Projective Unitary Representations Research Articles

On operator systems generated by reducible projective unitary representations of compact groups

We study reducible projective unitary representations $(U_g)_{g\in G}$ of a compact group $G$ in separable Hilbert spaces $H$. It is shown that there exist the projections $Q$ and $P$ for which ${\mathcal V}=\overline {span(U_gQU_g^*,\ g\in G)}$ is the operator system and $P{\mathcal V}P=\{{\mathbb C}P\}$. As an example, a bipartite Hilbert space $H={\mathfrak {H}}\otimes {\mathfrak {H}}$ is considered. In this case, the action of $(U_g)_{g\in G}$ has the property of transforming separable vectors to entangled.

Projective unitary representations of infinite-dimensional Lie groups

For an infinite-dimensional Lie group G modeled on a locally convex Lie algebra g, we prove that every smooth projective unitary representation of G corresponds to a smooth linear unitary representation of a Lie group extension G♯ of G. (The main point is the smooth structure on G♯.) For infinite-dimensional Lie groups G which are 1-connected, regular, and modeled on a barreled Lie algebra g, we characterize the unitary g-representations which integrate to G. Combining these results, we give a precise formulation of the correspondence between smooth projective unitary representations of G, smooth linear unitary representations of G♯, and the appropriate unitary representations of its Lie algebra g♯.

Phase-retrievable operator-valued frames and representations of quantum channels

We examine some connections among phase-retrievable (not necessarily self-adjoint) operator-valued frames, projective group representation frames and representations of quantum channels. We first present some characterizations of phase-retrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces, which generalize the known results for vector-valued frames, fusion frames and frames of Hermitian matrices. For an irreducible projective unitary representation of a finite group, the image system is automatically phase-retrievable and, moreover, it is a point-wisely tight operator-valued frames. We generalize this notion to more general operator-valued frames, and prove that point-wise tight operator-valued frames are exactly the ones that are right equivalent to operator-valued tight frames. For an operator system that represent a quantum channel, we show that phase-retrievability of the system is independent of the choices of the representations of the quantum channel.

Covariant projective representation of symplectic group on discrete phase space

The phase point operator Δ(q,p) is the quantum mechanical counterpart of the classical phase point (q,p). The discrete form of Δ(q,p) was formulated for an odd number of lattice points by Cohendet et al. and for an even number of lattice points by Leonhardt. Both versions have symplectic covariance, which is of fundamental importance in quantum mechanics. However, an explicit form of the projective unitary representation of the symplectic group that appears in the covariance relation is not yet known. We show in this paper the existence and uniqueness of the representation, and describe a method to construct it using the Euclidean algorithm.

Current superalgebras and unitary representations

In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is g=A⊗k, where k is a compact simple Lie superalgebra and A is a supercommutative associative (super)algebra; the crucial case is when A=Λs(R) is a Graßmann algebra. Since we are interested in projective representations, the first step consists in determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if k is a simple compact Lie superalgebra with k1≠{0}, then each (projective) unitary representation of Λs(R)⊗k factors through a (projective) unitary representation of k itself, and these are known by Jakobsen's classification. If k1={0}, then we likewise reduce the classification problem to semidirect products of compact Lie groups K with a Clifford–Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo and Varadarajan.

What Chern–Simons theory assigns to a point

We answer the questions, "What does Chern-Simons theory assign to a point?" and "What kind of mathematical object does Chern-Simons theory assign to a point?" Our answer to the first question is representations of the based loop group. More precisely, we identify a certain class of projective unitary representations of the based loop group [Formula: see text] We define the fusion product of such representations, and we prove that, modulo certain conjectures, the Drinfel'd center of that representation category of [Formula: see text] is equivalent to the category of positive energy representations of the free loop group [Formula: see text]† The abovementioned conjectures are known to hold when the gauge group is abelian or of type [Formula: see text] Our answer to the second question is bicommutant categories. The latter are higher categorical analogs of von Neumann algebras: They are tensor categories that are equivalent to their bicommutant inside [Formula: see text], the category of bimodules over a hyperfinite [Formula: see text] factor. We prove that, modulo certain conjectures, the category of representations of the based loop group is a bicommutant category. The relevant conjectures are known to hold when the gauge group is abelian or of type [Formula: see text].

Phase Retrievable Projective Representation Frames for Finite Abelian Groups

We consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recovering for the phase-retrieval problem. For a given multiplier $$\mu $$ of a finite abelian group G, we show that the representation dimension of any irreducible $$\mu $$ -projective representation of G is exactly the rank of the symmetric multiplier matrix associated with $$\mu $$ . With the help of this result we are able to prove that every irreducible $$\mu $$ -projective representation of a finite abelian group G admits a frame vector with the maximal span property, and obtain a complete characterization for all such frame vectors. Consequently the complement of the set of all the maximal span frame vectors for any projective unitary representation of any finite abelian group is Zariski-closed. These generalize some of the recent results about phase-retrieval with Gabor (or STFT) measurements.

Loop groups and noncommutative geometry

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level [Formula: see text] projective unitary positive-energy representations of any given loop group [Formula: see text]. The construction is based on certain supersymmetric conformal field theory models associated with [Formula: see text] in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.

Quantisation of Super Teichmüller Theory

We construct a quantisation of the Teichmuller spaces of super Riemann surfaces using coordinates associated to ideal triangulations of super Riemann surfaces. A new feature is the non-trivial dependence on the choice of a spin structure which can be encoded combinatorially in a certain refinement of the ideal triangulation. By constructing a projective unitary representation of the groupoid of changes of refined ideal triangulations we demonstrate that the dependence of the resulting quantum theory on the choice of a triangulation is inessential.

Local Equivalence of Representations of $${{\rm Diff}^+(S^1)}$$ Diff + ( S 1 ) Corresponding to Different Highest Weights

Let $c,h$ and $c,\tilde{h}$ be two admissible pairs of central charge and highest weight for ${\rm Diff}^+(S^1)$. It is shown here that the positive energy irreducible projective unitary representations $U_{c,h}$ and $U_{c,\tilde{h}}$ of the group ${\rm Diff}^+(S^1)$ are locally equivalent. This means that for any $I\Subset S^1$ open proper interval, there exists a unitary operator $W_I$ such that $W_I U_{c,h}(\gamma)W_I^* = U_{c,\tilde{h}}(\gamma)$ for all $\gamma \in {\rm Diff}^+(S^1)$ which act identically on $I^c\equiv S^1\setminus I$ (i.e. which can "displace" or "move" points only in $I$). This result extends and completes earlier ones that dealt with only certain regions of the "$c,h$-plane", and closes the gap in the full classification of superselection sectors of Virasoro nets.

A note on the density theorem for projective unitary representations

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.

Holomorphic automorphisms of noncommutative polyballs

In this paper, we study free holomorphic functions on regular polyballs and provide analogues of several classical results from complex analysis such as: Abel theorem, Hadamard formula, Cauchy inequality, Schwarz lemma, and maximum principle. These results are used together with a class of noncommutative Berezin transforms to obtain a complete description of the group Aut(B_n) of all free holomorphic automorphisms of the polyball. The abstract polyball B_n has a universal model S consisting of left creation operators acting on tensor products of full Fock spaces. We determine: the group of automorphisms of the Cuntz-Toeplitz algebra C*(S) which leaves invariant the noncommutative polyball algebra A_n; the group of unitarily implemented automorphisms of the polyball algebra A_n and the noncommutative Hardy algebra F_n^\infty, respectively. We prove that the free holomorphic automorphism group Aut(B_n) is a sigma-compact, locally compact topological group with respect to the topology induced by an appropriate metric. Finally, we obtain a concrete unitary projective representation of the topological group Aut(B_n)in terms of noncommutative Berezin kernels associated with regular polyballs.

Topological properties of spaces of projective unitary representations

<div data-canvas-width="10.602647058244418">Sea G un grupo de Lie compacto y conexo y PU(H) el grupo de operadores proyectivos e unitarios en un espacio de Hilbert separable e infinito dimensional H, provisto de la topología fuerte de operadores. Estudiamos el espacio homst(G,PU(H)) de homomorfismos continuos desde G a PU (H) que son estables, es decir homomorfismos cuyas representaciones inducidas contienen cada representación irreducible un número infinito de veces. Demostramos que las componentes conexas del espacio homst(G,PU(H)) están parametrizadas por las clases de isomorfía de extensiones centrales de</div><div>G por el grupo S1, y que cada componente conexa tiene por grupo fundamental al grupo hom(G,S1) y sus grupos de homotopía superiores son triviales. Estudiamos la aplicación conjugación PU(H)→homst(G,PU(H)),F→FαF−1 , demostramos que no tiene secciones locales y demostramos que para cualquier aplicación continua B→homst(G,PU(H)) con B paracompacto de dimensión paracompacta finita, los levantamientos locales a PU (H) sí existen. © Acad. Colomb. Cienc. Ex. Fis. Nat. 2016.</div>

The Higson–Mackey analogy for finite extensions of complex semisimple groups

In the 1970s, George Mackey pointed out an analogy that exists between tempered representations of semisimple Lie groups and unitary representations of associated semidirect product Lie groups. More recently, Nigel Higson refined Mackey’s analogy into a one-to-one correspondence for connected complex semisimple groups, and in doing so obtained a novel verification of the Baum–Connes conjecture with trivial coefficients for such groups. Here we extend Higson’s results to any Lie group with finitely many connected components whose connected component of the identity is complex semisimple. Our methods include Mackey’s description of unitary representations of group extensions involving projective unitary representations, as well as the notion of twisted crossed product C^* -algebra introduced independently by Green and Dang Ngoc.

Wilson loops on Riemann surfaces, Liouville theory and covariantization of the conformal group

The covariantization procedure is usually referred to the translation operator, that is the derivative. Here we introduce a general method to covariantize arbitrary differential operators, such as the ones defining the fundamental group of a given manifold. We focus on the differential operators representing the sl(2,R) generators, which in turn, generate, by exponentiation, the two-dimensional conformal transformations. A key point of our construction is the recent result on the closed forms of the Baker-Campbell-Hausdorff formula. In particular, our covariantization receipt is quite general. This has a deep consequence since it means that the covariantization of the conformal group is {\it always definite}. Our covariantization receipt is quite general and apply in general situations, including AdS/CFT. Here we focus on the projective unitary representations of the fundamental group of a Riemann surface, which may include elliptic points and punctures, introduced in the framework of noncommutative Riemann surfaces. It turns out that the covariantized conformal operators are built in terms of Wilson loops around Poincar\'e geodesics, implying a deep relationship between gauge theories on Riemann surfaces and Liouville theory.

Quantum Bochner’s theorem for phase spaces built on projective representations

Bochner’s theorem gives the necessary and sufficient conditions on a function such that its Fourier transform corresponds to a true probability density function. In the Wigner phase space picture, quantum Bochner’s theorem gives the necessary and sufficient conditions on a function such that it is a quantum characteristic function of a valid (and possibly mixed) quantum state and such that its Fourier transform is a true probability density. We extend this theorem to discrete phase space representations which possess enough symmetry. More precisely, we show that discrete phase space representations that are built on projective unitary representations of abelian groups, with a slight restriction on admissible two-cocycles, enable a quantum Bochner’s theorem.

Recurrence relations for finite-temperature correlators via AdS2/CFT1

This note is aimed at presenting a new algebraic approach to momentum-space correlators in conformal field theory. As an illustration we present a new Lie-algebraic method to compute frequency-space two-point functions for charged scalar operators of CFT$_{1}$ dual to AdS$_{2}$ black hole with constant background electric field. Our method is based on the real-time prescription of AdS/CFT correspondence, Euclideanization of AdS$_{2}$ black hole and projective unitary representations of the Lie algebra $\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{sl}(2,\mathbb{R})$. We derive novel recurrence relations for Euclidean CFT$_{1}$ two-point functions, which are exactly solvable and completely determine the frequency- and charge-dependences of two-point functions. Wick-rotating back to Lorentzian signature, we obtain retarded and advanced CFT$_{1}$ two-point functions that are consistent with the known results.

Fictitious Forces and Simulated Magnetic Fields in Rotating Reference Frames

We show that the Wigner-Bargmann program of grounding nonrelativistic quantum mechanics in the unitary projective representations of the Galilei group can be extended to include all noninertial reference frames. The key concept is the Galilean line group, the group of transformations that ties together all accelerating reference frames, and its representations. These representations are constructed under the natural constraint that they reduce to the well-known unitary, projective representations of the Galilei group when the transformations are restricted to inertial reference frames. This constraint can be accommodated only for a class of representations with a sufficiently rich cocycle structure. Unlike the projective representations of the Galilei group, these cocycle representations of the Galilean line group do not correspond to central extensions of the group. Rather, they correspond to a class of nonassociative extensions, known as loop prolongations, that are determined by three-cocycles. As an application, we show that the phase shifts due to the rotation of Earth that have been observed in neutron interferometry experiments and the rotational effects that lead to simulated magnetic fields in optical lattices can be rigorously derived from the representations of the loop prolongations of the Galilean line group.

POSITIVE ENERGY REPRESENTATIONS AND CONTINUITY OF PROJECTIVE REPRESENTATIONS FOR GENERAL TOPOLOGICAL GROUPS

AbstractLetGandTbe topological groups, α :T→ Aut(G) a homomorphism defining a continuous action ofTonGandG♯:=G⋊αTthe corresponding semidirect product group. In this paper, we address several issues concerning irreducible continuous unitary representations (π♯,${\mathcal{H}}$) ofG♯whose restriction toGremains irreducible. First, we prove that, forT=${\mathbb R}$, this is the case for any irreducible positive energy representation ofG♯, i.e. for which the one-parameter groupUt:= π♯(1,t) has non-negative spectrum. The passage from irreducible unitary representations ofGto representations ofG♯requires that certain projective unitary representations are continuous. To facilitate this verification, we derive various effective criteria for the continuity of projective unitary representations. Based on results of Borchers forW*-dynamical systems, we also derive a characterization of the continuous positive definite functions onGthat extend toG♯.

Duality properties in von Neumann algebras of projective unitary representations

Let π be a projective representation of a countable discrete group G on a Hilbert space H. If the set Bπ of Bessel vectors for π is dense in H, then for any vector x ϵ H, the analysis operator θx makes sense as a densely defined operator from Bπ to l2(G)-space. If a projection e ϵ M is equivalent to a projection ƒ1 ϵ M with ƒ1 ≤ ƒ ϵ M, then we write e ⋨ ƒ. Let Px (resp. Py) be the orthogonal projection from ℯ2(G) onto [θx(Bπ)] (resp. [θy(Bπ)]). Han and Larson have proved the duality properties of projective unitary representations, i.e. Px ≤ Py is equivalent to Qx ≤ Qy. In this paper we prove that a similar result is true in the sense of von Neumann equivalence of projections, i.e. Px ⋨ Py in λ(G)' is equivalent to Qx ⋨ Qy in π(G)'.