scholarly journals Homogeneous 3-Dimensional Permutation Structures

10.37236/7506 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Samuel Braunfeld
Keyword(s):  
Electronic Journal ◽  
Linear Orders ◽  
3 Dimensional ◽  
Finite Dimensional ◽  
Homogeneous Structures ◽  
Natural Description

We provide a classification of the homogeneous 3-dimensional permutation structures, i.e. homogeneous structures in a language of 3 linear orders, partially answering a 2002 question of Cameron. We also arrive at a natural description of all known homogeneous finite-dimensional permutation structures by modifying the language used in the construction from [Samuel Braunfeld, Electronic Journal of Combinatorics, 2016], completing the catalog begun there.

scholarly journals The Lattice of Definable Equivalence Relations in Homogeneous $n$-Dimensional Permutation Structures

10.37236/5980 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Samuel Braunfeld
Keyword(s):  
Distributive Lattice ◽  
Equivalence Relations ◽  
Electronic Journal ◽  
Finite Distributive Lattice ◽  
Linear Orders ◽  
Finite Dimensional ◽  
Homogeneous Structures ◽  
Special Case

In Homogeneous permutations, Peter Cameron [Electronic Journal of Combinatorics 2002] classified the homogeneous permutations (homogeneous structures with 2 linear orders), and posed the problem of classifying the homogeneous $n$-dimensional permutation structures (homogeneous structures with $n$ linear orders) for all finite $n$. We prove here that the lattice of $\emptyset$-definable equivalence relations in such a structure can be any finite distributive lattice, providing many new imprimitive examples of homogeneous finite dimensional permutation structures. We conjecture that the distributivity of the lattice of $\emptyset$-definable equivalence relations is necessary, and prove this under the assumption that the reduct of the structure to the language of $\emptyset$-definable equivalence relations is homogeneous. Finally, we conjecture a classification of the primitive examples, and confirm this in the special case where all minimal forbidden structures have order 2. 

scholarly journals The Classification of Homogeneous Finite-Dimensional Permutation Structures

10.37236/8321 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Samuel Braunfeld ◽  
Pierre Simon
Keyword(s):  
Linear Orders ◽  
Complete Answer ◽  
Finite Dimensional ◽  
Homogeneous Structures

We classify the homogeneous finite-dimensional permutation structures, i.e. homogeneous structures in a language of finitely many linear orders, giving a nearly complete answer to a question of Cameron, and confirming the classification conjectured by the first author. The primitive case was proven by the second author using model-theoretic methods, and those methods continue to appear here.

scholarly journals Classification of Degenerate Verma Modules for E(5, 10)

2021 ◽  
Author(s):  
Nicoletta Cantarini ◽  
Fabrizio Caselli ◽  
Victor Kac
Keyword(s):  
Infinite Number ◽  
Verma Module ◽  
Lie Superalgebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Finite Dimensional ◽  
De Rham ◽  
Via Classification

AbstractGiven a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

scholarly journals Complete Bredon cohomology and its applications to hierarchically defined groups

2016 ◽  
Vol 161 (1) ◽  
pp. 143-156
Author(s):  
BRITA E. A. NUCINKIS ◽  
NANSEN PETROSYAN
Keyword(s):  
Integral Characteristic ◽  
Linear Groups ◽  
Dimensional Model ◽  
Classifying Space ◽  
Soluble Groups ◽  
3 Dimensional ◽  
Cyclic Groups ◽  
Bredon Cohomology ◽  
Finite Dimensional ◽  
The Family

AbstractBy considering the Bredon analogue of complete cohomology of a group, we show that every group in the class$\cll\clh^{\mathfrak F}{\mathfrak F}$of type Bredon-FP∞admits a finite dimensional model for$E_{\frak F}G$.We also show that abelian-by-infinite cyclic groups admit a 3-dimensional model for the classifying space for the family of virtually nilpotent subgroups. This allows us to prove that for$\mathfrak {F}$, the class of virtually cyclic groups, the class of$\cll\clh^{\mathfrak F}{\mathfrak F}$-groups contains all locally virtually soluble groups and all linear groups over${\mathbb{C}}$of integral characteristic.

Classification of 3-dimensional left-invariant almost paracontact metric structures

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Giovanni Calvaruso ◽  
Antonella Perrone
Keyword(s):  
Lie Groups ◽  
Three Dimensional ◽  
Complete Classification ◽  
Natural Assumption ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Metric Structures

AbstractWe study left-invariant almost paracontact metric structures on arbitrary three-dimensional Lorentzian Lie groups. We obtain a complete classification and description under a natural assumption, which includes relevant classes as normal and almost para-cosymplectic structures, and we investigate geometric properties of these structures.

scholarly journals k-step nilpotent Lie algebras

2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Michel Goze ◽  
Elisabeth Remm
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Deformation Process ◽  
Nilpotent Lie Algebras ◽  
Early Stages ◽  
Finite Dimensional ◽  
Contraction Process ◽  
Finite Dimensional Lie Algebras

AbstractThe classification of complex or real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example, the nilpotent Lie algebras are classified only up to dimension 7. Moreover, to recognize a given Lie algebra in the classification list is not so easy. In this work, we propose a different approach to this problem. We determine families for some fixed invariants and the classification follows by a deformation process or a contraction process. We focus on the case of 2- and 3-step nilpotent Lie algebras. We describe in both cases a deformation cohomology for this type of algebras and the algebras which are rigid with respect to this cohomology. Other

Sign in / Sign up

Export Citation Format

Share Document