The restriction of indecomposable modules of group algebras and the quasi green correspondence

1998 ◽  
Vol 26 (1) ◽  
pp. 83-95 ◽  
Author(s):  
L. Héthelyi ◽  
M. Szöke ◽  
K. Lux
Keyword(s):  
Group Algebras ◽  
Indecomposable Modules ◽  
Green Correspondence

Analogs of the Green and Puig correspondences for twisted group algebras

2016 ◽  
Vol 19 (1) ◽  
pp. 1-24
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Groups ◽  
Modular Representation ◽  
Group Algebras ◽  
Modular Representation Theory ◽  
Twisted Group ◽  
Standard Derivation ◽  
Green Correspondence

AbstractIn the modular representation theory of finite groups, we show that the standard derivation of the Green correspondence lifts to a derivation of a Green correspondence for twisted group algebras (Theorem 1.3). Then, from these results we derive a lift of the Puig correspondences for twisted group algebras (Theorem 1.6).Clearly twisted group algebras arise naturally in finite group modular representation theory. We conclude with some suggestions for applications in this mathematical area.

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.

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.

Roots of Simple Modules

2006 ◽  
Vol 49 (1) ◽  
pp. 96-107 ◽  
Author(s):  
Burkhard Külshammer
Keyword(s):  
Finite Groups ◽  
Group Algebras ◽  
Indecomposable Modules ◽  
Simple Modules

AbstractWe introduce roots of indecomposable modules over group algebras of finite groups, and we investigate some of their properties. This allows us to correct an error in Landrock's book which has to do with roots of simple modules.

Some Remarks on the Tensor Product of Algebras and Applications II

2007 ◽  
Vol 14 (01) ◽  
pp. 143-154 ◽  
Author(s):  
Morton E. Harris
Keyword(s):  
Tensor Product ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Modular Representation ◽  
Group Algebras ◽  
Closed Field ◽  
Modular Representation Theory ◽  
Complete Discrete Valuation Ring ◽  
Algebra Theory ◽  
Green Correspondence

We apply the G-algebra theory to the tensor product of algebras. These considerations are applied to extend the results of Alghamdi and Khammash [1], Khammash [4] and Külshammer [5, Proposition 1.2] on the tensor product of group algebras and modules over an algebraically closed field to lattices over a complete discrete valuation ring. This places these results in the standard integral finite group modular representation theory of G-algebras as pioneered by Puig (cf. [8]). We also study some aspects of covering homomorphisms and the Green correspondence in this context (cf. [8, Sections 20 and 25]).

Complex group algebras of the direct product of non-isomorphic Suzuki groups

2019 ◽  
Vol 19 (02) ◽  
pp. 2050036
Author(s):  
Morteza Baniasad Azad ◽  
Behrooz Khosravi
Keyword(s):  
Direct Product ◽  
Group Algebra ◽  
Irreducible Character ◽  
Prime Numbers ◽  
Product Formula ◽  
Character Degrees ◽  
Group Algebras ◽  
Complex Group ◽  
Complex Group Algebra ◽  
Suzuki Groups

In this paper, we prove that the direct product [Formula: see text], where [Formula: see text] are distinct numbers, is uniquely determined by its complex group algebra. Particularly, we show that the direct product [Formula: see text], where [Formula: see text]’s are distinct odd prime numbers, is uniquely determined by its order and three irreducible character degrees.

scholarly journals EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

2020 ◽  
pp. 1-14
Author(s):  
NICOLÁS ANDRUSKIEWITSCH ◽  
DIRCEU BAGIO ◽  
SARADIA DELLA FLORA ◽  
DAIANA FLÔRES
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Group Algebras ◽  
Finite Abelian Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Characteristic 2 ◽  
Nichols Algebras ◽  
Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

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