q-Ultraspherical polynomials for q a root of unity

Properties of the q-ultraspherical polynomials for q being a primitive root of unity, are derived using a formalism of the so _q (3) algebra. The orthogonality condition for these polynomials provides a new class of trigonometric identities representing discrete finite-dimensional analogues of q-beta integrals of Ramanujan.     

The centre of a quantum affine algebra at a root of unity

Using a recent extension of the Lusztig braid group automorphisms of a quantum affine algebra, I prove that at an oddl-th root of unity, thel-th power of every real root vector lies in the centre of the quantum affine algebra. The centre of a quantum affine algebra at a root of unity is infinite dimensional: nevertheless it is infinite dimensional over its centre.     

Quantum invariants at the sixth root of unity

A general topological formula is given for theSU (2) quantum invariant of a 3-manifoldM at the sixth root of unity. It is expressed in terms of the homology, Witt invariants and signature defects of the various 2-fold covers ofM, and thus ties in with basic 4-dimensional invariants. A discussion of the range of values of these quantum invariants is included, and explicit evaluations are made for lens spaces.     

Quantum relativistic Toda chain at root of unity: invariant approach

Recent results, concerning the quantum relativistic Toda chain at root of unity, are interpreted in this note from the point of view of invariant auxiliary linear problem. Such form of the linear problem arises in the three-dimensional integrable systems, and several results from the three dimensional models are applied to the relativistic Toda chain. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. This work was supported in part by the grant INTAS OPEN 00-00055. S.P.’s work was supported by the grants RFBR 00-02-16477 and grant for support of scientific schools RFBR 00-15-96557, S.S’s work was supported by the grant RFBR 01-01-00201.     

The cyclic representations of the quantum superalgebra U q osp(2, 1) withq a root of unity

By generalizing De Concini and Kac's cyclic representation theory of quantum groups at roots of unity, the cyclic representations of the quantum superalgebra U _q osp(2, 1) are constructed in three classes: irreducible representations with single multiplicities, irreducible representations with the multiplicities larger than one, and indecomposable representations.This work is supported in part by the National Sciene Foundation in China.     

Decomposition of the step operators of theλ √q (n)-covariant oscillator algebra withq as a root of unity

In this paper the decomposition of the creation and annihilation operators ofgl _√q (n)-covariant oscillator algebra is discussed when the deformation parameterq is a (s+1)-th primitive root of unity.     

Massive 3-loop Feynman diagrams reducible to SC^* primitives of algebras of the sixth root of unity

In each of the 10 cases with propagators of unit or zero mass, the finite part of the scalar 3-loop tetrahedral vacuum diagram is reduced to 4-letter words in the 7-letter alphabet of the 1-forms and , where is the sixth root of unity. Three diagrams yield only . In two cases combines with the Euler-Zagier sum ; in three cases it combines with the square of Clausen's . The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: . The previously unidentified term in the 3-loop rho-parameter of the standard model is merely . The remarkable simplicity of these results stems from two shuffle algebras: one for nested sums; the other for iterated integrals. Each diagram evaluates to 10 000 digits in seconds, because the primitive words are transformable to exponentially convergent single sums, as recently shown for and , familiar in QCD. Those are SC constants, whose base of super-fast computation is 2. Mass involves the novel base-3 set SC. All 10 diagrams reduce to SCSC constants and their products. Only the 6-mass case entails both bases. Received: 13 March 1998 / Published online: 22 March 1999     

Some remarks onU q (\mathfrak{s}\mathfrak{l}(2,\mathbb{R})) at root of unityat root of unity

We discuss a modification ofU _q and a class of its irreducible representations whenq is a root of unity. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

The generalized product partition of unity for the meshless methods

The partition of unity is an essential ingredient for meshless methods named by GFEM, PUFEM (partition of unity FEM), XFEM (extended FEM), RKPM (reproducing kernel particle method), RPPM (reproducing polynomial particle method), the method of hp clouds in the literature. There are two popular choices for partition of unity: a piecewise linear FEM mesh and the Shepard-type partition of unity. However, the partition of unity (PU) by a FEM mesh leads to the singular (or nearly singular) matrices and non-smooth approximation functions. The Shepard-type partition of unity requires lengthy computing time and its implementation is difficult. In order to alleviate these difficulties, Oh et al. introduced the smooth piecewise polynomial PU functions with flat-top, that lead to small matrix condition numbers, and almost everywhere partition of unity, that can handle essential boundary conditions. Nevertheless, we could not have the smooth closed form PU functions with flat-top for general polygonal patches (2D) and general polyhedral patches (3D). In this paper, we introduce one of the most simple and efficient partition of unity, called the (generalized) product partition of unity. The product PU functions constructed by this method are the closed form smooth piecewise polynomials with flat-top and could handle background meshes (general polygonal patches as well as general polyhedral patches) arising in practical applications of meshless methods.     

On universal R-matrices at roots of unity

It is well-known that quantum algebras at roots of unity are not quasi-triangular. They indeed do not possess an invertible universalR-matrix. They have, however, families of quotients, on which no obstructiona priori forbids the existence an universalR-matrix. In particular, the universalR-matrix of the so-called finite dimensional quotient is already known. We try here to answer the following questions: are most of these quotients equivalent (or Hopf equivalent)? Can the universalR-matrix of one be transformed to the universalR-matrix of another using isomorphisms?     

Roots of unity: Representations of quantum groups

Representations of Quantum GroupsU (g _n ), g _n any semi-simple Lie algebra of rankn, are constructed from arbitrary representations of rankn–1 quantum groups for a root of unity. Representations which have the maximal dimension and number of free parameters for irreducible representations arise as special cases.Supported by the Japan Society for the Promotion of ScienceDeceased     

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.     

Two-rowed Hecke algebra representations at roots of unity

In this paper, we initiate a study into the explicit construction of irreducible representations of the Hecke algebraH _n (q) of typeA _n-1 in the non-generic case whereq is a root of unity. The approach is via the Specht modules ofH _n (q) which are irreducible in the generic case, and possess a natural basis indexed by Young tableaux. The general framework in which the irreducible non-genericH _n (q)-modules are to be constructed is set up and, in particular, the full set of modules corresponding to two-part partitions is described. Plentiful examples are given.Presented at the 4th International Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.     

The two-dimensional Euclidean quantum algebra at roots of unity

The two-dimensional Euclidean quantum algebra is considered at roots of unity. The algebraic properties and the Poisson-Hopf structure are first investigated. We then determine the irreducible representations and study the reduction of the tensor product of two of them.     

Self-similar potentials and theq-oscillator algebra at roots of unity

Properties of the simplest class of self-similar potentials are analyzed. Wave functions of the corresponding Schrödinger equation provide bases of representations of theq-deformed Heisenberg-Weyl algebra. When the parameterq is a root of unity, the functional form of the potentials can be found explicitly. The generalq ³ = 1 and the particularq ⁴ = 1 potentials are given by the equi-anharmonic and (pseudo) lemniscatic Weierstrass functions, respectively.     

On automorphisms and universal ℛ-matrices at roots of unity

Invertible universal ?-matrices of quantum Lie algebras do not exist at roots of unity. However, quotients exist for which intertwiners of tensor products of representations always exist, i.e. ?-matrices exist in the representations. One of these quotients, which is finite-dimensional, has a universal ?-matrix. In this Letter we answer the following question: under which condition are the different quotients of U _q (sl(2)) (Hopf)-equivalent? In the case when they are equivalent, the universal ?-matrix of the one can be transformed into a universal ?-matrix of the other. We prove that this happens only whenq ⁴ = 1, and we explicitly give the expressions for the automorphisms and for the transformed universal ?-matrices in this case.     

Centre and representations of small quantum superalgebras at roots of unity

When the deformation parameter is a root of unity, the centre of a quantum group can be described by a set of generators and non trivial relations. In the case ofU _q (sl(N)), these relations simply derive from the expressions of the deformed Casimir operators. In the case ofU _q (osp(1|2)), the relation is simple if we use an operator which anticommutes with the fermionic generators and whose square is the quadratic Casimir. This operator also simplifies the classification of finite dimensional irreducible representations. In the case ofU _q (sl(1|2)), the relations derive from the (infinite set of) standard Casimir operators.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.