Finite dimensional representations of the quantum Lorentz group

All finite dimensional irreducible representations of the quantum Lorentz group SL _q (2,) are described explicitly and it is proved all finite dimensional representations of SL _q (2,) are completely reducible. The conjecture of Podle and Woronowicz will be answered affirmatively.     

Duality for a Lorentz quantum group

We give the algebra _q ^/* dual to the matrix Lorentz quantum group _q of Podles-Woronowicz, and Watamuraet al. As a commutation algebra, it has the classical form _q ^/* U _q (sl(2, )) U _q (sl(2, )). However, this splitting is not preserved by the coalgebra structure which we also give. For the derivation, we use a generalization of the approach of Sudbery, viz. tangent vectors at the identity.     

Finslerian N-Spinors: Algebra

New mathematical objects called Finslerian N-spinors are discussed. The Finslerian N-spinor algebra is developed. It is found that Finslerian N-spinors are associated with an N²-dimensional flat Finslerian space. A generalization of the epimorphism SL(2, ) O ₊ (1, 3) to a case of the group SL(N,) is constructed. Particular examples of Finslerian N-spinors for N = 2, 3 are considered in detail.     

A borel-weil-bott approach to representations of sl q (2, ℂ)

We use a quite concrete and simple realization of sl _q (2, ) involving finite difference operators. We interpret them as derivations (in the noncommutative sense) on a suitable graded algebra, which gives rise to the noncommutative scheme ¹ II ^1* as the counterpart of the standard ¹ = Sl(2, )/B.     

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U _q ( _n+1) (whereq ^× is not a root of unity) which are irreducible as representations of U _q (sl _n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev _a from U _q (sl _n+1) to (an enlargement of) U _q (sl_,n+1), depending on a parametera ^·. A second family,ev ^a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U _q ( _n+1) is obtained by pulling back an irreducible representation of U _q (sl _n+1) byev _a orev ^a for somea ^·.     

Differential algebras for quantum groups of type B, C, and D

We study higher order bicovariant differential calculi on the quantum groups O_q(N) and Sp _q (N). We show that the second antisymmetrizer exterior algebra _u is the quotient of the universal exterior algebra _u by the principal ideal generated by . Here denotes the unique up to scalars biinvariant 1-form. Moreover is central in _u and _u is an inner differential calculus. We show that the quadratic dual to the left-invariant algebra _{s L} is isomorphic to the reflection equation algebra. Let be an arbitrary left-covariant first order differential calculus. We show that the dimension of the space of left-invariant 2-forms in the universal exterior algebra equals the number of linearly independent quadratic-linear relations in the quantum tangent space.     

Some examples ofq-regularization

An introduction to Hopf algebras as a tool for the regularization of relevant quantities in quantum field theory is given. We deform algebraic spaces by introducingq as a regulator of a noncommutative and noncocommutative Hopf algebra. Relevant quantities are finite providedq 1 and diverge in the limitq 1. We discussq-regularization on differentq-deformed spaces for ⁴ theory as examples to illustrate the idea.     

Quantum Riemann surfaces: II. The discrete series

We continue our study of noncommutative deformations of two-dimensional hyperbolic manifolds which we initiated in Part I. We construct a sequence of ^*-algebras which are quantizations of a compact Riemann surface of genus g corresponding to special values of the Planck constant. These algebras are direct integrals of finite-dimensional ^*-algebras.Supported by DOE under Grant DE-FG02-88ER25065.     

Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

Common tangent space ℝ 1 4 fromU(2) charges

It is discussed how a common space-time can be constructed from a proposed hiddenU(2) world. Schrödinger's idea to obtain discrete eigenvalues by solving the Maxwell equations for the fieldF on compact spaces without boundaries is modified by orthogonality and identification concepts for the potentialsA. Using residue classes with respect to the metric (Clifford algebra), a common spinor space ⁴=RL and a common Minkowski tangent space ₁ ⁴ are bilinearly constructed from tangent spaces ofU(2) individuals [U(2) manifolds with orthogonal potentials]. The space constructed has the following properties. (1) There are algebraic elements for the identification ofU(2) individuals from ₁ ⁴ as spinors and vectorsA. (2) The transfer of the potentials fromU(2) via ⁴ to ₁ ⁴ is linear. (3) The hiddenU(2) content of the left- and right-handed spaces (L, R) is quite different. The potentials on U(2) individuals are transformed into complex wave functions on the spinor space and into 1-formsA on ₁ ⁴ that can be enlarged to gauge potentials. The construction is discussed from an old point of view of Einstein's, starting with the electric charge as the primary concept for quantum theory. The construction of the tangent space ₁ ⁴ does not depend on a preceding introduction of any points (uncertainty). The identity problem of the interpretation of the quantum theory is discussed in some detail. It is indicated how the algebraic, partiallyad hoc constructions can give a rigid frame for further analytical work.     

The explicit form of the vertex operator fields in two-dimensional quantum SL(2, ℂ) field theory

The explicit form of the vertex operator fields in two-dimensional quantum sl(2, ) field theory is found.     

Zero value of the Schwarzschild mass for an asymptotically Euclidian system of gravitational waves

A class of the asymptotically Euclidian space-times is shown to exist for which the Schwarzschild mass is equal to zero. The coordinate atlases of these space-times satisfy two additional conditions: _k (-gg ^0k )=0 and _ik ⁰ _0g ^ik - _ik ^k _0g ⁰ⁱ =0. In aT-orthogonal metricgs ²=g ₀₀ dt ² -g dx dx these conditions take a simple form: ₀(detg )=0 and (₀ g )(₀ g )=0.     

The Lie algebra of the sl(2, ℂ)-valued automorphic functions on a torus

It is shown that the Lie algebra of the automorphic, meromorphic sl(2, )-valued functions on a torus is a geometric realization of a certain infinite-dimensional finitely generated Lie algebra. In the trigonometric limit, when the modular parameter of the torus goes to zero, the former Lie algebra goes over into the sl(2, )-valued loop algebra, while the latter goes into the Lie algebra (A ₁ ⁽¹⁾ )/(centre).     

Noncommutative Geometry and the Regularization Problem of 4D Quantum Field Theory

We give a noncommutative version of the complex projective space ² and show that scalar QFT on this space is free of UV divergencies. The tools necessary to investigate quantum fields on this fuzzy ² are developed and several possibilities to introduce spinors and Dirac operators are discussed.     

Direct evaluation of iwasawa and triangle decompositions for the real forms of lie algebrasgl(n + 1, ℂ)

The Iwasawa and triangle decompositions for any real form of Lie algebrasgl(n + 1, ) are given. Construction of these decompositions is based on the explicit calculation of the Cartna automorphisms with the help of which the real forms of Lie algebrasgl(n + 1, ) are defined.     

Disentangling q-Exponentials: A General Approach

We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson q-exponential of the sum of two non-q-commuting operators as an (in general) infinite product of q-exponential operators involving repeated q-commutators of increasing order, E_q(A+B) = E_q0(A)E_q1 (B) _i=2 E_qi. By systematically transforming the q-exponentials into exponentials of series and using the conventional Baker–Campbell–Hausdorff formula, we prove that one can make any choice for the bases ^qi, i=0, 1, 2, ..., of the q-exponentials in the infinite product. An explicit calculation of the operators C _i in the successive factors, carried out up to sixth order, also shows that the simplest q-Zassenhaus formula is obtained for ₀ = ₁ =1, and ₂ = 2, and ₃ = 3. This confirms and reinforces a result of Sridhar and Jagannathan, on the basis of fourth-order calculations.     

Derivations commuting with abelian gauge actions on lattice systems

Let be an action of a compact abelian groupG on aC*-algebraA, and assume that the fixed-point subalgebraA is an AF-algebra. We show that if is a closed *-derivation onA commuting with , and the restriction of toA generates a one-parameter group of *-automorphisms, then itself is a generator. In particular, the result applies if is an infinite product action ofG on a UHF algebra. Furthermore, if in this situation ₁ and ₂ are two derivations both satisfying the hypotheses on , and ₁ and ₂ have the same restriction toA , then there exists a one-parameter subgroup of the action with generator ₀ such thatD(₁)D(₂)D(₀) is a joint core for the three derivations, and ₂=₁+₀ on this core.     

Period doubling bifurcations for families of maps on ℝ n

Infinite sequences of period doubling bifurcations in one-parameter families (1-pf) of maps enjoy very strong universality properties: This is known numerically in a multitude of cases and has been shown rigorously for certain 1-pf of maps on the interval. These bifurcations occur in 1-pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 /4.6692 ....In this paper we indicate the main steps of a proof that the same is true for 1-pf of analytic maps from ⁿ to ⁿ , whose restriction to ⁿ is real.Work supported by the Fonds National Suisse, and by the National Science Foundation under Grant PHY-79-16812.     

Non-linear multi-plane wave solutions of self-dual Yang-Mills theory

New solutions of self-dual Yang-Mills (SDYM) equations are constructed in Minkowski space-time for the gauge groupSL(2, ). After proposing a Lorentz covariant formulation of Yang's equations, a set of Ansätze for exact non-linear multiplane wave solutions are proposed. The gauge fields are rational functions ofe ^x·ki (K _i ² =0, 1iN) for these Ansätze. At least, three families of multisoliton type solutions are derived explicitly. Their asymptotic behaviour shows that non-linear waves scatter non-trivially in Minkowski SDYM.On leave from LPTHE Université Paris VI, 4, Place Jussieu, Tour 16, ler étage, F-75230 Paris Cedex 05, France     

P-polarized nonlinear surface polaritons near the surface of a saturable self-defocusing medium: Threshold case

We have obtained the solution of the stationary problem for P-polarized nonlinear surface polaritons (NLSP) near the interface of two media, one of which is optically linear and the other having an optical nonlinearity with saturation and a dielectric function of the form: =_xx=_zz=₀+a|E|²/(1+b|E|²), where E is the amplitude of the electric field. On this basis the NLSP energy and propagation properties, the polarization structure and the localization depth of the NLSP field as well as the physically allowed region in the parameter space have been investigated for the case in which the parameter ₀ is positive, b is non-negative, and a and ₁ are negative (₁ is the dielectric function of the linear medium); ₀>|₁|.