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).     

The Hilbert–Poincaré Series for Some Algebras of Invariants of Cyclic Groups

Let be a linear representation of a finite group over a field of characteristic 0. Further, let R be the corresponding algebra of invariants, and let P (t) be its Hilbert–Poincaré series. Then the series P (t) represents a rational function (t)/(t). If R is a complete intersection, then (t) is a product of cyclotomic polynomials. Here we prove the inverse statement for the case where is an almost regular (in particular, regular) representation of a cyclic group. This yields an answer to a question of R. Stanley in this very special case. Bibliography: 3 titles.     

On the Skitovich–Darmois Theorem for Compact Abelian Groups

Let X be a separable compact Abelian group, Aut(X) the group of topological automorphisms of X, f _n: XX a homomorphism f _n(x)=nx, and X ⁽ⁿ⁾=Im f _n. Denote by I(X) the set of idempotent distributions on X and by (X) the set of Gaussian distributions on X. Consider linear statistics L ₁= ₁( ₁)+ ₂( ₂) and L ₂= ₁( ₁)+ ₂( ₂), where _j are independent random variables taking on values in X and with distributions _j, and _j, _jAut(X). The following results are obtained. Let X be a totally disconnected group. Then the independence of L ₁ and L ₂ implies that ₁, ₂I(X) if and only if X possesses the property: for each prime p the factor-group X/X ^(p) is finite. If X is connected, then there exist independent random variables _j taking on values in X and with distributions _j, and _j, _jAut(X) such that L ₁ and L ₂ are independent, whereas ₁, ₂(X) * I(X).     

Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups

Journal of Fourier Analysis and Applications - A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses...     

Cyclic Cohomology of étale Groupoids: The General Case

We give a general method for computing the cyclic cohomology of crossed products by étale groupoids, extending the Feigin–Tsygan–Nistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor, and Tsygan for the convolution algebra C _c (G) of an étale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered.     

A Quantum Modification of Relative Chen–Ruan Cohomology

In this paper, by using the de Rham model of Chen–Ruan cohomology, we define the relative Chen–Ruan cohomology ring for a pair of almost complex orbifold(G, H) with H being an almost sub-orbifold of G. Then we use the Gromov–Witten invariants ofG, the blow-up of G along H,to give a quantum modification of the relative Chen–Ruan cohomology ring H*CR(G, H) when H is a compact symplectic sub-orbifold of the compact symplectic orbifold G.     

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.     

On Hardy’s Uncertainty Principle for Solvable Locally Compact Groups

We establish analogues of Hardy’s uncertainty principle for the Fourier transform on ? for locally compact Abelian groups and for some classes of solvable Lie groups, such as diamond groups and exponential Lie groups with non-trivial centre.     

The Compact Quantum Group Uq（2） （Ⅱ）

In this paper, we first prove that the θ-deformation Uθ（2） of U（2） constructed by Connes and Violette is our special case of the quantum group Uq（2） constructed in our previous paper. Then we will show that the set of truces on the C^＊ -algebra Uθ, θ irrational, is determined by the set of the truces on a subalgebra of Uθ.     

Ordinary p-adic étale Cohomology Groups Attached to Towers of Elliptic Modular Curves

Fix a prime number p 5 and a positive integer N prime to p. We consider the projective limits of p-adic étale cohomology groups of the modular curves X1(Npr) and Y1(Npr) (r 1), which are denoted by ESp(N) Z _p and GES p(N)Z _p , respectively. Let e* be the projector to the direct sum of the i-eigenspaces of the ordinary part, for i 0, -1 mod p-1. Our main result states that e* GESp (N)Z _p has a good p-adic Hodge structure, which can be described in terms of -adic modular forms, extending the previously known result for e* ESp (N)Z _p . We then apply the method of Harder and Pink to the Galois representation on e* ESp(N) Z _p to construct large unramified abelian p-extensions over cyclotomic Z _p -extensions of abelian number fields.     

White Noise Approach Construction of Ф4 4 Quantum to The Fields （Ⅱ）

In this paper a complete proof for the existence of generalized operators satisfying abstract 02 dynamical equations of quantum motions δ^2/δt^2Ф（t, x） ＋ （ △- m^2）Ф（t, x） = -λ ：Ф^3（t, x）, subject to a suitable initial condition, is given under the framework of white noise analysis. Also some important commutation relations related to Ф44 quantum fields are discussed and proved in detail.     

The Entries of Haar-Invariant Matrices from the Classical Compact Groups

Let Γ _n =(γ _ij ) _n×n be a random matrix with the Haar probability measure on the orthogonal group O(n), the unitary group U(n), or the symplectic group Sp(n). Given 1≤m<n, a probability inequality for a distance between (γ _ij ) _n×m and some mn independent F-valued normal random variables is obtained, where F=ℝ, ℂ, or ℍ (the set of real quaternions). The result is universal for the three cases. In particular, the inequality for Sp(n) is new.     

Abelianness of Mumford–Tate Groups Associated to Some Unitary Groups

In this paper, we investigate the action of the -cohomology of the compact dual of a compact Shimura Variety S() on the -cohomology of S()> under a cup product. We use this to split the cohomology of S() into a direct sum of (not necessarily irreducible) -Hodge structures. As an application, we prove that for the class of arithmetic subgroups of the unitary groups U(p,q) arising from Hermitian forms over CM fields, the Mumford–Tate groups associated to certain holomorphic cohomology classes on S() are Abelian. As another application, we show that all classes of Hodge type (1,1) in H² of unitary four-folds associated to the group U(2,2) are algebraic.     

Regular Balanced Cayley Maps for Cyclic, Dihedral and Generalized Quaternion Groups

A Cayley map is a Cayley graph embedded in an orientable surface such that. the local rotations at every vertex are identical. In this paper, balanced regular Cayley maps for cyclic groups, dihedral groups, and generalized quaternion groups are classified.     

A Generalized Fejér’s Theorem for Locally Compact Groups

The classical Fejér’s theorem is a criterion for pointwise convergence of Fourier series on the unit circle. We generalize it to locally compact groups.     

The Baum–Connes Conjecture and Discrete Group Actions on Trees

We introduce a property (BC) for discrete groups, which we prove to imply the Baum–Connes conjecture with coefficients and the K-amenability of the group. Then, we show that if is a discrete group which acts on a tree X such that X/ is compact, and the stabilizers of the vertices and the stabilizers of the edges satisfy (BC), then itself satisfies (BC) Finally, we indicate a couple of applications.     

Cibils–Rosso's Theorem for Quantum Groupoids

Abstract

We generalize the Cibils–Rosso's theorem for categories of Sweedler's Hopf bimodules to the one for categories of weak entwined bimodules. We show that the weak entwined bimodules are modules over a certain algebra. Our best results are attained for categories of weak Hopf bimodules over quantum groupoids (weak Hopf algebras), as special cases of weak Doi–Hopf bimodules.