scholarly journals Classification of Roberts actions of strongly amenable C∗-tensor categories on the injective factor of type III1

2017 ◽  
Vol 28 (07) ◽  
pp. 1750052
Author(s):  
Toshihiko Masuda
Keyword(s):  
Classification Theorem ◽  
Finitely Generated ◽  
Tensor Categories ◽  
Type Iii ◽  
Type Iii1 ◽  
Injective Factor

In this paper, we generalize Izumi’s result on uniqueness of realization of finite C[Formula: see text]-tensor categories in the endomorphism category of the injective factor of type III1 for finitely generated strongly amenable C[Formula: see text]-tensor categories by applying Popa’s classification theorem of strongly amenable subfactors of type III1.

The Nielsen-Thurston Classification

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Hyperbolic Geometry ◽  
Mapping Class Groups ◽  
Classification Theorem ◽  
Mapping Class ◽  
Fundamental Result ◽  
Class Groups ◽  
Mapping Torus ◽  
Higher Genus ◽  
Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

Morphological types of spermatheca in Coreidae: bearing on intra-familial classification and tribal-groupings (Hemiptera: Heteroptera)

Zootaxa ◽  
2020 ◽  
Vol 4834 (4) ◽  
pp. 451-501
Author(s):  
DOMINIQUE PLUOT-SIGWALT ◽  
PIERRE MOULET
Keyword(s):  
Western Hemisphere ◽  
Type I ◽  
Type Ii ◽  
Type Iii ◽  
Spermathecal Duct ◽  
Eastern Hemisphere ◽  
Intermediate Part ◽  
Muscular Pump ◽  
Familial Classification

The morphology of the spermatheca is described in 109 species of 86 genera representing all four currently recognised subfamilies of Coreidae, covering the undivided Hydarinae, both tribes of Pseudophloeinae, all three tribes of Meropachyinae and 27 of the 32 tribes of Coreinae. Three types of spermatheca are recognised. Type I is bipartite, consisting only of a simple tube differentiated into distal seminal receptacle and proximal spermathecal duct and lacks the intermediate part present in most Pentatomomorpha, in which it serves as muscular pump. Type II is also bipartite but more elaborate in form with the receptacle generally distinctly wider than the duct. Type III is tripartite, with receptacle, duct and an often complex intermediate part. Four subtypes are recognised within type III. Type I is found only in Hydarinae and type II only in Pseudophloeinae. Type III is found in both Coreinae and Meropachyinae. Subtype IIIA (“Coreus-group”) unites many tribes from the Eastern Hemisphere and only one (Spartocerini) from the Western Hemisphere. Subtypes IIIB (“Nematopus-group”) and IIID (“Anisoscelis-group”) are confined to taxa from the Western Hemisphere and subtype IIIC (“Chariesterus-group”) is found in tribes from both hemispheres. The polarity of several characters of the intermediate part and some of the spermathecal duct is evaluated, suggesting autapomorphies or apomorphies potentially relevant to the classification of Coreidae at the sufamilial and tribal levels. Characters of the intermediate part strongly indicate that the separation of Meropachyinae and Coreinae as currently constituted cannot be substantiated. The tribes Anisoscelini, Colpurini, Daladerini and Hyselonotini are heterogeneous, each exhibiting two subtypes of spermatheca, and probably polyphyletic. Two tribes, Cloresmini and Colpurini, requiring further investigation remain unplaced. This study demonstrates the great importance of characters of the spermatheca, in particular its intermediate part, for research into the phylogeny and taxonomy of Pentatomomorpha. 

scholarly journals ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

A First Approximation to {X, Y}

1969 ◽  
Vol 21 ◽  
pp. 702-711 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Homotopy Theory ◽  
Abelian Groups ◽  
Classification Theorem ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
First Approximation ◽  
Image Position

If ℭ and ℭ′ are classes of finite abelian groups, we write ℭ + ℭ′ for the smallest class containing the groups of ℭ and of ℭ′. For any positive number r, ℭ < r is the smallest class of abelian groups which contains the groups Zp for all primes p less than r.Our aim in this paper is to prove the following theorem.THEOREM. Iƒ ℭ is a class of finite abelian groups and(i) πi(Y) ∈ℭ for i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z)∈ ℭ for i > n + k,ThenThis statement contains many of the classical results of homotopy theory: the Hurewicz and Hopf theorems, Serre's (mod ℭ) version of these theorems, and Eilenberg's classification theorem. In fact, these are all contained in the case k = 0.

FREE INVERSE MONOIDS AND GRAPH IMMERSIONS

1993 ◽  
Vol 03 (01) ◽  
pp. 79-99 ◽  
Author(s):  
STUART W. MARGOLIS ◽  
JOHN C. MEAKIN
Keyword(s):  
Formal Language ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
Rational Subset ◽  
The Relationship ◽  
Subgroup Theorem ◽  
Free Inverse Monoid ◽  
Natural Way

The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

scholarly journals Realizing the Braided Temperley–Lieb–Jones C*-Tensor Categories as Hilbert C*-Modules

2020 ◽  
Vol 380 (1) ◽  
pp. 103-130
Author(s):  
Andreas Næs Aaserud ◽  
David E. Evans
Keyword(s):  
Orthonormal Basis ◽  
Full Subcategory ◽  
Tensor Category ◽  
Compact Operators ◽  
Finitely Generated ◽  
Tensor Categories ◽  
C Algebra

Abstract We associate to each Temperley–Lieb–Jones C*-tensor category $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ T L J ( δ ) with parameter $$\delta $$ δ in the discrete range $$\{2\cos (\pi /(k+2)):\,k=1,2,\ldots \}\cup \{2\}$$ { 2 cos ( π / ( k + 2 ) ) : k = 1 , 2 , … } ∪ { 2 } a certain C*-algebra $${\mathcal {B}}$$ B of compact operators. We use the unitary braiding on $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ T L J ( δ ) to equip the category $$\mathrm {Mod}_{{\mathcal {B}}}$$ Mod B of (right) Hilbert $${\mathcal {B}}$$ B -modules with the structure of a braided C*-tensor category. We show that $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ T L J ( δ ) is equivalent, as a braided C*-tensor category, to the full subcategory $$\mathrm {Mod}_{{\mathcal {B}}}^f$$ Mod B f of $$\mathrm {Mod}_{{\mathcal {B}}}$$ Mod B whose objects are those modules which admit a finite orthonormal basis. Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories.

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