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

Leslie E Shader
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.  ... 
doi:10.1016/0097-3165(74)90047-8 fatcat:gjwkqmkonfahzm2n7kih3vag6q

Frieze patterns as root posets and affine triangulations

M. Cuntz
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] ).  ... 
doi:10.1016/j.ejc.2014.06.005 fatcat:wv3raqzc5fgu5cbk3jv6e2np4y

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

Robert Laugwitz
2016 arXiv   pre-print
First, we focus 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.  ... 
arXiv:1605.00227v1 fatcat:dpugfyytqjaqxaxhm6wnqd3jiu

Finite primes in simple algebras

Hoyt Warner
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 iΓ-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.  ... 
doi:10.2140/pjm.1971.36.245 fatcat:slereqbqxzd5xnq6fpaxjjpuzq

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

Sava Krstić
1989 Journal of Algebra  
If J is a base-point preserving inner endomorphism 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.  ... 
doi:10.1016/0021-8693(89)90154-3 fatcat:w2vbeljpdjbp5iop642gwlsdzi

Almost Finiteness for General Étale Groupoids and Its Applications to Stable Rank of Crossed Products

Yuhei Suzuki
2018 International mathematics research notices  
We extend Matui's notion 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.  ... 
doi:10.1093/imrn/rny187 fatcat:wrz5niqvinaa7nqg7rtkanb7ri

Groupoid representations and modules over the convolution algebras [article]

Jure Kalisnik
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.  ... 
arXiv:0806.1832v1 fatcat:htqitqb5ljdv5klnlydwh2uxi4

Weyl groupoids of rank two and continued fractions

Michael Cuntz, István Heckenberger
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 iI , 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.  ... 
doi:10.2140/ant.2009.3.317 fatcat:nzvxch7m5ndidohqjnjbsyiqke

A simplicial complex of Nichols algebras [article]

Michael Cuntz, Simon Lentner
2016 arXiv   pre-print
As applications, some 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).  ... 
arXiv:1503.08117v2 fatcat:kk7u53poyzf6fauafclsx6flia

Nichols algebras over groups with finite root system of rank two II

István Heckenberger, Leandro Vendramin
2014 Journal of group theroy  
AbstractWe classify 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] .  ... 
doi:10.1515/jgth-2014-0024 fatcat:eaxt6yclljcyxhcfle4tadbwfm

Groupoids, root systems and weak order I [article]

Matthew Dyer
2011 arXiv   pre-print
This is 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 ) iI of elements of L and B ∈ L satisfying A i ∩B = ∅ for all iI. If the join A = i A i exists in L, then A ∩ B = ∅.  ... 
arXiv:1110.3217v1 fatcat:wmr3to6lljfubomsx3lv43f27y

Central stability for the homology of congruence subgroups and the second homology of Torelli groups [article]

Jeremy Miller, Peter Patzt, Jennifer C. H. Wilson
2019 arXiv   pre-print
These results follow from a general theorem 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  ... 
arXiv:1704.04449v2 fatcat:emjw77a3vvgplkykp23rdizxw4

Property FW, differentiable structures, and smoothability of singular actions [article]

Yash Lodha, Nicolás Matte Bon, Michele Triestino
2020 arXiv   pre-print
is conjugate to an action by true diffeomorphisms 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.  ... 
arXiv:1803.08567v4 fatcat:hovovb6b5nezvh5qiquy7bjqlq

Finite symmetry group actions on substitution tiling C*-algebras [article]

Charles Starling
2013 arXiv   pre-print
For a 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.  ... 
arXiv:1207.6301v3 fatcat:chx2xdppl5e2dbny63jj2fu3pq

Groupoids, imaginaries and internal covers [article]

Ehud Hrushovski
2011 arXiv   pre-print
As a corollary, we describe 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  ... 
arXiv:math/0603413v3 fatcat:zpkkbwvgvfhl3ptsevgksqgg7m
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