[continued from previous message]   
      
   quantum groups, I think I'll home in on my main topic at last:   
   r-commutative geometry.   This is a particular approach to   
   noncommutative geometry that generalizes what mathematicians and   
   physicists call "supergeometry".  So first I should say a brief word   
   about supergeometry.  First, recall that the plain old geometry of   
   manifolds can be cast into the language of commutative algebra by   
   considering not the manifold itself as a set of points, but the ALGEBRA   
   of (smooth, complex) functions ON the manifold.  (In this language, for   
   example, vector fields are derivations, vector bundles are projective   
   modules, and so on - every geometric construct has an algebraic analog.)   
   Now in the 70's physicists really caught on to the fact that considering   
   only commutative algebras was horribly unfair to fermions, which like to   
   anticommute: i.e. they have   
      
   ab = -ba   
      
   instead of   
      
   ab = ba.   
      
   So while the phase space of bosonic system is a manifold, the phase   
   space of a system containing both bosonic and fermionic degrees of   
   freedom is a "supermanifold" - a (particular kind of) algebra which has   
   "even" and "odd" elements, such that the even, or bosonic, elements   
   commute, while the odd elements anticommute.  (The even elements commute   
   with the odd elements, by the way.)  Generalizing lots of concepts from   
   commutative algebras to supercommutative algebras simply amounts to   
   sticking in appropriate minus signs!  The rule of thumb is that whenever   
   one switches two elements a and b, one should stick in a factor of   
   (-1)^{deg a deg b}, where  deg a = 0 if a is even (bosonic) and deg a =   
   1 if a is odd (fermionic).  One may extend the notions of vector field,   
   differential form, metric, curvature, and all your favorite concepts   
   from geometry to "supergeometry" in this manner.   
      
   This turned out to be fascinating (it's a bit premature to say "useful")   
   in particle physics, where it goes by the name of supersymmetry.  The   
   idea is that there should be a symmetry between bosons and fermions.   
   While this is NOT AT ALL observed in nature, it would be nice if it *were*   
   true, so people have developed clever ways of rigging up their theories   
   so that you never see the "superpartners" every particle has: for the   
   photon, the photino, for the gluon, the gluino, for the leptons,   
   schleptons, for the quarks, squarks... you get the pattern.  (Don't   
   complain to *me* if you think this is silly, it wasn't my idea!)   
   Superstrings are the latest of these "super" ideas in particle physics.   
      
   Supersymmetry has actually proven itself in a more practical manner in   
   nuclear physics, where it lets one model resonances in nuclei, relating   
   the properties of fermionic and bosonic nuclei.   
      
   Where supergeometry really shines, though, is in mathematics.  For   
   example, Ed Witten came up with beautiful proofs of the Atiyah-Singer   
   index theorem and the positive mass theorem (a theorem about general   
   relativity) using supergeometry.  (As usual, he left it to others to   
   make his arguments rigorous.)   
      
   While I personally don't think that bosons and fermions were created   
   equal in the manner postulated by supersymmetry, I do favor an   
   approach to physics which doesn't take bosons, or commuting variables,   
   to be somehow superior to fermions, or anticommuting variables.  This   
   demands supergeometry.  For example, a decent treatment of *classical*   
   fermions requires a supermanifold for the "phase space".   
      
   My own personal twist (motivated by the work of many people on anyons,   
   the braid group, quantum groups, etc.) is to try to take a look at what   
   geometry would be like if one wanted to be fair to ANYONS as well as   
   bosons and fermions.  This is "r-commutative geometry."   
      
   Recall that given a vector space  V , an invertible linear   
   transformation R: V x V -> V x V, where "x" denotes the tensor product,   
   the Yang-Baxter equations say that   
      
   (R x I)(I x R)(R x I) = (I x R)(R x I)(I x R)   
      
   where I is the identity on V.	The idea is that R "switches" two   
   elements of V, mapping  v x w to R(V x w), and if we draw  R  as a   
   "crossing," as follows:   
      
   \   /   
    \ /   
     \   
    / \   
   /   \   
      
   The Yang-Baxter equations say that   
      
      
   \   /   |	|     \ /   
    \ /    |	|      \   
     \     |	|     / \   
    / \    |	|    /   \   
   /   \   |	\   /    |   
   |    \ / 	 \ /     |   
   |     \     =	  \      |	   
   |    / \         / \     |   
   \   /   |	|   \   /   
    \ /    |	|    \ /   
     \     |  	|     \   
    / \    |	|    / \   
   /   \   |	|   /   \   
      
   Now suppose that our vector space is really an algebra - let's call it   
   A.  (I mean an associative algebra with unit.)  The product in the   
   algebra defines a multiplication map m: A x A -> A, given by   
      
   m(a x b) = ab.   
      
   We can draw this as the *joining* of two strands:   
      
   \   /   
    \ /   
     |   
     |   
     |   
      
   Associativity simply says that (ab)c = a(bc), or in terms of diagrams,   
      
   \   /     /   
    \ /     /   
     |     /   
      \   /   
       \ /   
        |   
        |   
        |   
      
   is equal to   
      
   \    \   /   
    \    \ /   
     \    |   
      \   /   
       \ /   
        |   
        |   
        |   
      
   Now these diagrams aren't really braids because of the "fusion of   
   strands" that's taking place, but they fit in well with the braid group   
   philosophy.  For example, there is a sense in which the two trees I've   
   drawn above are topologically the same, just as the Yang-Baxter equation   
   expresses a topological identity.   There are generalizations of braids,   
   e.g. the "ribbon graphs" of Reshetikhin and Turaev, that make all this   
   precise.   
      
   Now: an "r-algebra" is an algebra A equipped with   
   a solution R: A x A -> A x A of the Yang-Baxter equations such that   
   1) R(a x 1) = 1 x a and R(1 x a) = a x 1 for all a in A -- i.e., one   
   "switches" the identity 1 with a in the usual manner, and 2) the   
   following conditions hold -- I'll draw them pictoriallly:   
      
   \   /     /		\   \   /   
    \ /     /		 \   \ /   
     |     /                 \   \   
      \   /                   \ / \   
       \ /         =           \   \   
        \                     / \   |   
       / \                   /   \ /   
      /   \                 /     |   
     /     \               /      |   
      
   and   
      
   \   \   /		\   /   /   
    \   \ /                 \ /   /   
     \   |                   \   /   
      \ /                   / \ /   
       \                   /   \   
      / \                 |   / \   
     /   \                 \ /   \   
    /     \                 |     \   
   /       \                |      \   
      
      
   If you like equations instead of pictures, these are   
      
   R(m x I) = (I x m)(R x I)(I x R)   
      
   and   
      
   R(I x m) = (m x I)(I x R)(R x I),   
      
   as maps from A x A x A to A x A, respectively.  The first one tells you   
   how to compute R(ab x c), and the second one tells you how to compute   
   R(a x bc), so both tell you how to switch a product of two elements past   
   a third element.  These are called the "quasitriangularity" conditions   
   and are crucial in the theory of quantum groups (usually in a   
   slightly disguised form).   
      
   Now commutativity says that ab = ba -- in other words, you can multiply   
   two elements of A, or you can switch them first and then multiply them,   
   and you'll get the same result.  Generalizing this, we say that an   
   r-algebra is "r-commutative" if   
      
   m = mR   
      
   as maps from A x A to A.  In diagrams:   
      
   \   /     \     /   
    \ /       \   /   
     \         \ /   
    / \         |   
    \ /    =    |   
     |          |   
     |          |   
      
   In words: "switch, then multiply, equals multiply."   
      
   It turns out that one can do a fair amount of geometry for r-commutative   
   algebras.  And there are lots of examples of r-commutative algebras.  In   
   fact, many of the algebras obtained by *quantization* are r-commutative:   
      
   [continued in next message]   
      
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