Products of infinite upper triangular quadratic matrices

Let F be a field and $q\left(x\right)$ a quadratic polynomial in $F\left[x\right]$ with $q\left(0\right)\ne 0$. We denote by $T_{\infty }\left(F\right)$ the algebra of all infinite upper triangular matrices over the field F. A matrix $A\in T_{\infty }\left(F\right)$ is called a quadratic matrix with respect to $q\left(x\right)$ if $q\left(A\right)=0$. In this paper, we first investigate the subgroup in $T_{\infty }\left(F\right)$ generated by all quadratic matrices with respect to $q\left(x\right)$ and then present some applications.