Lines in space-times

A family of Riemannian metrics on the moduli space of irreducible selfdual connections of instanton numberk=1 overCP ² is considered. We find explicit formulas for these metrics and deduce conclusions concerning the geometry of the instanton space.     

The Quantized Enveloping Algebra ? q (sl(n))¶at the Roots of Unity

In this paper we classify the irreducible, subregular representations of the quantum group at a primitive, -root of unity ɛ, for with p prime and k∈N. We show that every such a representation is induced from an irreducible -module and prove the De Concini, Kac, Procesi conjecture about the dimension of the -modules. Received: 30 December 1997 / Accepted: 15 November 1999     

Realization ofU q (so(N)) within the differential algebra onR q N

We realize the Hopf algebraU _q–1 (so(N)) as an algebra of differential operators on the quantum Euclidean spaceR _q ^N . The generators are suitableq-deformed analogs of the angular momentum components on ordinaryR ^N . The algebra Fun(R _q ^N ) of functions onR _q ^N splits into a direct sum of irreducible vector representations ofU _q–1 (so(N)); the latter are explicitly constructed as highest weight representations.     

Polynomial deformations of enveloping algebra U(sl(2,2102))

We define the polynomial deformation, U _t ^p, of enveloping algebra U(sl(2,)) as an analogue of quantum group and construct a nice set D _p ^even of parameters on which we described irreducible representations and classify the irreducible ones in terms of the highest weights. Also, we construct the Casimir elements and using them we see that every finite dimensional U _t ^p-module is semisimple.     

Symmetry aspect of the phase transitions in boracites

The phase transitions in boracites are analysed by using the group-theoretical formulation of the Landau theory of phase transitions. It is shown that the orthorhombic, monoclinic and trigonal phase transitions could be induced by the same irreducible representation of the space groupT _d ⁵ with the star determined by the wave vectork=1/2(b ₁+b ₂). The corresponding free energy function is constructed and the symmetry of normal modes is discussed.The authors thank Dr. V. Janovec of the Institute of Physics for valuable remarks to this paper.     

Irreducible representations of the quantum analogue of SU(2)

We give the complete set of irreducible representations of U(SU(2))_q when q is a mth root of unity. In particular, we show that their dimensions are less or equal to m. Some of them are not highest-weight representations.     

Equivalent rotations associated with the permutation inversion group revisited: symmetry projection of the rovibrational functions of methane

In this work the analysis of the equivalent rotations from the permutation inversion group formalism is revisited. We emphasize that explicit knowledge of changes in the Euler angles are not required in order to determine the transformation that a given symmetry operation causes to the rotational functions when dealing with the permutation inversion group formalism. Indeed, matrix elements of the equivalent rotations are provided by a single Wigner's D ^(j)(R) function. Taking advantage of this, we propose a symmetry projection approach to build the rovibrational functions of methane. This approach focuses on the relevance of the isomorphism between permutations and equivalent rotations. In our method, symmetry adapted functions are obtained by simultaneous diagonalization of a set of commuting operators, whose representation is given in terms of direct products of Wigner's D functions and vibrational matrix representations provided by a local scheme. The proposed approach is general and permits us to obtain in a systematic fashion an orthonormal set of symmetry-projected functions, with good total angular momentum, and carrying the irreducible representations of the molecular symmetry group.     

Simple Current Actions of Cyclic Groups

Permutation actions of simple currents on the primaries of a Rational Conformal Field Theory are considered in the framework of admissible weighted permutation actions. The solution of admissibility conditions is presented for cyclic quadratic groups: an irreducible WPA corresponds to each subgroup of the quadratic group. As a consequence, the primaries of a RCFT with an order n integral or half-integral spin simple current may be arranged into multiplets of length k² (where k is a divisor of n) or 3k² if the spin of the simple current is half-integral and k is odd.Mathematical Subject Classifications (2000). 81T40.Work supported by grant OTKA TS044839.     

Representations of quantumso(8) and related quantum algebras

We study irreducible representations of the quantum groupU (so(8)) when ^* is a primitivel ^th root of unity. By a theorem of De Concini and Kac, there is a finite number of such representations associated to each point of a complex algebraic variety of dimension 28 and the generic representation has dimensionl ¹².We give explicit constructions of essentially all the irreducible representations whose dimension is divisible byl ⁸. In addition, we construct all cyclic representations of minimal dimension. This minimal dimension isl ⁵, in accordance with a conjecture of De Concini, Kac and Procesi.Partially supported by the NSF, DMS-9115984     

Projective indecomposable modules over the small quantum osp(1|2)

In this paper, we construct the finite dimensional Hopf superalgebra u _q (osp(1|2)) arising from U _q(osp(1|2)) when q is a root of unity and describe the projective objects and the irreducible morphisms in a category of Z-graded u _q (osp(l|2))-modules.     

A q-deformed differential calculus at roots of unity

We present main results obtained and published recently, concerning the derivation satisfying d ^N = 0, and the algebras defined by N-ary relations imposed by such condition. We show the covariant version of this calculus, and interpret the irreducible parts of the tensorial products of N such differentials in terms of generalized curvature and torsion.     

Derivation of the B (3) Field and Concomitant Vacuum Energy Density from the Sachs Theory of Electrodynamics

The archetypical and phaseless vacuum magnetic flux density of O(3) electrodynamics, the B ⁽³⁾ field, is derived from the irreducible representation of the Einstein group and is shown to be accompanied by a vacuum energy density which depends directly on the square of the scalar curvature R of curved spacetime. The B ⁽³⁾ field and the vacuum energy density are obtained respectively from the non-Abelian part of the field tensor F and the non-Abelian part of the metrical field equation. Both of these terms are given by Sachs [5].     

Vector Bundle Moduli

We give the general presciption for calculating the number of moduli of irreducible, stable U(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi–Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B = F _r. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory.     

New Weyl groups forA 1 (1) and characters of singular highest weight modules

We consider singular Verma modules overA ₁ ⁽¹⁾ , i.e., Verma modules for which the central charge is equal to minus the dual Coxeter number. We calculate the characters of certain factor modules of these Verma modules. In one class of cases we are able to prove that these factor modules are actually the irreducible highest modules for those highest weights. We introduce new Weyl groups which are infinitely generated abelian groups and are proper subgroups or isomorphic between themselves. Using these Weyl groups we can rewrite the character formulae obtained in the paper in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semisimple Lie algebras (respectively Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras) so that the new Weyl groups play the role of the usual Weyl group (respectively affine Weyl group).     

Positive Energy Representations of the Loop Groups of Non-Simply Connected Lie Groups

We classify and construct all irreducible positive energy representations of the loop group of a compact, connectedand simple Lie group and show that they admit an intertwining action of Diff(S ¹). Received: 7 April 1998 / Accepted: 26 April 1999     

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Abstract

We classify the Noether point symmetries of the generalized Lane-Emden equation y″+ ny′/x+ f(y)?=?0 with respect to the standard Lagrangian L = xⁿy′²/2 — xⁿ ∫f(y)dy for various functions f(y). We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f(y). One of them is f(y) = αy^r , where r ≠ 0,1. The case r?=?5 was considered previously and only a one-parameter family of solutions was presented. Here we provide a complete integration not only for r?= 5 but for other r values. We also give the Lie point symmetries for each case. In two of the new cases, the single Noether symmetry is also the only Lie point symmetry.     

Derivation of O(3) Electrodynamics from the Irreducible Representations of the Einstein Group

By considering the irreducible representations of the Einstein group (the Lie group of general relativity), Sachs [1] has shown that the electromagnetic field tensor can be developed in terms of a metric q , which is a set of four quaternion-valued components of four-vector. Using this method, it is shown that the electromagnetic field vanishes [1] in flat spacetime, and that electromagnetism in general is a non-Abelian field theory. In this paper the non-Abelian component of the field tensor is developed to show the presence of the B ⁽³⁾ field of the O(3) electrodynamics, and the basic structure of O(3) electrodynamics is shown to be a sub-structure of general relativity as developed by Sachs. The extensive empirical evidence for both theories is summarized.     

A Weyl group for the Virasoro andN=1 super-Virasoro algebras

We introduce a Weyl group for the highest weight modules over the Virasoro algebra and the Neveu-Schwarz and Ramond superalgebras. Using this group we rewrite the character formulae for the irreducible highest weight modules over these algebras in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semi-simple Lie algebras (and also of the Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras). This is the same group we introduced recently in order to rewrite in a similar manner the characters of the singular highest weight modules over the affine Kac-Moody algebraA ₁ ⁽¹⁾ .