabilizing regulator is designed which drives the process output to a reference trajectory. The second main aim is to develop a dynamical controller that uses the tracking error as an input. Furthermore, it has been demonstrated that the closed-loop plant is exponentially stable and the tracking error asymptotically goes to zero. The developed regulators are evaluated through numerical simulations for the case study of methane combustion.

The Gelfand–Tsetlin basis for infinite-dimensional representations of

We consider the problem of determination of the Gelfand–Tsetlin basis for unitary principal series representations of the Lie algebra . The Gelfand–Tsetlin basis for an infinite-dimensional representation can be defined as the basis of common eigenfunctions of corner quantum minors of the corresponding L-operator. The construction is based on the induction with respect to the rank of the algebra: an element of the basis for is expressed in terms of a Mellin–Barnes type integral of an element of the basis for . The integration variables are the parameters (in other words, the quantum numbers) setting the eigenfunction. Explicit results are obtained for ranks 3 and 4, and the orthogonality of constructed sets of basis elements is demonstrated. For the kernel of the integral is expressed in terms of gamma-functions of the parameters of eigenfunctions, and in the case of —in terms of a hypergeometric function of the complex field at unity. The formulas presented for an arbitrary rank make it possible to obtain the system of finite-difference equations for the kernel. They include expressions for the quantum minors of L-operator via the minors of L-operator for the principal series representations, as well as formulas for action of some non-corner minors on the eigenfunctions of corner ones. The latter hold for any representation of (not only principal series) in which the corner minors of the L-operator can be diagonalized.

Alpert multi-wavelets for functional inverse problems: direct optimization and deep learning

Computational engineering models often contain unknown entities (e.g. parameters, initial and boundary conditions) that require estimation from other measured observable data. Estimating such unknown entities is challenging when they involve spatio-temporal fields because such functional variables often require an infinite-dimensional representation. We address this problem by transforming an unknown functional field using Alpert wavelet bases and truncating the resulting spectrum. Hence the problem reduces to the estimation of few coefficients that can be performed using common optimization methods. We apply this method on a one-dimensional heat transfer problem where we estimate the heat source field varying in both time and space. The observable data is comprised of temperature measured at several thermocouples in the domain. This latter is composed of either copper or stainless steel. The optimization using our method based on wavelets is able to estimate the heat source with an error between 5% and 7%. We analyze the effect of the domain material and number of thermocouples as well as the sensitivity to the initial guess of the heat source. Finally, we estimate the unknown heat source using a different approach based on deep learning techniques where we consider the input and output of a multi-layer perceptron in wavelet form. We find that this deep learning approach is more accurate than the optimization approach with errors below 4%.

Multiplicity in restricting small representations

We give a geometric criterion for the bounded multiplicity property of “small” infinite-dimensional representations of real reductive Lie groups in both induction and restrictions. In particular, for a reductive symmetric pair $(G,H)$, we determine the reductive subgroups $G'$ having the property that any irreducible $H$-distinguished admissible representations of $G$ are of bounded multiplicity when restricted to $G'$.