On tensor product decomposition of positive representations of Uqq~(sl(2,R))\mathcal{U}_{q\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))

We study the tensor product decomposition of the split real quantum group $U_{q\tilde{q}}(sl(2,R))$ from the perspective of finite dimensional representation theory of compact quantum groups. It is known that the class of positive representations of $U_{q\tilde{q}}(sl(2,R))$ is closed under taking tensor product. In this paper, we show that one can derive the corresponding Hilbert space decomposition, given explicitly by quantum dilogarithm transformations, from the Clebsch-Gordan coefficients of the tensor product decomposition of finite dimensional representations of the compact quantum group $U_q(sl_2)$ by solving certain functional equations and using normalization arising from tensor products of canonical basis. We propose a general strategy to deal with the tensor product decomposition for the higher rank split real quantum group $U_{q\tilde{q}}(g_R)

Positive Representations of Split Real Simply-laced Quantum Groups

We construct the positive principal series representations for $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ where $\mathfrak{g}$ is of simply-laced type, parametrized by $\mathbb{R}_{\geq 0}^r$ where $r$ is the rank of $\mathfrak{g}$. We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group $\mathbf{U}_{\mathfrak{q}\tilde{\mathfrak{q}}}(\mathfrak{g}_\mathbb{R})$ of the modular double and show that the representations of both parts of the modular double commute weakly with each other, there is an embedding into a quantum torus algebra, and the commutant contains its Langlands dual.Comment: Finalized published version. Introduction has been rewritten to reflect recent progress and references added. Some typos fixe

Positive representations, multiplier Hopf algebra, and continuous canonical basis

We introduce the language of multiplier Hopf algebra in the context of positive representations of split real quantum groups, and discuss its applications with a continuous version of Lusztig-Kashiwara's canonical basis, which may provide a key to prove the closure of the positive representations under tensor products, and harmonic analysis of quantized algebra of functions in the sense of locally compact quantum groups.Comment: Revised version for publication to Proceedings of 2013 RIMS Conference "String theory, integrable systems and representation theory" Extended Section 2, added Section 3 and 5, updated reference

Positive representations of non-simply-laced split real quantum groups

We construct the positive principal series representations for $U_q(g_R)$ where $g$ is of type $B_n$, $C_n$, $F_4$ or $G_2$, parametrized by $R^r$ where $r$ is the rank of $g$. We show that under the representations, the generators of the Langlands dual group $U_{\tilde{q}}({}^Lg_R)$ are related to the generators of $U_q(g_R)$ by the transcendental relations. We define the modified quantum group of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the $q$-tori polynomials.Comment: Title changed. Fixed typos in the representations of C_n and F_4. Fixed typos in the matrix at (4.48) and signs of lambda. Add a remark on the representation of type G

On tensor products of positive representations of split real quantum Borel subalgebra Uqq~(bR)U_{q\tilde{q}}(b_R)

We studied the positive representations $P_\lambda$ of split real quantum groups $U_{q\tilde{q}}(g_R)$ restricted to the Borel subalgebra $U_{q\tilde{q}}(b_R)$. We proved that the restriction is independent of the parameter $\lambda$. Furthermore, we prove that it can be constructed from the GNS-representation of the multiplier Hopf algebra $U_{q\tilde{q}}^{C^*}(b_R)$ constructed earlier, which enables us to decompose their tensor product using the theory of the "multiplicative unitary". This will be an essential ingredient in the construction of quantum higher Teichm\"{u}ller theory from the perspective of representation theory, generalizing earlier work by Frenkel-Kim.Comment: Revised version for publication. Introduction is rewritten. Section 6.3 is removed to shorten the pape

Gauss-Lusztig Decomposition for GLq+(N,R)GL_q^+(N,R) and Representation by q-Tori

We found an explicit construction of a representation of the positive quantum group $GL_q^+(N,\R)$ and its modular double GL_{q\til[q]}^+(N,\R) by positive essentially self-adjoint operators. Generalizing Lusztig's parametrization, we found a Gauss type decomposition for the totally positive quantum group $GL_q^+(N,\R)$ for $|q|=1$, parametrized by the standard decomposition of the longest element $w_0\in W=S_{N-1}$. Under this parametrization, we found explicitly the relations between the standard quantum variables, the relations between the quantum cluster variables, and realizing them using non-compact generators of the $q$-tori $uv=q^2 vu$ by positive essentially self-adjoint operators. The modular double arises naturally from the transcendental relations, and an L^2(GL_{q\til[q]}^+(N,\R)) space in the von Neumann setting can also be defined.Comment: Reorganizing the contents involving positivity. Renewed reference

Positive Casimir and Central Characters of Split Real Quantum Groups

We describe the generalized Casimir operators and their actions on the positive representations $P_{\lambda}$ of the modular double of split real quantum groups $U_{q\tilde{q}}(g_R)$. We introduce the notion of virtual highest and lowest weights, and show that the central characters admit positive values for all parameters $\lambda$. We show that their image defines a semi-algebraic region bounded by real points of the discriminant variety independent of $q$, and we discuss explicit examples in the lower rank cases.Comment: 33 pages, 6 figures Expanded introduction. Minor typo fixe

Q-operator and fusion relations for Cq(2)(2)C^{(2)}_q(2)

The construction of the Q-operator for twisted affine superalgebra $C^{(2)}_q(2)$ is given. It is shown that the corresponding prefundamental representations give rise to evaluation modules some of which do not have a classical limit, which nevertheless appear to be a necessary part of fusion relations.Comment: 22 p, published versio

Supersymmetry and the Modular Double

A counterpart of the modular double for quantum superalgebra \cU_q(\osp(1|2)) is constructed by means of supersymmetric quantum mechanics. We also construct the $R$-matrix operator acting in the corresponding representations, which is expressed via quantum dilogarithm.Comment: 21 page

Positive representations of split real quantum groups and future perspectives

We construct a special principal series representation for the modular double $U_{q\tilde{q}}(g_R)$ of type $A_r$ representing the generators by positive essentially self-adjoint operators satisfying the transcendental relations that also relate $q$ and $\tilde{q}$. We use the cluster variables parametrization of the positive unipotent matrices to derive the formulas in the classical case. Then we quantize them after applying the Mellin transform. Our construction is inspired by the previous results for $g_R=sl(2,R)$ and is expected to have a generalization to other simply-laced types. We conjecture that our positive representations are closed under the tensor product and we discuss the future perspectives of the new representation theory following the parallel with the established developments of the finite-dimensional representation theory of quantum groups.Comment: 35 pages, 3 figure