The Ramifications of the Centres: Quantised Function Algebras at Roots of Unity

This paper continues the study of quantised function algebrasO[G] of a semisimple group G at an lth root of unity . Thesealgebras were introduced by De Concini and Lyubashenko in 1994,and studied further by De Concini and Procesi and by Gordon,amongst others. Our main purpose here is to increase understandingof the finite-dimensional factor algebras O[G](g), for g G.We determine the representation type and block structure ofthese factors, and (for many g) describe them up to isomorphism.A series of parallel results is obtained for the quantised Borelalgebras and . 2000 Mathematical Subject Classification: 16W35,17B37.     

The Proalgebraic Completion of Rigid Groups

A finitely generated group is called representation rigid (briefly, rigid) if for every n, has only finitely many classes of simple representations in dimension n. Examples include higher rank S-arithmetic groups. By Margulis super rigidity, the latter have a stronger property: they are representation super rigid; i.e., their proalgebraic completion is finite dimensional. We construct examples of nonlinear rigid groups which are not super rigid, and which exhibit every possible type of infinite dimensionality. Whether linear representation rigid groups are super rigid remains an open question.     

Blow-up of semilinear pde's at the critical dimension. A probabilistic approach

We present a probabilistic approach which proves blow-up of solutions of the Fujita equation in the critical dimension . By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as . In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Escobedo and Levine (1995) to the case of -Laplacians with possibly different parameters .

     

Differential operators on a polarized abelian variety

Let be an ample line bundle over a complex abelian variety . We show that the space of all global sections over of and are both of dimension one. Using this it is shown that the moduli space of rank one holomorphic connections on a compact Riemann surface does not admit any nonconstant algebraic function. On the other hand, is biholomorphic to the moduli space of characters of , which is an affine variety. So is algebraically distinct from the character variety if is of genus at least one.

     

Stability of the C *-Algebra Associated with Twisted CCR

The universal enveloping C ^*-algebra A of twisted canonical commutation relations is considered. It is shown that, for any (–1,1), the C ^*-algebra A is isomorphic to the C ^*-algebra A ₀ generated by partial isometries t _i ,t _i ^*,i=1,¨,d satisfying the relations t _i ^* t _j = _ij (1– _k<i t _k t _k ^*), t _j t _i =0, ij and it is proved that the Fock representation of A is faithful.     

Complex Hyperspherical Equations, Nodal-Partitioning, and First-Excited-State Theorems in ℝ n

Three problems related to the spherical quantum billiard in are considered. In the first, a compact form of the hyperspherical equations leads to their complex contracted representation. Employing these contracted equations, a proof is given of Courant's nodal-symmetry intersection theorem for diagonal eigenstates of spherical-like quantum billiards in . The second topic addresses the first-excited-state theorem for the spherical quantum billiard in . Wavefunctions for this system are given by the product form, ( )Z _q+()Y ⁽ⁿ⁾ , where is dimensionless displacement, is angular-momentum number, qis an integer function of dimension, Z() is either a spherical Bessel function (nodd) or a Bessel function of the first kind (neven) and represents (n– 1) independent angular components. Generalized spherical harmonics are written . It is found that the first excited state (i.e., the second eigenstate of the Laplacian) for the spherical quantum billiard in is n-fold degenerate and a first excited state for this quantum billiard exists which contains a nodal bisecting hypersurface of mirror symmetry. These findings establish the first-excited-state theorem for the spherical quantum billiard in . In a third study, an expression is derived for the dimension of the th irreducible representation (irrep) of the rotation group O(n) in by enumerating independent degenerate product eigenstates of the Laplacian.     

集对Fuzzy格及其在格表示论中的应用

用幂集格构造了集对 Ｆｕｚｚｙ 格（这与用整数对构造有理数集有相似之处）,并用它证明了完整的软代数表示定理,即定义了到自身的映射且有最大元和最小元的格为软代数的充要条件是它与某个集对 Ｆｕｚｚｙ 格的子格同构．这样,与分配格在幂集 Ｂｏｏｌｅ 格中表示相对应,软代数在集对 Ｆｕｚｚｙ 格中有表示,在理论上是很完美的     

Finite-dimensional left ideals in some algebras associated with a locally compact group

Let be a locally compact group, let be its group algebra, let be its usual measure algebra, let be the second dual of with an Arens product, and let be the conjugate of the space of bounded, left uniformly continuous, complex-valued functions on with an Arens-type product. We find all the finite-dimensional left ideals of these algebras. We deduce that such ideals exist in and if and only if is compact, and in (except those generated by right annihilators of ) and if and only if is amenable.

     

Two-dimensional asymptotic expansions for large deviations of spherically distributed random vectors if the dominating point degenerates asymptotically

A geometric approach to asymptotic expansions for large-deviation probabilities, developed for the Gaussian law by Breitung and Richter [J. Multivariate Anal.,58, 1–20 (1996)], will be extended in the present paper to the class of spherical measures by utilizing their common geometric properties. This approach consists of rewriting the probabilities under consideration as large parameter values of the Laplace transform of a suitably defined function, expanding this function in a power series, and then applying Watson’s lemma. A geometric representation of the Laplace transform allows one to combine the global and local properties of both the underlying measure and the large-deviation domain. A special new type of difficulty is to be dealt with because the so-called dominating points of the large-deviation domain degenerate asymptotically. As is shown in Richter and Schumacher (in print), the typical statistical applications of large-deviation theory lead to such situations. In the present paper, consideration is restricted to a certain two-dimensional domain of large-deviations having asymptotically degenerating dominating points. The key assumption is a parametrized expansion for the inverse $\bar g^{ - 1} $ of the negative logarithm of the density-generating function of the two-dimensional spherical law under consideration.