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| In mathematics, a '''Bratteli diagram''' is a [[graph (mathematics)|graph]] composed of vertices having positive integer 'level's and unoriented edges between vertices having levels differing by one. It is used in operator algebra to describe directed sequences of finite dimensional algebras. The notion was introduced by [[Ola Bratteli]]<ref>Bratteli, Ola; Inductive limits of finite dimensional C<sup>*</sup>-algebras. Trans. Amer. Math. Soc. 171 (1972), 195–234.</ref> in 1972, and played an important role in Elliott's classification of [[approximately finite dimensional C*-algebra|AF-algebras]] and the theory of [[subfactor]]s.
| | <br><br>The Chilly Metal Recon 1 is a large folder which happens to have the most effective grip on this gallery. The closely checkered G-10 handle scales virtually feel sticky. It flicks open sweetly with both of the thumb studs to disclose a 4-inch tanto blade of black Japanese AUS 8A Stainless with Tuff-Ex coating. It has a reversible belt clip, and the blade locks into open place with a tricky lever lock on the back of handle. This model weighs 5.2 ounces. There are various variations of the Recon 1 to choose from, including micro variations, mini versions and the impressive XL versions.<br><br>Total, it’s spectacular to see simply how much Spyderco has sunk into this child for such a low price. This knife has nearly 550 critiques on Amazon with a mean rating of 4.7. The Tenacious will usually set you again no more than $forty! >> Click here for present pricing and critiques on the Spyderco Tenacious. SOG Flash II I believe Gerber made a sensible choice not adding a rubberized grip and compensated for this nicely by designing an over sized finger notch into the handle. And because of the these two cutouts, this knife fits small and medium sized grownup hands in addition to the arms of younger Scouts.<br><br>There’s a cause this is likely one of the finest selling knives in existence at this time. Extremely regarded knife maker Ken Onion has designed this stunning yet inexpensive pocket knife which is a part of Kershaw’s Leek series. It employs the patented SpeedSafe opening system which ensures a smooth blade deployment with only a single hand. The Sandvik 14C28N steel blade is extremely sharp and will keep sharp for a very long time however do note the blade tip is considerably delicate as in comparison with different knives so we don’t advocate this for heavy obligation applications.<br><br>Total this can be a nice knife but may be very angular and huge — too much so for it to take a seat comfortably in my pocket. It isn't uncomfortable in my hand, however it is simply one other solid knife — not what I used to be searching [http://en.wikipedia.org/wiki/Blade pocket knife brands] for. I wouldn’t feel screwed if I used to be stuck within the woods with this knife, however once more not my first choice amongst this group of folders. It has extremely easy and fluid opening, the double sided lock launch is a pleasant contact for a lefty like me.<br><br>Welcome to Finest Pocket Knife At this time! The final word buyer’s guide with tons of useful info for each knife fans and newbies alike. Pocket knives are essential everyday instruments that no responsible person should be with out. At the moment you’ll find actually a whole lot of folding knives obtainable to purchase from a bunch of various manufacturers ranging from an inexpensive $5 penknife to effectively over $500 for prime of the line models. The handle is just a tad over a quarter inch extensive, which is actually skinny compared to most pocket knives. And truthfully, it’s thinness and low weight make the knife nearly unnoticeable in my pocket.<br><br>Cutting an apple. I love consuming an apple that I’ve lower with my pocket knife, slice by slice. You're feeling like a foul ass doing it. You maintain the apple in your non-dominant hand after which make a slice with the knife using your dominant hand. After you make the slice, pinch it between your thumb and knife blade. Convey the blade to your mouth and [https://Bergelmir.iki.His.se/trac/deeds/ticket/12303 deposit] the apple slice. There are three keys to wrapping your head around carrying a pocket knife each day. You have to de-weaponize, utilize, and familiarize your knife.<br><br>I had mentioned today's tool on the Fb group last week however didn't have a finished one on hand to submit a picture. At any rate, I'm more than happy to convey this one again from the useless. A number of years in the [http://www.thebestpocketknifereviews.com/best-automatic-knife-discover-the-pefect-model/ best automatic knives] past I made a sequence of tools known as the Atwrench which lots of you in all probability bear in mind. I did a bunch of SAE model instruments in steel and in addition made a few titanium items. Then I made a decision to do a metric providing and I made only one small batch of them in S30v metal. |
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| ==Definition==
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| A Bratteli diagram is given by the following objects:
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| * A sequence of sets ''V''<sub>''n''</sub> ('the vertices at level ''n'' ') labeled by positive integer set '''N'''. In some literature each element v of ''V''<sub>''n''</sub> is accompanied by a positive integer ''b''<sub>''v''</sub> > 0.
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| * A sequence of sets ''E''<sub>''n''</sub> ('the edges from level ''n'' to ''n'' + 1 ') labeled by '''N''', endowed with maps ''s'': ''E''<sub>''n''</sub> → ''V''<sub>''n''</sub> and ''r'': ''E''<sub>''n''</sub> → ''V''<sub>''n''+1</sub>, such that:
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| ** For each ''v'' in ''V''<sub>''n''</sub>, the number of elements ''e'' in ''E''<sub>''n''</sub> with ''s''(''e'') = ''v'' is finite.
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| ** So is the number of ''e'' ∈ ''E''<sub>''n''−1</sub> with ''r''(''e'') = ''v''.
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| ** When the vertices have markings by positive integers ''b''<sub>''v''</sub>, the number ''a''<sub>''v'', ''v'' '</sub> of the edges with ''s''(''e'') = ''v'' and ''r''(''e'') = v' for ''v'' ∈ ''V''<sub>''n''</sub> and v' ∈ ''V''<sub>''n''+1</sub> satisfies ''b''<sub>''v''</sub> ''a''<sub>v, v'</sub> ≤ ''b''<sub>v'</sub>.
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| A customary way to pictorially represent Bratteli diagrams is to align the vertices according to their levels, and put the number ''b''<sub>''v''</sub> beside the vertex ''v'', or use that number in place of ''v'', as in
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| : 1 = 2 − 3 − 4 ...
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| : \ 1 ∠ 1 ∠ 1 ... .
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| An '''Ordered Bratteli diagram''' is a Bratteli diagram together with a partial order on ''E''<sub>''n''</sub> such that for any ''v'' ∈ ''V''<sub>''n''</sub> the set { ''e'' ∈ ''E''<sub>''n-1''</sub> : ''r''(''e'')=''v'' } is totally ordered. Edges that do not share a common range vertex are incomparable. This partial order allows us to define the set of all maximal edges ''E''<sub>max</sub> and the set of all minimal edges ''E''<sub>min</sub>. A Bratteli diagram with a unique infinitely long path in ''E''<sub>max</sub> and ''E''<sub>min</sub> is called ''essentially simple''.
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| <ref>Herman, Richard H. and Putnam, Ian F. and Skau, Christian F.''Ordered Bratteli diagrams, dimension groups and topological dynamics''. International Journal of Mathematics, volume 3, number 6. 1992, pp. 827-864.</ref>
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| ==Sequence of finite-dimensional algebras==
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| Any [[semisimple algebra]] over the [[complex numbers]] '''C''' of finite dimension can be expressed as a [[direct sum of algebras|direct sum]] ⊕<sub>k</sub> M<sub>n<sub>k</sub></sub>('''C''') of [[matrix ring|matrix algebras]], and the '''C'''-algebra homomorphisms between two such algebras up to inner automorphisms on both sides are completely determined by the multiplicity number between 'matrix algebra' components. Thus, an injective homomorphism of ⊕<sub>''k''=1</sub><sup>''i''</sup> ''M''<sub>''n''<sub>''k''</sub></sub>('''C''') into ⊕<sub>''l''=1</sub><sup>''j''</sup> ''M''<sub>''m''<sub>''l''</sub></sub>('''C''') may be represented by a collection of numbers ''a''<sub>''k'', ''l''</sub> satisfying ∑ ''n''<sub>''k''</sub> ''a''<sub>''k'', l</sub> ≤ ''m''<sub>''l''</sub> (the equality holds if and only if the homomorphism is unital). This can be illustrated as a bipartite graph having the vertices marked by numbers (''n''<sub>''k''</sub>)<sub>''k''</sub> on one hand and the ones marked by (''m''<sub>''l''</sub>)<sub>''l''</sub> on the other hand, and having ''a''<sub>''k'', ''l''</sub> edges between the vertex ''n''<sub>''k''</sub> and the vertex ''m''<sub>''l''</sub>.
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| Thus, when we have a sequence of finite-dimensional semisimple algebras ''A''<sub>''n''</sub> and injective homomorphisms ''φ''<sub>''n''</sub> : ''A''<sub>''n'''</sub> → ''A''<sub>''n''+1</sub>: between them, we obtain a Bratteli diagram by putting
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| : ''V''<sub>''n''</sub> = the set of simple components of ''A''<sub>''n''</sub>
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| (each isomorphic to a matrix algebra), marked by the size of matrices.
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| : (''E''<sub>''n''</sub>, ''r'', ''s''): the number of the edges between ''M''<sub>''n''<sub>''k''</sub></sub>('''C''') ⊂ ''A''<sub>''n''</sub> and ''M''<sub>''m''<sub>''l''</sub></sub>('''C''') ⊂ ''A''<sub>''n''+1</sub> is equal to the multiplicity of ''M''<sub>''n''<sub>''k''</sub></sub>('''C''') into ''M''<sub>''m''<sub>''l''</sub></sub>('''C''') under ''φ''<sub>''n''</sub>.
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| ==Sequence of split semisimple algebras==
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| Any [[semisimple algebra]] (possibly of infinite dimensiona) is one whose [[Module (mathematics)|modules]] are completely reducible, i.e. they decompose into the direct sum of [[Module_(mathematics)#Types_of_modules|simple modules]]. Let <math> A_0 \subseteq A_1 \subseteq A_2 \subseteq \cdots </math> be a chain of split semisimple algebras, and let <math>\hat A_i</math> be the indexing set for the irreducible representations of <math>A_i</math>. Denote by <math>A_i^\lambda</math> the irreducible module indexed by <math>\lambda \in \hat A_i</math>. Because of the inclusion <math>A_i \subseteq A_{i+1}</math>, any <math>A_{i+1}</math>-module <math>M</math> restricts to a <math>A_i</math>-module. Let <math>g_{\lambda, \mu}</math> denote the decomposition numbers <br>
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| <math> A_{i+1}^{\mu} \downarrow^{A_{i+1}}_{A_i} = \bigoplus_{\lambda \in \hat A_i} g_{\lambda,\mu} A_i^{\lambda}.</math><br>
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| The '''Bratteli diagram''' for the chain <math> A_0 \subseteq A_1 \subseteq A_2 \subseteq \cdots </math> is obtained by placing one vertex for every element of <math>\hat A_i</math> on level <math>i</math> and connecting a vertex <math>\lambda</math> on level <math>i</math> to a vertex <math>\mu</math> on level <math>i+1</math> with <math>g_{\lambda,\mu}</math> edges.
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| ===Examples===
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| [[File:Brauer-Bratteli.jpg|thumb|Bratteli diagram for Brauer and BMW algebras on i=0,1,2,3, and 4 strands.]]
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| (1) If <math>A_i = S_i</math>, the i<sup>th</sup> [[symmetric group]], the corresponding Bratteli diagram is the same as [[Young's lattice]].{{citation needed|date=July 2013}}
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| (2) If <math>A_i</math> is the [[Brauer algebra]] or the [[Birman-Wenzl algebra]] on ''i'' strands, then the resulting Bratteli diagram has partitions of ''i-2k'' (for <math>k=0,1,2,\ldots,\lfloor i/2 \rfloor</math>) with one edge between partitions on adjacent levels if one can be obtained from the other by adding or subtracting 1 from a single part.
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| (3) If <math>A_i</math> is the [[Temperley-Lieb algebra]] on ''i'' strands, the resulting Bratteli has integers i-2k (for <math>k=0,1,2,\ldots,\lfloor i/2 \rfloor</math>) with one edge between integers on adjacent levels if one can be obtained from the other by adding or subtracting 1.
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| * {{cite journal | zbl=0856.16038 | last1=Halverson | first1=Tom | last2=Ram | first2=Arun | title=Characters of algebras containing a Jones basic construction: The Temperley-Lieb, Okada, Brauer, and Birman-Wenzl algebras | journal=Adv. Math. | volume=116 | number=2 | pages=263–321 | year=1995 | issn=0001-8708 }}
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| * {{cite book | zbl=0958.46029 | last=Davidson | first=Kenneth R. | title=C*-algebras by example | series=Fields Institute Monographs | volume=6 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1996 | isbn=0-8218-0599-1 }}
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| * {{cite book | zbl=0967.19001 | first1=Mikael | last1=Rørdam | first2=Flemming | last2=Larsen | first3=Niels | last3=Laustsen | title=An introduction to K-theory for C*-algebras | series=London Mathematical Society Student Texts | volume=49 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2000 | isbn=0-521-78334-8 }}
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| * {{cite book | last=Durand | first=Fabien | chapter=6. Combinatorics on Bratteli diagrams and dynamical systems | pages=324–372 | editor1-last=Berthé | editor1-first=Valérie | editor2-last=Rigo | editor2-first=Michael | title=Combinatorics, automata, and number theory | location=Cambridge | publisher=[[Cambridge University Press]] | series=Encyclopedia of Mathematics and its Applications | volume=135 | year=2010 | isbn=978-0-521-51597-9 | zbl=pre05879512 }}
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| {{refend}}
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| [[Category:Application-specific graphs]]
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| [[Category:Operator algebras]]
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The Chilly Metal Recon 1 is a large folder which happens to have the most effective grip on this gallery. The closely checkered G-10 handle scales virtually feel sticky. It flicks open sweetly with both of the thumb studs to disclose a 4-inch tanto blade of black Japanese AUS 8A Stainless with Tuff-Ex coating. It has a reversible belt clip, and the blade locks into open place with a tricky lever lock on the back of handle. This model weighs 5.2 ounces. There are various variations of the Recon 1 to choose from, including micro variations, mini versions and the impressive XL versions.
Total, it’s spectacular to see simply how much Spyderco has sunk into this child for such a low price. This knife has nearly 550 critiques on Amazon with a mean rating of 4.7. The Tenacious will usually set you again no more than $forty! >> Click here for present pricing and critiques on the Spyderco Tenacious. SOG Flash II I believe Gerber made a sensible choice not adding a rubberized grip and compensated for this nicely by designing an over sized finger notch into the handle. And because of the these two cutouts, this knife fits small and medium sized grownup hands in addition to the arms of younger Scouts.
There’s a cause this is likely one of the finest selling knives in existence at this time. Extremely regarded knife maker Ken Onion has designed this stunning yet inexpensive pocket knife which is a part of Kershaw’s Leek series. It employs the patented SpeedSafe opening system which ensures a smooth blade deployment with only a single hand. The Sandvik 14C28N steel blade is extremely sharp and will keep sharp for a very long time however do note the blade tip is considerably delicate as in comparison with different knives so we don’t advocate this for heavy obligation applications.
Total this can be a nice knife but may be very angular and huge — too much so for it to take a seat comfortably in my pocket. It isn't uncomfortable in my hand, however it is simply one other solid knife — not what I used to be searching pocket knife brands for. I wouldn’t feel screwed if I used to be stuck within the woods with this knife, however once more not my first choice amongst this group of folders. It has extremely easy and fluid opening, the double sided lock launch is a pleasant contact for a lefty like me.
Welcome to Finest Pocket Knife At this time! The final word buyer’s guide with tons of useful info for each knife fans and newbies alike. Pocket knives are essential everyday instruments that no responsible person should be with out. At the moment you’ll find actually a whole lot of folding knives obtainable to purchase from a bunch of various manufacturers ranging from an inexpensive $5 penknife to effectively over $500 for prime of the line models. The handle is just a tad over a quarter inch extensive, which is actually skinny compared to most pocket knives. And truthfully, it’s thinness and low weight make the knife nearly unnoticeable in my pocket.
Cutting an apple. I love consuming an apple that I’ve lower with my pocket knife, slice by slice. You're feeling like a foul ass doing it. You maintain the apple in your non-dominant hand after which make a slice with the knife using your dominant hand. After you make the slice, pinch it between your thumb and knife blade. Convey the blade to your mouth and deposit the apple slice. There are three keys to wrapping your head around carrying a pocket knife each day. You have to de-weaponize, utilize, and familiarize your knife.
I had mentioned today's tool on the Fb group last week however didn't have a finished one on hand to submit a picture. At any rate, I'm more than happy to convey this one again from the useless. A number of years in the best automatic knives past I made a sequence of tools known as the Atwrench which lots of you in all probability bear in mind. I did a bunch of SAE model instruments in steel and in addition made a few titanium items. Then I made a decision to do a metric providing and I made only one small batch of them in S30v metal.