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), |q| = 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 Teichmüllerspaces. 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 ∈ 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 {E,F,K} and {E,e F,e Ke}. 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 operators, (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 variables, 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 representations 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 representations? 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 ∈ 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 ∈ 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 determinant, 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 parameters 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. For the compact quantum group SUq(2), the matrix coefficients are expressed in terms of q-numbers, which have well-defined limits as q −→ 1. However in the split real case the situation is more complicated, where the expressions are expressed in terms of the quantum dilogarithm Gb which does not possess a classical limit. In [Ip10] however, I showed that we can indeed talk about certain kind of classical limit of the quantum dilogarithm Gb(x), and that it tends to the classical Gamma function Γ(x). This is quite remarkable because various representations for non-compact quantum groups can be considered in the classical limits. Using this fact, I studied in [Ip10] the classical limits of the intertwiners of representations of the quantum plane, which yields those of the classical ax + b group of affine transformations on R, while in [Ip11a] I found + that the matrix coefficients of the fundamental corepresentation of SLq (2, R) tends to the hypergeometric functions 2F1 under the classical limit, and the resulting formulas give precisely the expressions corresponding to the prinpical series representations restricted to the positive semigroup SL+(2, R). These are also expressed in terms of m,n the generalized associated Legendre functions of the second kind Qk (z). The quantum dilogarithm function Gb(x) and its variants play a very prominant + role in the representation theory of GLq (2, R). It is the central tool in the analysis of unbounded self-adjoint operators, where it provides a unitary map for studying positive operators of the form e2πbx + e2πbp, as well as the tau-beta theorem generalizing the beta integral, and the binomial formula

1 1 1 (u + v) b2 = u b2 + v b2 , (3) which is extremely useful in the study of modular double. As mentioned above, 2 + it gives the Plancherel measure of the decomposition of the L (GLq (2, R)) space,

4 Research Statement Ivan Chi-Ho Ip November 2011 as well as the multiplicity of the decomposition of the tensor product of positive + representations. Furthermore, in the theory of GLq (2, R) it also gives the explicit expressions for the matrix coeﬃcients, the multiplicative unitaries in the C∗-algebraic setting, the analytic structure of the multiplier Hopf algebra, and much more. As a side project, in [Ip11b] I surveyed the relationship between Gb(x) and other variants of it introduced e.g. by Faddeev-Kashaev [FK10], Fock-Goncharov [FG07], Volkov [Vo05], and Ruijinaars [Ru05], and also presented several visualizations of the complex graph of Gb(z) using Mathematica programming (see e.g. Figure 1).

Figure 1: The quantum dilogarithm Gb(x) for b = 0.7

In analogy to the quantum case, we would like to seek the following result

Problem 2.2. Do the L2 functions on the positive semigroup L2(SL+(2, R)) de- 2 2 compose naturally as the part Lcont(SL(2, R)) of L (SL(2, R)), corresponding to the continuous principal series representations of U(sl(2, R))? There are several classical results in the literature that relate the continuous part 2 Lcont(SL(2, R)) to various spaces. It is for example related to the harmonic functions on GL+(2, R) as O(2, 2) representations [St73], and its restriction to the boundary 2 L (C) gives the kernel of the ultra-harmonic operators 2,2 as O(3, 3) representations [KO03], while it is also identiﬁed with the vector bundle Γ(RP3, (−2)) by the half Fourier transform [Wi04] and the X-Ray (split Penrose) transform [Ar08]. Generalizing the idea for the positive representations of the quantum groups of higher rank, we can also ask for any arbitrary simple real Lie group GR, whether the space L2(G+) decomposes into the part L2 (G ) of L2(G ) corresponding to R mc R R the most continuous (or minimal parabolic) series representation of U(gR), which are studied in [vBS97]. I believe that we can gain more insight into physical questions by understanding the relationship between the quantum and classical picture. For example, we know

5 Research Statement Ivan Chi-Ho Ip November 2011 that the classical semi-group is closely related to causality in physics, where the time parameter cannot go backwards. This is discussed e.g. in [HN93, HLV95]. On the other hand, the non-existence of classical limits of the principal series representations of U (sl(2, )) suggested certain kind of renormalization [BS59], that separates the qqe R discrete series representations of the classical group.

3 Further applications

As discussed in the introduction, I am interested in understanding the categorification of the newly constructed positive principal series representations. In the compact finite dimensional case, the highest weight representation Vλ is decomposed into direct sum of weight spaces ⊕Vλ(n), and roughly speaking categorification is done by replacing these spaces with categories such that the quantum generators become functors between categories which satisfy certain conditions involving natural transforms. In the case of the positive representations discussed in section 1, the representations are now infinite dimensional, and the decomposition of weight spaces resembles that of a Fourier transformation on L2(R). As proposed in [FIp11], it will be very interesting to solve the following problem: Problem 3.1. Find the notion of a “continuous categorification” for the family of positive representations of U (g ). qqe R Another direction of my research prospect is related to quaternionic analysis in the quantum setting. In [FL08], the algebra of quaternions H is studied from the point of view of representation theory of the quaternionic conformal group SL(2, H). In particular the regular functions on H can be expressed in terms of the matrix coefficients of the standard representations of SU(2). An important formula on harmonic functions, called the Poisson formula, is proved. Roughly speaking, given a harmonic function φ in H defined on a closed ball BR of radius R, it can be expressed as an φ(z) 3 integral of N(z−w) over the surface SR, where N(z − w) is the quaternionic determinant. This result generalized the Poisson formula for the usual harmonic functions on the complex plane. In an upcoming work with I. Frenkel [FIp], we generalize this analysis to the quantum SUq(2) case, using the quantum Minkowski spacetime Mq introduced in [FJ10]. In particular, we try to prove the Poisson-type formula where one of the classical quanternion variable is replaced by the quantum SUq(2) matrix. This will give us a natural quantization formula, where we obtain the quantized harmonic functions on Mq from classical functions on SU(2). The difficulty arises from the invertibility of the determinant operator N(z −w), where now z ∈ SUq(2) is represented as operators on a Fock space. We found a proof of the invertibility based on a transformation using the Al-Salam-Chihara polynomials [KS98]. A quantum analogue of the decomposition

6 Research Statement Ivan Chi-Ho Ip November 2011

−1 l of N(z −w) into q-matrix coeﬃcients Mmn(z) of SUq(2) is also found. A remaining issue in the proof of the quantum Poisson formula is to tackle the unboundedness of this series expansion, so that we can complete the proof by replacing the integration in the formula with a quantum trace formula, which is a certain summation over the q-matrix coeﬃcients. In a recent work [FL11], the quaternionic analysis is extended to the split real case, and the corresponding representation theory for SL(2, R) is studied. There the Poisson formula is used to solve the problem of separation of the discrete and continuous series, and the minimal representation of the conformal group SL(4, R) is also studied. Hence a natural extension to the quantum case will follow from the + analysis of GLq (2, R) studied extensively in [Ip11a] that gives the quantum Minkowski spacetime in the split real setting. Therefore two of the future projects are:

+ Problem 3.2. Find the quantum Poisson formula for GLq (2, R) in the split real case, generalizing [FL11].

Problem 3.3. Give the quantum counterpart of the minimal representation for SUq(2, 2) + on F un(Mq(2)), the minimal representation of the positive quantum semigroup SLq (4, R) 2 + on L (GLq (2, R)), and discuss its relationship with causality. Finally, as remarked in [FL08], it is known that the Feynman integrals of the four point diagrams can be expressed in terms of products of the determinants N(z− w)−1. The result can be expressed in terms of the dilogarithm funciton [UD93] and has a remarkable relationship with the volume of the ideal tetrahedra in the three-dimensional hyerbolic space. Since we now know how to quantize N(z − w), together with my ex- perience on the quantum dilogarithm function, I would like to solve the following problem:

Problem 3.4. Find the relationship between the “quantum Feynman integral” and the quantum dilogarithm, and study its noncommutative-geometric interpretation.

References

[Ar08] M. Aryapoor, The Penrose transform in the split signature, arXiv:0812.3692, (2008) [BS59] N.N. Bogoliubov, D.V. Shirkov, Introduction to the theory of quantized ﬁelds, translated from the Russian by G. M. Volkoﬀ, Interscience Publishers, Inc., New York, (1959),

[BT03] A.G. Bytsko, K. Teschner, R-operator, co-product and Haar-measure for the modular double of Uq(sl(2, R)), Comm. Math. Phys. 240, 171-196, (2003)

7 Research Statement Ivan Chi-Ho Ip November 2011

[BFZ96] A. Berenstein, S. Fomin, A. Zelevinsky, Parametrizationf of canonical bases and totally positive matrices, Adv. in Math., 122, 49-149, (1996) [BFZ05] A. Berenstein, S. Fomin, A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J., 126, no. 1, 1-52, (2005) [BZ05] A. Berenstein, A. Zelevinsky, Quantum cluster algebras, Adv. in Math., 195, 405-455, (2005) [FK94] L.D. Faddeev, R.M. Kashaev, Quantum dilogarithm, Modern Phys. Lett. A9, 427-434, (1994) [FG07] V. V.Fock, A.B. Goncharov, The quantum dilogarithm and representations of quantized cluster varieties, Inv. Math., 175 (2), 223-286, (2007) [FIp11] I. Frenkel, I. Ip, Positive representation of split real quantum groups and future perspectives, arXiv:1111:1033, (2011)

[FIp] I. Frenkel, I. Ip, Quantum Poisson formula, in preparation [FJ10] I. Frenkel, M. Jardim, Quantum instantons with classical moduli spaces, Comm. Math. Phys., 237(3), 471-505, (2003) [FK10] I. Frenkel, H. Kim, Quantum Teichmuller space from quantum plane, arXiv:1006.3895v1, (2010) [FL08] I. Frenkel, M. Libine, Quaternionic analysis, representation theory and physics, Adv. in Math., 218 (6), 1806-1877, (2008) [FL11] I. Frenkel, M. Libine, Split Quaternionic Analysis and Separation of the Series for SL(2, R) and SL(2, C)/SL(2, R), Adv. in Math., 228 (2), 678-763, (2011) [HN93] J. Hilgert, K.H. Neeb, Lie semigroups and their applications, Lecture Notes in Mathematics 1552, Springer-Verlag, (1993) [HLV95] K. Hofmann, J. Lawson, E. Vinberg, Semigroups in algebra, geometry, and analysis, Exp. in Math., 20, De Gruyter, Berlin, (1995)

[Ip10] I. Ip, The classical limit of representation theory of the quantum plane, arXiv:1012.4145 [math.RT], (2010) [Ip11a] I. Ip, Representation of the quantum plane, its quantum double and harmonic + analysis on GLq (2, R), arXiv:1108.5365, (2011) [Ip11b] I. Ip, Graphs of quantum dilogarithm, arXiv:1108.5376 (2011)

8 Research Statement Ivan Chi-Ho Ip November 2011

+ [Ip11c] I. Ip, Gauss-Lusztig decomposition of GLq (N, R) and representations by q- tori, arXiv:1111.4025 (2011) [Kn86] A. Knapp, Representation theory of semisimple groups: An overview based on examples, Princeton University Press, (1986) [KO03] T. Kobayashi, B. Orsted Analysis on the minimal representations of O(p,q) III, Adv. in Math., 180 2, 486-512, (2003) [KS98] R.Koekoek, R.F.Swarttouw, The Askey-Scheme of hypergeometric orthogonal polynomials and its q-analogue, arXiv:math/9602214, (1998) [PT99] B. Ponsot, J. Teschner, Liouville bootstrap via harmonic analysis on a non- compact quantum group, arXiv: hep-th/9911110, (1999) [PT01] B. Ponsot, J. Teschner, Clebsch-Gordan and Racah-Wigner coeﬃcients for a continuous series of representations of Uq(sl(2, R)), Comm. Math. Phys 224, 613-655, (2001) [Ru05] S.N.M. Ruijsenaars, A unitary joint eigenfunction transform for the A∆O’s exp(ia±d/dz) + exp(2πz/a∓), J. Nonlinear Math. Phys. 12 Suppl. 2, 253-294, (2005) [St73] R. Strichartz, Harmonic analysis on hyperboloids, Journal of Functional Anal- ysis, 12 Issue 4, 341-383, (1973) [UD93] N. I. Ussyukina, A. I. Davydychev, An approach to the evaluation of three- and four-point ladder diagrams, Phys. Lett. B298, 363370, (1993) [vBS97] E.P. van den Ban, H. Schlichtkrull, The most-continuous part of the Plancherel decomposition for a reductive symmetric space, Ann. of Math., 145, 267-364, (1997) [Vo05] A.Yu. Volkov, Noncommutative hypergeometry, Comm. Math. Phys. 258(2), 257-273, (2005) [Wi04] E. Witten, Perturbative gauge theory as a string theory in twistor space, Comm. in Math. Phys., 252, no.1-3, 189-258, (2004)