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 Semisimplification for subgroups of reductive algebraic groups

2020 ◽  
Vol 8 ◽  
Author(s):  
Michael Bate ◽  
Benjamin Martin ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Invariant Theory ◽  
Algebraic Groups ◽  
Geometric Invariant ◽  
Reductive Algebraic Group ◽  
Complete Reducibility ◽  
Special Cases ◽  
Closed Fields ◽  
Completely Reducible ◽  
Algebraically Closed Fields

Let G be a reductive algebraic group—possibly non-connected—over a field k, and let H be a subgroup of G. If $G= {GL }_n$ , then there is a degeneration process for obtaining from H a completely reducible subgroup $H'$ of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a G-completely reducible subgroup $H'$ of G, unique up to $G(k)$ -conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL _n$ and with Serre’s ‘G-analogue’ of semisimplification for subgroups of $G(k)$ from [19]). We also show that under some extra hypotheses, one can pick $H'$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf, and Rousseau.

scholarly journals Complete reducibility: variations on a theme of Serre

2021 ◽  
Author(s):  
Maike Gruchot ◽  
Alastair Litterick ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Spherical Building ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
General Properties ◽  
Complete Reducibility ◽  
Special Cases ◽  
Variations On A Theme

AbstractIn this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and $$\sigma $$ σ -complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

Complete reducibility of subgroups of reductive algebraic groups over non-perfect fields IV: An 𝐹4 example

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Falk Bannuscher ◽  
Alastair Litterick ◽  
Tomohiro Uchiyama
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Complete Reducibility ◽  
Completely Reducible ◽  
Rationality Problems ◽  
Reductive Algebraic Groups

Abstract Let 𝑘 be a non-perfect separably closed field. Let 𝐺 be a connected reductive algebraic group defined over 𝑘. We study rationality problems for Serre’s notion of complete reducibility of subgroups of 𝐺. In particular, we present the first example of a connected non-abelian 𝑘-subgroup 𝐻 of 𝐺 that is 𝐺-completely reducible but not 𝐺-completely reducible over 𝑘, and the first example of a connected non-abelian 𝑘-subgroup H ′ H^{\prime} of 𝐺 that is 𝐺-completely reducible over 𝑘 but not 𝐺-completely reducible. This is new: all previously known such examples are for finite (or non-connected) 𝐻 and H ′ H^{\prime} only.

On Automorphisms and Derivations of a Lie Algebra

2006 ◽  
Vol 13 (01) ◽  
pp. 119-132 ◽  
Author(s):  
V. R. Varea ◽  
J. J. Varea
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Nilpotent Lie Algebra ◽  
Identity Component ◽  
Trivial Center

We study automorphisms and derivations of a Lie algebra L of finite dimension satisfying certain centrality conditions. As a consequence, we obtain that every nilpotent normal subgroup of the automorphism group of L is unipotent for a very large class of Lie algebras. This result extends one of Leger and Luks. We show that the automorphism group of a nilpotent Lie algebra can have trivial center and have yet a unipotent identity component.

scholarly journals On Cocharacters Associated to Nilpotent Elements of Reductive Groups

2008 ◽  
Vol 190 ◽  
pp. 105-128 ◽  
Author(s):  
Russell Fowler ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Nilpotent Element ◽  
Maximal Rank ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Reductive Groups ◽  
Algebraically Closed Field ◽  
Reductive Subgroup ◽  
Good For

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we consider particular classes of connected reductive subgroups H of G and show that the cocharacters of H that are associated to a given nilpotent element e in the Lie algebra of H are precisely the cocharacters of G associated to e that take values in H. In particular, we show that this is the case provided H is a connected reductive subgroup of G of maximal rank; this answers a question posed by J. C. Jantzen.

scholarly journals On Commuting Varieties of Nilradicals of Borel Subalgebras of Reductive Lie Algebras

2014 ◽  
Vol 58 (1) ◽  
pp. 169-181 ◽  
Author(s):  
Simon M. Goodwin ◽  
Gerhard Röhrle
Keyword(s):  
Finite Number ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Borel Subgroup ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Commuting Variety ◽  
Irreducible Components

AbstractLet G be a connected reductive algebraic group defined over an algebraically closed field of characteristic 0. We consider the commuting variety of the nilradical of the Lie algebra of a Borel subgroup B of G. In case B acts on with only a finite number of orbits, we verify that is equidimensional and that the irreducible components are in correspondence with the distinguishedB-orbits in . We observe that in general is not equidimensional, and determine the irreducible components of in the minimal cases where there are infinitely many B-orbits in .

scholarly journals Some Groups of Type E7

2006 ◽  
Vol 182 ◽  
pp. 259-284 ◽  
Author(s):  
T. A. Springer
Keyword(s):  
Automorphism Group ◽  
Irreducible Representation ◽  
Vector Space ◽  
Algebraic Group ◽  
Automorphism Groups ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Identity Component ◽  
Field Of Definition ◽  
Quartic Form

AbstractAn algebraic group of type E7 over an algebraically closed field has an irreducible representation in a vector space of dimension 56 and is, in fact, the identity component of the automorphism group of a quartic form on the space. This paper describes the construction of the quartic form if the characteristic is ≠ 2, 3, taking into account a field of definition F. Certain F-forms of E7 appear in the automorphism groups of quartic forms over F, as well as forms of E6. Many of the results of the paper are known, but are perhaps not easily accessible in the literature.

scholarly journals On the automorphism group of a symplectic half-flat 6-manifold

2019 ◽  
Vol 31 (1) ◽  
pp. 265-273
Author(s):  
Fabio Podestà ◽  
Alberto Raffero
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Group Action ◽  
Cohomogeneity One ◽  
Cohomogeneity One Action

Abstract We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat {\mathrm{SU}(3)} -structure has Abelian Lie algebra with dimension bounded by {\min\{5,b_{1}(M)\}} . Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on {T\mathbb{S}^{3}} which are invariant under a cohomogeneity one action of {\mathrm{SO}(4)} .

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.

Automorphism Group of a Special Type Lie Algebra I

2010 ◽  
Vol 17 (spec01) ◽  
pp. 815-828
Author(s):  
Seul Hee Choi ◽  
Ki-Bong Nam
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Algebra I

In an earlier paper, we defined a degree stable Lie algebra, and determined the Lie algebra automorphism group AutLie(S+(2)) of the Lie algebra S+(2). In this paper, we determine the Lie algebra automorphism group AutLie(S(1,0,2)) of the Lie algebra S(1,0,2).

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