The embedding ofU q(sl (2)) and sine algebras in generalized Clifford algebras

In this note, we establish the connection between certain quantum algebras and generalized Clifford algebras (GCA). Precisely, we embed the quantum tori Lie algebra andU _q(sl (2)) in GCA.     

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.     

Correction of CT burst array errors in the generalized-lee-RT spaces

In [Jain, S.： Array codes in the generalized-Lee-RT-pseudo-metric （the GLRTP-metric）, to appear in Algebra Colloq.], Jain introduced a new pseudo-metric on the space Matm×s（Zq）, the module space of all m × s matrices with entries from the finite ring Zq, generalized the classical Lee metric [Lee, C. Y.： Some properties of non-binary error correcting codes. IEEE Trans. Inform. Theory, IT-4, 77- 82 （1958）] and array RT-metric [Rosenbloom, M. Y., Tsfasman, M. A.： Codes for m-metric. Prob. Inf. Transm., 33, 45-52 （1997）] and named this pseudo-metric as the Generalized-Lee-RT-Pseudo-Metric （or the GLRTP-Metric）. In this paper, we obtain some lower bounds for two-dimensional array codes correcting CT burst array errors [Jain, S.： CT bursts from classical to array coding. Discrete Math., 308-309, 1489-1499 （2008）] with weight constraints under the GLRTP-metric.     

Onq-difference equations andZ n decompositions of exp q function

Theq-extended hyperbolic functions ofn-th order {h _q,s(z)}s∈ Z _n which areZ _n-components of expq function form the set fundamental solutions of a simpleq-difference equation. Against the background ofq-deformed hyperbolic functions ofn-th order other natural extensions and related topics are considered. Apart from easy general solution of homogenous ordinaryq-difference equations ofn-th order main trigonometric-like identity known for hyperbolic functions ofn-th order is given itsq-commutative counterpart. Hint how to arrive at other identities is implicit in theq-treatment proposed. The paper constitutes an example of the application of the method of projections presented in Advances in Applied Clifford Algebras publication [19]; see also references to Ben Cheikh’s papers.     

Cyclic group actions on odd-dimensional spheres

We show that for any simple (2q−1)-knotk, q>1, and any positive integern, the knot # ₁ ⁿ k is the fixed-point set of aZ _n -action onS ^2q+1. Further, we show that for many values ofn there are examples of (2q−1)-knots,q≥2, which are the fixed-point sets of inequivalentZ _n -actions. This paper was written whilst the first author was in receipt of a Research Grant from the Science Research Council of Great Britain.     

Quantum groups and quantum shuffles

Let U _q ⁺ be the “upper triangular part” of the quantized enveloping algebra associated with a symetrizable Cartan matrix. We show that U _q ⁺ is isomorphic (as a Hopf algebra) to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra associated with a suitable Hopf bimodule on the group algebra of Z ⁿ . This method gives supersymetric as well as multiparametric versions of U _q ⁺ in a uniform way (for a suitable choice of the Hopf bimodule). We give a classification result about the Hopf algebras which can be obtained in this way, under a reasonable growth condition. We also show how the general formalism allows to reconstruct higher rank quantized enveloping algebras from U _q sl(2) and a suitable irreducible finite dimensional representation. Oblatum 21-III-1997 & 12-IX-1997     

Finite-dimensional crystals B 2,s for quantum affine algebras of type D (1) n

The Kirillov–Reshetikhin modules W^r,s are finite-dimensional representations of quantum affine algebras U’_q labeled by a Dynkin node r of the affine Kac–Moody algebra and a positive integer s. In this paper we study the combinatorial structure of the crystal basis B^2,s corresponding to W^2,s for the algebra of type D⁽¹⁾_n. 2000 Mathematics Subject Classification Primary—17B37; Secondary—81R10 Supported in part by the NSF grants DMS-0135345 and DMS-0200774.     

Higher order recurrences for analytical functions of Tchebysheff type

Relation of hyperbolons of volume one to generalized Clifford algebras is described in [1b] and there some applications are listed. In this note which is an extension of [8] we use the one parameter subgroups of the group of hyperbolons of volume one in order to define and investigate generalization of Tchebysheff polynomial system. Parallely functions of roots of polynomials of any degree are studied as possible generalization of symmetric functions considered by Eduard Lucas. It is found how functions of roots of polynomial of any degree are related to this generalization of Tchebysheff polynomials. The relation is explicit. In a primary sense the considered generalization is in passing fromZ ₂ toZ _n group decomposition of the exponential. We end up with an application of the discovered generalization to quite large class of dynamical systems with iteration.     

Derivation Algebras and 2-Cocycles of the Algebras of q-Differential Operators

Abstract

In the paper, the deriviation algebras of the associative algebras of the one-variable (resp. multivariable) q-differential operators and of their corresponding Lie algebras are determined. The completeness of the derivation algebras of the algebras of q-differential operators is also discussed. Finally, we calculate H ²(𝒟 _q (n)^?, C) for n ≥ 1, as well as H ²(g l _n (𝒟 _q ), C) under the assumption that q is transcendental over the rational numbers field Q.     

On Appell Sets and the Fueter-Sce Mapping

It is proved in Clifford algebras generated by an odd number of basis vectors e ₁, ... , e _n , that the recently discussed Appell polynomials in Clifford algebras are the Fueter-Sce extension of the complex monomials z ^k . Furthermore, it is shown, for which complex functions the Fueter-Sce extension and the extension method using Appell polynomials coincide.      

Idempotent structure of Clifford algebras

Spinor spaces can be represented as minimal left ideals of Clifford algebras and they are generated by primitive idempotents. Primitive idempotents of the Clifford algebras R _{p, q} are shown to be products of mutually nonannihilating commuting idempotent % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaIXaaabaGaaGOmaaaaaaa!3DBD!\[{\textstyle{1 \over 2}}\]2}}\](1+e _T ), where the k=q–r _q–p basis elements e _T satisfy e _T ²=1. The lattice generated by a set of mutually annihilating primitive idempotents is examined. The final result characterizes all Clifford algebras R _{p, q} with an anti-involution such that each symmetric elements is either a nilpotent or then some right multiple of it is a nonzero symmetric idempotent. This happens when p+q<-3 and (p, q)(2, 1).     

Between quantum Virasoro algebra $$\mathcal{L}_c $$ and generalized Clifford algebras

In this paper we construct the quantum Virasoro algebra generators in terms of operators of the generalized Clifford algebras C_n^k. Precisely, we show that can be embedded into generalized Clifford algebras. Junior Associate at The Abdus Salam ICTP, Trieste, Italy.     

Approximations simultanées de nombres algébriques de Q p par des rationnels

 In this paper, we prove that if β₁,…, β _n are p-adic numbers belonging to an algebraic number field K of degree n + 1 over Q such that 1, β₁,…,β _n are linearly independent over Z, there exist infinitely many sets of integers (q ₀,…, q _n ), with q ₀ ≠ 0 and

Clifford algebras and geometric algebra

The geometric algebra as defined by D. Hestenes is compared with a constructive definition of Clifford algebras. Both approaches are discussed and the equivalence between a finite geometric algebra and the universal Clifford algebra R _{p, q} is shown. Also an intermediate way to construct Clifford algebras is sketched. This attempt to conciliate two separated approaches may be useful taking into account the recognized importance of Clifford algebras in theoretical and applied physics.     

The Birman-Murakami-Wenzl algebras of type E n

The Birman-Murakami-Wenzl algebras (BMW algebras) of type E _n for n = 6; 7; 8 are shown to be semisimple and free over the integral domain \mathbbZ[ d^±1,l^±1,m ]

Linear quivers and the geometric setting of quantum GLn

This paper presents a connection between the defining basis presented by Beilinson-Lusztig-MacPherson [1] in their geometric setting for quantum GL_n and the isomorphism classes of linear quiver representations. More precisely, the positive part of the basis in [1] identifies with the defining basis for the relevant Ringel-Hall algebra; hence, it is a PBW basis in the sense of quantum groups. This approach extends to q-Schur algebras, yielding a monomial basis property with respect to the Drinfeld-Jimbo type presentation for the positive (or negative) part of the q-Schur algebra. Finally, the paper establishes an explicit connection between the canonical basis for the positive part of quantum GL_n and the Kazhdan-Lusztig basis for q-Schur algebras.     

A generalization of a result of R. Lyons about measures on [0, 1)

Letμ be a probability measure on [0, 1), invariant underS:x ↦px mod 1, and for which almost every ergodic component has positive entropy. Ifq is a real number greater than 1 for which logq/ logp is irrational, andT _n sendsx toq ⁿx mod 1, then for any ε>0 the measureμT _n ⁻¹ will — for a set ofn of positive lower density — be within ε of Lebesgue measure.     

Sur la singularité des produits de Riesz et des mesures spectrales associées à la somme des chiffres

Using a property of generalized characters, we first prove that two Riesz products with constant coefficients and distinct Fourier spectra are mutually singular. IfS _r (n) denotes the sum of digits in ther-adic representation of the integern, the same technique allows us to establish the mutual singularity of the spectral measures of the sequences: α(n)=exp[2iπaS _p (n)],β(n)=exp[2iπbS _q (n)], for every pair of integersp≠q, a, b being real numbers such thata(p−1)∉ {tiZ} andb(q−1)∉Z. This result has been proved by T. Kamae whenp andq are two relatively prime integers.      

Expansion in SLd(Z/qZ), q arbitrary

Let S be a fixed finite symmetric subset of SL _d (Z), and assume that it generates a Zariski-dense subgroup G. We show that the Cayley graphs of π _q (G) with respect to the generating set π _q (S) form a family of expanders, where π _q is the projection map Z→Z/q Z.