Representation Theory of Artin Algebras I

This is the first of a series of papers dealing with the representation theory of artin algebras, where by an artin algebra we mean an artin ring having the property that its center is an artin ring and λ is a finitely generated module over its center. The over all purpose of this paper is to develop terminology and background material which will be used in the rest of the papers in the series. While it is undoubtedly true that much of this material can be found in the literature or easily deduced from results already in the literature, the particular development presented here appears to be new and is especially well suited as a foundation for the papers to come.     

Representation Theory of Reductive Normal Algebraic Monoids

New results in the representation theory of ``semisimple' algebraic monoids are obtained, based on Renner's monoid version of Chevalley's big cell. (The semisimple algebraic monoids have been classified by Renner.) The rational representations of such a monoid are the same thing as ``polynomial' representations of the associated reductive group of units in the monoid, and this representation category splits into a direct sum of subcategories by ``homogeneous' degree. We show that each of these homogeneous subcategories is a highest weight category, in the sense of Cline, Parshall, and Scott, and so equivalent with the module category of a certain finite-dimensional quasihereditary algebra, which we show is a generalized Schur algebra in S. Donkin's sense.

     

群与代数的表示理论（迎接ICM2002特约文章）

有限群模表示论,代数群与量子群,代数表示论,同调代数与代数K-理论是当前国际数学研究的前沿重点领域,在20世纪得到巨大发展,它们之间的相互交叉融合与综合统一在数学发展中具有鲜明特色。本文简要概述上述研究领域的发展历史与最新研究进展。     

Topologically Algebraic Algebras

We show that a locally convex algebra is topologically algebraic if, and only if, it is algebraic.Thanks are offered to Professor M. Oudadess for many remarks which allowed an improvement of the first version of this paper. We are also indebted to Professor A. Mallios for valuable suggestions.     

代数表示论的某些新进展

代数表示理论是代数学的一个新的重要分支，在近二十五年的时间里，这一理论有很大的发展，关于代数表示的基础理论的介绍可参见文献（１０１），本文主要从Ｈａｌｌ代数和拟遗传代数两个方面介绍代数表示论的一些最新进展，第一章给出了Ｈａｌｌ代数的基本理论及其方法，并且着重指出了利用这一理论和方法通过代数表示论去实现Ｋａｃ－Ｍｏｏｄｙ李代数及相应的量子包络代数，第二章介绍了拟遗传代数及其表示理论，以及这一理论与复     

代数表示论简介与综述

代数表示论是本世纪七十年代初兴起的代数学的一个新的分支。它的基本内容是研究一个Ａｒｔｉｎ代数上的模范畴。由于各国代数学家的共同努力，这一理论于最近二十年代有了异常迅猛的发展并逐步趋于完善。本文介绍了代数表示论的理论基础；几乎可裂序列；箭图和赋值箭图的表示；Ｃｏｘｅｔｅｒ函子；ＡＲ－箭图的覆盖及代数的Ｇａｌｏｉｓ覆盖。并简单介绍了在代数表示论中普遍应用的工具：Ｔｉｌｔ理论，以及著名的Ｄｐｏｚｇ定理证     

Algebraic K-theory of Normed Algebras

Quillen's algebraic K-theory of discrete rings is extended to the category of normed algebras over a commutative Banach ring k with unit and its relationship with topological K-theory is established. Sufficient conditions for the isomorphism of algebraic and topological K-groups on the category of real normed algebras are given. The isomorphism of algebraic and topological K-functors on the category of polynomial extensions of stable C-algebras is proved.     

A Note on Periodicity in Representation Theory of Finite Dimensional Algebras

Let Λ be an algebra over an algebraically closed field K of infinite transcendence degree over its prime subfield. We prove in particular that if every s-dimensional indecomposable nonprojective Λ-module is Ω-periodic (resp. DTr-periodic) then there exists a common bound for the Ω-periods (resp. DTr-periods) of s-dimensional indecomposable nonprojective Λ-modules.     

A Tableau Approach to the Representation Theory of 0-Hecke Algebras

A 0-Hecke algebra is a deformation of the group algebra of a Coxeter group. Based on work of Norton and Krob-Thibon, we introduce a tableau approach to the representation theory of 0-Hecke algebras of type A, which resembles the classic approach to the representation theory of symmetric groups by Young tableaux and tabloids. We extend this approach to types B and D, and obtain a correspondence between the representation theory of 0-Hecke algebras of types B and D and quasisymmetric functions and noncommutative symmetric functions of types B and D. Other applications are also provided.     

Jordan Algebras Over Algebraic Varieties

General results on the module structure of Jordan algebras over locally ringed spaces are obtained. Albert algebras over a Brauer–Severi variety with associated central simple algebra of degree 3 are constructed using generalizations of the Tits process and the first Tits construction.     

Identities on Units of Algebraic Algebras

Let be an algebraic algebra over an infinite field K and let ( ) be its group of units. We prove a stronger version of Hartley's conjecture for , namely, if a Laurent polynomial identity (LPI, for short) f = 0 is satisfied in ( ), then satisfies a polynomial identity (PI). We also show that if is non-commutative, then is a PI-ring, provided f = 0 is satisfied by the non-central units of . In particular, is locally finite and, thus, the Kurosh problem has a positive answer for K-algebras whose unit group is LPI. Moreover, f = 0 holds in ( ) if and only if the same identity is satisfied in . The last fact remains true for generalized Laurent polynomial identities, provided that is locally finite.     

The Algebraic K-Theory of Operator Algebras

We the study the algebraic K-theory of C *-algebras, forgetting the topology. The main results include a proof that commutative C*-algebras are K-regular in all degrees (that is, all theirN ^T K _iand extensions of the Fischer-Prasolov Theorem comparing algebraic and topological K-theory with finite coefficients.     

Virtual Algebraic Lie Theory: Tilting Modules and Ringel Duals for Blob Algebras

In this paper we describe a tensor space representation of theblob algebra over a ring allowing base change to every interesting(that is, non-semisimple) specialisation. We show that in quasihereditaryspecialisations this passes to a full tilting module. We considerthe nature of the resultant Ringel dual algebra. 2000 MathematicsSubject Classification 20C08.     

Representation Type of Borel-Schur Algebras

Algebras and Representation Theory - In our previous work (Erdmann et al. J. Algebra Appl. 17(2),1850028, 2018), we found all Borel-Schur algebras of finite representation type. In the present...     

Linear Representation of Medial-Like Algebras

In this paper we characterize the paramedial, comedial, and coparamedial pairs of quasigroup operations and paramedial, comedial, and coparamedial algebras with the quasigroup operations. Then we characterize paramedial, comedial, and coparamedial algebras with a unit element and an idempotent regular element.     

Division Algebras with Left Algebraic Commutators

Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a ₀ + a ₁ x + ? + a _n x ⁿ (resp. right polynomial a ₀ + x a ₁ + ? + x ⁿ a _n ) over K such that a ₀ + a ₁ a + ? + a _n a ⁿ = 0 (resp. a ₀ + a a ₁ + ? + a ⁿ a _n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions.