We introduce two-parameter quantum toroidal algebras of simply laced types and provide their group theoretic realization using finite subgroups of SL2(C) via McKay correspondence. In particular our construction contains a realization of the vertex representation of the two-parameter quantum affine algebras of ADE types.

String theory requires two kinds of loop expansion: classical (α ′) world-sheet loops with expansion parameter < T > where T is a modulus field, and quantum (¯ h) spacetime loops with expansion parameter < S > where S is the dilaton field. Four-dimensional string/string duality (a corollary of ten-dimensional string/fivebrane duality) interchanges the roles of S and T and hence interchanges cla...

We construct twisting elements for module algebras of restricted two-parameter quantum groups from factors of their R-matrices. We generalize the theory of Giaquinto and Zhang to universal deformation formulas for categories of module algebras and give examples arising from R-matrices of two-parameter quantum groups.

We propose a proof of the recent observation due to Varagnolo and Vasserot that the q-deformed Fock spaces are modules of the quantum toroidal algebra U ′ q(sln,tor) (n ≥ 3) with the level (0, 1). The quantum toroidal action on the Fock space depends on a certain parameter κ. We find that with a specific choice of this parameter the action on the Fock spaces gives rise to the toroidal action on...

Precise parameter estimation plays a central role in science and technology. The statistical error in estimation can be decreased by repeating measurement, leading to that the resultant uncertainty of the estimated parameter is proportional to the square root of the number of repetitions in accordance with the central limit theorem. Quantum parameter estimation, an emerging field of quantum tec...

Recently Varagnolo and Vasserot established that the q-deformed Fock spaces due to Hayashi, and Kashiwara, Miwa and Stern, admit actions of the quantum toroidal algebra U ′ q(sln,tor) (n ≥ 3) with the level (0, 1). In the present article we propose a more detailed proof of this fact then the one given by Varagnolo and Vasserot. The proof is based on certain non-trivial properties of Cherednik’s...

An efficient geometric formulation of the problem of parameter estimation is developed, based on Hilbert space geometry. This theory, which allows for a transparent transition between classical and quantum statistical inference, is then applied to the analysis of exponential families of distributions (of relevance to statistical mechanics) and quantum mechanical evolutions. The extension to qua...

A simple model of coupled oscillators is investigated from the perspective of quantum mechanics. The classical model of two oscillators connected by a common platform can be easily solved analytically, but the quantum system requires a numerical approach. We assume that both the oscillators and the platform are quantum objects in their respective ground states at start and they evolve in time a...

An exact correspondence is established between a N-body classical interacting system and a (N-1)-body quantum system with respect to the partition function. The resulting Hermitian quantum potential is a (N-1)-body one. Inversely the Kelbg potential is reproduced which describes quantum systems at a quasi-classical level. The correspondence found between classical and quantum systems allows als...

I perform the factorization(s) of the Wheeler-DeWitt equation for the FRW minisuperspace universes of arbitrary Hartle-Hawking factor ordering introducing by this means a new, Darboux-type, free parameter in “quantum” cosmology. This parameter can be thought of as giving the time scale of a force due to a cosmological field. Mathematically speaking, the parameter is a Riccati integration consta...