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Research Statement 1 Positive Representations of Split Real Research Statement Ivan Chi-Ho Ip November 2011 Research Statement My major research interest is on the representation theory of quantum groups and its relations to the classical matrix groups. It is motivated by the study of various classical limits and analogies between the four basic objects SU(2) ! SUq(2); 0 < q < 1 x x ? ? y? y + + SL(2; R) ! SLq (2; R); jqj = 1 and its generalization to higher rank, as well as Lie groups of arbitrary types. Here SL+(2; R) stands for the Lie sub-semigroup of SL(2; R) where each entry is strictly + positive, while SLq (2; R) is the quantized function space where each matrix element is represented by a positive self-adjoint operator. We will discuss below that there is a strong parallel between the representation + theory of SUq(2) and a special class of representations of SLq (2; R). This suggests a vast program to generalize those applications that use the compact quantum group to the split real case. The C∗-algebraic framework that is needed to study the split real quantum group may lead to new results in noncommutative geometry. Applications to three-dimensional topology include for example the geometric approach to TQFT known as the Chern-Simons-Witten (CSW) model for compact groups, which leads in the split real case to a new class of TQFT's that arises from the quantization of the Teichm¨ullerspaces. The categorification of Uq(sl(2)), giving rise to the Khovanov homology, may lead in the split real case to a notion of \continuous categorification” which may provide new four-dimensional topological invariants. In this research statement, I will briefly describe several directions of my current research projects that will remain active in the coming few years. 1 Positive representations of split real quantum groups The starting point of this research project is the work of Teschner et al. [BT03, PT99, PT01], who studied extensively a very special class of \q-deformation" of the principal 2 series of representations of the quantum group Uq(sl(2; R)) in the space L (R). In my work [Ip11a], this class of representations and its relations to the quantum double + ∗ GLq (2; R) of the quantum plane in the C -algebraic framework is also studied in detail (cf. section 2). This family of representations have several remarkable properties. The formula resembles a perturbed version of the classical formula for the principal series representations of U(sl(2; R)), parametrized by positive real numbers λ 2 R+ as in the classical case, however it has no classical limit. What has been obtained is a duality between the quantum parameters πib2 πib−2 q = e ! qe = e ; 1 Research Statement Ivan Chi-Ho Ip November 2011 providing us with a representation of the modular double U (sl(2; )), generated by qqe R two mutally commuting sets of generators fE; F; Kg and fE;e F;e Keg. For generic q, this family of representations possesses the following properties: ± ± (i) the operators e; f; K and e;e f;e Ke are represented by positive self-adjoint op- erators, (ii) the generators satisfy the transcendental relations 1 1 1 2 2 2 e b = e;e f b = f;e K b = K;e (1) where e; f are certain rescaled versions of the generators E; F , respectively. In the work with I. Frenkel [FIp11], we generalized this family of representations to that of U (sl(n; )) for arbitrary n. These representations, called the positive qqe R principal series representations or positive representations in short, are constructed using again a special q-deformation on the minimal principal series representations for U(sl(n; R)) induced by the Borel subgroup, parametrized by the real span of the positive weights P + ⊂ h∗ , where h is the real form of the Cartan subalgebra. We R R R were able to construct the representation so that the generators are realized by positive self-adjoint operators, and they also satisfy satisfy the transcendental relations exactly as in the case of U (sl(2; )). Furthermore, these operators can be represented using qqe R the non-compact q-tori explicitly. The above construction was obtained from the Lie algebra action on the totally + positive unipotent subgroup U>0, which is naturally parametrized by the cluster vari- ables, first introduced in [BFZ96], which for type Ar corresponds to the determinants of various minors of the matrices. Therefore by studying the parametrization using the cluster variables for arbitrary simply-laced type Lie algebra, as proposed in [BFZ05], and possibly also their exchange relations, we can try to solve the following problem Problem 1.1. Find the positive representations of U (g ) for simply-laced type g qqe R R by positive self-adjoint operators satisfying the transcendental relations (1). For compact quantum groups, it is known that the finite dimensional represen- tations are closed under the tensor product. Although the class of continuous series representation for the classical real group is not closed under the tensor product, the class of positive representations for U (sl(2; )) is shown in [PT01] to be closed un- qqe R der the tensor product in the sense of the direct integral decomposition. This is in strong analogy with the compact case, which therefore provides us with a continuous version of a braided tensor category that certainly will have potential in other fields of mathematics. Since the positivity plays a prominant role in the analysis, we believe that this is also true in the higher rank case: Conjecture 1.2. The class of positive representations of U (sl(n; )) is closed under qqe R tensor product. 2 Research Statement Ivan Chi-Ho Ip November 2011 This conjecture can be proved if we can answer the following question: Problem 1.3. Do properties (i) and (ii) characterize the family of positive represen- tations? This is because the tensor product representation is easily seen to satisfy these properties. Finally in the classical case, there is a family of intertwiners corresponding to the Weyl group elements w 2 W between representations of principal series parametrized by h∗ [Kn86]. In the case of U(sl(2; )), the intertwining operator corresponding to R R the nontrivial Weyl element becomes multiplication by ratios of gamma functions, and the q-deformed intertwiner for Uq(sl(2; R)) is given by ratios of quantum dilogarithm functions [PT99]. It would be an interesting problem to write down explicit formulas for the intertwining operators in the q-deformed setting for higher rank, and show that the positive representations are only parametrized by the positive R+-span of the positive weights. 2 Harmonic analysis and classical limits of positive quantum groups For a compact group G, by Peter-Weyl theorem we know that the functions on G can be decomposed into the finite dimensional regular representation of U(g) M ∗ F un(G) = Vλ ⊗ Vλ ; (2) λ parametrized by the positive weights λ 2 P + ⊂ h∗ . For the split real group G this R R becomes more complicated. For example for G = SL(2; R), we know that the regular representation of U(sl(2; R)) on L2(SL(2; R)) is decomposed into both the continuous and discrete series. + However, in the quantum case, things are more interesting. Let us define GLq (2; R) to be the quantum group GLq(2) such that each entry, as well as the quantum de- terminant, are realized by positive self-adjoint operators. In [Ip11a] I studied this group in the C∗-algebraic and von Neumann setting, where I proved that it can be constructed as the quantum double group of the quantum plane, and furthermore a 2 + new Haar functional is found such that it induces an L structure on GLq (2; R), also compatible with its modular double. Then using the theory of multiplicative unitary, and various transformations involving the non-compact q-torus, I go on to prove that 2 + L (GLq (2; R)) is decomposed into direct integral of the positive representations de- fined in the previous section, with the Plancherel measure expressed in terms of the 2 + quantum dilogarithm Gb. The corresponding result for L (SLq (2; R)) is announced in [PT99] but the proof was never published. 3 Research Statement Ivan Chi-Ho Ip November 2011 For higher rank, I studied in [Ip11c] the Gauss-Lusztig decomposition for the pos- + itive quantum group GLq (n; R). What I did in this work is to take the Lusztig + decomposition of the totally positive unipotent matrix U>0 by positive real param- eters ai, and quantize the relations so that they are realized by variables that only commute up to a factor of q2. This also gives a more transparent relationship between the quantum cluster variables introduced in [BZ05] for type Ar quantum group. Fur- thermore a representation using only the standard q-tori was found explicitly, and this enabled me to define the group in the C∗-algebraic setting, and also talk about an L2(GL+ (n; )) space of \functions" over the modular double. In analogy to the qqe R case of n = 2, it is then natural to ask Conjecture 2.1. Does the regular representation of U (sl(n; )) on the space qqe R L2(GL+ (n; )) decompose into direct integral of positive representations? qqe R On the other hand, it is natural to ask if the above results in the quantum case de- scend to certain new structural results for the classical semigroup SL+(2; R). This is given by several evidences on the classical limit of special functions.
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