Algebraic Integrability of Macdonald Operators and Representations of Quantum Groups

In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper [ES] to the q-deformed case. A generalized Baker–Akhiezer function is realized as a matrix character of a Verma module and is a common eigenfunction for a commutative ring of difference operators. In particular, we obtain the following result in Macdonald theory: at integer values of the Macdonald parameter k, there exist difference operators commuting with Macdonald operators which are not polynomials of Macdonald operators. This result generalizes an analogous result of Chalyh and Veselov for the case q=1, to arbitrary q. As a by-product, we prove a generalized Weyl character formula for Macdonald polynomials (= Conjecture 8.2 from [FV]), the duality for the -function, and the existence of shift operators.     

Borel Subalgebras Redux with Examples from Algebraic and Quantum Groups

Both building upon and revising previous literature, this paper formulates the general notion of a Borel subalgebra B of a quasi-hereditary algebra A. We present various general constructions of Borel subalgebras, establish a triangular factorization of A, and relate the concept to graded Kazhdan–Lusztig theories in the sense of Cline et al. (Tôhoku Math. J. 45 (1993), 511–534). Various interesting types of Borel subalgebras arise naturally in different contexts. For example, `excellent" Borel subalgebras come about by abstracting the theory of Schubert varieties. Numerous examples from algebraic groups, q-Schur algebras, and quantum groups are considered in detail.     

Module Categories Over Representations of SL q (2) in the Non-Semisimple Case

We classify semisimple module categories over the tensor category of representations of quantum SL(2). Received: May 2006, Revised: December 2006, Accepted: December 2006     

Galois Reconstruction of Finite Quantum Groups

Let be a (small) category and let F:  →  alg_f be a functor, where alg_f is the category of finite-dimensional measured algebras over a field k (or Frobenius algebras). We construct a universal Hopf algebra A_aut(F) such that F factorizes through a functor :  →  coalg_f(A_aut(F)), where coalg_f(A_aut(F)) is the category of finite-dimensional measured A_aut(F)-comodule algebras. This general reconstruction result allows us to recapture a finite-dimensional Hopf algebra A from the category coalg_f(A) and the forgetful functor ω: coalg_f(A) →  alg_f: we have A  A_aut(ω). Our universal construction is also done in a C*-algebra framework, and we get compact quantum groups in the sense of Woronowicz.     

Flat Connections and Quantum Groups

We review the Kohno–Drinfeld theorem and a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection _C on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes and values in any g-module V. We sketch our proof of this conjecture for the cases when g=sl _n or when g is arbitrary and V is a vector, spin or adjoint representation. We also establish a precise link between the connection _C and Cherednik's generalisation of the Knizhnik–Zamolodchikov connection to finite reflection groups.     

量子群的基变换与范畴同构

令Ｍ是Ｚ［ｖ］的由ｖ－１和奇素数ｐ生成的理想，Ｕ是Ａ＝Ｚ［ｖ］Ｍ上相伴于对称Ｃａｒｔａｎ矩阵的量子群， Ａ－Γ是环同态， Ｕг＝ＵＡΓ［Ｕг］是Ｕг的量子坐标代数，本文建立了量子坐标代数的基变换：即在相关约束条件下有Г－Ｈｏｐｆ同构 Ａ［Ｕ］ＡГ≌Г［Ｕг］．我们证明了有限秩 Ａ自由 １型可积 Ｕ模范畴和有限秩 Ａ自由 Ａ［Ｕ］余模范畴是同构的．特别，当 Г是域时，局部有限 １型 Ｕг模范畴和Г［Ｕг］余模范畴是同构的．最后，我们还证明了在［１］中定义的诱导函子和Ｂ．Ｐａｒｓｈａｌｌ与王建磐博士在［２］中研究的诱导函子的一致性．     

The Tilting Tensor Product Theorem and Decomposition Numbers for Symmetric Groups

We show how the tilting tensor product theorem for algebraic groups implies a reduction formula for decomposition numbers of the symmetric group. We use this to prove generalisations of various theorems of Erdmann and of James and Williams. Supported by Nuffield grant scheme NUF-NAL 02. Preliminary work on this paper was undertaken at the Isaac Newton Institute as part of the programme on Symmetric Functions and Macdonald Polynomials.     

Spin Representations and Powers of 2

The elementary divisors of the reduced spin 2–decomposition matrix for the double covers of the finite symmetric groups are described. In addition, the maximal power of 2 dividing all spin character values on a fixed conjugacy class corresponding to a cycle type with odd parts only is determined.     

Harish-Chandra Vertices and Steinberg's Tensor Product Theorems for Finite General Linear Groups

Representations of Hecke and q-Schur algebras are closely relatedto those of finite general linear groups G in non-describingcharacteristics. Such a relationship can be described by certainfunctors. Using these functors, we determine the Harish-Chandravertices and sources of certain indecomposable G-modules. TheGreen correspondence is investigated in this context. As a furtherapplication of our theory, we establish Steinberg's tensor producttheorems for irreducible representations of G in non-describingcharacteristics. 1991 Mathematics Subject Classification: 20C20,20C33, 20G05, 20G40.     

Directed Algebraic Topology,Categories and Higher Categories

Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A ‘directed space’ has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). On the other hand, directed homotopy can give geometric models for lax higher categories. Applications have been mostly developed in the theory of concurrency. Unexpected links with noncommutative geometry and the modelling of biological systems have emerged. Work partially supported by MIUR Research Projects.     

有限群的某些定理

本文给出了有限群的如下结果：１）成为可解群与超可解群的条件；２）推广了Ｉｔｏ定理，３）超可解且内幂零的群结构。     

Crossed Modules and Quantum Groups in Braided Categories

Let A be a Hopf algebra in a braided category . Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group the corresponding braided category of modules is identified with a full subcategory in . The connection with cross products is discussed and a suitable cross product in the class of quantum braided groups is built. Majid–Radford theorem, which gives equivalent conditions for an ordinary Hopf algebra to be such a cross product, is generalized to the braided category. Majid's bosonization theorem is also generalized.     

Adaptive Surface Reconstruction Based on Tensor Product Algebraic Splines

Surface reconstruction from unorganized data points is a challenging problem in Computer Aided Design and Geometric Modeling. In this paper, we extend the mathematical model proposed by Juttler and Felis (Adv. Comput. Math., 17 (2002), pp. 135-152) based on tensor product algebraic spline surfaces from fixed meshes to adaptive meshes. We start with a tensor product algebraic B-spline surface defined on an initial mesh to fit the given data based on an optimization approach. By measuring the fitting errors over each cell of the mesh, we recursively insert new knots in cells over which the errors are larger than some given threshold, and construct a new algebraic spline surface to better fit the given data locally. The algorithm terminates when the error over each cell is less than the threshold. We provide some examples to demonstrate our algorithm and compare it with Jiittler's method. Examples suggest that our method is effective and is able to produce reconstruction surfaces of high quality.AMS subject classifications: 65D17     

Reductions of Tensor Categories Modulo Primes

We study good (i.e., semisimple) reductions of semisimple rigid tensor categories modulo primes. A prime p is called good for a semisimple rigid tensor category 𝒞 if such a reduction exists (otherwise, it is called bad). It is clear that a good prime must be relatively prime to the Müger squared norm |V|² of any simple object V of 𝒞. We show, using the Ito–Michler theorem in finite group theory, that for group-theoretical fusion categories, the converse is true. While the converse is false for general fusion categories, we obtain results about good and bad primes for many known fusion categories (e.g., for Verlinde categories). We also state some questions and conjectures regarding good and bad primes.     

Structure of Solvable Rational Groups

R. Gow proved that the order of a solvable rational group isdivisible only by the primes 2, 3 and 5. In this paper it isproved that in a solvable rational group the Sylow 5-subgroupis always normal and elementary Abelian. Moreover, the structureof rational {2, 5}-groups is described in detail. 2000 MathematicsSubject Classification 20C15, 20C20, 20E34, 20E45.     

Slices of Essentially Algebraic Categories

This paper is a contribution to the theory of functor slices of J. Sichler and V. Trnková. For every ordinal α we introduce a basket , prove that every essentially algebraic category of height α is a slice of , characterize small slices of and give a common generalization of known results about slices of the algebraic basket .      

De Rham Cohomology and Hodge Decomposition For Quantum Groups

Let be one of the N²-dimensionalbicovariant first order differential calculi for the quantumgroups GL_q(N), SL_q(N), SO_q(N), or Sp_q(N), where q is a transcendentalcomplex number and z is a regular parameter. It is shown thatthe de Rham cohomology of Woronowicz' external algebra coincides with the de Rham cohomologiesof its left-coinvariant, its right-coinvariant and its (two-sided)coinvariant subcomplexes. In the cases GL_q(N) and SL_q(N) thecohomology ring is isomorphic to the coinvariant external algebra and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in thesecases. The main technical tool is the spectral decompositionof the quantum Laplace-Beltrami operator. 2000 MathematicalSubject Classification: 46L87, 58A12, 81R50.     

The k-Orbit Reconstruction for Abelian and Hamiltonian Groups

Let G be a permutation group on a set . Then G acts in the natural way on the collection ^{k} of all k-element subsets. Orbits under this action are called k-orbits. A problem similar to the Edge-Reconstruction Conjecture in graph theory can be posed for k-orbits of a general group G. Here the k-orbit reconstruction problem is solved for transitive Abelian and Hamiltonian groups: all k-orbits of Abelian groups are reconstructible if k>3 and the same is true for Hamiltonian groups if k>4.     

Obstruction Theory for Objects in Abelian and Derived Categories

ABSTRACT

In this article, we develop the obstruction theory for lifting complexes, up to quasi-isomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular case we also obtain the corresponding obstruction theory for lifting of objects in terms of Yoneda Ext-groups. In an appendix we prove the existence of miniversal derived deformations of complexes.