The Irreducible Unitary Representation of SU(2) and Its Lie algebra Representations

We study the three dimensional special unitary group whose the Lie algebra is given by . The research aims to construct a representation of and realized on the inner product space of all homogeneous polinomials of degree and of all homogeneous polinomials of degree which satisfying irreducibility and unitarity conditions. Namely, The action of and are presented on the spaces and respectively. In the first step, we computed all representations of on and . Furthermore, in the second step, by simply connectedness property of then the irreducible unitary representation of Lie algebra realized on can be obtained from the representation by using derived representation. The results showed the explicit formulas of representations of and