The algebra of matrices of the form
$M= \left[ {\begin{array}{cc} \alpha_1 & \overline{v} \\ 0 & \alpha_2 \\ \end{array} } \right] $
Where the values $\alpha$ are real scalars and $\overline{v}$ is a vector in $\mathbb{R}^2$ with basis consisting of the following matrices:
$e_1= \left[ {\begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} } \right] $, $e_2= \left[ {\begin{array}{cc} 0 & 0 \\ 0 & 1 \\ \end{array} } \right] $, $B= \left[ {\begin{array}{cc} 0 & <1, 0> \\ 0 & 0 \\ \end{array} } \right] $, $C = \left[ {\begin{array}{cc} 0 & <0, 1> \\ 0 & 0 \\ \end{array} } \right]$
Does it have a special name or applications or is it just a random example of an algebra.
I would like to look up some properties of this particular algebra and its representations.
The product is $$ \begin{bmatrix} \alpha_1 & v \\ 0 & \alpha_2 \end{bmatrix} \begin{bmatrix} \beta_1 & w \\ 0 & \beta_2 \end{bmatrix} = \begin{bmatrix} \alpha_1\beta_1 & \alpha_1v+w\beta_2 \\ 0 & \alpha_2\beta_2 \end{bmatrix} $$ This is a particular case of a trivial extension. If $R$ is a ring and $_RM_R$ a bimodule, the trivial extension of $R$ by $M$ is the ring $R\ltimes M$ consisting of pairs $(r,x)$, where $r\in R$ and $x\in M$, with componentwise addition and multiplication $$ (r,x)(s,y)=(rs,xs+ry) $$ (this should remind the Dorroh extension of a ring). In your case $R=\mathbb{R}\times\mathbb{R}$ (product ring) and $M=\mathbb{R}^2$ (vector space) with the bimodule structure $$ (\alpha,\beta)x=\alpha x \qquad x(\alpha,\beta)=x\beta $$ Rings of this type are good sources for example, because it's rather easy to classify modules over them.