Construction of fuzzy S^4

Abstract

We construct a fuzzy $S^4$, utilizing the fact that ${\mathbf{CP}}^3$ is an $S^2$ bundle over $S^4$. We find that a fuzzy $S^4$ can be described by a block-diagonal form whose embedding square matrix represents a fuzzy ${\mathbf{CP}}^3$. We discuss some pending issues on fuzzy $S^4$, i.e., precise matrix-function correspondence, associativity of the algebra, and etc. Similarly, we also obtain a fuzzy $S^8$, using the fact that ${\mathbf{CP}}^7$ is a ${\mathbf{CP}}^3$ bundle over $S^8$.

17 pages; v2. extensively rewritten; v3. published version