Related papers: Hofstadter's Butterfly and Langlands Duality
In this report I review some aspects of the algebraic structure of QFT related with the doubling of the degrees of freedom of the system under study. I show how such a doubling is related to the characterizing feature of QFT consisting in…
The competition between the length scales associated with the periodicity of a lattice potential and the cyclotron radius of a uniform magnetic field is known to have dramatic effects on the single-particle properties of a quantum particle,…
We study an extension of Bernstein's theorem to the setting of quantum groups. For a d-tuple of free, identically distributed random variables we consider a problem of preservation of freeness under the action of a quantum subset of the…
Let $k$ be an algebraically closed field of characteristic two and let $Q_8$ be the quaternion group of order $8$. We determine the Gerstenhaber Lie algebra structure and the Batalin-Vilkovisky structure on the Hochschild cohomology ring of…
When the Lyapunov exponent $\lambda_L$ in a quantum chaotic system saturates the bound $\lambda_L\leqslant 2\pi k_BT$, it is proposed that this system has a holographic dual described by a gravity theory. In particular, the butterfly effect…
Phenomena associated with topological properties of physical systems are naturally robust against perturbations. This robustness is exemplified by quantized conductance and edge state transport in the quantum Hall and quantum spin Hall…
The "quantum duality principle" states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie…
A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles (`quantum bundles') over the spacetime manifold yield a quantum ``formalism'' along…
We address the energy spectrum of honeycomb lattice with various defects or impurities under a perpendicular magnetic field. We use a tight-binding Hamiltonian including interactions with the nearest neighbors and investigate its energy…
Based on the vanishing of the second Hochschild cohomology group of the enveloping algebra of the Heisenberg algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum…
The studies of quantum states and optics in a {\it p}-type heterojunction with lateral surface quantum dot (antidot) superlattice and in the presence of perpendicular magnetic field are performed. For the first time the Azbel'-- Hofstadter…
In some recent theories including Quantum SuperString theory we encounter duality - it arises due to a non commutative geometry which in effect adds an extra term to the Heiserberg Uncertainity Principle. The result is that the micro world…
In the first part of this paper, we implement the multiplier algebra of the dual of an algebraic quantum group (A,Delta) as a space of linear functionals on A. In the second part, we construct the universal corepresentation of (A,Delta) and…
For a Hopf algebra B, we endow the Heisenberg double H(B^*) with the structure of a module algebra over the Drinfeld double D(B). Based on this property, we propose that H(B^*) is to be the counterpart of the algebra of fields on the…
The structure of quantum principal bundles is studied, from the viewpoint of Tannaka-Krein duality theory. It is shown that if the structure quantum group is compact, principal G-bundles over a quantum space M are in a natural…
The "quantum duality principle" states that the quantization of a Lie bialgebra - via a quantum universal enveloping algebra (QUEA) - provides also a quantization of the dual Lie bialgebra (through its associated formal Poisson group) - via…
We show that for a system of two entangled particles, there is a dual description to the particle equations in terms of classical theory of conformally stretched spacetime. We also connect these entangled particle equations with Finsler…
This is the last part of a series of three papers on the subject. In the first part we have considered the duality of algebraic quantum groups. In that paper, we use the term algebraic quantum group for a regular multiplier Hopf algebra…
The (small) quantum cohomology ring of a flag manifold F encodes enumerative geometry of rational curves on F. We give a proof of the presentation of the ring and of a quantum Giambelli formula, which is more direct and geometric than the…
We prove the Berenstein-Zelevinsky conjecture that the quantized coordinate rings of the double Bruhat cells of all finite dimensional simple algebraic groups admit quantum cluster algebra structures with initial seeds as specified by [4].…