Nakayama Twisted Centers and Dual Bases of Frobenius Cellular Algebras

For a Frobenius cellular algebra, we prove that, if the left (right) dual basis of a cellular basis is again cellular, then the algebra is symmetric. Moreover, some ideals of the center are constructed by using the so-called Nakayama twisted center.     

具1-关系的正规乘基的有限表示型连通基本代数

本文确定了所有的具有１ 关系的正规乘基的有限表示型连通基本代数．     

胞腔代数和仿射胞腔代数简介

表示论中一个最基本的问题是确定不可约表示的参数集,这个问题至今没有完全解决.对于Graham和Lehrer引入的有限维胞腔代数,这个问题得到了完满解答,并被成功地应用于数学和物理中出现的许多代数.近来,人们引入仿射胞腔代数,将Graham和Lehrer有限维胞腔代数的表示理论框架推广到一类无限维代数上.仿射胞腔代数不仅包括有限维胞腔代数,也包括无限维的仿射Temperley-Lieb代数和Lusztig的A-型仿射Hecke代数.本文将对胞腔代数的发展历史和主要研究成果做一些综述,同时,对新引入的仿射胞腔代数及其最新成果做一点简介.     

Crystal Bases for Quantum Generalized Kac-Moody Algebras

In this paper, we develop the crystal basis theory for quantumgeneralized Kac–Moody algebras. For a quantum generalizedKac–Moody algebra U_q(g), we first introduce the categoryO_int of U_q(g)-modules and prove its semisimplicity. Next, wedefine the notion of crystal bases for U_q(g)-modules in thecategory O_int and for the subalgebra . We then prove the tensor product rule and the existence theoremfor crystal bases. Finally, we construct the global bases forU_q(g)-modules in the category O_int and for the subalgebra . 2000 Mathematics Subject Classification17B37, 17B67.     

Quasiequational Bases of Cantor Algebras

There are continuum many quasivarieties of Cantor algebras having an ω-independent quasiequational basis but no independent quasiequational basis whose intersection does have an independent quasiequational basis.     

On Primitive Cellular Algebras

First we define and study the exponentiation of a cellular algebra by a permutation group that is similar to the corresponding operation (the wreath product in primitive action) in permutation group theory. Necessary and sufficient conditions for the resulting cellular algebra to be primitive and Schurian are given. This enables us to construct infinite series of primitive non-Schurian algebras. Also we define and study, for cellular algebras, the notion of a base, which is similar to that for permutation groups. We present an upper bound for the size of an irredundant base of a primitive cellular algebra in terms of the parameters of its standard representation. This produces new upper bounds for the order of the automorphism group of such an algebra and in particular for the order of a primitive permutation group. Finally, we generalize to 2-closed primitive algebras some classical theorems for primitive groups and show that the hypothesis for a primitive algebra to be 2-closed is essential. Bibliography: 16 titles.     

On a Filtered Multiplicative Bases of Group Algebras II

We give an explicit list of all p-groups G with a cyclic subgroup of index p ², such that the group algebra KG over the field K of characteristic p has a filtered multiplicative K-basis. We also prove that such a K-basis does not exist for the group algebra KG, in the case when G is either a non-Abelian powerful p-group or a two generated p-group (p2) with a central cyclic commutator subgroup. This paper is a continuation of the paper which appeared in Arch. Math. (Basel) 74 (2000), 217–285.     

Partition Algebras are Cellular

The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.     

Diagonal Bases in Orlik-Solomon Type Algebras

To encode an important property of the no broken circuit bases of the Orlik- Solomon-Terao algebras, Szenes has introduced a particular type of bases, the so called diagonal basis. We prove that this definition extends naturally to a large class of algebras, the so called - algebras. Our definitions make also use of an iterative residue formula based on the matroidal operation of contraction. This formula can be seen as the combinatorial analogue of an iterative residue formula introduced by Szenes. As an application we deduce nice formulas to express a pure element in a diagonal basis.AMS Subject Classification: 52C35, 05B35, 14F40.     

Double Basic Algebras

To any bounded lattice with section antitone bijections can be assigned an algebra with two binary, two unary and a nullary operations satisfying similar axioms as a basic algebra. Due to the doubled similarity type, this algebra is called a double basic algebra. Also conversely, every double basic algebra induces a bounded lattice equipped with antitone bijections in every section. We study properties of double basic algebras, their interval algebras and several conclusions of the so-called pseudo-commutativity. This work is supported by the Research Project MSM 6198959214 by Czech Government.     

Perfect Bases for Integrable Modules over Generalized Kac-Moody Algebras

We introduce the notion of perfect bases for integrable highest weight modules over generalized Kac-Moody algebras and show that the colored oriented graphs arising from perfect bases are isomorphic to the highest weight crystals B(λ) over quantum generalized Kac-Moody algebras.     

On Double Basic Algebras and Pseudo-effect Algebras

Double basic algebras are a counterpart of bounded lattices with order-antiautomorphisms on principal filters. In the paper, an independent axiomatization of double basic algebras is given and lattice pseudo-effect algebras are characterized in the setting of double basic algebras.     

Representations of Ariki-Koike Algebras and Grobner-Shirshov Bases

In this paper, we investigate the structure of Ariki–Koikealgebras and their Specht modules using Gröbner–Shirshovbasis theory and combinatorics of Young tableaux. For a multipartition, we find a presentation of the Specht module S given by generatorsand relations, and determine its Gröbner–Shirshovpair. As a consequence, we obtain a linear basis of S consistingof standard monomials with respect to the Gröbner–Shirshovpair. We show that this monomial basis can be canonically identifiedwith the set of cozy tableaux of shape . 2000 Mathematics SubjectClassification 16Gxx, 05Exx.     

Filtered Multiplicative Bases of Restricted Enveloping Algebras

We study the problem of the existence of filtered multiplicative bases of a restricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotent restricted Lie algebra over a field of positive characteristic p.     

Cellular Structure of q-Brauer Algebras

Let R be a commutative noetherian domain. The q-Brauer algebras over R are shown to be cellular algebras in the sense of Graham and Lehrer. In particular, they are iterated inflations of Hecke algebras of type A. When R is a field of arbitrary characteristic, we determine for which parameters the q-Brauer algebras are quasi-hereditary. Then, using the general theory of cellular algebras we parametrize all irreducible representations of q-Brauer algebras.     

Extremal Properties of Bases for Representations of Semisimple Lie Algebras

Let be a complex semisimple Lie algebra with specified Chevalley generators. Let V be a finite dimensional representation of with weight basis . The supporting graph P of is defined to be the directed graph whose vertices are the elements of and whose colored edges describe the supports of the actions of the Chevalley generators on V. Four properties of weight bases are introduced in this setting, and several families of representations are shown to have weight bases which have or are conjectured to have each of the four properties. The basis can be determined to be edge-minimizing (respectively, edge-minimal) by comparing P to the supporting graphs of other weight bases of V. The basis is solitary if it is the only basis (up to scalar changes) which has P as its supporting graph. The basis is a modular lattice basis if P is the Hasse diagram of a modular lattice. The Gelfand-Tsetlin bases for the irreducible representations of sl(n, ) serve as the prototypes for the weight bases sought in this paper. These bases, as well as weight bases for the fundamental representations of sp(2n, ) and the irreducible one-dimensional weight space representations of any semisimple Lie algebra, are shown to be solitary and edge-minimal and to have modular lattice supports. Tools developed here are used to construct uniformly the irreducible one-dimensional weight space representations. Similar results for certain irreducible representations of the odd orthogonal Lie algebra o(2n + 1, ), the exceptional Lie algebra G ₂, and for the adjoint and short adjoint representations of the simple Lie algebras are announced.     

Complete Commutative Basic Algebras

By a basic algebra is meant an MV-like algebra (A, ⊕, ¬, 0) of type 〈2, 1, 0〉 derived in a natural way from bounded lattices having antitone involutions on their principal filters. In the previous paper (Botur and Hala?, Mult. Valued Log. Soft Comp., 2007) we have shown that finite basic algebras for which the operation ⊕ is commutative are MV-algebras. In this paper we generalize this result by considering commutative basic algebras for which the underlying lattice is complete.