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We construct a non-polynomial generalization of the q-Askey scheme. Whereas the elements of the q-Askey scheme are given by q-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose integrands are built from Ruijsenaars' hyperbolic gamma function. Alternatively, the integrands can be expressed in terms of Faddeev's quantum dilogarithm, Woronowicz's quantum exponential, or Kurokawa's double sine function. We present the basic properties of all the elements of the scheme, including their integral representations, joint eigenfunction properties, and polynomial limits. [ABSTRACT FROM AUTHOR]

We consider two recent solutions of the generalized Faddeev–Volkov model, which is an exactly solvable Ising-type lattice spin model. The first solution is obtained by using the noncompact quantum dilogarithm, and the second one is constructed in a recent study via the gauge/YBE correspondence. We show that the weight functions of these models obtained by different techniques are the same upto a constant. [ABSTRACT FROM AUTHOR]

We analyze the structure of Feigin-Stoyanovsky's principal subspaces of affine Lie algebra from the jet algebra viewpoint. For type A level one principal subspaces, we show that their shifted multi-graded Hilbert series can be expressed either using the quantum dilogarithm or as certain generating functions "counting" finite-dimensional representations of A -type quivers. This notably results in novel fermionic character formulas for these principal subspaces. Moreover, our result implies that all level one principal subspaces of type A are "classically free" as vertex algebras. We also analyze infinite jet algebras associated to principal subspaces of affine vertex algebras L 1 (so 5) , L 1 (so 8) and L 1 (g 2). We derive a new character formula for the principal subspace of L 1 (so 5) , proving that it is classically free, and present evidence that the principal subspaces of L 1 (so 8) and of L 1 (g 2) are also classically free. [ABSTRACT FROM AUTHOR]

In Dimofte, Gukov, and Soibelman (Lett. Math. Phys. 95 (2011), 1-25), four quantum dilogarithm identities containing infinitely many factors are proposed as wall-crossing formulas for the refined BPS invariant. We give an algebraic proof of these identities using the formula for the universal R-matrix of the quantum affine algebra developed by Ito (Hiroshima Math. J. 40 (2010), 133-183), which yields various product presentations of the universal R-matrix by choosing various convex orders on an affine root system. By the uniqueness of the universal R-matrix and appropriate degeneration, we can construct various quantum dilogarithm identities, including the ones proposed in Dimofte, Gukov, and Soibelman (Lett. Math. Phys. 95 (2011), 1-25), which turn out to correspond to convex orders of multiple row type. [ABSTRACT FROM AUTHOR]

We study the tensor product decomposition of the split real quantum group U q 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 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 arising from analytic continuation and using normalization arising from tensor products of canonical basis. [ABSTRACT FROM AUTHOR]

We construct the positive principal series representations for Uq(gR) where g is of simply-laced type, parametrized by Rr≥0 where r is the rank of 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 Uqq~(gR) 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. [ABSTRACT FROM AUTHOR]

We study the positive representations Pλ of split real quantum groups uqq̃(gR) restricted to the Borel subalgebra uqq̃(bR). We prove that the restriction is independent of the parameter λ. Furthermore, we prove that it can be constructed from the GNS-representation of the multiplier Hopf algebra uC*qq̃ (bR) defined earlier, which allows us to decompose their tensor product using the theory of the “multiplicative unitary”. In particular, the quantum mutation operator can be constructed from the multiplicity module, which will be an essential ingredient in the construction of quantum higher Teichmüller theory from the perspective of representation theory, generalizing earlier work by Frenkel-Kim. [ABSTRACT FROM AUTHOR]

We study the positive representations Pλ of split real quantum groups uqq̃(gR) restricted to the Borel subalgebra uqq̃(bR). We prove that the restriction is independent of the parameter λ. Furthermore, we prove that it can be constructed from the GNS-representation of the multiplier Hopf algebra uC*qq̃ (bR) defined earlier, which allows us to decompose their tensor product using the theory of the “multiplicative unitary”. In particular, the quantum mutation operator can be constructed from the multiplicity module, which will be an essential ingredient in the construction of quantum higher Teichmüller theory from the perspective of representation theory, generalizing earlier work by Frenkel-Kim. [ABSTRACT FROM AUTHOR]

The 3D index of Dimofte-Gaiotto-Gukov is a partially defined function on the set of ideal triangulations of 3-manifolds with r tori boundary components. For a fixed 2r tuple of integers, the index takes values in the set of q-series with integer coefficients. Our goal is to give an axiomatic definition of the tetrahedron index and a proof that the domain of the3Dindex consists precisely of the set of ideal triangulations that support an index structure. The latter is a generalization of a strict angle structure. We also prove that the 3D index is invariant under 3-2 moves, but not in general under 2-3 moves. [ABSTRACT FROM AUTHOR]

We extend the notion of quantizing the coefficients of ordinary cluster algebras to the generalized cluster algebras of Chekhov and Shapiro. In parallel to the ordinary case, it is tightly integrated with certain generalizations of the ordinary quantum dilogarithm, which we call the quantum dilogarithms of higher degrees. As an application, we derive the identities of these generalized quantum dilogarithms associated with any period of quantum Y -seeds. [ABSTRACT FROM AUTHOR]

We derive a family of solutions to the tetrahedron equation using the RTT presentation of a two parametric quantized algebra of regular functions on an upper triangular subgroup of GL( n). The key ingredients of the construction are the longest element of the Weyl group, the quantum dilogarithm function, and central elements of the quantized division algebra of rational functions on the subgroup in question. [ABSTRACT FROM AUTHOR]

The (quantum) pentagon relation underlies the existing constructions of three-dimensional quantum topology in the combinatorial framework of triangulations. We discuss a special class of integral pentagon relations and their relations to the Faddeev-type operator pentagon relations. [ABSTRACT FROM AUTHOR]

The problem of identification of zero modes for the quantum Liouville model is discussed and the corresponding Hilbert space representation is constructed. [ABSTRACT FROM AUTHOR]

We give complete detail of the description of the GNS representation of the quantum plane $$\mathcal{A }$$ and its dual $${\widehat{\mathcal{A }}}$$ as a von Neumann algebra. In particular, we obtain a rather surprising result that the multiplicative unitary $$W$$ is manageable in this quantum semigroup context. We study the quantum double group construction introduced by Woronowicz, and using Baaj and Vaes’ construction of the multiplicative unitary $$\mathbf{W}_m$$ , we give the GNS description of the quantum double $$\mathcal{D }(\mathcal{A })$$ which is equivalent to $$GL_q^+(2,\mathbb{R })$$ . Furthermore, we study the fundamental corepresentation $$T^{\lambda ,t}$$ and its matrix coefficients, and show that it can be expressed by the $$b$$ -hypergeometric function. We also study the regular corepresentation and representation induced by $$\mathbf{W}_m$$ and prove that the space of $$L^2$$ functions on the quantum double decomposes into the continuous series representation of $$U_{q\widetilde{q}}(\mathfrak{gl }(2,\mathbb{R }))$$ with the quantum dilogarithm $$|S_b(Q+2i\alpha )|^2$$ as the Plancherel measure. Finally, we describe certain representation theoretic meaning of integral transforms involving the quantum dilogarithm function. [ABSTRACT FROM AUTHOR]

Abstract: We study the density of the supremum of a strictly stable Lévy process. Our first goal is to investigate convergence properties of the series representation for this density, which was established recently by Hubalek and Kuznetsov (2011) [24]. Our second goal is to investigate in more detail the important case when is rational: we derive an explicit formula for the Mellin transform of the supremum. We perform several numerical experiments and discuss their implications. Finally, we state some interesting connections that this problem has to other areas of Mathematics and Mathematical Physics and we also suggest several open problems. [Copyright &y& Elsevier]

The new form of pentagon equations suggested by Volkov (Int. Math. Res. Notices (2011); ) for the q-exponential on the basis of formal series is derived within the Hilbert space framework for the modular version of the quantum dilogarithm. [ABSTRACT FROM AUTHOR]

A review of the recent formulation of the quantum discrete Liouville model in the strongly coupled regime (corresponding to the Virasoro central charge 1

Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturally emerge from quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method.

Quantization of the Teichmüller space of a non-compact Riemann surface has emerged in 1980s as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston's shear coordinate functions on the edges form a coordinate system for the Teichmüller space, and they should be replaced by suitable self-adjoint operators on a Hilbert space. Upon a change of triangulations, one must construct a unitary operator between the Hilbert spaces intertwining the quantum coordinate operators and satisfying the composition identities up to multiplicative phase constants. In the well-known construction by Chekhov, Fock and Goncharov, the quantum coordinate operators form a family of reducible representations, and the phase constants are all trivial. In the present paper, we employ the harmonic–analytic theory of the Shale–Weil intertwiners for the Schrödinger representations, as well as Faddeev–Kashaev's quantum dilogarithm function, to construct a family of irreducible representations of the quantum shear coordinate functions and the corresponding intertwiners for the changes of triangulations. The phase constants are explicitly computed and described by the Maslov indices of the Lagrangian subspaces of a symplectic vector space, and by the pentagon relation of the flips of triangulations. The present work may generalize to the cluster X -varieties. [ABSTRACT FROM AUTHOR]