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We study the homotopy type of the space of the unitary group [Formula: see text] of the uniform Roe algebra [Formula: see text] of [Formula: see text]. We show that the stabilizing map [Formula: see text] is a homotopy equivalence. Moreover, when [Formula: see text], we determine the homotopy type of [Formula: see text], which is the product of the unitary group [Formula: see text] (having the homotopy type of [Formula: see text] or [Formula: see text] depending on the parity of [Formula: see text]) of the Roe algebra [Formula: see text] and rational Eilenberg–MacLane spaces.

We study the elementary C*-algebra whose elements are the sum of a diagonal plus a compact operator. We describe the structure of the unitary group, the sets of ideals, automorhisms and projections., 24 pages

The third part of our survey series concerning the evaluation of matrix elements (MEs) of the unitary group generators and of their products in the electronic Gel’fand–Tsetlin basis of the two-column irreps of U(n)—which are essential in the unitary group approach (UGA) to the many-electron correlation problem as handled by the configuration interaction (CI) and the coupled cluster (CC) approaches—relies on what we refer to as the Green-Gould (G-G) approach. In addition to the CI and CC methods, the G-G formalism proved to be very helpful in a number of other tasks, particularly in handling of the spin-dependent operators, the density matrices, or partitioned basis sets adapted to a chosen group chain.

The objective of this series of papers is to survey important techniques for the evaluation of matrix elements (MEs) of unitary group generators and their products in the electronic Gel’fand–Tsetlin basis of two-column irreps of U(n) that are essential to the unitary group approach (UGA) to the many-electron correlation problem as handled by configuration interaction and coupled cluster approaches. Attention is also paid to the MEs of one-body spin-dependent operators and of their relationship to a standard spin-independent UGA formalism. The principal goal is to outline basic principles, concepts, and ideas without getting buried in technical details and thus to help an interested reader to follow the detailed developments in the original literature. In this first instalment we focus on tensorial techniques, particularly those designed specifically for UGA purposes, which exploit the spin-adapted tensorial analogues of the standard creation and annihilation operators of the ubiquitous second-quantization formalism. Subsequent instalments will address techniques based on the graphical methods of spin-algebras and on the Green–Gould polynomial formalism. In the “Appendix A” we then provide a succinct historical outline of the origins of the Lie group and Lie algebra concepts.

The purpose of this article is to give a new, explicit and elementary construction of the Fi22 in 2E6(2), using the notion of M-sets introduced by the second author, and properties of the generaliz...

We will introduce the notion of inductive limits of compact quantum groups as $W^*$-bialgebras equipped with some additional structures. We also formulate their unitary representation theories. Those give a more explicit representation-theoretic meaning to our previous study of quantized characters associated with a given inductive system of compact quantum groups. As a byproduct, we will give an explicit representation-theoretic interpretation to some transformations that play an important role in the analysis of $q$-central probability measures on the paths in the Gelfand-Tsetlin graph., 17 pages, second version (added an appendix)

String structures have played an important role in algebraic topology, via elliptic genera and elliptic cohomology, in differential geometry, via the study of higher geometric structures, and in physics, via partition functions. We extend the description of String structures from connected covers of the definite-signature orthogonal group ${\rm O}(n)$ to the indefinite-signature orthogonal group O(p, q), i.e. from the Riemannian to the pseudo-Riemannian setting. This requires that we work at the unstable level, which makes the discussion more subtle than the stable case. Similar, but much simpler, constructions hold for other noncompact Lie groups such as the unitary group U(p, q) and the symplectic group Sp(p, q). This extension provides a starting point for an abundance of constructions in (higher) geometry and applications in physics., 28 pages, major revisions, published version

In recent works, a link between group actions and information metrics on the space of faithful quantum states has been highlighted in particular cases. In this contribution, we give a complete discussion of this instance for the particular case of the qubit.

Algebraic approaches based on a discrete variable representation scheme (DVR) are proposed to describe interatomic effective potentials. The recently proposed unitary group approach (UGA) is identified as a discrete variable representation method among other more general methods discussed in this work. The basic idea consists in establishing an algebraic realisation of the coordinate and momentum in the framework of a dynamical group associated with a complete basis. In the context of the algebraic DVR methods, the discrete coordinate and momentum representations are obtained by diagonalising their matrix representation in a basis associated with a specific potential, in particular, harmonic oscillator, Morse and Pöschl-Teller potentials. This feature provides useful tools to obtain the matrix representation of the Hamiltonian in a simple form through the use of the transformation coefficients connecting the different bases. The proposed algebraic DVR approaches are applied and compared in the framework of 1,2 and 3-D systems. The different approaches are evaluated by considering the Morse, Pöschl-Teller, Kratzer-Fues, Deng-Fan, Varshni and Coulomb potentials., This work is partially supported by DGAPA-UNAM, Mexico (Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México), under project IN-212020, DGAPA UNAM for postdoctoral scholarship (Facultad de Química) to second author, by Spanish Consejería de Economía, Innovación, Ciencia y Empleo, Junta de Andalucía [grant numbers FQM-160], and by the Spanish Ministerio de Ciencia e Innovación, ref. FIS2017-88410-P, PID2019-104002GB-C22, and FPA2016-77689-C2-1-R and by the European Commission, ref. H2020-INFRAIA-2014-2015 (ENSAR2). First author is also grateful for the scholarship (Posgrado en Ciencia e Ingeniería de Materiales) provided by CONACyT, México.

The exact solution of the interacting many-body system is important and is difficult to solve. In this paper, we introduce a group method to solve the interacting many-body problem using the relation between the permutation group and the unitary group. We prove a group theorem first, then using the theorem, we represent the Hamiltonian of the interacting many-body system by the Casimir operators of unitary group. The eigenvalues of Casimir operators could give the exact values of energy and thus solve those problems exactly. This method maps the interacting many-body system onto an intermediate statistical representation. We give the relation between the conjugacy-class operator of permutation group and the Casimir operator of unitary group in the intermediate statistical representation, called the Gentile representation. Bose and Fermi cases are two limitations of the Gentile representation. We also discuss the representation space of symmetric and unitary group in the Gentile representation and give an example of the Heisenberg model to demonstrate this method. It is shown that this method is effective to solve interacting many-body problems., Comment: 11 pages, 0 figures

Identifying symmetries in data sets is generally difficult, but knowledge about them is crucial for efficient data handling. Here we present a method how neural networks can be used to identify symmetries. We make extensive use of the structure in the embedding layer of the neural network which allows us to identify whether a symmetry is present and to identify orbits of the symmetry in the input. To determine which continuous or discrete symmetry group is present we analyse the invariant orbits in the input. We present examples based on rotation groups $SO(n)$ and the unitary group $SU(2).$ Further we find that this method is useful for the classification of complete intersection Calabi-Yau manifolds where it is crucial to identify discrete symmetries on the input space. For this example we present a novel data representation in terms of graphs., 20 pages, 9 figures

In this paper, we give an affirmative answer to Tingley's problem between finite von Neumann algebras. To do this, we see that every norm closed proper face of the unit ball of a unital C ⁎ -algebra can be represented as the intersection of a family of maximal proper faces.

In this paper we consider the new concept of the quaternionic unitary representation of a locally compact group to the unitary group of a quaternionic Hilbert space, and study its properties. Also, we establish a continuous wavelet transform by means of a special case of such representations to extend the continuous wavelet transform related to semidirect product groups. The results may be useful in development of wavelet theory.

We show that, for a quadratic extension of p-adic fields, no cuspidal representation of the quasi-split unitary group admits a non-trivial linear form invariant by the symplectic subgroup. Our proof is purely local.

Twirling operations, which average a quantum state with respect to a unitary subgroup, have become a frequently-employed tool in quantum information processing. We investigate the efficient implementation of twirling operations with minimal resources, without necessitating the ability to perform all possible unitary operations on the quantum system of interest. We present a general algebraic method allowing us to choose a set of - typically very few - unitary operators which, when applied randomly and repeatedly, produce the given twirling operation exponentially quickly. The method is applied to twirling operations for bipartite quantum systems with respect to the unitary group U ( d ) ⊗ U ( d ) , an essential ingredient in entanglement distillation protocols. In particular, we provide a complete classification of sets of unitary operators capable of performing twirling on two qubits. Moreover, we construct a generic set containing at most three unitary operators achieving the twirling operation for a general two-qudit system.

We provide a computer algebra package called Random Tensor Network Integrator (RTNI). It allows to compute averages of tensor networks containing multiple Haar-distributed random unitary matrices and deterministic symbolic tensors. Such tensor networks are represented as multigraphs, with vertices corresponding to tensors or random unitaries and edges corresponding to tensor contractions. Input and output spaces of random unitaries may be subdivided into arbitrary tensor factors, with dimensions treated symbolically. The algorithm implements the graphical Weingarten calculus and produces a weighted sum of tensor networks representing the average over the unitary group. We illustrate the use of this algorithmic tool on some examples from quantum information theory, including entropy calculations for random tensor network states as considered in toy models for holographic duality. Mathematica and Python implementations are supplied., Comment: Code available (for Mathematica and python) at https://github.com/MotohisaFukuda/RTNI

In this paper, firstly we explore some properties of K-frames in terms of decompositions of operators. Then we pay more attention to characterize the dual K-Bessel sequences of a given K-frames. Since the frame operator for a K-frame may not be invertible, there is no classical canonical dual for a K-frame. We introduce the concept of canonical dual K-Bessel sequence of a K-frame which generalizes the classical dual of a frame. We prove its existence and uniqueness and some other properties. We also give a simple way to construct new K-frames from given ones. For the K-frame generators of a unitary system, we give a condition such that a K-frame generator for a unitary group has a dual K-Bessel generator so that both of them share the same group structure. We characterize the K-frame generators in terms of operators and we also consider the construction of new K-frame generators from given ones.

It is well known that a Gabor representation on L 2 ( R d ) L^{2}(\mathbb {R}^{d}) admits a frame generator h ∈ L 2 ( R d ) h\in L^{2}(\mathbb {R}^{d}) if and only if the associated lattice satisfies the Beurling density condition, which in turn can be characterized as the “trace condition” for the associated von Neumann algebra. It happens that this trace condition is also necessary for any projective unitary representation of a countable group to admit a frame vector. However, it is no longer sufficient for general representations, and in particular not sufficient for Gabor representations when they are restricted to proper time-frequency invariant subspaces. In this short note we show that the condition is also sufficient for a large class of projective unitary representations, which implies that the Gabor density theorem is valid for subspace representations in the case of irrational types of lattices.

We present a family of unitary representations of a group of diffeomorphisms of a finite-dimensional real Euclidean space using a family of quasi-invariant measures. In the one-dimensional case, for a special kind of group diffeomorphisms of the halfline, we prove the irreducibility of the representations thus obtained.

Let F be a field with char F ≠ 2 and |F| > 9, and let B2n(F) be the standard Borel subgroup of the unitary group U2n(F) over F. For n ≥ 3, we obtain a complete description of all bijective maps preserving commutators on B2n(F).

In this paper we determine the structure of all so-called generalized isometries of the special unitary group which are transformations that respect any member of a large collection of generalized distance measures.

We study Whittaker--Fourier coefficients of automorphic forms on a quasi-split unitary group. We reduce the analogue of the Ichino--Ikeda conjectures to a conjectural local statement using the descent method of Ginzburg--Rallis--Soudry., Comment: In honor of Jim Cogdell for his 60th birthday

International audience; We provide a mathematically proven, simple and efficient algorithm to build localised Wannier functions, with the only requirement that the system has vanishing Chern numbers. Our algorithm is able to build localised Wannier for topological insulators such as the Kane-Mele model. It is based on an explicit and constructive proof of homotopies for the unitary group. We provide numerical tests validating the methods for several systems, including the Kane-Mele model.

The Langlands classification theorem describes all admissible representations of a reductive group $G$ in terms of the tempered representations of Levi subgroups of $G$. I will describe work with Susana Salamanca-Riba that provides (conjecturally) a similar description of the unitary representations of $G$ in terms of certain very special unitary representations of Levi subgroups.

The Killing form induces a 2-plectic structure on a compact semisimple Lie group. The associated Lie group of canonical transformations (2-plectomorphisms) is compact. This 2-plectic structure induces a Cartan connection on the Lie group. The curvature and torsion tensor of this connection have been calculated for the special unitary Lie group S U ( 3 ) . It is shown that the homogeneous spaces S U ( 3 ) S U ( 2 ) , S U ( 3 ) S 1 and S U ( 3 ) T 2 also, admit 2-plectic structures, which are induced by closed left invariant 3-forms on S U ( 3 ) , whereas S U ( 3 ) U ( 2 ) and S U ( 3 ) S O ( 3 ) do not admit such 2-plectic structures.

Multiple basic hypergeometric series associated with the unitary group U ( n + 1 ) have been investigated by various authors. The purpose of this paper is to obtain a U ( n + 1 ) analogue of the Agarwal–Andrews–Bressoud Bailey lattice and to give some applications.

In this paper, we construct noncommutative algebras over a square-closed field such that the unitary linear groups over these algebras decompose to nontrivial free products. In particular, we give an example in which the elementary unitary subgroup belongs to one of these factors.

We introduce a twisted relative trace formula which simultaneously generalizes the twisted trace formula of Langlands et.al. (in the quadratic case) and the relative trace formula of Jacquet and Lai. Certain matching statements relating this twisted relative trace formula to a relative trace formula are also proven (including the relevant undamental lemma in the "biquadratic case"). Using recent work of Jacquet, Lapid and their collaborators and the Rankin-Selberg integral representation of the Asai $L$-function (obtained by Flicker using the theory of Jacquet, Piatetskii-Shapiro, and Shalika), we give the following application: Let $E/F$ be a totally real quadratic extension with $\langle \sigma \rangle=\mathrm{Gal}(E/F)$, let $U^{\sigma}$ be a quasi-split unitary group with respect to a CM extension $M/F$, and let $U:=\mathrm{Res}_{E/F}U^{\sigma}$. Under suitable local hypotheses, we show that a cuspidal cohomological automorphic representation $\pi$ of $U$ whose Asai $L$-function has a pole at the edge of the critical strip is nearly equivalent to a cuspidal cohomological automorphic representation $\pi'$ of $U$ that is $U^{\sigma}$-distinguished in the sense that there is a form in the space of $\pi'$ admitting a nonzero period over $U^{\sigma}$. This provides cohomologically nontrivial cycles of middle dimension on unitary Shimura varieties analogous to those on Hilbert modular surfaces studied by Harder, Langlands, and Rapoport., Comment: This is an old paper uploaded for arXival purposes. In section 10.4 of the published version the Jacobson density theorem was erroneously invoked where the Dixmier-Malliavin lemma should have been invoked. Moreover the proof of the (standard) argument proving Lemma 10.5 in the published version has been explicated

A significant background in linear algebra will be assumed in later chapters, and we’ll need a range of specific facts from that subject.

Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite group can be realized as a subrepresentation of a permutation representation. Thus, quantum-mechanical problems can be expressed in terms of permutation groups. This approach allows us to clarify the meaning of a number of physical concepts. Combining methods of computational group theory with Monte Carlo simulation we study a model based on representations of permutation groups., Comment: 8 pages, based on plenary lecture at Mathematical Modeling and Computational Physics 2017, Dubna, July 3--7, 2017

Let be the set of Toeplitz matrices. We describe the matrices in the unitary group such that

Number fields and global function fields have many similar properties. Both have many applications to cryptography and coding theory, and the main computational problems for number fields, such as computing the ring of integers and computing the class group and the unit group, have analogues over function fields. The complexity of the number field problems has been studied extensively and these problems have been the source of some exponential speedups by quantum computation. In this paper we study the analogous problems in function fields. We show that there are efficient quantum algorithms for computing the unit group, the class group and for solving the principal ideal problem in function fields of arbitrary degree. We show that compact representations exist, which allows us to show that the principal ideal problem is in NP. Unlike the number field case, we are also able to show that these compact representations can be computed efficiently.

The paper describes subgroups of the general linear group GL 4(D) over a quaternion division ring D that contain the special unitary group of degree 4 over a non-central subfield of D. The connections between these subgroups and the spinor groups of appropriate symmetric bilinear forms are established.

We show that finite elation Laguerre planes with a group of automorphisms acting two-transitively on the set of generators are Miquelian. In this paper we discuss in detail three series of groups of (twisted) Lie type that may possibly occur as socles of the two-transitive group induced on the set of generators; all other cases have been treated in a separate paper.

In this paper we describe all isometries on the special orthogonal group. As an application we give a form of spectrally multiplicative map on the special orthogonal group.

We present a symbolic manipulation algorithm for the efficient automated implementation of rigorously spin-free coupled cluster (CC) theories based on a unitary group parametrization. Due to the lack of antisymmetry of the unitary group generators under index permutations, all quantities involved in the equations are expressed in terms of non-antisymmetric tensors. Given two tensors, all possible contractions are first generated by applying Wick's theorem. Each term is then put down in the form of a non-antisymmetric Goldstone diagram by assigning its contraction topology. The subsequent simplification of the equations by summing up equivalent terms and their factorization by identifying common intermediates is performed via comparison of these contraction topologies. The definition of the contraction topology is completely general for non-antisymmetric Goldstone diagrams, which enables our algorithm to deal with noncommuting excitations in the cluster operator that arises in the unitary group based CC formulation for open-shell systems. The resulting equations are implemented in a new code, in which tensor contractions are performed by successive application of matrix-matrix multiplications. Implementation of the unitary group adapted CC equations for closed-shell systems and for the simplest open-shell case, i.e., doublets, is discussed, and representative calculations are presented in order to assess the efficiency of the generated codes.