Continuous Irreducible Representations Research Articles

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.

Edge modes of gravity. Part III. Corner simplicity constraints

In the tetrad formulation of gravity, the so-called simplicity constraints play a central role. They appear in the Hamiltonian analysis of the theory, and in the Lagrangian path integral when constructing the gravity partition function from topological BF theory. We develop here a systematic analysis of the corner symplectic structure encoding the symmetry algebra of gravity, and perform a thorough analysis of the simplicity constraints. Starting from a precursor phase space with Poincaré and Heisenberg symmetry, we obtain the corner phase space of BF theory by imposing kinematical constraints. This amounts to fixing the Heisenberg frame with a choice of position and spin operators. The simplicity constraints then further reduce the Poincaré symmetry of the BF phase space to a Lorentz subalgebra. This picture provides a particle-like description of (quantum) geometry: the internal normal plays the role of the four-momentum, the Barbero-Immirzi parameter that of the mass, the flux that of a relativistic position, and the frame that of a spin harmonic oscillator. Moreover, we show that the corner area element corresponds to the Poincaré spin Casimir. We achieve this central result by properly splitting, in the continuum, the corner simplicity constraints into first and second class parts. We construct the complete set of Dirac observables, which includes the generators of the local mathfrak{sl}left(2,mathrm{mathbb{C}}right) subalgebra of Poincaré, and the components of the tangential corner metric satisfying an mathfrak{sl}left(2,mathrm{mathbb{R}}right) algebra. We then present a preliminary analysis of the covariant and continuous irreducible representations of the infinite-dimensional corner algebra. Moreover, as an alternative path to quantization, we also introduce a regularization of the corner algebra and interpret this discrete setting in terms of an extended notion of twisted geometries.

Irreducible continuous representations of the simple linearly compact n-Lie superalgebra of type S

In the present paper, we classify all irreducible continuous representations of the simple linearly compact [Formula: see text]-Lie superalgebra of type [Formula: see text]. The classification is based on a bijective correspondence between the continuous representations of the [Formula: see text]-Lie algebras [Formula: see text] and continuous representations of the Lie algebra of Cartan type [Formula: see text], on which some two-sided ideal acts trivially.

Regular characters of groups of type [formula omitted] over discrete valuation rings

Let o be a complete discrete valuation ring with finite residue field k of odd characteristic. Let G be a general or special linear group or a unitary group defined over o and let g denote its Lie algebra. For every positive integer ℓ, let Kℓ be the ℓ-th principal congruence subgroup of G(o). A continuous irreducible representation of G(o) is called regular of level ℓ if it is trivial on Kℓ+1 and its restriction to Kℓ/Kℓ+1≃g(k) consists of characters with G(k‾)-stabiliser of minimal dimension. In this paper we construct the regular characters of G(o), compute their degrees and show that the latter satisfy Ennola duality. We give explicit uniform formulae for the regular part of the representation zeta functions of these groups.

Irreducible continuous representations of the simple linearly compact n-Lie superalgebra of type W

In the present paper we classify all irreducible continuous representations of the simple linearly compact n-Lie superalgebra of type W. The classification is based on a bijective correspondence between the continuous representations of the n-Lie algebras Wn and continuous representations of the Lie algebra of Cartan type Wn−1, on which some two-sided ideal acts trivially.

The inverse deformation problem

Given a commutative complete local noetherian ring $A$ with finite residue field $\boldsymbol{k}$, we show that there is a topologically finitely generated profinite group $\unicode[STIX]{x1D6E4}$ and an absolutely irreducible continuous representation $\overline{\unicode[STIX]{x1D70C}}:\unicode[STIX]{x1D6E4}\rightarrow \text{GL}_{n}(\boldsymbol{k})$ such that $A$ is a universal deformation ring for $\unicode[STIX]{x1D6E4},\overline{\unicode[STIX]{x1D70C}}$.

Commutators, commutativity and dimension in the socle of a Banach algebra: A generalized Wedderburn–Artin and Shoda's theorem

As a follow-up to work done in [7], some new insights to the structure of the socle of a semisimple Banach algebra are obtained. In particular, it is shown that the socle is isomorphic as an algebra to the direct sum of tensor products of corresponding left and right minimal ideals. Remarkably, the finite-dimensional case here reduces to the classical Wedderburn–Artin Theorem, and this approach does not use any continuous irreducible representations of the algebra in question. Furthermore, the structure of the socles for which the classical Shoda's Theorem for matrices can be extended, is characterized exactly as those socles which are minimal two-sided ideals. It is then shown that the set of commutators in the socle (i.e. {xy−yx:x,y∈SocA}) is a vector subspace. Finally, we characterize those socles which belong to the center of a Banach algebra and obtain results which suggest that the dimension of certain subalgebras of the socle in fact provides a measure, to some extent, of commutativity.

Unitary Representations of Oligomorphic Groups

We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group S ∞, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and, moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the Gelfand–Raikov theorem holds for topological subgroups of S ∞: for all such groups, continuous irreducible representations separate points in the group.

Erratum to “Mass equidistribution for automorphic forms of cohomological type on $GL_{2}$”

Corollary 3 of [4] is not known unconditionally, as cohomological automorphic forms on GL2 over an imaginary quadratic field are not known to satisfy the Ramanujan conjecture. We shall briefly describe the reason for this and discuss what information Theorem 1 of [4] does give in the case of imaginary quadratic fields. Let K be an imaginary quadratic field with nontrivial automorphism c, and let π be a cuspidal automorphic representation of GL2(AK) with unitary central character ω. Suppose that ω = ω and that π∞ has Langlands parameter WC = C× → GL2(C) given by z → diag(z1−k, z1−k) for some integer k ≥ 2 (i.e. so that π is any cohomological representation up to twist). It is then known (see Theorem 1.1 of [1]) that for any one may associate a continuous irreducible representation ρ : Gal(K/K) → GL2(Q ) to π such that the characteristic polynomial of ρ(Frobv) agrees with the Hecke polynomial of πv at all places v which do not divide and at which K/Q, π, and π are unramified. However, because ρ is constructed via an -adic limiting process, it is not known to arise from a motive and so is not known to be pure. To construct ρ, one first makes a theta lift from π to a holomorphic limit of discrete series representation Π on Sp4/Q as in [3]. Weissauer [6] has proven that if Π′ is a holomorphic discrete series representation of Sp4/Q which is not a CAP representation, then one may associate a Galois representation to it which is pure and locally compatible with Π′ at all unramified places, so that Π′ satisfies Ramanujan wherever it is unramified. These results are not known for the limit of discrete series representation Π, and to associate a Galois representation to it one must apply techniques of Taylor [5] which are similar to those used by Deligne and Serre to associate Galois representations to classical weight 1 modular forms. These involve multiplying a holomorphic form in Π by a well understood regular holomorphic form of large weight, applying the results of Weissauer and recovering ρ from these products by an -adic limiting process. Any Archimedean information about the Frobenius eigenvalues of ρ is lost during this, and neither does one know that ρ arises from a motive. As a result, the algebraic information we obtain about π is insufficient to deduce Ramanujan for it. We may still draw interesting conclusions from Theorem 1 of [4] in the imaginary quadratic case. Cohomological forms on GL2/K which are base changes from Q will satisfy Ramanujan, and so Theorem 1 establishes their equidistribution as their weight becomes large. Moreover, the experimental results of [2] suggest that all

Square Integrable Representation of Groupoids

A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur’s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle ℋ = {Hu}u∈¸G0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle (G0, {Hu}, μ), then dimHu = 1 (u ∈¸ G0). Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.

THE NON-EXISTENCE OF CERTAIN MOD p GALOIS REPRESENTATIONS

The non-existence is proved of continuous irreducible representations p : Gal(Q/Q) \longrightarrow GL(F) with Artin conductor N outside p for a few small values of p and N.N.

Serre's Conjecture for Imaginary Quadratic Fields

We study an analog over an imaginary quadratic field K of Serre's conjecture for modular forms. Given a continuous irreducible representation ρ:Gal(Q/K) →GL2(Fl) we ask if ρ is modular. We give three examples of representations ρ obtained by restriction of even representations of Gal(Q/Q). These representations appear to be modular when viewed as representations over K, as shown by the computer calculations described at the end of the paper.

The Jacobson radical of a CSL algebra

Extrapolating from Ringrose's characterization of the Jacobson radical of a nest algebra, Hopenwasser conjectured that the radical of a CSL algebra coincides with the Ringrose ideal (the closure of the union of zero diagonal elements with respect to finite sublattices). A general interpolation theorem is proved that reduces this conjecture for completely distributive lattices to a strictly combinatorial problem. This problem is solved for all width two lattices (with no restriction of complete distributivity), verifying the conjecture in this case. In [9] (cf. [3, Chapter 6]), Ringrose characterizes the Jacobson radical of a nest algebra. In [6], Hopenwasser described the appropriate generalization for reflexive algebras with commutative subspace lattice (CSL algebras), and verified his conjecture in a few special cases. The analogy with the nest case was pushed further in [8], clarifying the role of the carrier space. In this paper, we verify this conjecture for all width two CSL algebras. Moreover, we establish a framework for attacking the general problem and make significant progress in this direction. Recall that the Jacobson radical of a Banach algebra is the intersection of the kernels of all irreducible continuous representations. It also coincides with those elements T such that AT is quasinilpotent for every A in the algebra. Thus when 9 is a finite CSL, Alg(SF) is a finite matrix algebra with certain coefficients running over W(X) (and the rest 0). There is a unique contractive projection A, of the algebra onto the diagonal = Alg(S,) nAlg(F)* given by A_(T) = ZE ETE as E runs over the atoms of S . It is easy to see that the radical of Alg(Sr) is precisely the set of zero diagonal elements (T such that A_(T) = 0), which we denote Algo(SF). Indeed, this is a nilpotent ideal of order at most 1 I. Furthermore, it is easy to verify that Algo(F) n Alg(2') is an ideal in Alg(2') for all finite sublattices F of 2'; and indeed, it is a nilpotent ideal. Thus it is contained in the radical. It follows that the norm closure

Representations of Compact Right Topological Groups

AbstractCompact right topological groups arise naturally as the enveloping semigroups of distal flows. Recently, John Pym and the author established the existence of Haar measure μ on such groups, which invites the consideration of the regular representations. We start here by characterizing the continuous representations of a compact right topological group G, and are led to the conclusion that the right regular representation r is not continuous (unless G is topological). The domain of the left regular representation l is generally taken to be the topological centreor a tractable subgroup of it, furnished with a topology stronger than the relative topology from G (the goals being to have l both defined and continuous). An analysis of l and r on H = L2(G) for some non-topological compact right topological groups G shows, among other things, that: (i)for the simplest (perhaps) G generated by ℤ, (l, H) decomposes into one copy of each irreducible representation of ℤ and c copies of the regular representation.(ii)for the simplest (perhaps) G generated by the euclidean group of the plane , (l, H) decomposes into one copy of each of the continuous one-dimensional representations of and c copies of each continuous irreducible representation Ua,a > 0.(iii)when Λ(G) is not dense in G, it can seem very reasonable to regard r as a continuous representation of a related compact topological group, and also, G can be almost completely "lost" in the measure space (G, μ).

Complete Irreducibility and X-Spherical Representations

Let G be a locally compact group, K a compact subgroup, and χ: K → {| z| = 1} a continuous homomorphism. Let π be a continuous irreducible representation of G on a complete locally convex space V such that the subspace { v ∈ V | π( k) v = χ( k) v, ∀ k ∈ K} has dimension 1. Then π is completely irreducible. Furthermore π is Naimark related to a representation on a reflexive Fréchet space which is a closed subspace of C ( G) of the form span { L( g)φ | g ∈ G} where φ is a χ -spherical function. A corollary is that irreducible eigenspace representations are completely irreducible.

On the irreducible representations of the lorentz group

All continuous irreducible representations of the SL(2, C) group (as given by Naimark) are obtained by means of methods developed by Harish-Chandra and Kihlberg. The analysis is done in the SU(2) basis and a single closed expression for the matrix elements of the noncompact generators for an arbitrary irreducible representation of SL(2, C) is given. For the unitary irreducible representations the scalar product for each irreducible Hilbert space is found explicitly. The connection between the unitary irreducible representations of SL(2, C) and those of ▪ is discussed by means of Inönü and Wigner contraction procedure and the Gell-Mann formula. Finally, due to physical interest, the addition of a four-vector operator to SL(2, C) unitary irreducible representations in a minimal way is considered; and all group extensions of the parity and time reversal operators by SL(2, C) are explicitly obtained and some aspects of their representations are treated.

Compatible topologies and continuous irreducible representations

Compatible topologies and continuous irreducible representations

The structure of certain unitary representations of infinite symmetric groups

Let S be an infinite set, β \beta an infinite cardinal number, and G β ( S ) {G_\beta }(S) the group of those permutations of S whose support has cardinal number less than β \beta . If T is any nonempty set, S T {S^T} is the set of functions from T to S. The canonical representation Λ β T \Lambda _\beta ^T of G β ( S ) {G_\beta }(S) on L 2 ( S T ) {L^2}({S^T}) is the direct sum of factor representations. Factor representations of types I ∞ , II 1 {{\text {I}}_\infty },{\text {II}_1} , and II ∞ {\text {II}_\infty } occur in this decomposition, depending upon S, β \beta , and T; the type II 1 {\text {II}_1} factor representations are quasi-equivalent to the left regular representation. Let G β ( S ) {G_\beta }(S) have the topology of pointwise convergence on S. G β ( S ) {G_\beta }(S) is a topological group but is not locally compact. Every continuous representation of G β ( S ) {G_\beta }(S) is the direct sum of irreducible representations. Let Γ \Gamma be a nontrivial continuous irreducible representation of G β ( S ) {G_\beta }(S) . Then Γ \Gamma is continuous iff Γ \Gamma is equivalent to a subrepresentation of Λ β T \Lambda _\beta ^T for some nonempty finite set T iff there is a nonempty finite subset Z of S such that the restriction of Γ \Gamma to the subgroup of those permutations which leave Z pointwise fixed contains the trivial representation of this subgroup.

Irreducible algebras of operators which contain a minimal idempotent.

We prove that when A is a closed subalgebra of the bounded operators on a reflexive Banach space X, which acts irreducibly on X and contains a minimal idempotent, then every bounded operator with finite dimensional range on X is in A. We use this result to prove that every continuous irreducible representation of a GCR-algebra on a Hilbert space H \mathcal {H} is similar to a ∗ ^ \ast -representation on H \mathcal {H} .

Locally compact topologies on a group and the corresponding continuous irreducible representations

Locally compact topologies on group G and corresponding continuous irreducible unitary representations, noting existence of subgroup