Right Coideal Subalgebras of Quantized Universal Enveloping Algebras of Type G 2*

In this article we describe the right coideal subalgebras containing all group-like elements of the two-parameter quantum group U _q (𝔤), where 𝔤 is a simple Lie algebra of type G ₂, while the main parameter of quantization q is not a root of 1. As a consequence, we determine that there are precisely 60 different right coideal subalgebras containing all group-like elements. If the multiplicative order t of q is finite, t > 4, t ≠ 6, then the same classification remains valid for homogeneous right coideal subalgebras of the two-parameter version of the small Lusztig quantum group u _q (𝔤).     

The Triangulated Hopf Category n +SL(2)

We discuss an example of a triangulated Hopf category related to SL(2). It is an equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. We prove some coherence equations for structure isomorphisms. In particular, the Hopf category is monoidal.     

Factorization of simple modules for certain pointed Hopf algebras

We study the representations of two types of pointed Hopf algebras: restricted two-parameter quantum groups, and the Drinfel'd doubles of rank one pointed Hopf algebras of nilpotent type. We study, in particular, under what conditions a simple module can be factored as the tensor product of a one-dimensional module with a module that is naturally a module for the quotient by central group-like elements. For restricted two-parameter quantum groups, given θ a primitive ℓth root of unity, the factorization of simple -modules is possible, if and only if gcd((y−z)n,ℓ)=1. For rank one pointed Hopf algebras, given the data , the factorization of simple -modules is possible if and only if |χ(a)| is odd and |χ(a)|=|a|=|χ|. Under this condition, the tensor product of two simple -modules is completely reducible, if and only if the sum of their dimensions is less than or equal to |χ(a)|+1.     

Yetter–Drinfeld modules over bosonizations of dually paired Hopf algebras

Let (R^∨,R)$(R^{\vee },R)$ be a dual pair of Hopf algebras in the category of Yetter–Drinfeld modules over a Hopf algebra H$H$ with bijective antipode. We show that there is a braided monoidal isomorphism between rational left Yetter–Drinfeld modules over the bosonizations of R$R$ and of R^∨$R^{\vee }$, respectively. As an application of this very general category isomorphism we obtain a natural proof of the existence of reflections of Nichols algebras of semi-simple Yetter–Drinfeld modules over H$H$.     

Right coideal subalgebras of the quantum Borel algebra of type

In this paper we describe the right coideal subalgebras containing all group-like elements of the multiparameter quantum group , where is a simple Lie algebra of type G₂, while the main parameter of quantization q is not a root of 1. If the multiplicative order t of q is finite, t>4, t≠6, then the same classification remains valid for homogeneous right coideal subalgebras of the positive part of the multiparameter version of the small Lusztig quantum group.     

SL（2，q）的一个特征性质

本文证明了非单群系列ＳＬ（２，ｑ）（ｑ＝ｐ￣ｎ＞３）可以仅用其极大子群阶之集来刻划，从而得到了ＳＬ（２，ｑ）的一个特征性质．     

Twisted Comodule Algebras and the Generalized Quantum Double

In this article we show that the notion of twisted comodule algebras is a generalization of Drinfeld's cocycle twisting construction of bialgebras or Hopf algebras, and that several other twisting constructions, such as Ferrer's twisting product and Lu's one-sided twisting, are also special examples of twisted comodule algebras. Some general properties on twisted comodule algebras are investigated, and applications to the generalized quantum double are given.     

Divisible Designs Admitting GL(3, q) as an Automorphism Group

Starting from a 3-dimensional projective space we construct divisible designs admitting GL(3,q) as an automorphism group.     

2-Fusion Systems with Components of Type SL2(q)

We study a problem in fusion systems designed to mimic, simplify, and generalize parts of the Classification of Finite Simple Groups. A 2-fusion system is a 2-group S with a family of injective homomorphisms on subgroups of S. Fusion systems arise in the study of modular representations and classifying spaces, and our results have potential ramifications beyond finite group theory. One problem is to determine S or, whenever possible, the entire 2-fusion system from the knowledge of certain subgroups and maps. We consider the case where S contains a subgroup and maps that arise in the Classification with standard component of type SL₂(q).     

2-Cohomologies of the groups SL(n, q)

Let G = SL(n, q), where q is odd, V be a natural module over G, and L = S²(V) be its symmetric square. We construct a 2-cohomology group H²(G, L). The group is one-dimensional over F _q if n = 2 and q ≠ 3, and also if (n, q) = (4, 3). In all other cases H²(G, L) = 0. Previously, such groups H²(G, L) were known for the cases where n = 2 or q = p is prime. We state that H²(G, L) are trivial for n ⩾ 3 and q = p^m, m ⩾ 2. In proofs, use is made of rather elementary (noncohomological) methods. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 687–704, November–December, 2008.     

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     

An Algorithm for Computing the Global Basis of a Finite Dimensional Irreducible U q ( so 2 n + 1 ) or U q ( so 2 n )-Module

Abstract

We describe a simple algorithm for computing the canonical basis of any irreducible finite-dimensional U _q (s o _2n+1) or U _q (s o _2n )-module.     

The product of lattice covolume and discrete series formal dimension: p-adic GL(2)

Let $F$ be a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. We show how to calculate the product of the covolume of a torsion-free lattice in $PGL(2,F)$ and the formal dimension of a discrete series representation of $GL(2,F)$. The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.     

有限域上F4和G2型Chevalley群的二元生成

本文证明了有限域上F4和G2型Chevalley群均可由两个元素生成.     

The peak algebra and the descent algebras of types B and D

We show the existence of a unital subalgebra of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that is the image of the descent algebra of type B under the map to the descent algebra of type A which forgets the signs, and also the image of the descent algebra of type D. The algebra contains a two-sided ideal which is defined in terms of interior peaks. This object was introduced in previous work by Nyman (2003); we find that it is the image of certain ideals of the descent algebras of types B and D. We derive an exact sequence of the form . We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions and by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces and over and explain the connection with previous work of Stembridge (1997); we also obtain new properties of his descents-to-peaks map and construct a type B analog.

     

Verifying Huppert's Conjecture for PSL3(q) and PSU3(q 2)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ? H × A, where A is an abelian group. We examine arguments to verify this conjecture for the simple groups of Lie type of rank two. To illustrate our arguments, we extend Huppert's results and verify the conjecture for the simple linear and unitary groups of rank two.     

A conjecture of Abe-Iiyori and the Ree groups 2F4(q),Ⅱ

In this paper the author has solved a problem of Abe and liyori for the finite simple groups 2F4(q) and 2F4(2)'.