Let $\text{U}$ be an associative $\mathbb{C}$-algebra with three generators $E$, $H$, $F$, and three defining relations$$HE - EH = 2E,\text{ }Hf - FH = -2F,\text{ }EF - FE = H.$$Let $\mathbb{C}^m[x, y]$ denote the space of degree $m$ homogeneous polynomials in $2$ variables. Show that the formulas$$E(f) := x{{\partial f}\over{\partial y}},\text{ }H(f) := x{{\partial f}\over{\partial x}} - y{{\partial f}\over{\partial y}},\text{ }F(f) := y{{\partial f}\over{\partial x}}$$give a irreducible representation of $\text{U}$ in $\mathbb{C}^m[x, y]$.
Progress so far. I want to verify$$HE - EH = 2E,\text{ }HF = FH = -2F,\text{ }EF - FE = H$$for operations on $\mathbb{C}[x, y]$. We have$$(HE - EH) (f) = H\left( x{{\partial f}\over{\partial y}}\right) - E\left(x {{\partial f}\over{\partial x}} - y {{\partial f}\over{\partial y}}\right)$$$$= x\left({{\partial f}\over{\partial y}} + x {{\partial ^2 f}\over{\partial x \,\partial y}}\right) - yx {{\partial^2f}\over{\partial y^2}} - x^2 {{\partial^2 f}\over{\partial x\,\partial y}} + xy {{\partial^2 f}\over{\partial y^2}} + x {{\partial f}\over{\partial y}} = 2x {{\partial f}\over{\partial y}} = 2E(f),$$$$(HF - FH)(f) = H\left(y {{\partial f}\over{\partial x}}\right) - F\left(x {{\partial f}\over{\partial x}} - y{{\partial f}\over{\partial y}}\right)$$$$= xy {{\partial ^2f}\over{\partial x^2}} - y {{\partial f}\over{\partial x}} - y^2{{\partial^2 f}\over{\partial x\,\partial y}} - y{{\partial f}\over{\partial x}} - yx {{\partial^2 f}\over{\partial x^2}} + y^2 {{\partial^2 f}\over{\partial x\,\partial y}} = -2y {{\partial f}\over{\partial x}} = -2F(f),$$$$(EF - FE)(f) = E\left(y {{\partial f}\over{\partial x}}\right) - F\left( x {{\partial f}\over{\partial y}}\right)$$$$= x{{\partial f}\over{\partial x}} + xy{{\partial^2f}\over{\partial x\,\partial y}} - y {{\partial f}\over{\partial y}} - xy{{\partial^2 f}\over{\partial x\,\partial y}} = x{{\partial f}\over{\partial x}} - y{{\partial f}\over{\partial y}} = H(f).$$Therefore, the formulas give the vector space $\mathbb{C}[x, y]$ the structure of an $\text{U}$-module, i.e. a representation, but how do I see that it is simple $\text{U}$-module, i.e. irreducible?