Finite Dimensional Hopf Algebras Over the Dual Group Algebra of the Symmetric Group in Three Letters

We classify finite-dimensional Hopf algebras whose coradical is isomorphic to the algebra of functions on 𝕊₃. We describe a new infinite family of Hopf algebras of dimension 72.     

Z n -Orthograded Monocomposition Algebras

We study NQM algebras A having an orthogonal automorphism of finite order n 3 (called Z _n -orthograded NQM algebras). The Z ₃-orthograded NQM algebras of dimension 7 are treated in more detail. In particular, we find all algebras A which are not bi-isotropic in this class, and for every algebra A, determine an automorphism group Aut,A and an orthogonal automorphism group Ortaut,A. In constructing and classifying (up to isomorphism) NQM algebras, use is made of orthogonal decompositions of the algebras.     

Various Structures Associated to the Representation Categories of Eight-Dimensional Nonsemisimple Hopf Algebras

We determine various additional structures on all nonsemisimple Hopf algebras of dimension 8 over an algebraically closed field k of characteristic 0, including their representation rings and quasitriangular structures. As a consequence, it is shown that for two such Hopf algebras, the tensor categories of their representations are monoidally equivalent if and only if the representation rings of them are isomorphic as rings. An erratum to this article is available at .     

Induced Projective Modules Over Group von Neumann Algebras

Let G be a group and k a subring of the field of complex numbers. In this paper we study the additive map in reduced K-theory, which is associated with the inclusion of the group algebra kG into the group von Neumann algebra G, and obtain necessary and sufficient conditions for it to be identically zero (or zero modulo torsion). Our results complete the work of Eckmann [Comm. Math. Helvet. 71 (1996), 453–462; Arch. Math. 76 (2001), 241–249] and Schafer [K-theory 19 (2000), 211–217], while reducing to Swan’s theorem on induced representations [Ann. Math. 71 (1960), 552–578], in the case where the group G is finite. (Received: January 2005) Research supported by University of Athens grant Pythagoras 70/3/7298.     

Quasitriangular Pointed Hopf Algebras Constructed by Ore Extensions

We study the quasitriangular structures for a family of pointed Hopf algebras which is big enough to include Taft's Hopf algebras H _n ², Radford's Hopf algebras H _N,n,q, and E(n). We give necessary and sufficient conditions for the Hopf algebras in our family to be quasitriangular. For the case when they are, we determine completely all the quasitriangular structures. Also, we determine the ribbon elements of the quasitriangular Hopf algebras and the quasi-ribbon elements of their Drinfel'd double.     

The Second Indicator and the Trace of the Antipode for Representations of a Class of Hopf Algebras of Andruskiewitch-Schneider Type

We give definition of the second indicator for any finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0 with the square of antipode an inner automorphism. We derive a closed formula for the trace of the antipode on endomorphism algebras of simple self-dual modules of nilpotent type liftings of quantum linear spaces and compute the value of their second indicators. Uniqueness of indicators is examined.     

Conjugacy Classes,Class Sums and Character Tables for Hopf Algebras

We extend the notion of conjugacy classes and class sums from finite groups to semisimple Hopf algebras and show that the conjugacy classes are obtained from the factorization of H as irreducible left D(H)-modules. For quasitriangular semisimple Hopf algebras H, we prove that the product of two class sums is an integral combination of the class sums up to d ^?2 where d = dim H. We show also that in this case the character table is obtained from the S-matrix associated to D(H). Finally, we calculate explicitly the generalized character table of D(kS ₃), which is not a character table for any group. It moreover provides an example of a product of two class sums which is not an integral combination of class sums.     

Conjugacy in Permutation Representations of the Symmetric Group

Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e., that conjugacy classes of S _n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S _n  × S _n . We also consider the permutation representations of S _n which arise from the action of S _n on ordered tuples and on unordered subsets, and classify which of them unite conjugacy classes and which do not.     

Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class A n , II

Up to derived equivalence, the representation-finite self-injective algebras of class A _n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.     

On Some Finite Dimensional Representations of Symplectic Reflection Algebras Associated to Wreath Products

Let Γ _N : = S _N  ? Γ ^N be the wreath product of Γ, a finite subgroup of SL(2,C), by the symmetric group of degree N. In this article we classify all the irreducible representations of S _N  ? Γ ^N that can be extended to a representation of the associated symplectic reflection algebra H _1,k,c (Γ _N ) (where k is a complex number and c a class function on the nontrivial elements of Γ) for nonzero values of k.     

Specht Modules and Simple Modules for Centralizer Algebras of the Symmetric Group

Let Σ_n be the symmetric group on n letters. For l ≤ n identify Σ_l with a subgroup of Σ_n in the natural way. Let k be an algebraically closed field of characteristic p. This article begins to develop a theory for modules over the centralizer algebras kΣ_n^Σ_l that is analogous to James's theory of permutation modules, Specht modules, and simple modules over kΣ_n. We make a conjecture about how to construct all simple kΣ_n^Σ_l-modules, we develop tools to test the conjecture, and we prove that it is correct for all n when l < p.     

The Bass Conjecture and Group von Neumann Algebras

The precise relationship between the Bass conjecture for the Hattori–Stallings trace for projective ZG-modules and the map from reduced K-theory of ZG to reduced K-theory of the von Neumann algebra, NG, of G, of G is determined. As a consequence it is shown this map is zero for all groups G. It is also shown that the map induced on K-theory from the inclusion of NG to the ring of closed, densely defined operators affiliated to NG is an isomorphism. Together with the above result, this gives some positive evidence for the validity of the Division Ring Conjecture for torsion free groups.     

相应于型$B_n$的非标准形变的弱Hopf代数

我们引入了型$B_n$的非标准量子群$X_q(B_n)$, 它具有Hopf代数结构,然后我们替换$X_q(B_n)$的类群元得到对应的弱Hopf代数${\mathfrak{w}X_q(B_{n})}$. 最后我们描述了${\mathfrak{w}X_q(B_{n})}$作为余代数的Ext--箭图.     

Ore Extensions of Hopf Algebras

In this paper, Ore extensions in the class of Hopf algebras are studied. The classification theorem enables one to describe the Hopf--Ore extensions for the group algebras, for the algebras and , and for the quantum ax + b group.     

Characters of Fundamental Representations of Quantum Affine Algebras

We give closed formulae for the q-characters of the fundamental representations of the quantum loop algebra of a classical Lie algebra, in terms of a family of partitions satisfying some simple properties. We also give the multiplicities of the eigenvalues of the imaginary subalgebra in terms of these partitions.     

Symmetry Classification for Jackson Integrals Associated with the Root System BC n

The Jackson integrals associated with the non-reduced root system are defined as multiple sums which are generalization of the Bailey's very-well-poised ₆₆ sum. They are classified by the number of their parameters when they can be expressed as a product of the Jacobi elliptic theta functions. The sums which appear in the classification list coincide with those investigated individually by Gustafson and van Diejen.     

An Simple Elementary Proof for the Inequality dn〈3^n

In this paper, by using a variation of the Chebyshev＇s method, we present a very simple, elementary proof of an inequality which has applications in number theory.     

Commutative Group Algebras of Cardinality {\cal N} 1

We let FG be the group algebra of an abelian group G over a field F with characteristic p. Also, we define G_p and S(FG) as the groups of all p-primary normed elements in G and FG, respectively. We prove that if G_p is Hausdorff and both F and G have cardinalities not exceeding ₁, then S(FG)/G_p is a direct sum of cyclics. Thus G_p is a direct factor of S(FG), and in particular G is a direct factor of the group of all normalized units V(FG), provided that the torsion part of G is a p-group. This answers a question posed by us in Hokkaido Math. J. (2000). Moreover we establish that if G is p-splitting, then any F-isomorphism of the group algebras FG and FH implies that H is p-splitting. We also show that if G is of power ₁ whose p-component G_p is a direct sum of torsion-complete groups and F has power p, then the F-isomorphism of FG and FH for any group H yields an isomorphism between G_p and H_p. In particular, when G is of power ₁ and is p-mixed of torsion-free rank 1 whose G_p is torsion-complete, we have G H. If F is in power p and G, with cardinality ₁, is a direct sum of p-local algebraically compact groups such that FG FH as F-algebras for some group H, then G H. These statements extend results due to Beers-Richman-Walker (1983), and also partially solve a well-known question raised by May in 1979.     

Interval Valued Intuitionistic （S, T）-fuzzy Hv-submodules

On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K. Atanassov, the notion of interval valued intuitionistic fuzzy Hv-submodules of an Hv-module with respect to a t-norm T and an s-norm S is given and the characteristic properties are described. The homomorphic image and the inverse image are investigated. In particular, the connections between interval valued intuitionistic （S, T）-fuzzy Hv-submodules and interval valued intuitionistic （S, T）-fuzzy submodules are discussed.