Matrix Configurations for Spherical 4-branes and Non-commutative Structures on S⁴

We present a Matrix theory action and Matrix configurations for spherical 4-branes. The dimension of the representations is given by N=2(2j+1) (j=1/2,1,3/2,...). The algebra which defines these configurations is not invariant under SO(5) rotations but under SO(3) \otimes SO(2). We also construct a non-commutative product for field theories on S^4 in terms of that on S^2. An explicit formula of the non-commutative product which corresponds to the N=4 dim representation of the non-commutative S^4 algebra is worked out. Because we use S^2 \otimes S^2 parametrization of S^4, our S^4 is doubled and the non-commutative product and functions on S^4 are indeterminate on a great circle (S^1) on S^4. We will however, show that despite this mild singularity it is possible to write down a finite action integral of the non-commutative field thoery on S^4. NS-NS B field background on S^4 which is associated with our Matrix S^4 configurations is also constructed.