The Structure of Normal Algebraic Monoids

We show that any normal algebraic monoid is an extension of an abelian variety by a normal affine algebraic monoid. This extends (and builds on) Chevalley's structure theorem for algebraic groups.     

Algebraic Representation Theory of Partial Algebras

. This paper develops an algebraic representation theory of partial algebras. We introduce the notion of irreducibility for representations of a partial algebra, prove necessary and sufficient conditions for a representation to be (a) irreducible or (b) completely reducible, and develop aspects of the decomposition theory of completely reducible representations. An application to boson quantum field theory is furnished.     

Various Forms of Generating Subsemigroups in Algebraic Monoids

Let M be an irreducible affine algebraic monoid over an algebraically closed field, G its unit group, and E(M) the set of idempotents of M. We study various forms of subsemigroup generating in affine algebraic monoids and relevant generating problems with kernel data. We determine the structure of minimal irreducible algebraic submonoids containing the kernel, in particular, of M = G ∪ ker(M). We also prove that M with a dense unit group is regular if and only if M = ? E(M), G ? and ? E(M) ? is regular.     

The Cross Section Lattices and Renner Monoids of the Odd Special Orthogonal Algebraic Monoids

Abstract. Let MSO _n (n is odd) be the special orthogonal algebraic monoid and M _n the monoid of all n × n matrices over an algebraically closed field. We will explicitly determine the cross section lattices Λ and the Renner monoids R of MSO _n by using admissible subsets (see Definition 3.1) and the Weyl group. It turns out that Λ is a sublattice of the cross section lattice of M_n and that R is a submonoid of the Renner monoid M _n . Also, we obtain some interesting properties of the submonoid (MSO _n ) _e ={y∈ MSO _n | ye = ey =e} of MSO _n where e is an idempotent in MSO _n .     

Algebraic Representation of Mappings Between Submodule Lattices

We discuss algebraic representations of mappings preserving arbitrary joins between submodule lattices. For a given join-preserving mapping $ \bar{g}:\mathfrak{L}\left( {_RM} \right)\to \mathfrak{L}\left( {_SN} \right) $ between submodule lattices, a representation is an R-balanced mapping h : B × M → N, where _S B _R is a bimodule such that $ \left\langle {h\left( {B\times U} \right)} \right\rangle =\bar{g}(U) $ for all $ U\in \mathfrak{L}\left( {_RM} \right) $ . We begin by posing the question in a general abstract context and by defining the canonical subrepresentation, which is a representation if and only if there exists a representation. The problem is to give easy and natural conditions for the existence of a representation. We consider a very general situation for the mappings and give sufficient criteria for the existence of a representation. We also consider lattice isomorphisms.     

Affine Algebraic Varieties Relative to an Algebraic Theory

Affine algebraic varieties relative to an algebraic theory are introduced and described as irreducible components of affine algebraic sets. Their category is shown to be dually equivalent to the category of irreducible functional algebras.     

Algebraic Theory of Two-Grid Methods

About thirty years ago, Achi Brandt wrote a seminal paper providing a convergence theory for algebraic multigrid methods [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, this theory has been improved and extended in a number of ways, and these results have been used in many works to analyze algebraic multigrid methods and guide their developments. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflation-based preconditioners and two-level domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is self-contained thanks to a collection of new proofs, often shorter than the original ones.     

Algebraic Representation of Correlation Functions in Integrable Spin Chains

Taking the XXZ chain as the main example, we give a review of an algebraic representation of correlation functions in integrable spin chains obtained recently. We rewrite the previous formulas in a form which works equally well for the physically interesting homogeneous chains. We discuss also the case of quantum group invariant operators and generalization to the XYZ chain. Communicated by Vincent Rivasseau Dedicated to the memory of Daniel Arnaudon Submitted: January 18, 2006; Accepted: February 28, 2006     

Categorical Abstract Algebraic Logic: Subdirect Representation of Pofunctors

Pa?asińska and Pigozzi developed a theory of partially ordered varieties and quasi-varieties of algebras with the goal of addressing issues pertaining to the theory of algebraizability of logics involving an abstract form of the connective of logical implication. Following their lead, the author has abstracted the theory to cover the case of algebraic systems, systems that replace algebras in the theory of categorical abstract algebraic logic. In this note, an order subdirect representation theorem for partially ordered algebraic systems is proven. This is an analog of the Order Subdirect Representation Theorem of Pa?asińska and Pigozzi, which, in turn, generalizes the well-known Subdirect Representation Theorem of Universal Algebra.     

The Patch Construction is Dual to Algebraic DCPO Representation

Using the parallel between the preframe and the suplattice approach to locale theory it is shown that the patch construction, as an action on topologies, is the same thing as the process of recovering a discrete poset from its algebraic dcpo (ideal completion).     

Algebraic Potential Theory on Graphs

This paper encompasses a motley collection of ideas from severalareas of mathematics, including, in no particular order, randomwalks, the Picard group, exchange rate networks, chip-firinggames, cohomology, and the conductance of an electrical network.The linking threads are the discrete Laplacian on a graph andthe solution of the associated Dirichlet problem. Thirty yearsago, this subject was dismissed by many as a trivial specialisationof cohomology theory, but it has now been shown to have hiddendepths. Plumbing these depths leads to new theoretical advances,many of which throw light on the diverse applications of thetheory. 1991 Mathematics Subject Classification 05C50.     

Algebraic Theory of Causal Double Products

Corresponding to each “rectangular” double product in the form of a formal power series R[h] with coefficients in the tensor product ?(?)⊙ ? (?) with itself of the Itô Hopf algebra, we construct “triangular” elements T[h] of ?(?) satisfying ΔT[h] = T[h]⁽¹⁾ R[h]T{h]⁽²⁾. In Fock space representations of ?(?) by iterated quantum stochastic integrals when ? is the algebra of Itô differentials of the calculus, these correspond to “causal” double product integrals in a single Fock space.     

Normal Forms of Maps: Formal and Algebraic Aspects

We survey and discuss Poincaré–Dulac normal forms of maps near a fixed point. The presentation is accessible with no particular prerequisites. After some introductory material and general results (mostly known facts) we turn to further normalization in the simple resonance case and to formal and analytic infinitesimal symmetries.Mathematics Subject Classifications (2000) 37G05, 39A11.Todor Gramchev: The author is supported by a NATO grant PST.CLG.979347 and GNAMPA–INDAM, Italy.     

Representation Theory of Polyadic Groups

In this article, we introduce the notion of representations of polyadic groups and we investigate the connection between these representations and those of retract groups and covering groups.     

纠错编码中的代数理论与方法

纠错编码是保证通信可靠性的重要手段.本文首先介绍了纠错编码的基本概念和主要数学问题,然后基于线性代数、抽象代数的理论与方法,讨论了两类常见的纠错编码——线性码、循环码.     

New Directions in Representation Theory

The Iwahori?CHecke algebra H(G, B) of a finite Chevalley group G with respect to a Borel subgroup B is described as a deformation of the group algebra of the Weyl group of G Similarly, the +-part of the quantized enveloping algebra ${{U^+_v (\mathfrak{g})}}$ associated with a semisimple Lie algebra ${{\mathfrak{g}}}$ can be viewed as a deformation of the +-part of the universal enveloping algebra ${{U(\mathfrak{g})}}$ . In both cases it is shown how information concerning the deformed algebras H(G, B) and ${{U^+_v (\mathfrak{g})}}$ can be used to obtain results about the representation theory of the Chevalley group G and the semisimple Lie algebra ${{\mathfrak{g}}}$ .     

李群表示论和Schubert条件

本文将Grassmann流形上的Schubert子簇所满足的经典的Schubert条件推广到一般的复半单李群G的广义旗流形.利用复半单李群的表示理论,我们首先在李群的权空间上引入自然的Ehresman偏序.这一偏序可以导出李群的最高权表示的权系、Weyl群及其陪集空间上的Ehresman偏序.然后我们对一般的复表示定义了相应的射影空间,Grassmann流形和旗流形.这使得能够像经典的情形一样来分析广义旗流形的Schubert子簇满足的Schubert条件.在讨论中,我们还给出了李群G的Weyl群及其陪集空间中的Bruhat-Chevalley偏序的简单判别条件.我们的结果应用到例外群,给出了Fulton提出的关于例外群的Schubert分析的问题的部分回答.