scholarly journals Quivers of monoids with basic algebras

2012 ◽  
Vol 148 (5) ◽  
pp. 1516-1560 ◽  
Author(s):  
Stuart Margolis ◽  
Benjamin Steinberg
Keyword(s):  
Semigroup Theory ◽  
Cartan Matrix ◽  
Maximal Subgroups ◽  
Group Algebras ◽  
Closed Field ◽  
Good Characteristic ◽  
Indecomposable Modules ◽  
Complete Computation ◽  
Theoretic Description ◽  
Better Than

AbstractWe compute the quiver of any finite monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known asDO) to representation-theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of ℛ-trivial monoids, we also provide a semigroup-theoretic description of the projective indecomposable modules and compute the Cartan matrix.

scholarly journals Strongly simply connected Auslander algebras

1997 ◽  
Vol 39 (1) ◽  
pp. 21-27 ◽  
Author(s):  
Ibrahim Assem ◽  
Peter Brown
Keyword(s):  
Representation Theory ◽  
Closed Field ◽  
Interesting Problem ◽  
Indecomposable Modules ◽  
Finite Algebras ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Infinite Case ◽  
Simple Connectedness ◽  
Simply Connected

Letkbe an algebraically closed field. By an algebra is meant an associative finite dimensionalk-algebra A with an identity. We are interested in studying the representation theory of Λ, that is, in describing the category mod Λ of finitely generated right Λ-modules. Thus we may, without loss of generality, assume that Λ is basic and connected. For our purpose, one strategy consists in using covering techniques to reduce the problem to the case where the algebra is simply connected, then in solving the problem in this latter case. This strategy was proved efficient for representation-finite algebras (that is, algebras having only finitely many isomorphism classes of indecomposable modules) and representation-finite simply connected algebras are by now well-understood: see, for instance [5], [7],[8]. While little is known about covering techniques in the representation-infinite case, it is clearly an interesting problem to describe the representation-infinite simply connected algebras. The objective of this paper is to give a criterion for the simple connectedness of a class of (mostly representationinfinite) algebras.

On Degenerations of Modules With Nondirecting Indecomposable Summands

1996 ◽  
Vol 48 (5) ◽  
pp. 1091-1120 ◽  
Author(s):  
A. Skowroński ◽  
G. Zwara
Keyword(s):  
Natural Number ◽  
Partial Order ◽  
General Linear Group ◽  
Partial Orders ◽  
Connected Components ◽  
Closed Field ◽  
Affine Variety ◽  
Indecomposable Modules ◽  
Isomorphism Classes ◽  
Finite Dimensional

AbstractLet A be a finite dimensional associative K-algebra with an identity over an algebraically closed field K, d a natural number, and modA(d) the affine variety of d-dimensional A-modules. The general linear group Gld(K) acts on modA(d) by conjugation, and the orbits correspond to the isomorphism classes of d-dimensional modules. For M and N in modA(d), N is called a degeneration of M, if TV belongs to the closure of the orbit of M. This defines a partial order ≤deg on modA(d). There has been a work [1], [10], [11], [21] connecting ≤deg with other partial orders ≤ext and ≤deg on modA(d) defined in terms of extensions and homomorphisms. In particular, it is known that these partial orders coincide in the case A is representation-finite and its Auslander-Reiten quiver is directed. We study degenerations of modules from the additive categories given by connected components of the Auslander-Reiten quiver of A having oriented cycles. We show that the partial orders ≤ext, ≤deg and < coincide on modules from the additive categories of quasi-tubes [24], and describe minimal degenerations of such modules. Moreover, we show that M ≤degN does not imply M ≤ext N for some indecomposable modules M and N lying in coils in the sense of [4].

FREE GROUP ALGEBRAS IN DIVISION RINGS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250044 ◽  
Author(s):  
J. Z. GONÇALVES ◽  
E. TENGAN
Keyword(s):  
Free Group ◽  
Infinite Order ◽  
Algebraic Function ◽  
Group Algebras ◽  
Closed Field ◽  
Skew Polynomial Ring ◽  
Algebraic Function Field ◽  
Skew Field ◽  
Rank 2 ◽  
Skew Polynomial

Let k be an algebraically closed field of characteristic zero and let L be an algebraic function field over k. Let σ : L → L be a k-automorphism of infinite order, and let D be the skew field of fractions of the skew polynomial ring L[t; σ]. We show that D contains the group algebra kF of the free group F of rank 2.

Research on Microstructure and Tribological Properties of Cr-Based Hard Coatings

2013 ◽  
Vol 811 ◽  
pp. 126-130
Author(s):  
Bai Jun Shi ◽  
Yuan Ping Peng ◽  
Si Chi Wu ◽  
Hang Li
Keyword(s):  
Tribological Properties ◽  
Steel Substrate ◽  
Optical Microscope ◽  
Hard Coatings ◽  
Closed Field ◽  
Mechanical And Tribological Properties ◽  
Unbalanced Magnetron ◽  
Magnetron Sputter ◽  
Better Than ◽  
Ball On Disc

In this paper, three kinds of Cr-based hard coatings have been studied for its structural, mechanical and tribological properties at room temperature. By the technology of closed field unbalanced magnetron sputter ion plating (CFUMSIP), CrN CrAlN and CrTiAlN hard coatings were deposited onto DC53 cold die steel and silicon (100) substrate. The coatings were characterized by means of field emission scanning electron microscopy (FESEM), X-ray diffractometer (XRD), microhardness tester, optical microscope and ball-on-disc tribometer, in order to check their structural, as well as to determine the mechanical and tribological properties. The experimental results showed that the CrTiAlN coating performed better than the CrN coating and the CrAlN coating in terms of mechanical and tribological properties. The wear resistances of CrN, CrAlN and CrTiAlN coatings have been significantly improved compared with DC53 steel substrate.

scholarly journals The Centralizer of a Nilpotent Section

2008 ◽  
Vol 190 ◽  
pp. 129-181 ◽  
Author(s):  
George J. McNinch
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Local Ring ◽  
Nilpotent Element ◽  
Group Scheme ◽  
Closed Field ◽  
Good Characteristic ◽  
Component Group ◽  
Levi Factor ◽  
Subgroup Scheme

Let F be an algebraically closed field and let G be a semisimple F-algebraic group for which the characteristic of F is very good. If X ∈ Lie(G) = Lie(G)(F) is a nilpotent element in the Lie algebra of G, and if C is the centralizer in G of X, we show that (i) the root datum of a Levi factor of C, and (ii) the component group C/C° both depend only on the Bala-Carter label of X; i.e. both are independent of very good characteristic. The result in case (ii) depends on the known case when G is (simple and) of adjoint type.The proofs are achieved by studying the centralizer of a nilpotent section X in the Lie algebra of a suitable semisimple group scheme over a Noetherian, normal, local ring . When the centralizer of X is equidimensional on Spec(), a crucial result is that locally in the étale topology there is a smooth -subgroup scheme L of such that Lt is a Levi factor of for each t ∈ Spec ().

A Remark on the Krull-Schmidt-Azumaya Theorem

1970 ◽  
Vol 13 (4) ◽  
pp. 501-505 ◽  
Author(s):  
B. L. Osofsky
Keyword(s):  
Dedekind Domain ◽  
The Other ◽  
Group Algebras ◽  
Endomorphism Rings ◽  
Algebraic Integers ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Other Hand

It is well known that if a module M is expressible as a direct sum of modules with local endomorphism rings, then such a decomposition is essentially unique. That is, if M = ⊕i∊IMi = ⊕j∊JNj then there is a bijection f: I → J such that Mi is isomorphic to Nf(i) for all i∊I (see [1]). On the other hand, a nonprincipal ideal in a Dedekind domain provides an example where such a theorem fails in the absence of the local hypothesis. Group algebras of certain groups over rings R of algebraic integers is another such example, where even the rank as R-modules of indecomposable summands of a module is not uniquely determined (see [2]). Both of these examples yield modules which are expressible as direct sums of two indecomposable modules in distinct ways. In this note we construct a family of rings which show that the number of summands in a representation of a module M as a direct sum of indecomposable modules is also not unique unless one has additional hypotheses.

SOLUBLE MAXIMAL SUBGROUPS IN GLn(D)

2011 ◽  
Vol 10 (06) ◽  
pp. 1371-1382 ◽  
Author(s):  
H. R. DORBIDI ◽  
R. FALLAH-MOGHADDAM ◽  
M. MAHDAVI-HEZAVEHI
Keyword(s):  
Maximal Subgroup ◽  
Division Ring ◽  
Real Closed Field ◽  
Maximal Subgroups ◽  
Closed Field

Let D be an F-central non-commutative division ring. Here, it is proved that if GL n(D) contains a non-abelian soluble maximal subgroup, then n = 1, [D : F] < ∞, and D is cyclic of degree p, a prime. Furthermore, a classification of soluble maximal subgroups of GL n(F) for an algebraically closed or real closed field F is also presented. We then determine all soluble maximal subgroups of GL 2(F) for fields F with Char F ≠ 2.

scholarly journals Isomorphisms of subcategories of fusion systems of blocks and Clifford theory

2020 ◽  
Vol 23 (5) ◽  
pp. 925-930
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
London Math ◽  
Group Algebras ◽  
Closed Field ◽  
Prime Characteristic ◽  
Cambridge University ◽  
Finite Poset ◽  
Clifford Theory ◽  
A Finite Group ◽  
Brauer Pairs

AbstractLet k be an algebraically closed field of prime characteristic p. Let G be a finite group, let N be a normal subgroup of G, and let c be a G-stable block of kN so that {(kN)c} is a p-permutation G-algebra. As in Section 8.6 of [M. Linckelmann, The Block Theory of finite Group Algebras: Volume 2, London Math. Soc. Stud. Texts 92, Cambridge University, Cambridge, 2018], a {(G,N,c)}-Brauer pair {(R,f_{R})} consists of a p-subgroup R of G and a block {f_{R}} of {(kC_{N}(R))}. If Q is a defect group of c and {f_{Q}\in\operatorname{\textit{B}\ell}(kC_{N}(Q))}, then {(Q,f_{Q})} is a {(G,N,c)}-Brauer pair. The {(G,N,c)}-Brauer pairs form a (finite) poset. Set {H=N_{G}(Q,f_{Q})} so that {(Q,f_{Q})} is an {(H,C_{N}(Q),f_{Q})}-Brauer pair. We extend Lemma 8.6.4 of the above book to show that if {(U,f_{U})} is a maximal {(G,N,c)}-Brauer pair containing {(Q,f_{Q})}, then {(U,f_{U})} is a maximal {(H,C_{N}(c),f_{Q})}-Brauer pair containing {(Q,f_{Q})} and conversely. Our main result shows that the subcategories of {\mathcal{F}_{(U,f_{U})}(G,N,c)} and {\mathcal{F}_{(U,f_{U})}(H,C_{N}(Q),f_{Q})} of objects between and including {(Q,f_{Q})} and {(U,f_{U})} are isomorphic. We close with an application to the Clifford theory of blocks.

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