Finite Dimensional Unitary Representations of Quantum Anti–de Sitter Groups at Roots of Unity

We study irreducible unitary representations of U _q (SO(2,1)) and U _q (SO(2,?3)) for q a root of unity, which are finite dimensional. Among others, unitary representations corresponding to all classical one-particle representations with integral weights are found for , with M being large enough. In the “massless” case with spin bigger than or equal to 1 in 4 dimensions, they are unitarizable only after factoring out a subspace of “pure gauges” as classically. A truncated associative tensor product describing unitary many-particle representations is defined for . Received: 27 November 1996 / Accepted: 28 July 1997     

Matrix elements for the representations of SO(n) and SO0(n, 1)

Matrix elements of the unitary irreducible representations of the group SO(n) of class higher then 1 (with respect to SO(n−1)) in Gel'fand-Zetlin basis are obtained in explicit form. They are represented as polynomials in cosθ and sinθ of the order equal to the first coordinate of the highest weight. Making use of them the representation matrix elements for the group SO₀(n, 1) in SO(n) basis are calculated.     

Master Analytic Representations and unified representation theory of certain orthogonal and pseudo-orthogonal groups

The representation theory of the groupsSO(5),SO(4, 1),SO(6) andSO(5, 1) is studied using the method of Master Analytic Representations (MAR). It is shown that a single analytic expression for the matrix elements of the generators ofSO(n+1) andSO(n, 1) in anSO(n) basis yields all the unitary representations (forn=4,5); and that the compact and non-compact groups have essentially the same analytic representation. Once the MAR of a group is worked out, the search for the unitary irreducible representations is reduced to a purely arithmetic operation. The utmost care has been exercised to conduct the discussions at an elementary level: knowledge of simple angular momentum theory is the only prerequisite.Work supported in part by the National Science Foundation.Work supported in part by the U.S. Atomic Energy Commission.     

Localization of Unitary Braid Group Representations

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e ^πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N > 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science.     

A classification of the unitary irreducible representations ofSO 0(N, 1)

All inequivalent continuous unitary irreducible representations of the groupSO ₀(N, 1),N3, and its universal covering group are classified.     

The matrix elements of irreducible representations of the group SO(n,1) in integral form

Using the method of induced representations, the matrix elements of unitary irreducible representations of the group SO(n,1) are found in integral form.     

De Sitter Quasigroups

The Cayley–Klein parameters for the de Sitter groups SO(4, 1) and SO(3, 2) are introduced, and in an extension of the earlier investigation of quasigroups connected with Clifford groups, quasigroups connected with the SO(4, 1) and SO(3, 2) groups are determined. It is shown that these quasigroups have eight-dimensional, double-valued irreducible cracovian representations. The covariance of a five-dimensional form of the Dirac equation with respect to the quasi-rotations forming quasigroups connected with the groups SO(4, 1) and SO(3, 2) is demonstrated. An analogy is drawn between Weyl's hidden symmetry group and a quasigroup.     

Clebsch-Gordan coefficients of the groups SO(p, 1)

An explicit expression for the Clebsch-Gordan coefficients for the coupling of most degenerate unitary representations of SO(p, 1) is obtained.     

Algebraic Coset Conformal Field Theories

All unitary Rational Conformal Field Theories (RCFT) are conjectured to be related to unitary coset Conformal Field Theories, i.e., gauged Wess–Zumino–Witten (WZW) models with compact gauge groups. In this paper we use subfactor theory and ideas of algebraic quantum field theory to approach coset Conformal Field Theories. Two conjectures are formulated and their consequences are discussed. Some results are presented which prove the conjectures in special cases. In particular, one of the results states that a class of representations of coset W _N (N≥ 3) algebras with critical parameters are irreducible, and under the natural compositions (Connes' fusion), they generate a finite dimensional fusion ring whose structure constants are completely determined, thus proving a long-standing conjecture about the representations of these algebras. Received: 5 November 1998 / Accepted: 18 October 1999     

On the reduction of tensor products of representations of pseudo-orthogonal groups

We verify and generalize a conjecture of Fulling [1] that the Kronecker product of a finite number of unitary representations, not all of which possess an invariant vector, of the Lorentz group SO ₀(1, n), any n2, does not contain the trivial representation discretely.     

Contractions of representations of de Sitter groups

In order to construct the quantum field theory in a curved space with no old infinities as the curvature tends to zero, the problem of contraction of representations of the corresponding group of motions is studied. The definitions of contraction of a local group and of its representations are given in a coordinate-free manner. The contraction of the principal continuous series of the de Sitter groupsSO ₀(n, 1) to positive mass representations of both the Euclidean and Poincaré groups is carried out in detail. It is shown that all positive mass continuous unitary irreducible representations of the resulting groups can be obtained by this method. For the Poincaré groups the contraction procedure yields reducible representations which decompose into two non-equivalent irreducible representations.On leave of absence from the Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.     

Unitarization of a singular representation ofSO(p,q)

A geometric construction of a certain singular unitary representation ofSO _e(p,q), withp+q even is given. The representation is realized geometrically as the kernel of aSO _e(p,q)-invariant operator on a space of sections over a homogeneous space forSO _e(p,q). TheK-structure of these representations is elucidated and we demonstrate their unitarity by explicitly writing down anso(p,q) positive definite hermitian form. Finally, we demonstrate that the annihilator inU[g] of this representation is the Joseph ideal, which is the maximal primitive ideal associated with the minimal coadjoint orbit.     

A unified treatment of the groups SO(4) and SO(3,1)

The irreducible representations of the group SO(4) in which the SO(3) subgroup is reduced are studied by an explicit construction of the operators and the basis in the spinor representation. The basis function which is formally identical with that for the coupling of two angular momentaj ₁ andj ₂ is expressible in terms of a hypergeometric function and strongly resembles the one for the irreducible representations of the groups SO(3,1). For the Lorentz group, the bases for the unitary representations which require unphysical values ofj ₁ andj ₂ are found to be analytic continuation of those for SO(4). The realization of the unitary irreducible representations of the group SO(4) in the Hilbert space of these functions leads, for appropriate unphysical values ofj ₁,j ₂, to the Gelfand-Naimark formula for the principal and complementary series of the representations of SO(3;1). The matrix elements for finite transformations of SO(4) and SO(3,1) can be evaluated, in this approach, in a unified manner by using standard properties of the hypergeometric function. These turn out to be a finite sum of₃ F ₂-functions which, as expected, are polynomials for SO(4) and infinite series for SO(3,1). A number of special matrix elements are calculated from the general formula and these agree with the results obtained previously.The authors are deeply indebted to Professor S.Dutta Majumdar fo many important suggestions and clarifications.     

Discrete degenerate representations of non-compact unitary groups

Two degenerate principal series of irreducible unitary representations of an arbitrary non-compact unitary groupU(p,1) are derived. These series are determined by the eigenvalues of the first and second-order invariant operators, which are shown to possess a discrete spectrum. The explicit form of the corresponding harmonic functions is derived and the properties of the discrete representations are discussed in detail. Moreover, in the Appendix, we derive the properties of the corresponding degenerate representations of an arbitrary compactU(p) group.On leave of absence from Institute of Nuclear Research, Warsaw, Poland.On leave of absence from Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.     

Spinor-vector duality in heterotic string vacua

Classification of the N=1 space–time supersymmetric fermionic Z₂×Z₂ heterotic-string vacua with symmetric internal shifts, revealed a novel spinor-vector duality symmetry over the entire space of vacua, where the S_t↔V duality interchanges the spinor plus anti-spinor representations with vector representations. In this paper we demonstrate that the spinor-vector duality exists also in fermionic Z₂ heterotic string models, which preserve N=2 space–time supersymmetry. In this case the interchange is between spinorial and vectorial representations of the unbroken SO(12) GUT symmetry. We provide a general algebraic proof for the existence of the S_t↔V duality map. We present a novel basis to generate the free fermionic models in which the ten-dimensional gauge degrees of freedom are grouped into four groups of four, each generating an SO(8) modular block. In the new basis the GUT symmetries are produced by generators arising from the trivial and non-trivial sectors, and due to the triality property of the SO(8) representations. Thus, while in the new basis the appearance of GUT symmetries is more cumbersome, it may be more instrumental in revealing the duality symmetries that underly the string vacua.     

On unitary implementability of conformal transformations

We show that the only projective representations of the conformal group in a Hilbert space which, when restricted to the Poincaré subgroup, are unitary irreducible of mass zero and discrete helicity, are the usual unitary representations of SU(2, 2) often called ladder representations. Some physical consequences are also discussed.     

A Trinomial Analogue of Bailey's Lemma and N = 2 Superconformal Invariance

We propose and prove a trinomial version of the celebrated Bailey's lemma. As an application we obtain new fermionic representations for characters of some unitary as well as nonunitary models of N= 2 superconformal field theory (SCFT). We also establish interesting relations between N= 1 and N= 2 models of SCFT with central charges and . A number of new mock theta function identities are derived. Dedicated to Dora Bitman on her 70th birthday Received: 8 March 1997 / Accepted: 29 June 1997     

Duals of crystallographic groups. Band and quasi-band representations

Band representations are analyzed from a pure group theoretical point of view, with the aid of the dual of the crystallographic group (the set of equivalence classes of unitary irreducible representations). It is shown on the examples of the onedimensional crystallographic groups that we have to introduce a distinction between band and quasi-band representations, the wordband being reserved for induced representations.The dual of the groupF222 is explicitly constructed. It permits to show that two elementary band representations which have the same decompositions into unitary irreducible representations are not equivalent.     

Nuclear Algebraic Geometry and SO(10)

In this contribution nuclear representations of the Dirac ring, developed over many years, are shown to be a particular case of a theorem in algebraic geometry which at the same time associates them with a Hodge decomposition of a Kaehler manifold. This yields a shape that in some cases is independent of any appeal to a symmetry group. However, because the nuclear representations are in the infinitesimal ring of SO(4) and the internal space of each representation is in a Kaehler (even Calabi-Yau) manifold K; the group SO(10) = SO(4) × K can give additional information. This paper develops the very fruitful symbiosis between algebra and irreducible representations of SO(10) and covers some aspects of string theory.     

On Metaplectic Modular Categories and Their Applications

For non-abelian simple objects in a unitary modular category, the density of their braid group representations, the #P-hard evaluation of their associated link invariants, and the BQP-completeness of their anyonic quantum computing models are closely related. We systematically study such properties of the non-abelian simple objects in the metaplectic modular categories SO(m)₂ for an odd integer m ≥ 3. The simple objects with quantum dimensions \({\sqrt{m}}\) have finite image braid group representations, and their link invariants are classically efficient to evaluate. We also provide classically efficient simulations of their braid group representations. These simulations of the braid group representations can be regarded as qudit generalizations of the Knill–Gottesmann theorem for the qubit case. The simple objects of dimension 2 give us a surprising result: while their braid group representations have finite images and are efficiently simulable classically after a generalized localization, their link invariants are #P-hard to evaluate exactly. We sharpen the #P-hardness by showing that any sufficiently accurate approximation of their associated link invariants is already #P-hard.