Commutative diagrams for groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T22:15:23Z http://mathoverflow.net/feeds/question/55641 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55641/commutative-diagrams-for-groups Commutative diagrams for groups Hans Stricker 2011-02-16T16:48:29Z 2011-02-17T06:13:11Z <p>You can present a group in a Cayley-like manner, replacing colors by explicit assignment of nodes to edges: while in a Cayley graph \$x \circ y = z\$ is presented like this:</p> <p><img src="http://yoga-hebamme.de/mathoverflow/cayley.png" alt="alt text"></p> <p>you can also present it like this:</p> <p><img src="http://yoga-hebamme.de/mathoverflow/cayley2.png" alt="alt text"></p> <p>Now the group axioms can be stated like this:</p> <ul> <li>For each \$x, y\$ there is a unique \$z\$ with \$x \circ y = z\$, or: such that the following diagram holds ("commutes"):</li> </ul> <p><img src="http://yoga-hebamme.de/mathoverflow/cayley2.png" alt="alt text"></p> <ul> <li>There is an \$e\$ such that for all \$x\$ it holds that \$x\circ e = e \circ x = x\$, or: such that the following two diagrams commute:</li> </ul> <p><img src="http://yoga-hebamme.de/mathoverflow/neutral.png" alt="alt text"></p> <p>and for each \$x\$ there is a \$x^{-1}\$ such that \$x \circ x^{-1} = x^{-1} \circ x = e\$, or: such that the following diagram commutes:</p> <p><img src="http://yoga-hebamme.de/mathoverflow/inverse.png" alt="alt text"></p> <ul> <li>For each \$x, y, z\$ it holds that \$x \circ (y \circ z) = (x \circ y) \circ z\$, or: such that the following diagram commutes:</li> </ul> <p><img src="http://yoga-hebamme.de/mathoverflow/assoc.png" alt="alt text"></p> <p>The last diagram is somewhat ugly, even when drawn in this most balanced way (I didn't find a more appealing and symmetric one).</p> <p>But an astonishing symmetry arises, when we consider Abelian groups. Commutativity is expressed by the diagram:</p> <p><img src="http://yoga-hebamme.de/mathoverflow/commut.png" alt="alt text"></p> <p>and associativity becomes:</p> <p><img src="http://yoga-hebamme.de/mathoverflow/assoc2.png" alt="alt text"></p> <p>In the presence of commutativity, associativity seems to be related to commutativity (some sort of "second level commutativity").</p> <blockquote> <p>Can any use be made of this kind of diagrams, or is it just vain baublery?</p> </blockquote> http://mathoverflow.net/questions/55641/commutative-diagrams-for-groups/55697#55697 Answer by S. Carnahan for Commutative diagrams for groups S. Carnahan 2011-02-17T06:13:11Z 2011-02-17T06:13:11Z <p>These pictures are a way of writing a group as an algebra over an <a href="http://en.wikipedia.org/wiki/Operad_theory" rel="nofollow">operad</a>. The little square is the 2-ary operation, and the arrows are an indicator that makes the inputs distinguishable.</p> <p>John Baez has written about operads using pictures similar to yours. See for example <a href="http://math.ucr.edu/home/baez/week191.html" rel="nofollow">TWF week 191</a>.</p> <p>The relation between associativity and commutativity is similar to a fact seen in some books on vertex algebras, where one starts with an axiom like \$x(yz) = y(xz)\$ and after some power series manipulations deduces \$x(yz) = (xy)z\$. </p>