Finite Hopf Algebras and Bidifferential Structures

In this Letter we show the existence of bidifferential structures on finite Hopf algebras, which arise from quantum groups at roots of unity.     

Thermal evolution of ladder-like silsesquioxanes during formation of black glasses

Heat capacity of single-crystal samples of five chalcogenides (LiInS₂, LiInSe₂, LiGaS₂, LiGaSe₂, and LiGaTe₂) was measured with DSC in a temperature range from 180 to 460 K. The data for LiInS₂ and LiInSe₂ were compared with the literature data and shown to agree with the results of adiabatic calorimetry (Gmelin and Hönle in Thermochimica Acta 269: 575–590, 1995) better than with other DSC data (Kühn et al. in Cryst Res Technol 22: 265–269, 1987). Besides, the high-temperature fitting polynomial for C _P(T) published about 30 years ago for LiInS₂ is wrong. LiGaS₂, LiGaSe₂, and LiGaTe₂ were measured for the first time.     

Structure and thermal properties of the fritted glazes in SiO<Subscript>2</Subscript>–Al<Subscript>2</Subscript>O<Subscript>3</Subscript>–CaO–MgO–Na<Subscript>2</Subscript>O–K<Subscript>2</Subscript>O–ZnO system

In this work, the structure and thermal properties of aluminosilicate fritted glazes in SiO₂–Al₂O₃–CaO–MgO–Na₂O–K₂O–ZnO system with (4.0 mol%) and without addition of ZnO were examined by GIXRD, FTIR, MAS-NMR and thermal methods (DTA, DIL). It has been found that the all experimental glazes are amorphous material (transparent glazes). On the base of spectroscopic investigations, it was found that zinc ions exist in the network glazes in the octahedral coordination—Zn²⁺ ions play a network modifier role in structure of glazes. An analysis of the data obtained from thermal tests showed that addition of ZnO into chemical composition results in decrease in glass transition temperature value (T _g) for all glazes (DTA, DIL). The coefficient of thermal expansion (α) is decreased as the whole measurement range for one series of fritted glazes.     

Noncommutative Circle Bundles and New Dirac Operators

We study spectral triples over noncommutative principal U(1) bundles. Basing on the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and the Dirac operator on the total space and on the base space of the bundle. We analyze in details the example of the noncommutative three-torus viewed as a U(1) bundle over the noncommutative two-torus and find all connections compatible with an admissible Dirac operator. Conversely, we find a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection.     

Field Theory on a q = −1 Quantum Plane

We build a q = –1 deformation of a plane on a product of two copies of algebras of functions on the plane. This algebra contains a subalgebra of functions on the plane. We present a general scheme (which could be also used to construct a quaternion from pairs of complex numbers) and we use it to derive differential structures and metrics, and discuss sample field-theoretical models.Mathematics Subject Classifications (1991):46L87, 81T13, 17B37.     

Noncommutative differential geometry with higher-order derivatives

We build a toy model of differential geometry on the real line, which includes derivatives of the second order. Such construction is possible only within the framework of noncommutative geometry. We introduce the metric and briefly discuss two simple physical models of scalar field theory and gauge theory in this geometry.Partially supported by KBN grant 2 P302 168 4.     

Comparison of the impact of the addition of three alkaline earth metal oxides BaO,SrO and ZnO on sintering of glass–ceramic glazes from the SiO2–Al2O2–CaO–MgO–Na2O–K2O system

Journal of Thermal Analysis and Calorimetry - Glazes are the most important elements in the technology of producing the ceramic wares. They not only give aesthetic effects, like color and gloss,...     

On the Tensor Product Construction for q-Differential Algebras

We show that for q -1 there is no natural tensor product for q-differential algebras. In particular, the q-graded tensor product of q-differentials fails to satisfy the q-graded Leibniz rule.     

The Standard Model in noncommutative geometry: fundamental fermions as internal forms

Given the algebra, Hilbert space H, grading and real structure of the finite spectral triple of the Standard Model, we classify all possible Dirac operators such that H is a self-Morita equivalence bimodule for the associated Clifford algebra.     

Heat Trace and Spectral Action on the Standard Podleś Sphere

We give a new definition of dimension spectrum for non-regular spectral triples and compute the exact (i.e., not only the asymptotics) heat-trace of standard Podle? spheres \({S^2_q}\) for 0 < q < 1, study its behaviour when \({q\to 1}\) , and fully compute its exact spectral action for an explicit class of cut-off functions.