The non-commutative differential calculus on the quantum groups SL q (N) is constructed. The quantum external algebra proposed contains the same number of generators as in the classical case. The exterior derivative defined in the constructive way obeys the modified version of the Leibnitz rules.

We derive explicit expressions for the generating series of the fundamental solutions of the Ar quantum Q-system of Di Francesco and Kedem [Non-commutative integrability, paths and quasi-determinants, Adv. in Math. 228(1) (2011) 97–152], expressed in terms of any admissible initial data. These involve products of quantum multinomial coefficients, coded by the initial data structure.

We give a description of the (small) quantum cohomology ring of the flag variety as a certain commutative subalgebra in the tensor product of the Nichols algebras. Our main result can be considered as a quantum analog of a result by Y. Bazlov.

We report the results of infrared and magneto-optical spectroscopy study on electrodynamic response of bismuth doped with 1.5% of antimony. The spectra are presented for temperatures down to 4.2 K, and in magnetic fields as high as 18 T. The results reveal strong magneto-optical activity, similar to pure bismuth, however there are some differences introduced by antimony doping. Analysis of opti...

We study the Hochschild homology of algebras related via split pairs, and apply this to fibre products, trivial extensions, monomial algebras, graded-commutative algebras and quantum complete intersections. In particular, we compute lower bounds for the dimensions of both the Hochschild homology and cohomology groups of quantum complete intersections.

It will be shown that the Reeh-Schlieder property holds for states of quantum fields for ultrahyperfunctional Wightman theory. As by product, it is shown that the Reeh-Schlieder property also holds for states of quantum fields on a non-commutative Minkowski space in the setting ultrahyperfunctional.

A reconciliation of gravitation and electromagnetism has eluded physics for neearly a century. It is argued here that this is because both quantum physics and classical physics are set in differentiable space time manifolds with point particles. Once we consider extended particles as in Quantum Superstring theory, and the consequential underlying Non-Commutative geometry, then a reconciliation ...