The ideals of the homological Goldman Lie algebra

We determine all the ideals of the homological Goldman Lie algebra, which reflects the structure of an oriented surface.Comment: 8 page

Hopf Maps, Lowest Landau Level, and Fuzzy Spheres

This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-spheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an interesting hierarchical structure made of "compounds" of lower dimensional spheres. We give a physical interpretation for such particular structure of fuzzy spheres by utilizing Landau models in generic even dimensions. With Grassmann algebra, we also introduce a graded version of the Hopf map, and discuss its relation to fuzzy supersphere in context of supersymmetric Landau model.Comment: v2: note and references added; v3: references adde

Non-Commutative Geometry in Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator

We clarify relations between the higher dimensional quantum Hall effect and A-class topological insulator. In particular, we elucidate physical implications of the higher dimensional non-commutative geometry in the context of A-class topological insulator. This presentation is based on arXiv:1403.5066.Comment: 5 pages, 1 table; contribution to the proceedings of the Workshop on Noncommutative Field Theory and Gravity, Corfu, Greece, September 8-15, 2013, Fortschritte der Physik 201