QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

38RĂZVAN GELCA AND ALEJANDRO URIBEthe skein module, the map defined above can be extended linearly to theentire skein module.It should be noted that RT t (T 2 × [0,1]) is isomorphic to the Kauffmanbracket skein algebra of the torus (see [6] and [9]). However this is not thecase for higher genus surfaces, as it can be checked by looking at the productof a separating curve with a nonseparating curve.5.3. A Stone–von Neumann theorem on the pillow case. Weyl quantizationyields an irreducible representation of the Reshetikhin-Turaev skeinalgebra of the cylinder over the torus subject to the following conditions:1. t is mapped to the 4rth root of unity e iπ2r ,2. each simple closed curve on the torus γ is mapped to a self-adjointoperator,3. the skein S r−1 (γ) is mapped to zero, where S n (x) denotes the Chebyshevpolynomial of the second kind.This factors to an irreducible representation of the reduced Reshetikhin-Turaev skein algebra ˜RT t (T × [0,1]), which is the non-abelian analogue ofthe group ring of the finite Heisenberg group. Like for the finite Heisenberggroup, a Stone–von Neumann theorem holds.Theorem 5.2. The representation given by the Weyl quantization of themoduli space of flat SU(2)-connections on the torus is the unique irreduciblerepresentation of ˜RT t (T 2 × [0,1]) which maps simple closed curves to selfadjointoperators and t to multiplication by e2r.πiProof. Let us show that each vector is cyclic. Because the eigenspaces of eachquantized Wilson line are 1-dimensional, in particular those of op(2cos 2πx),it suffices to check this for the eigenvectors of this operator, namely for ζ τ j ,j = 1,2,... ,r − 1. And becauseop(2cos 2πy)ζ τ j = ζ τ j+1 + ζ τ j−1op(2cos 2π(x + y))ζ τ j = t −1 (t 2 ζ τ j−1 + t −2 ζ τ j+1),by taking linear combinations we see that from ζj τ we can generate both ζτ j+1and ζj−1 τ . Repeating, we can generate the entire basis. This shows that ζτ jis cyclic for each j = 1,2,... ,r − 1, hence the representation is irreducible.To prove uniqueness, consider an irreducible representation of ˜RT t (T 2 ×[0,1]) that has the properties from the statement. It is first important to notethat the condition S r−1 (γ) = 0 for all γ implies, by the spectral mappingtheorem, that the eigenvalues of the operator associated to γ are among thenumbers 2cos kπ r, k = 1,2,... ,r − 1.Inspired by the structure of the Kauffman bracket skein algebra of thetorus exhibited by Bullock and Przytycki in [3], we can write for the generatorsX = (1,0), Y = (0,1), and Z = (1,1) of the algebra ˜RT t (T 2 × [0,1])