QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

28RĂZVAN GELCA AND ALEJANDRO URIBErepresentation of the finite Heisenberg group has been written in 3.2 as theaction of the group ring of this group on a quotient of itself.We first replace the oriented framed graphs and links colored by irreduciblerepresentations of U (sl(2, C)) by formal sums of oriented framedlinks colored by the 2-dimensional irreducible representation. To this endwe use the Clebsch-Gordan theorem for U (sl(2, C)) to writen/2∑( ) n − jV n = (−1) j (V 2 ) n−2j = S n−1 (V 2 ), for n = 1,2, · · · ,r − 1jj=0where S n (x) is, as seen, the Chebyshev polynomial of second kind definedrecursively by S 0 (x) = 1, S 1 (x) = x, S n+1 (x) = xS n (x) − S n−1 (x). Thenwe replace a framed simple closed curve γ on the surface colored by V n byS n (γ) with components colored by V 2 . Here the sum is formal, while a kthpower means k parallel copies of the curve.For an oriented framed graph we first replace the edges using the recursiverelation described in Figure 11 (recall from the previous section that thecoupons stand for the isomorphisms between a representation and its dual).After doing this, at each vertex colored by V m ,V n ,V p (that stands for theprojection of V m ⊗V n onto V p ) there are p strands entering from above andtwo groups of m respectively n strands exiting below. Connect these strandsas shown in Figure 12, where x+y = p, x+z = m, y+z = n. Do the similarthing for the vertices corresponding to the inclusion of V p into V m ⊗V n . Thelink obtained this way has and even number of coupons on each component.Cancel the coupons on each link component in pairs, adding a factor of −1each time the two coupons are separated by an odd number of maxima onthe link component (for those familiar with the subject, note that the linkcomponent is colored by an even-dimensional representation).n n−1 2V V VVn−1V2V3==[n−2]V[n−1]VVV 2 V 2 V V1[2]n−2n−122 2Figure 11V2V2With these transformations, the computation of the matrices of operatorsof the quantum group quantization reduces to computations with links whosecomponents are colored by V 2 . Theorem 4.3 in [14] allows us to perform