adjoint group
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Author(s):  
B. Klingler ◽  
A. Otwinowska

AbstractGiven $${{\mathbb {V}}}$$ V a polarizable variation of $${{\mathbb {Z}}}$$ Z -Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$ V s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for $${{\mathbb {V}}}$$ V . Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$ V is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$ V whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$ A g of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$ A g is either a closed algebraic subvariety of S or is Zariski-dense in S.


2020 ◽  
Vol 60 (4) ◽  
pp. 1245-1260
Author(s):  
Toshiyuki Akita
Keyword(s):  

2020 ◽  
Vol 23 (5) ◽  
pp. 847-869
Author(s):  
Wolfgang Rump

AbstractBased on computing evidence, Guarnieri and Vendramin conjectured that, for a generalized quaternion group G of order {2^{n}\geqslant 32}, there are exactly seven isomorphism classes of braces with adjoint group G. The conjecture is proved in the paper.


2019 ◽  
Vol 62 (3) ◽  
pp. 733-738 ◽  
Author(s):  
Be'eri Greenfeld

AbstractWe prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated. We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated and, on the other hand, construct a finitely generated, infinite-dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.


2016 ◽  
Vol 12 (06) ◽  
pp. 1613-1624
Author(s):  
Manish Mishra
Keyword(s):  

We give the details of the construction of a map to restate a conjecture of Gan, Gross and Prasad about adjoint group action on generic representations in [Formula: see text]-packets. We give an application of the construction to give another proof of the classification of the Knapp–Stein [Formula: see text]-group associated to a unitary unramified character of a torus. Finally, we prove the conjecture for unramified [Formula: see text]-packets.


2011 ◽  
Vol 08 (06) ◽  
pp. 1169-1177 ◽  
Author(s):  
RUBEN FLORES ESPINOZA

In this paper, we study the existence problem of periodic first integrals for periodic Hamiltonian systems of Lie type. From a natural ansatz for time-dependent first integrals, we refer their existence to the existence of periodic solutions for a periodic Euler equation on the Lie algebra associated to the original system. Under different criteria based on properties for the Killing form or on exponential properties for the adjoint group, we prove the existence of Poisson algebras of periodic first integrals for the class of Hamiltonian systems considered. We include an application for a nonlinear oscillator having relevance in some modern physics applications.


2010 ◽  
Vol 95 (3) ◽  
pp. 213-224
Author(s):  
F. Catino ◽  
M. M. Miccoli ◽  
Ya. P. Sysak
Keyword(s):  

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