Harmonic Analysis on the SU(2) Dynamical Quantum Group

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang–Baxter equation, which is precisely the Yang–Baxter equation satisfied by 6j-symbols. We investigate one of the simplest examples, generalizing the standard SU(2) quantum group. The matrix elements for its corepresentations are identified with Askey–Wilson polynomials, and the Haar measure with the Askey–Wilson measure. The discrete orthogonality of the matrix elements yield the orthogonality of q-Racah polynomials (or quantum 6j-symbols). The Clebsch–Gordan coefficients for representations and corepresentations are also identified with q-Racah polynomials. This results in new algebraic proofs of the Biedenharn–Elliott identity satisfied by quantum 6j-symbols.     

Q-Operator and the Drinfeld Equation

We show that the TQ equation is satisfied by the trace over the quantum space of the product of R-matrices intertwining two representations of the quantum double of the Borel subalgebra of the affine algebra (the standard two-dimensional and the N-dimensional cyclic representations).     

Effective su q(2) Models and Polynomial Algebras for Fermion–Boson Hamiltonians

We show that schematic su(2)h ₃ interaction Hamiltonians, where su(2) plays the role of the pseudospin algebra of fermion operators and h ₃ is the Heisenberg algebra for bosons, are closely related to certain nonlinear models defined on a single quantum algebra su _q(2) of quasifermions. In particular, su _q(2) analogues of the Da Providencia–Schütte and extended Lipkin models are presented. We analyze the connection between q and the physical parameters of the fermion–boson system and, using polynomial algebras, discuss the integrability properties of the interaction Hamiltonians.     

Hodge and Laplace–Beltrami Operators for Bicovariant Differential Calculi on Quantum Groups

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace–Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite-dimensional. Using Jucys–Murphy elements of the Hecke algebra, the eigenvalues of the Laplace–Beltrami operator for the Hopf algebra (SL _q (N)) are computed.     

A Complete Formulation of the Baum–Connes Conjecture for the Action of Discrete Quantum Groups

We formulate a version of the Baum–Connes conjecture for a discrete quantum group, building on our earlier work. Given such a quantum group , we construct a directed family -algebras (F varying over some suitable index set), borrowing the ideas of Cuntz such that there is a natural action of satisfying the assumptions of Goswami and Kuku which makes it possible to define the analytical assembly map, say , i= 0, 1, as in our previous work, from the -equivariant K-homolgy groups of to the K-theory groups of the reduced dual (c.f. [9] and the references therein for more details). As a result, we can define the Baum–Connes maps , and in the classical case, i.e. when for a discrete group, the isomorphism of the above maps for i= 0, 1 is equivalent to the Baum–Connes conjecture. Furthermore, we verify its truth for an arbitrary finite-dimensional quantum group and obtain partial results for the dual of (2).     

Ring-Shaped Functions and Wigner 6j-Symbols

We obtain an explicit expression for the ring-shaped matrix relating the ring-shaped functions corresponding to different values of an axiality parameter and find the relation between this matrix and the Wigner 6j-symbols. We investigate the motion of a quantum particle in a ring-shaped model with a zero “bare” potential and find bases factored in spherical and cylindrical coordinates for this model. We derive a formula generalizing the Rayleigh expansion for a plane wave in terms of spherical waves in a ring-shaped model. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 299–310, February, 2006.     

Invariant Cyclic Homology

We define a noncommutative analogue of invariant de Rham cohomology. More precisely, for a triple (A, H, M) consisting of a Hopf algebra H, an H-comodule algebra A, an H-module M, and a compatible grouplike element in H, we define the cyclic module of invariant chains on A with coefficients in M and call its cyclic homology the invariant cyclic homology of A with coefficients in M. We also develop a dual theory for coalgebras. Examples include cyclic cohomology of Hopf algebras defined by Connes–Moscovici and its dual theory. We establish various results and computations including one for the quantum group SL(q,2).     

Some greedyt-intersecting families of finite sequences

Letn, s ₁,s ₂, ... ands _n be positive integers. Assume is an integer for eachi}. For , , and , denotes _p (a)={j|1jn,a _j p}, , and . is called anI _t ^p -intersecting family if, for any a,b ,a _i b _i =min(a _i ,b _i )p for at leastt i's. is called a greedyI _t ^P -intersecting family if is anI _t ^p -intersecting family andW _p (A)W _p (B+A ^c ) for anyAS _p ( ) and any with |B|=t–1.In this paper, we obtain a sharp upper bound of | | for greedyI _t ^p -intersecting families in for the case 2ps _i (1in) ands ₁>s ₂>...>s _n .This project is partially supported by the National Natural Science Foundation of China (No.19401008) and by Postdoctoral Science Foundation of China.     

An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic)

Elliptic 6j-symbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6j-symbols (or q-Racah polynomials) and Wilson’s biorthogonal ₁₀ W ₉ functions. We give an elementary construction of elliptic 6j-symbols, which immediately implies several of their main properties. As a consequence, we obtain a new algebraic interpretation of elliptic 6j-symbols in terms of Sklyanin algebra representations.     

On a conjecture concerningk-Hamilton-nice sequences

In this paper, we prove that a non-negative rational number sequence (a ₁,a ₂, ...,a _k+1) isk-Hamilton-nice, if (1)a _k+12, and (2) _j ^=1/h (i _j –1)k–1 implies for arbitraryi ₁,i ₂,...i _h {1,2,... ,k}. This result was conjectured by Guantao Chen and R.H. Schelp, and it generalizes several well-known sufficient conditions for graphs to be Hamiltonian.This project is supported by the National Natural Science Foundation of China.     

Semi-Invertible Extensions and Asymptotic Homomorphisms

We consider the semi-group Ext(A, B) of extensions of a separable C ^*-algebra A by a stable C ^*-algebra B modulo unitary equivalence and modulo asymptotically split extensions. This semi-group contains the group Ext^–1/2(A, B) of invertible elements (i.e. of semi-invertible extensions). We show that the functor Ext^–1/2(A, B) is homotopy invariant and that it coincides with the functor of homotopy classes of asymptotic homomorphisms from C A to M(B) that map S A C( ) A into B.     

The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

Distance-Regular Graphs Related to the Quantum Enveloping Algebra of sl(2)

We investigate a connection between distance-regular graphs and U _q(sl(2)), the quantum universal enveloping algebra of the Lie algebra sl(2). Let be a distance-regular graph with diameter d 3 and valency k 3, and assume is not isomorphic to the d-cube. Fix a vertex x of , and let (x) denote the Terwilliger algebra of with respect to x. Fix any complex number q {0, 1, –1}. Then is generated by certain matrices satisfying the defining relations of U _q(sl(2)) if and only if is bipartite and 2-homogeneous.     

A Graham-Sloane Type Construction for s-Surjective Matrices

We give a construction of (n–s)-surjective matrices with n columns over using Abelian groups and additive s-bases. In particular we show that the minimum number of rows ms _q(n,n–s) in such a matrix is at most s ^s q ^n–s for all q, n and s.     

Bell's Inequality for Two-Particle Mixed Spin States

We derive the Bell–Clauser–Horne–Shimony–Holt inequalities for two-particle mixed spin states both in the conventional quantum mechanics and in the hidden-variables theory. We consider two cases for the vectors , and specifying the axes onto which the particle spins of a correlated pair are projected. In the first case, all four vectors lie in the same plane, and in the second case, they are oriented arbitrarily. We compare the obtained inequalities and show that the difference between the predictions of the two theories is less for mixed states than for pure states. We find that the inequalities obtained in quantum mechanics and the hidden-variables theory coincide for some special states, in particular, for the mixed states formed by pure factorable states. We discuss the points of similarity and difference between the uncertainty relations and Bell's inequalities. We list all the states for which the right-hand side of the Bell–Clauser–Horne–Shimony–Holt inequality is identically equal to zero.     

Dual generalized Bernstein basis

The generalized Bernstein basis in the space Π_n of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced by Philips [Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518], is given by the formula [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT Numer. Math. 44 (2004) 63–78],

We give explicitly the dual basis functions for the polynomials , in terms of big q-Jacobi polynomials P_k(x;a,b,ω/q;q), a and b being parameters; the connection coefficients are evaluations of the q-Hahn polynomials. An inverse formula—relating big q-Jacobi, dual generalized Bernstein, and dual q-Hahn polynomials—is also given. Further, an alternative formula is given, representing the dual polynomial (0jn) as a linear combination of min(j,n-j)+1 big q-Jacobi polynomials with shifted parameters and argument. Finally, we give a recurrence relation satisfied by , as well as an identity which may be seen as an analogue of the extended Marsden's identity [R.N. Goldman, Dual polynomial bases, J. Approx. Theory 79 (1994) 311–346].     

A priori estimates of strong solutions of semilinear parabolic equations

We study an initial boundary value problem for the semilinear parabolic equation

Bounds for Non-Gaussian Approximations of U-Statistics

Let X, ,X ₁,...,X _n be i.i.d. random variables taking values in a measurable space ( ). Consider U-statistics of degree two

W-algebras related to parafermion algebras

We study a W-algebra of central charge 2(k−1)/(k+2), k=2,3,…, contained in the commutant of a Heisenberg algebra in a simple affine vertex operator algebra L(k,0) of type with level k. We calculate the operator product expansions of the W-algebra. We also calculate some singular vectors in the case k6 and determine the irreducible modules and Zhu's algebra. Furthermore, the rationality and the C₂-cofiniteness are verified for such k.