arXiv subjects
Publications and source records attributed to Boris Arm.
We study the Borel algebra de ne by [x a ; x b ] = 2 a;1 x b as a noncommutative manifold R 3 . We calculate its noncommutative di erential form relations. We deduce its partial derivative relations and the derivative of a plane wave. After calculating its de Rham cohomology, we deduce the wave operator and its corresponding magnetic solution.
We study the noncommutative geometry of the dihedral group D 6 using the tools of quantum group theory. We explicit the torsion free regular spin connection and the corresponding 'Levi-Civita' connection. Next, we nd the Riemann curvature and its Ricci tensor. The main result is the Dirac operator of a representation of the group which we nd the eigenvalues and the eigenmodes.
We calculate the torsion free spin connection on the quantum group B q [SU 2 ] at the fourth root of unity. From this we deduce the covariant derivative and the Riemann curvature. Next we compute the Dirac operator of this quantum group and we give numerical approximations of its eigenvalues.
A Manin triples (D; g; ~ g) is a bialgebra (g; ~ g which don't intersect each others and a direct sum of this bialgebra D = g ~ g). If the corresponding Lie groups have a Poisson structure, they are called Poisson-Lie groups. A Poisson-Lie sigma models is an action (3.13) calculated by a Poisson vector eld matrix. [3] have deduced the extremal eld which minimize the action of this models, which gives the motion equation (3.19). We calculate here the action and the equations of motion for some 6-dimensionals Manin triples and we give a general formula for each 4-dimensional Manin triples. The 6-dimensional Manin triples are (sl(2; C) sl(2; C) ; sl(2; C); sl(2; C)),(sl(2; C) sl(2; C) ; sl(2; C)); sl(2; C),(sl(2; C); su(2; C); sb(2; C)) and (sl(2; C); sb(2; C); su(2; C)).