Representations of the compact quantum groupSU q(2) and geometrical quantization

The Souriau—Kostant method of geometrical quantization is used to construct infinite-dimensional irreducible unitary representations of the algebra of functions of the compact quantum groupSU _q(2). The generalization to the case of the quantum groupSU _q (n) is discussed.V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 2, pp. 163–172, August, 1994.     

Exact calculability,semigroup of representations,and the stability property for representations of the algebra of functions on the quantum groupSU q(2)

An operation of the coproduct of representations of a bialgebra is defined. The coproduct operation of representations for the Hopf algebra of functions on the quantum groupSU _q (2) is investigated. For this Hopf algebra a representation II called the stable representation is constructed. This representation is invariant with respect to the coproduct with an arbitrary representation. An expression for the trace in the representation II is derived. The invariant Voronovich integal inSU _q (2) takes the form fdµ=tr(fcc ^*).V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 2, pp. 163–178, November, 1994.     

Clebsch-Gordan coefficients for the quantum algebrasu q (2)

A simple method for studying the Clebsch-Gordan coefficients for the algebrasu _q (2) is discussed.Institute of Physics, Academy of Sciences of Azerbaidzhan; I. V. Kurchatov Institute of Atomic Energy. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 1, pp. 3–11, January, 1994.     

On the bosonization ofL-operators for the quantum affine algebraU q (sl 2)

Some relations between different objects associated with quantum affine algebras are reviewed. It is shown that the Frenkel-Jing bosonization of a new realization of the quantum affine algebra as well as bosonization of L-operators for this algebra can be obtained from Zamolodchikov-Faddeev algebras defined by the quantum R-matrix satisfying unitarity and crossing-symmetry conditions.On leave of absence from the ITP, Kiev 252143, Ukraine. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 64–77, July, 1995.     

On (q 2 + q + 2, q + 2)-arcs in the Projective Plane PG(2, q)

A (k,n)-arc in PG(2,q) is usually defined to be a set of k points in the plane such that some line meets in n points but such that no line meets in more than n points. There is an extensive literature on the topic of (k,n)-arcs. Here we keep the same definition but allow to be a multiset, that is, permit to contain multiple points. The case k=q ²+q+2 is of interest because it is the first value of k for which a (k,n)-arc must be a multiset. The problem of classifying (q ²+q+2,q+2)-arcs is of importance in coding theory, since it is equivalent to classifying 3-dimensional q-ary error-correcting codes of length q ²+q+2 and minimum distance q ². Indeed, it was the coding theory problem which provided the initial motivation for our study. It turns out that such arcs are surprisingly rich in geometric structure. Here we construct several families of (q ²+q+2,q+2)-arcs as well as obtain some bounds and non-existence results. A complete classification of such arcs seems to be a difficult problem.     

On generalized Clifford algebraC 4 (n) andGL q (2;C) quantum group

The non commuting matrix elements of matrices from quantum groupGL _q (2;C) withq≡ω being then-th root of unity are given a representation as operators in Hilbert space with help ofC ₄ ⁽ⁿ⁾ generalized Clifford algebra generators appropriately tensored with unit 2×2 matrix infinitely many times. Specific properties of such a representation are presented. Relevance of generalized Pauli algebra to azimuthal quantization of angular momentum alà Lévy-Leblond [10] and to polar decomposition ofSU _q (2;C) quantum algebra alà Chaichian and Ellinas [6] is also commented. The case ofq∈C, |q|=1 may be treated parallely.     

Matching Realization of Uq（sln＋1） of Higher Rank in the Quantum Weyl Algebra Wq（2n）

In the paper, we further realize the higher rank quantized universal enveloping algebra Uq（sln＋1） as certain quantum differential operators in the quantum Weyl algebra Wq （2n） defined over the quantum divided power algebra Sq（n） of rank n. We give the quantum differential operators realization for both the simple root vectors and the non-simple root vectors of Uq（sln＋1）. The nice behavior of the quantum root vectors formulas under the action of the Lusztig symmetries once again indicates that our realization model is naturally matched.     

A different quantum stochastic calculus for the Poisson process

Summary We show that a gradient operator defined by perturbations of the Poisson process jump times can be used with its adjoint operator instead of the annihilation and creation operators on the Poisson-Charlier chaotic decomposition to represent the Poisson process. The quantum stochastic integration and the Itô formula are developed accordingly, leading to commutation relations which are different from the CCR. An analog of the Weyl representation is defined for a subgroup ofSL(2, ), showing that the exponential and geometric distributions are closely related in this approach.     

A New Family of 2-Designs over GF (q) Admitting SLm(q1)

Given a 2-(l,3,q³(q^l-5-1/q-1);q) design for an integer l 5 mod 6(q-1) which admits the action of a Singer cycle Z_l of GL_l(q), we construct a 2-(ml,3,q³(q^l-5-1/q-1);q) design for an arbitrary integer m 3 which admits the action of SL_m(q^l). The construction applied to Suzuki's designs actually provides a new family of 2-designs over GF(q) which admit the SL_m(q^l) action.     

SL（2，q）的一个特征性质

本文证明了非单群系列ＳＬ（２，ｑ）（ｑ＝ｐ￣ｎ＞３）可以仅用其极大子群阶之集来刻划，从而得到了ＳＬ（２，ｑ）的一个特征性质．     

A flutter function reveals the chaos of differential calculus

This paper disproves the widespread opinion that chaos may appear in non-linear connections only. The common differential operator, which assigns the first derivative to each function, is linear and chaotic in the sense of Li and Yorke.     

The KNTZ trick from arborescent calculus and the structure of the differential expansion

Theoretical and Mathematical Physics - The recently proposed Kameyama–Nawata–Tao–Zhang (KNTZ ) trick completed the long search for exclusive Racah matrices $$...     

Generalized quadrangles associated with G2(q)

If q ≡ 2 (mod 3), a generalized quadrangle with parameters q, q² is constructed from the generalized hexagon associated with the group G₂(q).     

2-Designs overGF(q)

We give a construction of a series of 2-(n, 3,q ²+q+1;q) designs of vector spaces over a finite fieldGF(q) of odd characteristic. These designs correspond to those constructed by Thomas and the author for even characteristic. As a natural generalization we give a collection ofm-dimensional subspaces which possibly become a 2-(n, m, λ; q) design.