Deformed Algebras from Elliptic sl(N) Algebras

We extend to the sl(N)sl(N) case the results that we previously obtained on the construction of W_q,p{\cal W}_{q,p} algebras from the elliptic algebra A_q,p([^(sl)](2)_c){\cal A}_{q,p}(\widehat{sl}(2)_{c}). The elliptic algebra \elp\elp at the critical level c= m N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N(Nу)/2 integers, defining q-deformations of the W_N{\cal W}_{N} algebra, are constructed. The operators t(z) also close an exchange algebra when (-p^\sfrac12)^NM = q^-c-N(-p^\sfrac{1}{2})^{NM} = q^{-c-N} for M ? \ZZM\in\ZZ. It becomes Abelian when in addition p= q^Nh, where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W_N{\cal W}_{N} algebras depending on the parity of h, characterizing the exchange structures at p p q^Nh as new W_q,p(sl(N)){\cal W}_{q,p}(sl(N)) algebras.     

On Endomorphisms of Quantum Tensor Space

We give a presentation of the endomorphism algebra ${\rm End}_{\mathcal {U}_{q}(\mathfrak {sl}_{2})}(V^{\otimes r})$ , where V is the three-dimensional irreducible module for quantum ${\mathfrak {sl}_2}$ over the function field ${\mathbb {C}(q^{\frac{1}{2}})}$ . This will be as a quotient of the Birman–Wenzl–Murakami algebra BMW _r (q) : =  BMW _r (q ^?4, q ² ? q ^?2) by an ideal generated by a single idempotent Φ _q . Our presentation is in analogy with the case where V is replaced by the two-dimensional irreducible ${\mathcal {U}_q(\mathfrak {sl}_{2})}$ -module, the BMW algebra is replaced by the Hecke algebra H _r (q) of type A _r-1, Φ _q is replaced by the quantum alternator in H ₃(q), and the endomorphism algebra is the classical realisation of the Temperley–Lieb algebra on tensor space. In particular, we show that all relations among the endomorphisms defined by the R-matrices on ${V^{\otimes r}}$ are consequences of relations among the three R-matrices acting on ${V^{\otimes 4}}$ . The proof makes extensive use of the theory of cellular algebras. Potential applications include the decomposition of tensor powers when q is a root of unity.     

From quantum to elliptic algebras

It is shown that the elliptic algebra at the critical level c = –2 has a multidimensional center containing some trace-like operators t(z). A family of Poisson structures indexed by a non-negative integer and containing the q-deformed Virasoro algebra is constructed on this center. We show also that t(z) close an exchange algebra when p ^m = q ^c+2 for , they commute when in addition p = q ^2k for k integer non-zero, and they belong to the center of when k is odd. The Poisson structures obtained for t(z) in these classical limits contain the q-deformed Virasoro algebra, characterizing the structures at p q ^2k as new algebras.     

Gleason's Theorem for Rectangular JBW-Triples

A JBW^*-triple B is said to be rectangular if there exists a W^*-algebra A and a pair (p,q) of centrally equivalent elements of the complete orthomodular lattice P(A)\mathcal{P}(A) of projections in A such that B is isomorphic to the JBW^*-triple pAq. Any weak^*-closed injective operator space provides an example of a rectangular JBW^*-triple. The principal order ideal CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} of the complete ^*-lattice CP(A)\mathcal{C}\mathcal{P}(A) of centrally equivalent pairs of projections in a W^*-algebra A, generated by (p,q), forms a complete lattice that is order isomorphic to the complete latticeI(B)\mathcal{I}(B) of weak^*-closed inner ideals in B and to the complete lattice S(B)\mathcal{S}(B) of structural projections on B. Although not itself, in general, orthomodular, CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} possesses a complementation that allows for definitions of orthogonality, centre, and central orthogonality to be given. A less familiar notion in lattice theory, that is well-known in the theory of Jordan algebras and Jordan triple systems, is that of rigid collinearity of a pair (e₂,f²) and (e₂,f²) of elements of CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)}. This is defined and characterized in terms of properties of P(A)\mathcal{P}(A). A W^*-algebra A is sometimes thought of as providing a model for a statistical physical system. In this case B, or, equivalently, pAq, may be thought of as providing a model for a fixed sub-system of that represented by A. Therefore, CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} may be considered to represent the set consisting of a particular kind of sub-system of that represented by pAq. Central orthogonality and rigid collinearity of pairs of elements of CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} may be regarded as representing two different types of disjointness, the former, classical disjointness, and the latter, decoherence, of the two sub-systems. It is therefore natural to consider bounded measures m on CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} that are additive on centrally orthogonal and rigidly collinear pairs of elements. Using results of J.D.M. Wright, it is shown that, provided that neither of the two hereditary sub-W^*-algebras pAp and qAq of A has a weak^*-closed ideal of Type I², such measures are precisely those that are the restrictions of bounded sesquilinear functionals {_m on pAp 2 qAq with the property that the action of the centroid Z(B) of B commutes with the adjoint operation. When B is a complex Hilbert space of dimension greater than two, this result reduces to Gleason's Theorem.     

Modules-at-Infinity for Quantum Vertex Algebras

This is a sequel to [Li4] and [Li5] in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian , denoted by DY _q (sl ₂) and with q a nonzero complex number. For each nonzero complex number q, we construct a quantum vertex algebra V _q and prove that every DY _q (sl ₂)-module is naturally a V _q -module. We also show that -modules are what we call V _q -modules-at-infinity. To achieve this goal, we study what we call -local subsets and quasi-local subsets of for any vector space W, and we prove that any -local subset generates a (weak) quantum vertex algebra and that any quasi-local subset generates a vertex algebra with W as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity.     

Screening Length of Rotating Heavy Meson from AdS/CFT

In this paper we consider a quark-antiquark (q[`(q)]q\bar{q}) pair which can be interpreted as a meson in N=4{\mathcal{N}}=4 SYM thermal plasma. We assume that the string moves at speed v and rotates around its center of mass simultaneously. By using the AdS/CFT correspondence, we obtain the momentum densities of the rotating string and determine its motion for small angular velocities. Then in general case, we calculate the screening length of q[`(q)]q\bar{q} pair numerically and show that its velocity dependance is in consistent with the well known formula L _s T∼(1−v ²)^1/4 in the literature.     

Highest Weight Modules Over Quantum Queer Superalgebra {U_q(\mathfrak {q}(n))}

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))}. The key ingredients are the triangular decomposition of U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))} and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))}-modules in the category O_q ^{3 0}{\mathcal {O}_{q}^{\geq 0}}.     

Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra

The quantized universal enveloping algebra U _q(q(n)) of the strange Lie superalgebra q(n) and a super-analogue HC _q (N) of the Hecke algebra H _q (N) are constructed. These objects are in a duality similar to the known duality between U _q (gl(n)) and H _q (N).     

Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on strip graphs G of the honeycomb lattice for a variety of transverse widths equal to L _y vertices and for arbitrarily great length, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form , where m denotes the number of repeated subgraphs in the longitudinal direction. We give general formulas for N _Z,G,j for arbitrary L _y . We also present plots of zeros of the partition function in the q plane for various values of v and in the v plane for various values of q. Plots of specific heat for infinite-length strips are also presented, and, in particular, the behavior of the Potts antiferromagnet at is investigated.     

Generalization of the $$\mathcal{U} $$ q (gl(N)) Algebra and Staggered Models

We develop a technique for the construction of integrable models with a ₂ grading of both the auxiliary (chain) and quantum (time) spaces. These models have a staggered disposition of the anisotropy parameter. The corresponding Yang–Baxter equations are written down and their solution for the gl(N) case is found. We analyze in details the N = 2 case and find the corresponding quantum group behind this solution. It can be regarded as the quantum group , with a matrix deformation parameter q such that (q )² = q ². The symmetry behind these models can also be interpreted as the tensor product of the (–1)-Weyl algebra by an extension of _q (gl(N)) with a Cartan generator related to deformation parameter –1.     

Asymptotics for the Covariance of the Airy<Subscript>2</Subscript> Process

In this paper we compute some of the higher order terms in the asymptotic behavior of the two point function \mathbbP(A₂(0) ￡ s₁,A₂(t) ￡ s₂)\mathbb{P}(\mathcal {A}_{2}(0)\leq s_{1},\mathcal{A}_{2}(t)\leq s_{2}), extending the previous work of Adler and van Moerbeke (; Ann. Probab. 33, 1326–1361, 2005) and Widom (J. Stat. Phys. 115, 1129–1134, 2004). We prove that it is possible to represent any order asymptotic approximation as a polynomial and integrals of the Painlevé II function q and its derivative q′. Further, for up to tenth order we give this asymptotic approximation as a linear combination of the Tracy-Widom GUE density function f ₂ and its derivatives. As a corollary to this, the asymptotic covariance is expressed up to tenth order in terms of the moments of the Tracy-Widom GUE distribution.     

Quantized affine Lie algebras and diagonalization of braid generators

Let U _q be a quantized affine Lie algebra. It is proven that the universal R-matrix R of U _q satisfies the celebrated conjugation relationR ⁺ =TR withT the usual twist map. As applications, the braid generator is shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight U _q -module and a spectral decomposition formula for the braid generator is obtained which is the generalization of Reshetikhin and Gould forms to the present affine case. Casimir invariants are constructed and their eigenvalues computed by means of the spectral decomposition formula. As a by-product, an interesting identity is found.     

A New Quantum Analog of the Brauer Algebra

We introduce a new algebra depending on two nonzero complex parameters z and q such that its specialization at z=q ⁿ and q=1 coincides the Brauer algebra. We show that the action of the new algebra commutes with the representation of the twisted deformation of the enveloping algebra U(o _n) in the tensor power of the vector representation.     

Singular vectors of quantum group representations for straight Lie algebra roots

We give explicit formulae for singular vectors of Verma modules over U_q(G), where G is any complex simple Lie algebra. The vectors we present correspond exhaustively to a class of positive roots of G which we call straight roots. In some special cases, we give singular vectors corresponding to arbitrary positive roots. For our vectors we use a special basis of U_q(G ^-), where G ^- is the negative roots subalgebra of G, which was introducted in our earlier work in the case q=1. This basis seems more economical than the Poincaré-Birkhoff-Witt type of basis used by Malikov, Feigin, and Fuchs for the construction of singular vectors of Verma modules in the case q=1. Furthermore, this basis turns out to be part of a general basis recently introduced for other reasons by Lusztig for U_q(^-), where ^- is a Borel subalgebra of G.A. v. Humboldt-Stiftung fellow, permanent address and after 22 September 1991: Bulgarian Academy of Sciences, Institute of Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria.     

On the Identity of Some von Neumann Algebras and Some Consequences

Among von Neumann algebras, the Weyl algebra W{\mathcal{W}} generated by two unitary groups {U(α)} and {V(β)}, the algebra U{\mathcal{U}} generated by a completely non-unitary semigroup of isometries {U ₊(α)} and the Weyl algebra W₊^h{\mathcal{W}_{+}^{h}} pertaining to a semi-bounded space with homogeneous spectrum of the generator of {V(β)}, all share the property that their representations are completely reducible and the irreducible representations are equivalent. We trace this fact to the identity of these algebras, in the sense that any of them contains a representation of any of the remaining two algebras, which in turn contains the original algebra. We prove this statement by explicit construction. The aforementioned results about the representations of the algebras follow immediately from the proof for any of them. Also, by the above construction we prove for W^h₊{\mathcal{W}^{h}_{+}} the analog of a theorem by Sinai for W{\mathcal{W}} : given {V(β)} with semi-bounded homogeneous spectrum, there exists a completely non-unitary semigroup {U ₊(α)} such that {V(β)} and {U ₊(α)} generate W₊^h{\mathcal{W}_{+}^{h}}.     

The 2009 world average of <Emphasis Type="Italic">α</Emphasis><Subscript><Emphasis Type="Bold">s</Emphasis></Subscript>

Measurements of α _s, the coupling strength of the Strong Interaction between quarks and gluons, are summarised and an updated value of the world average of a_s(M_Z⁰)\alpha_{\mathrm{s}}(M_{\mathrm{Z}^{0}}) is derived. Special emphasis is laid on the most recent determinations of α _s. These are obtained from τ-decays, from global fits of electroweak precision data and from measurements of the proton structure function F₂, which are based on perturbative QCD calculations up to O(a_s⁴)\mathcal{O}(\alpha_{\mathrm{s}}^{4}); from hadronic event shapes and jet production in e⁺e⁻ annihilation, based on O(a_s³)\mathcal{O}(\alpha_{\mathrm{s}}^{3}) QCD; from jet production in deep inelastic scattering and from ϒ decays, based on O(a_s²)\mathcal{O}(\alpha_{\mathrm{s}}^{2}) QCD; and from heavy quarkonia based on unquenched QCD lattice calculations. A pragmatic method is chosen to obtain the world average and an estimate of its overall uncertainty, resulting in

A superalgebra morphism of U q [osp(1/2N)] onto the deformed oscillator superalgebra W q (N)

We prove that the deformed oscillator superalgebra W _q (n) (which in the Fock representation is generated essentially byn pairs ofq-bosons) is a factor algebra of the quantized universal enveloping algebra U _q [osp(1/2n)]. We write down aq-analog of the Cartan-Weyl basis for the deformed osp(1/2n) and also give an oscillator realization of all Cartan-Weyl generators.