What is this associative algebra with two generators and one defining relation?

Question

Let $A$ be an associative $\mathbb{C}$-algebra with two generators $E$, $H$, and one defining relation $HE - EH = 2E$.

Question. Does this algebra have a name, and what is its significance?

Answer 1

Similar to this question, this algebra is the universal enveloping algebra of a Lie algebra, this time the unique nonabelian $2$-dimensional Lie algebra. One way to describe it is as the Borel subalgebra of $\mathfrak{sl}_2(\mathbb{C})$: explicitly, it is the Lie subalgebra of $\mathfrak{sl}_2(\mathbb{C})$ spanned by

$$H = \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right], E = \left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right].$$

You can check for yourself that $[H, E] = HE - EH = 2E$.

If we write $\mathfrak{g} = \mathfrak{sl}_2(\mathbb{C})$ then it's common to denote the Borel subalgebra by $\mathfrak{b}$ and the universal enveloping algebra by $U(-)$, so that this algebra would be called $U(\mathfrak{b})$. It naturally occurs in the study of the representation theory of $\mathfrak{g}$, or equivalently the study of modules over $U(\mathfrak{g})$; see Verma module for some details.