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   for example, the Clifford algebra, the Weyl algebra, noncommutative   
   tori, the quantum plane (and all other r-symmetric algebras), and   
   quantum groups.  Also, all supercommutative algebras are automatically   
   r-commutative.   
      
   In fact, I believe that there is a deep relation between   
   braids (or r-commutativity) and quantization.  This goes back to   
   Kaufmann's observation that the skein relation for the Conway polynomial   
      
   \   /    \   /               |   |   
    \ /      \ /                |   |   
     \   -    /     = -i hbar   |   |   
    / \      / \                |   |   
   /   \    /   \               |   |   
      
   and the canonical commutation relations   
      
      pq - qp = -i hbar.   
      
   It was made more clear by the discovery of the relation between quantum   
   groups and knot invariants (which I haven't really touched upon yet --   
   here I highly recommend Louis Kaufmann's book KNOTS AND PHYSICS, by   
   World Scientific Press).  It was made still more clear (in my opinion)   
   by the work of Frohlich, Gabbiani, Rehren, Schroer, and many others on   
   the appearance of braid group statistics in low-dimensional quantum   
   field theory.  (Essentially, these authors show that every nice quantum   
   field theory gives rise to a "fusion algebra" for the conserved charges,   
   and that these fusion algebras are r-algebras.)  But I suspect that at   
   the root of it all is something rather simpler which we haven't   
   understood yet.  A clue, I think, lies in the "classical   
   Yang-Baxter equations."   These are what physicists would call a   
   "semiclassical limit" of the Yang-Baxter equations.  Namely, take   
   R: V x V -> V x V of the form  R = T(1 + hbar r), where T is the usual   
   twist map   
      
   T(v x w) = w x v,   
      
   and r: V x V -> V x V, write down the Yang-Baxter equation for R, and   
   collect all terms of order hbar^2.  This equation is identical to the   
   equation that a "Poisson structure" must satisfy.  For details, see   
   Drinfeld's famous review paper on quantum groups.  The point here is   
   that a Poisson structure, which defines the Poisson bracket of classical   
   observables, is a semiclassical limit of the commutator which appears in   
   quantum mechanics.  Thus an infinitesimal deformation of the usual twist   
   map that is required to satisfy the Yang-Baxter equation is THE SAME   
   THING as an infinitesimal deformation of the commutative product in the   
   classical algebra of observables (i.e., a Poisson bracket).   
      
   More later...   
      
      
   r-Algebras   
      
   So... let's say we have an r-algebra.  That's an   
   algebra with an invertible linear map R: A x A -> A x A (again, "x"   
   denotes tensor product), which we draw as a right-crossing:   
      
   \   /   
    \ /   
     \   
    / \   
   /   \   
      
   which satisfies 1) the Yang-Baxter equations:   
      
      
   \   /   |	|     \ /   
    \ /    |	|      \   
     \     |	|     / \   
    / \    |	|    /   \   
   /   \   |	\   /    |   
   |    \ / 	 \ /     |   
   |     \     =	  \      |	   
   |    / \         / \     |   
   \   /   |	|   \   /   
    \ /    |	|    \ /   
     \     |  	|     \   
    / \    |	|    / \   
   /   \   |	|   /   \   
      
   2) R(1 x a) = a x 1 and R(a x 1) = 1 x a, and   
      
   3) the quasitriangularity conditions; writing multiplication as the   
   joining of strands these are   
      
      
   \   /     /		\   \   /   
    \ /     /		 \   \ /   
     |     /                 \   \   
      \   /                   \ / \   
       \ /         =           \   \   
        \                     / \   |   
       / \                   /   \ /   
      /   \                 /     |   
     /     \               /      |   
      
   and   
      
   \   \   /		\   /   /   
    \   \ /                 \ /   /   
     \   |                   \   /   
      \ /         =         / \ /   
       \                   /   \   
      / \                 |   / \   
     /   \                 \ /   \   
    /     \                 |     \   
   /       \                |      \   
      
   We say an r-algebra is "r-commutative" if   
      
   \   /     \     /   
    \ /       \   /   
     \         \ /   
    / \         |   
    \ /    =    |   
     |          |   
     |          |   
      
   and "strong" if R^2 is the identity, or R equals its inverse, i.e.:   
      
   \   /    \   /   
    \ /      \ /   
     \   =    /   
    / \      / \   
   /   \    /   \   
      
   (Note that a left-crossing   
      
   \   /   
    \ /   
     \   
    / \   
   /   \   
      
   is the inverse of a right-crossing.)   
      
   All sorts of noncommutative analogs of manifolds are r-commutative   
   algebras: quantum groups, noncommutative tori, quantum vector spaces,   
   the Weyl and Clifford algebras, certain universal enveloping algebras,   
   supermanifolds, etc..   It seems that the ones with direct relevance to   
   quantum theory in 4 dimensions are "strong," while the non-strong ones,   
   like quantum groups, are primarily relevant to 2- and 3-dimensional   
   physics.  I would now like to describe an analog of differential forms   
   for strong r-commutative algebras, and illustrate it for the case of the   
   Heisenberg algebra - i.e., the algebra defined by the canonical commutation   
   relations:  pq - qp = -i hbar.   
      
   What are differential forms?  Of course, they're the basis of a lot of   
   differential geometry, and there are lots of equivalent ways of defining   
   them, but let me take an algebraic viewpoint.  Let A be the algebra of   
   smooth functions on a manifold M.  We define differential forms as   
   follows.   Each function f in A has a "differential" df, and the   
   functions and their differentials generate an algebra in which we impose   
   the following relations:   
      
   1) Linearity:  d(f+g) = df + dg,  d(cf) = c(df) for any scalar c.   
      
   2) The product rule:   d(fg) = (df)g + f(dg).   
      
   3) The derivative of a constant vanishes:     d1 = 0.   
      
   4) Commutation relations:   f(dg) = (dg)f   and  (df)(dg) = -(dg)(df).   
      
   This algebra is called the algebra of differential forms on M.   
      
   That's all, folks!  If you've taken a course in differential geometry   
   you were probably exposed to tangent planes and all that stuff, but if   
   you want to get calculating with differential forms as soon as possible   
   this is all you need to start with.   
      
   Note that rule 4) is the only one in which we SWITCH elements of A --   
   moving f to the right of g.  This is the rule we'll need to modify for   
   an r-commutative algebra.  You could, of course, just leave out rule 4):   
   given any algebra A, the algebra whose relations are just given by 1)-3)   
   is called the "universal differential calculus" for A.  It's a   
   reasonable substitute for differential forms when A is any old   
   noncommutative algebra, and (from one viewpoint) it's the basis of Alain   
   Connes' approach to noncommutative differential geometry.  But when you   
   have a strong r-commutative algebra one can replace rule 3) with   
      
   4')  Commutation relations:  f(dg) = (dg^i)f_i   and   
   (df)(dg) = -(dg^i)(df_i).   
      
   Here I should explain that I'm writing  R(f x g)  as the sum over i of   
   tensor products of the form  g^i x f_i   (one can always do this), and   
   I'm using the Einstein summation convention (sum over repeated indices)   
   to avoid writing the summation sign.  What we're doing in rule 4') is   
   just what we should do: use R to "switch" f and g instead of "naively"   
   switching them as in 4).   
      
   Now let me show what this buys you in the case of the Heisenberg   
   algebra.   Actually I'm going to be sneaky and use a variant of the   
   Heisenberg algebra where we stick in a square root of hbar, which I'll   
   call h, just to confuse the heck out of the physicists.  (Actually, it's   
   because it's a pain to write square roots in ASCII!)  This is the   
   algebra over C generated by 3 formal variables, p, q, and h, subject to   
   the relations that   
      
   pq - qp = -i h^2   
      
   p h = h p   
      
   q h = h q   
      
   Note that we're NOT treating h (the square root of hbar) as a number   
   here, but as a variable, so I have to explicitly SAY that it commutes   
   with everything.   
      
      
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