Editing Representation theory of SU(2) (section)

===Action on polynomials===
Since SU(2) is simply connected, a general result shows that every representation of its (complexified) Lie algebra gives rise to a representation of SU(2) itself. It is desirable, however, to give an explicit realization of the representations at the group level. The group representations can be realized on spaces of polynomials in two complex variables.<ref>{{harvnb|Hall|2015}} Section 4.2</ref> That is, for each non-negative integer <math>m</math>, we let <math>V_m</math> denote the space of homogeneous polynomials <math>p</math> of degree <math>m</math> in two complex variables. Then the dimension of <math>V_m</math> is <math>m + 1</math>. There is a natural action of SU(2) on each <math>V_m</math>, given by
:<math>[U \cdot p](z) = p\left(U^{-1}z\right),\quad z\in\mathbb C^2, U\in\mathrm{SU}(2)</math>.

The associated Lie algebra representation is simply the one described in the previous section. (See [[Representation theory of semisimple Lie algebras#The case of sl(2,C)|here]] for an explicit formula for the action of the Lie algebra on the space of polynomials.)