[continued from previous message]   
      
   leave you with a nice solution of the Yang-Baxter equations: let V be   
   spanned by two vectors X and Y, and define R by   
      
   R(X x X) = X x X   
   R(Y x y) = Y x Y   
   R(X x Y) = q(Y x X)   
   R(Y x X) = q(X x Y) + (1 - q^2)(Y x X)   
      
   Exercise: check that R satisfies the Yang-Baxter equations and   
   R^2 = (1 - q^2)R + q^2.  This quadratic equation is called a Hecke   
   relation.   
      
   If you are interested in learning about quantum groups, the   
   Yang-Baxter equations, and braids, I suggest taking a look at the   
   following books:   
      
   L. Kauffman, Knots and Physics, World Scientific, New Jersey, 1991.   
      
   C. Yang and M. Ge, Braid Group, Knot Theory, and Statistical   
   Mechanics, World Scientific, New Jersey, 1989.   
      
   T. Kohno, New Developments in the Theory of Knots, World   
   Scientific, New Jersey, 1990.   
      
   Yu. I. Manin, Quantum Groups and Noncommutative Geometry,   
   Les Publ. du Centre de R'echerches Math., Universite de   
   Montreal, Montreal, 1988.   
      
      
   The Quantum Plane   
      
   Ever so slowly I'm creeping towards a description of my own research on   
   the relation between braids and quantization.  There are so many inviting   
   byways...  Anyway, recall where we were last time.  (I never knew I'd   
   become so professorial in my old age.)   
      
   Given an 1-1 and onto linear transformation R: V x V -> V x V, where   
   "x" (as always in this article) the TENSOR product, not the Cartesian   
   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 most famous example of a solution   
   of the Yang-Baxter equations is where V is spanned by two vectors X and Y, and   
      
   R(X x X) = X x X   
   R(Y x Y) = Y x Y   
   R(X x Y) = q(Y x X)   
   R(Y x X) = q(X x Y) + (1 - q^2)(Y x X)   
      
   Now, given any solution (R,V) of the Yang-Baxter equations, or "Yang-Baxter   
   operator," we can define a "quantum vector space" S_R V as follows.   
   Well, wait -- mathematicians like to ladle on the definitions and   
   explain the point later, but I really should give you some clue about   
   "quantum vector spaces" before proceeding.   
      
   Quantum vector spaces were basically invented by Manin in his book,   
   "Noncommutative Geometry and Quantum Groups," and I will only be talking   
   about a subclass of *his* quantum vector spaces.  To get at the concept,   
   you have to think like an algebraic geometer -- or a quantum field   
   theorist.  To an algebraic geometer, what counts about a space is the   
   algebra of functions on that space (to be a bit more precise, algebraic   
   functions from that space to the complex numbers... and please, algebraic   
   geometers, don't bother telling me how much I'm oversimplifying - I know).   
   To a field theorist, what counts about spacetime is the fields on   
   spacetime, which are again functions from spacetime to the complex   
   numbers.  (And again, I'm vastly oversimplifying things.)  There's a   
   common theme here: for the algebraic geometer POINTS may be regarded as   
   secondary in importance to FUNCTIONS, while for the physicist PARTICLES   
   may be regarded as secondary in importance to FIELDS.  This makes a lot   
   of sense when you pay attention to what you're doing as you calculate:   
   most of the time you are playing around with functions, and only at the   
   end (if ever) do you bother to evaluate them at a given point.   
      
   Well, regardless of how *true* this actually is, it's a fruitful point   
   of view!  It gives rise to noncommutative geometry, in which points are   
   not the point; rather, all attention is concentrated on the algebra of   
   "functions on space," which is allowed to have a noncommutative product.   
   Here I should mention Alain Connes, whose paper "Non-commutative   
   differential geometry" really got the subject rolling in the early   
   1980's, though the notion of replacing functions on an honest space with   
   a noncommutative algebra is as old as Heisenberg's matrix mechanics.   
   In quantum mechanics, phase space disappears, but the analog of the   
   functions on phase space, the algebra of observables (typically operators   
   on a Hilbert space) lives on.   
      
   So now let us define our quantum vector space: what we'll *really* be   
   defining is an abstract algebra, but in the back of our heads we'll   
   always secretly pretend it consists of functions on some (nonexistent)   
   space, a "quantum space."   
      
   First, notice that a Yang-Baxter operator R on V gives rise to a   
   Yang-Baxter operator (R^{-1})* on the dual space V* - the inverse of the   
   adjoint of R.  Recall that V* is none other than the linear functions on   
   V.  Normally we define the algebra of polynomial functions on V, or the   
   "symmetric algebra" SV*, to be the quotient of the tensor algebra TV*   
   by the ideal generated by elements of the form   
      
   f x g - g x f   
      
   In other words, we take all (formal) products of linear functions on V   
   and impose on them the "commutation relations" that f times g should   
   equal g times f.  Note that here we are SWITCHING f and g; in our   
   quantum vector space we will use the Yang-Baxter operator to do this   
   switching.  Thus we define the "r-symmetric algebra" S_R V* to be the   
   quotient of TV* by the ideal generated by elements of the form   
      
   f x g - (R^{-1})*(f x g).   
      
   That's all there is to it - the philosophy is that we should never   
   switch two objects blindly in the usual manner, since they might be   
   "anyonic": we should always switch things with the appropriate   
   Yang-Baxter operator.   
      
   Now, all these duals and adjoints really just clutter things up a bit;   
   let's just define the r-symmetric algebra S_R V to be TV modulo the   
   ideal generated by   
      
   v x w - R(v x w)   
      
   So if we use the example of a Yang-Baxter operator given up above, that   
   I said was so famous, we get the following r-symmetric algebra, called   
   the QUANTUM PLANE: it has coordinates X and Y, which do not commute, but   
   satisfy   
      
                      XY = qYX !!!!!!!   
      
   The less jaded of you will be dazzled by the mere appearance of this   
   bizarre entity.  The more jaded will wonder what it's good for.   
   Patience, patience.  Let me just say for now that many   
   papers have been written about this thing, in respectable physics   
   journals, so it must be good for something. (?)   
      
   One can do all sorts of geometry on the quantum plane just as one does   
   on the good old classical plane.   There is a "quantum group" of   
   "quantum matrices" which acts as linear transformations of the quantum   
   plane.  One can differentiate and integrate functions on the quantum   
   plane, and so on.   In fact, MIT just installed new blackboards in   
   some of the math classrooms which have a dial with which the instructor   
   can select the value of q.   
      
   All but the last sentence of the previous paragraph is true.  In fact,   
   the simplest quantum group, SL_q(2), consists of 2-by-2 "quantum   
   matrices" with "quantum determinant" equal to 1.  It is a prototype for the   
   quantum groups SL_q(n).  People are having a ball these days coming up   
   with q-analogs of everything they know about.  For exampe, all   
   semisimple Lie groups admit quantizations of this general type, and such   
   quantum groups turn out to arise as symmetries of field theories in 2 or   
   3 dimensions (where anyonic statistics arise), and to be crucial for   
   understanding the HOMFLY polynomial and other knot invariants.   I'm   
   running ahead of myself here and should quit and go to bed; next time   
   I'll either define quantum matrix algebras and SL_q(2), or start on what   
   I'm *really* interested in, the differential geometry of certain "quantum   
   spaces" (more precisely, algebras equipped with Yang-Baxter operators).   
      
      
   r-Commutative Geometry   
      
   While there are plenty of things to say about the quantum plane, and   
      
   [continued in next message]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)