Unitary Representations Research Articles

Covariant projective representations of Hilbert–Lie groups

Abstract Hilbert–Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. In this paper, we study unitary representations of these groups from various perspectives. First, we address norm-continuous, also called bounded, representations. These are well known for simple groups, but the general picture is more complicated. Our first main result is a characterization of the discrete decomposability of all bounded representations in terms of boundedness of the set of coroots. We also show that bounded representations of type II and III exist if the set of coroots is unbounded. Second, we use covariance with respect to a one-parameter group of automorphisms to implement some regularity. Here we develop some perturbation theory based on half-Lie groups that reduces matters to the case where a “maximal torus” is fixed, so that compatible weight decompositions can be studied. Third, we extend the context to projective representations which are covariant for a one-parameter group of automorphisms. Here important families of representations arise from “bounded extremal weights”, and for these, the corresponding central extensions can be determined explicitly, together with all one-parameter groups for which a covariant extension exists.

Crossing symmetry and the crossing map

We introduce and study the crossing map, a closed linear map acting on operators on the tensor square of a given Hilbert space that is inspired by the crossing property of quantum field theory. This map turns out to be closely connected to Tomita–Takesaki modular theory. In particular, crossing symmetric operators, namely those operators that are mapped to their adjoints by the crossing map, define endomorphisms of standard subspaces. Conversely, such endomorphisms can be integrated to crossing symmetric operators. We also investigate the relation between crossing symmetry and natural compatibility conditions with respect to unitary representations of certain symmetry groups, and furthermore introduce a generalized crossing map defined by a real object in an abstract [Formula: see text]-tensor category, not necessarily consisting of Hilbert spaces and linear maps. This latter crossing map turns out to be closely related to the (unshaded, finite-index) subfactor theoretical Fourier transform. Lastly, we provide families of solutions of the crossing symmetry equation, solving in addition the categorical Yang–Baxter equation, associated with an arbitrary Q-system.

Covariant Representation of Spin and Entanglement—A Review and Reformulation

A consistent theory of quantum entanglement requires that constituent single-particle states belong to the same Hilbert space, the coherent eigenstates of a complete set of operators in a given representation, defined with respect to a shared continuous parameterization. Formulating such eigenstates for a single relativistic particle with spin, and applying them to the description of many-body states, presents well-known challenges. In this paper, we review the covariant theory of relativistic spin and entanglement in a framework first proposed by Stueckelberg and developed by Horwitz, Piron, et al. This approach modifies Wigner’s method by introducing an arbitrary timelike unit vector nμ and then inducing a representation of SL(2,C), based on pμ rather than on the spacetime momentum. Generalizing this approach, we construct relativistic spin states on an extended phase space {(xμ,pμ),(ζμ,πμ)}, inducing a representation on the momentum πμ, thus providing a novel dynamical interpretation of the timelike unit vector nμ=πμ/M. Studying the unitary representations of the Poincaré group on the extended phase space allows us to define basis quantities for quantum states and develop the gauge invariant electromagnetic Hamiltonian in classical and quantum mechanics. We write plane wave solutions for free particles and construct stable singlet states, and relate these to experiments involving temporal interference, analogous to the spatial interference known from double slit experiments.

The $p$-colorable subgroup of Thompson’s group

V. F. R. Jones introduced a method of constructing knots and links from elements of Thompson’s group F by using its unitary representations. He also defined several subgroups of F as the stabilizer subgroups and some researchers studied them algebraically. One of the subgroups is called the 3 -colorable subgroup \mathcal{F} , and the authors proved that all knots and links obtained from non-trivial elements of \mathcal{F} are 3 -colorable. In this paper, for any odd integer p greater than two, we define the p -colorable subgroup of F whose non-trivial elements yield p -colorable knots and links and show it is isomorphic to a certain Brown–Thompson group.

Groups without unitary representations, submeasures, and the escape property

Groups without unitary representations, submeasures, and the escape property

Quantum Computational Complexity and Symmetry

Testing the symmetries of quantum states and channels provides a way to assess their usefulness for different physical, computational, and communication tasks. Here, we establish several complexity-theoretic results that classify the difficulty of symmetry-testing problems involving a unitary representation of a group and a state or a channel that is being tested. In particular, we prove that various such symmetry-testing problems are complete for BQP (bounded-error quantum polynomial time), QMA (quantum Merlin-Arthur), QSZK (quantum statistical zero-knowledge), QIP(2) (two-message quantum interactive proofs), QIP_EB(2) (two-message quantum interactive proofs restricted to entanglement-breaking provers), and QIP (quantum interactive proofs), thus spanning the prominent classes of the quantum interactive proof hierarchy and forging a non-trivial connection between symmetry and quantum computational complexity. Finally, we prove the inclusion of two Hamiltonian symmetry-testing problems in QMA and QAM, while leaving it as an intriguing open question to determine whether these problems are complete for these classes.

Quantum scalar field on fuzzy de Sitter space. Part I. Field modes and vacua

We study a scalar field on a noncommutative model of spacetime, the fuzzy de Sitter space, which is based on the algebra of the de Sitter group SO(1, d) and its unitary irreducible representations. We solve the Klein-Gordon equation in d = 2, 4 and show, using a specific choice of coordinates and operator ordering, that all commutative field modes can be promoted to solutions of the fuzzy Klein-Gordon equation. To explore completeness of this set of modes, we specify a Hilbert space representation and study the matrix elements (integral kernels) of a scalar field: in this way the complete set of solutions of the fuzzy Klein-Gordon equation is found. The space of noncommutative solutions has more degrees of freedom than the commutative one, whenever spacetime dimension is d > 2. In four dimensions, the new non-geometric, internal modes are parametrised by S2 × W, where W is a discrete matrix space. Our results pave the way to analysis of quantum field theory on the fuzzy de Sitter space.

A note on the standard zero-free region for [formula omitted]-functions

In this short note, we establish a standard zero-free region for a general class of L-functions for which their logarithms have coefficients with nonnegative real parts, including the Rankin–Selberg L-functions for unitary cuspidal automorphic representations.

Van der Corput’s difference theorem for amenable groups and the left regular representation

We establish a connection between two variants of van der Corput’s Difference Theorem (vdCDT) for countably infinite amenable groups G and the ergodic hierarchy of mixing properties of a unitary representation U of G. In particular, we show that one variant of vdCDT corresponds to subrepresentations of the left regular representation, and another variant of vdCDT corresponds to the absence of finite dimensional subrepresentations. We then obtain applications for measure preserving actions of countably infinite abelian groups.

Hankel wavelet multiplier associated with the unitary representation

In this paper, the boundedness and compactness of the Hankel wavelet multiplier associated with the unitary representation on [Formula: see text] space for [Formula: see text], are obtained by using the Hankel transform technique. The application of Hankel wavelet multipliers in form of the Landau–Pollak–Slepian operator is given. With the help of the Hankel wavelet multiplier, the Sobolev space associated with the Hankel transform is constructed and its various properties are studied.

Quasi-invariant states

In this paper, we develop the theory of quasi-invariant (respectively, strongly quasi-invariant) states under the action of a group [Formula: see text] of normal ∗-automorphisms of a ∗-algebra (or von Neumann algebra) [Formula: see text]. We prove that these states are naturally associated to left-[Formula: see text]-[Formula: see text]-cocycles. If [Formula: see text] is compact, the structure of strongly [Formula: see text]-quasi-invariant states is determined. For any [Formula: see text]-strongly quasi-invariant state [Formula: see text], we construct a unitary representation associated to the triple [Formula: see text]. We prove, under some conditions, that any quantum Markov chain with commuting, invertible and Hermitian conditional density amplitudes on a countable tensor product of type I factors is strongly quasi-invariant with respect to the natural action of the group [Formula: see text] of local permutations and we give the explicit form of the associated cocycle. This provides a family of nontrivial examples of strongly quasi-invariant states for locally compact groups obtained as inductive limit of an increasing sequence of compact groups.

Localised analytic torsion and relative analytic torsion for non compact Lie groups of type I

Let G be a (non compact) connected, simply connected, locally compact, second countable Lie group, either abelian or unimodular of type I, and let ρ be an irreducible unitary representation of G. Then, we define the analytic torsion of G localised at the representation ρ. The idea of considering localised invariants is due to Brodzki, Niblo, Plymen and Wright [5], and was exploited in [31] to define a localised eta function. Next, let Γ be a discrete co compact subgroup of G. We use the localised analytic torsion to define the relative analytic torsion of the pair (G,Γ), and we prove that the last coincides with the Lott L2 analytic torsion of a covering space. We illustrate these constructions analysing in some details two examples: the abelian case, and the case G=H, the Heisenberg group.

Decompositions of Hyperbolic Kac–Moody Algebras with Respect to Imaginary Root Groups

We propose a novel way to define imaginary root subgroups associated with (timelike) imaginary roots of hyperbolic Kac–Moody algebras. Using in an essential way the theory of unitary irreducible representation of covers of the group SO(2, 1), these imaginary root subgroups act on the complex Kac–Moody algebra viewed as a Hilbert space. We illustrate our new view on Kac–Moody groups by considering the example of a rank-two hyperbolic algebra that is related to the Fibonacci numbers. We also point out some open issues and new avenues for further research, and briefly discuss the potential relevance of the present results for physics and current attempts at unification.

Wave front sets of nilpotent Lie group representations

Let G be a nilpotent, connected, simply connected Lie group with Lie algebra g, and π a unitary representation of G. In this article we prove that the wave front set of π coincides with the asymptotic cone of the orbital support of π, i.e. WF(π)=AC(⋃σ∈supp(π)Oσ), where Oσ⊂ig⁎ is the coadjoint Kirillov orbit associated to the irreducible unitary representation σ∈Gˆ.

The G-invariant graph Laplacian part II: Diffusion maps

The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which is closed under the action of a continuous matrix group. An example of such a data set is images whose planar rotations are arbitrary. The G-invariant graph Laplacian, introduced in Part I of this work, admits eigenfunctions in the form of tensor products between the elements of the irreducible unitary representations of the group and eigenvectors of certain matrices. We employ these eigenfunctions to derive diffusion maps that intrinsically account for the group action on the data. In particular, we construct both equivariant and invariant embeddings, which can be used to cluster and align the data points. We demonstrate the utility of our construction in the problem of random computerized tomography.

Decomposition of the Unitary Representation of $SL_{2}(\mathbb{R})$ on the Upper Half Plane into Irreducible Components

The main purpose of the paper is to find the inversion formula for the covariant transform. This formula is equivalent to the decomposition of the unitary representation of $SL_{2}(\mathbb{R})$ on the upper half plane into irreducible components. We consider an eigenvalue $1+s^{2}$ of the Casimir operator: $$d\rho_{k}(C)=-4v^{2}\left(\partial_{u}^{2}+\partial_{v}^{2}\right),\quad \text{where}\, k=0.$$ To find the inversion formula, first we study the representation of $SL_{2}(\mathbb{R})$, $\rho_{k}$ and $\rho_{\tau}$, induced from the complex characters of $K$ and $N$ respectively. Then, we find the induced covariant transform $\mathcal{W}_{\varphi_{0}}^{\rho_{k}}$ with $N$-eigenvector to obtain a transform in the space $\FSpace{L}{2}(SL_{2}(\mathbb{R}/N)$. Thereafter, we compute the contravariant transform with $K$-eigenvector $$\mathcal{M}_{\phi_{0}}^{\rho_{\tau}}:\FSpace{L}{2}(SL_{2}(\mathbb{R})/N)\rightarrow \FSpace{L}{2}(SL_{2}(\mathbb{R})/K).$$

A universal framework for entanglement detection under group symmetry

One of the fundamental questions in quantum information theory is determining entanglement of quantum states, which is generally an NP-hard problem. In this paper, we prove that all PPT (π―A⊗πB) -invariant quantum states are separable if and only if all extremal unital positive (πB,πA) -covariant maps are decomposable where πA,πB are unitary representations of a compact group and π A is irreducible. Moreover, an extremal unital positive (πB,πA) -covariant map L is decomposable if and only if L is completely positive or completely copositive. We then apply these results to prove that all PPT quantum channels of the form Φ(ρ)=aTr(ρ)dIdd+bρ+cρT+(1−a−b−c)diag(ρ) are entanglement-breaking, and that there is no A-BC PPT-entangled (U⊗U―⊗U) -invariant tripartite quantum state. The former strengthens some conclusions in (Vollbrecht and Werner 2001 Phys. Rev. A 64 062307; Kopszak et al 2020 J. Phys. A: Math. Theor. 53 395306), and the latter resolves some open questions raised in (Collins et al 2018 Linear Algebra Appl. 555 398–411).

Genuine pro-$p$ Iwahori–Hecke algebras, Gelfand–Graev representations, and some applications

We study the Iwahori-component of the Gelfand–Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro- p level. Thus to begin we study the structure of a genuine pro- p Iwahori–Hecke algebra, establishing a Bernstein presentation. With this structure theory we first describe the pro- p part of the Gelfand–Graev representation and then the more subtle Iwahori part. For the first application we relate the Gelfand–Graev representation to the metaplectic representation of Sahi–Stokman–Venkateswaran, which conceptually realizes the Chinta–Gunnells action from the theory of Weyl group multiple Dirichlet series. For the second we compute the Whittaker dimension of the constituents of regular unramified principal series representations; for the third we do the same for unitary unramified principal series representations.

Mini-Workshop: Standard Subspaces in Quantum Field Theory and Representation Theory

Real standard subspaces of complex Hilbert spaces are long known to provide the right language for Tomita–Takesaki modular theory of von Neumann algebras. In recent years they have also become an object of prominent interest in mathematical quantum field theory (QFT) and unitary representation theory of Lie groups. This workshop brought together mathematicians and physicists working with standard subspaces, particularly in QFT (construction of QFT models, characterization of entropy, information-theoretic aspects), nets of standard subspaces on causal homogeneous spaces and aspects of reflection positivity and euclidean models related to standard subspaces and modular theory.

Unipotent Representations, Theta Correspondences, and Quantum Induction

In this paper, we construct unipotent representations for the real orthagonal groups and the metaplectic groups in the sense of Vogan. Our construction is based on quantum induction which involves the compositions of even number of theta correspondences. In particular, our results imply that there are irreducible unitary representations attached to each special nilpotent orbit.