scholarly journals Computing the Braid Monodromy of Completely Reducible n -gonal Curves

2019 ◽  
Vol 45 (1) ◽  
pp. 1-11
Author(s):  
Mehmet E. Aktas ◽  
Esra Akbas
Keyword(s):  
Completely Reducible

scholarly journals RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MAIKE GRUCHOT ◽  
ALASTAIR LITTERICK ◽  
GERHARD RÖHRLE
Keyword(s):  
Automorphism Group ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
Complete Reducibility ◽  
Closed Fields ◽  
Completely Reducible ◽  
Reductive Subgroup ◽  
The Absolute

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

scholarly journals COMPLETELY REDUCIBLE SETS

2013 ◽  
Vol 23 (04) ◽  
pp. 915-941 ◽  
Author(s):  
DOMINIQUE PERRIN
Keyword(s):  
Closure Properties ◽  
Linear Representations ◽  
Complete Reducibility ◽  
The Family ◽  
Syntactic Representation ◽  
Completely Reducible ◽  
Rational Sets ◽  
Cyclic Sets

We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets.

Rings of Quotients of Group Rings

1969 ◽  
Vol 21 ◽  
pp. 865-875 ◽  
Author(s):  
W. D. Burgess
Keyword(s):  
Group Ring ◽  
Group Rings ◽  
Maximum Condition ◽  
Regular Group ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Completely Reducible ◽  
Rings Of Quotients ◽  
Prime Case

The group ring AG of a group G and a ring A is the ring of all formal sums Σg∈G agg with ag ∈ A and with only finitely many non-zero ag. Elements of A are assumed to commute with the elements of G. In (2), Connell characterized or completed the characterization of Artinian, completely reducible and (von Neumann) regular group rings ((2) also contains many other basic results). In (3, Appendix 3) Connell used a theorem of Passman (6) to characterize semi-prime group rings. Following in the spirit of these investigations, this paper deals with the complete ring of (right) quotients Q(AG) of the group ring AG. It is hoped that the methods used and the results given may be useful in characterizing group rings with maximum condition on right annihilators and complements, at least in the semi-prime case.

On Extending Projectives of Finite Group-Graded Algebras

1991 ◽  
Vol 34 (2) ◽  
pp. 224-228
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Group Representation ◽  
Sufficient Conditions ◽  
Projective Cover ◽  
Graded Algebras ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet G be a finite group, let k be a field and let R be a finite dimensional fully G-graded k-algebra. Also let L be a completely reducible R-module and let P be a projective cover of R. We give necessary and sufficient conditions for P|R1 to be a projective cover of L|R1 in Mod (R1). In particular, this happens if and only if L is R1-projective. Some consequences in finite group representation theory are deduced.

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