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On the existence of finite central groupoids of all possible ranks. I

1974
*
Journal of combinatorial theory. Series A
*

*The*problem

*of*determining

*the*number

*of*

*finite*

*central*

*groupoids*(an algebraic system satisfying

*the*identity (x . y) . ... Since these are

*the*only

*possible*values,

*the*question

*of*

*existence*solutions

*of*

*all*

*possible*

*ranks*is completely answered.

*The*techniques and proofs are

*of*a constructive nature. ... For example, for n2 x n2 solutions

*of*

*rank*IZ, there is

*the*unique canonical form for

*the*natural

*central*

*groupoid*[2, p. 3861 . In [

*I*] A. .

*I*. ...

##
###
Frieze patterns as root posets and affine triangulations

2014
*
European journal of combinatorics (Print)
*

*The*entries

*of*frieze patterns may be interpreted as coordinates

*of*roots

*of*a

*finite*Weyl

*groupoid*

*of*

*rank*two. ... We prove

*the*

*existence*

*of*maximal elements in their root posets and classify those frieze patterns which can be used to build an affine simplicial arrangement. ...

*The*root posets

*of*

*finite*Weyl

*groupoids*

*of*

*rank*two.

*Finite*crystallographic arrangements

*of*

*rank*two are in

*one*-to-

*one*correspondence with η-sequences (see for example [8] ). ...

##
###
On Fomin--Kirillov Algebras for Complex Reflection Groups
[article]

2016
*
arXiv
*
pre-print

First, we focus

arXiv:1605.00227v1
fatcat:dpugfyytqjaqxaxhm6wnqd3jiu
*on**the*case*of*cyclic groups where*the*corresponding Nichols algebras are only*finite*-dimensional up to order four, and we include results about*the**existence**of*Weyl*groupoids*and*finite*-dimensional ... Second, recent results by Heckenberger--Vendramin [ArXiv e-prints, 1412.0857 (December 2014)]*on**the*classification*of*Nichols algebras*of*semisimple group type can be used to find that these algebras ... Special thanks go to thank István Heckenberger for teaching*the*author about Weyl*groupoids*and providing many helpful insights and guidance which made this work*possible*. ...##
###
Finite primes in simple algebras

1971
*
Pacific Journal of Mathematics
*

*The*situation when J contains a

*finite*prime invariant under

*all*

*i*£-automorphisms is studied closely; when K is

*the*rational numbers or char (IT) Φ 0, and 2 has prime power degree, such a prime

*exists*... Specifically, if K is a global field

*the*"

*finite*primes"

*of*2 (in Harrison's sense) which contain a

*i*Γ-basis for 2 are

*the*generators

*of*

*the*Brandt

*Groupoids*

*of*normal ^-lattices, R ranging over

*the*nontrivial ... Let Σ be a

*central*simple K-algebra, and P a discrete

*rank*

*one*

*finite*prime

*of*K. ...

##
###
Actions of finite groups on graphs and related automorphisms of free groups

1989
*
Journal of Algebra
*

If J is a base-point preserving inner endomorphism

doi:10.1016/0021-8693(89)90154-3
fatcat:w2vbeljpdjbp5iop642gwlsdzi
*of*n( f ), where r is a reduced pointed G-graph*of**rank*> 2 then*the*restriction*of*J*on*x,(T, *) is*the*conjugation by*I*"*. ... To find an inverse for J, we argue by induction*on**the*sum total*of*lengths*I*(&,), UE V(f). COROLLARY 2 . 2 Let r be a reduced G-graph ef*rank*22. ...##
###
Almost Finiteness for General Étale Groupoids and Its Applications to Stable Rank of Crossed Products

2018
*
International mathematics research notices
*

We extend Matui's notion

doi:10.1093/imrn/rny187
fatcat:wrz5niqvinaa7nqg7rtkanb7ri
*of*almost*finiteness*to general etale*groupoids*and show that*the*reduced*groupoid*C*-algebras*of*minimal almost*finite**groupoids*have stable*rank**one*. ... (ii) Any countable amenable group admits a minimal action*on**the*Cantor set*all*whose minimal extensions form*the*crossed product*of*stable*rank**one*. ...*The*author is grateful to*the*second reviewer*of*a journal for his/her careful reading and valuable comments. ...##
###
Groupoid representations and modules over the convolution algebras
[article]

2008
*
arXiv
*
pre-print

*of*constant

*rank*. ... We extend these results to obtain a correspondence between

*the*category

*of*representations

*of*an etale Lie

*groupoid*and

*the*category

*of*modules over its convolution algebra that are

*of*

*finite*type and ...

*I*would like to thank J. Mrčun for many helpful discussions and advice related to

*the*paper. ...

##
###
Weyl groupoids of rank two and continued fractions

2009
*
Algebra & Number Theory
*

*The*functor ᐃ → ᐃ(), which is

*the*identity

*on*

*the*objects, and

*on*

*the*set

*of*morphisms is given by s a

*i*→ σ a

*i*for

*all*

*i*∈

*I*, a ∈ A, is an isomorphism

*of*

*groupoids*. Definition 2.7. ... For

*all*

*i*, j ∈

*I*with

*i*= j

*the*restriction | {

*i*, j} (see [Cuntz and Heckenberger 2008, Definition 4 .1]) is a

*finite*root system

*of*

*rank*two. ...

##
###
A simplicial complex of Nichols algebras
[article]

2016
*
arXiv
*
pre-print

As applications, some

arXiv:1503.08117v2
fatcat:kk7u53poyzf6fauafclsx6flia
*of*these Nichols algebras provide Weyl*groupoids*which do not arise for Nichols algebras over*finite*groups and in fact we realize*all*root systems*of**finite*Weyl*groupoids**of**rank*... We translate*the*concept*of*restriction*of*an arrangement in terms*of*Hopf algebras. ...*The*cardinality*of**I*is*the**rank**of*W(C). Definition 2.9. A Cartan graph is called connected if its Weyl*groupoid*is connected, that is, if for*all*a, b ∈ A there*exists*w ∈ Hom(a, b). ...##
###
Nichols algebras over groups with finite root system of rank two II

2014
*
Journal of group theroy
*

AbstractWe classify

doi:10.1515/jgth-2014-0024
fatcat:eaxt6yclljcyxhcfle4tadbwfm
*all*non-abelian groups ... Weyl*groupoids**of**rank*two Let us consider*the*map η : Z → SL(2, Z), η(c) = c −1 1 0 . ... Then m*i*≥ 2 for at least*one**i*. By applying (3.1) several times we obtain that Cartan schemes*of**rank*two, their Weyl*groupoids*and their root systems were studied in [10] . ...##
###
Groupoids, root systems and weak order I
[article]

2011
*
arXiv
*
pre-print

This is

arXiv:1110.3217v1
fatcat:wmr3to6lljfubomsx3lv43f27y
*the*first*of*a series*of*papers which define and study structures called rootoids, which are*groupoids*equipped with a representation in*the*category*of*Boolean rings and with an associated 1-cocycle ... They imply that each*of**the*weak orders*of*a rootoid embeds as an order ideal in a complete ortholattice. ... (a) Suppose given a family (A*i*)*i*∈*I**of*elements*of*L and B ∈ L satisfying A*i*∩B = ∅ for*all**i*∈*I*. If*the*join A =*i*A*i**exists*in L, then A ∩ B = ∅. ...##
###
Central stability for the homology of congruence subgroups and the second homology of Torelli groups
[article]

2019
*
arXiv
*
pre-print

These results follow from a general theorem

arXiv:1704.04449v2
fatcat:emjw77a3vvgplkykp23rdizxw4
*on*syzygies*of*certain modules with*finite*polynomial degree. ... We also prove a new representation stability result for*the*homology*of*certain congruence subgroups, partially improving upon*the*work*of*Putman-Sam. ... A module A over FI, VIC H (R) for R*of**finite*stable*rank*, or SI(R) for R*of**finite*(symplectic) unitary stable*rank*is presented in*finite*degrees if and only if it has*finite**central*stability degree ...##
###
Property FW, differentiable structures, and smoothability of singular actions
[article]

2020
*
arXiv
*
pre-print

is conjugate to an action by true diffeomorphisms

arXiv:1803.08567v4
fatcat:hovovb6b5nezvh5qiquy7bjqlq
*of*class C^r*on*a homeomorphic (*possibly*non-diffeomorphic) manifold. ... As applications, we show that Navas's result for actions*of*Kazhdan groups*on**the*circle, as well as*the*recent solutions to Zimmer's conjecture, generalise to aperiodic actions by diffeomorphisms with ... Acknowledgements M.T. thanks Christian Bonatti for*the*helpful discussion about germs*of*singular diffeomorphisms. N.M.B. thanks Peter Feller for useful clarifications about exotic spheres. ...##
###
Finite symmetry group actions on substitution tiling C*-algebras
[article]

2013
*
arXiv
*
pre-print

For a

arXiv:1207.6301v3
fatcat:chx2xdppl5e2dbny63jj2fu3pq
*finite*symmetry group G*of*an aperiodic substitution tiling system (,ω), we show that*the*crossed product*of**the*tiling C*-algebra by G has real*rank*zero, tracial*rank**one*, a unique trace, and that ... We also show that*the*action*of*G*on*satisfies*the*weak Rokhlin property, and that it also satisfies*the*tracial Rokhlin property provided that has tracial*rank*zero. ... In particular, great gratitude is due Thierry Giordano who supervised work*on**the*author's PhD thesis from which this paper is derived. ...##
###
Groupoids, imaginaries and internal covers
[article]

2011
*
arXiv
*
pre-print

As a corollary, we describe

arXiv:math/0603413v3
fatcat:zpkkbwvgvfhl3ptsevgksqgg7m
*the*imaginary elements*of*families*of**finite*-dimensional vector spaces over pseudo-*finite*fields. ...*of*T^, and definable*groupoids*. ...*all**finite**central*extensions.Hence, after naming parameters for a further*finite*definable set, and passing to correspond-ing subgroups*of**finite*index in H, H, there*exists*a T -definable D (containing ...
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