Lower triangular matrix algebra
= Lower triangular matrix algebra
{title2=$T_n(k)$}
The algebra $T_n(k)$ consists of the lower triangular $n$ by $n$ matrices over $k$. Its <Jacobson radical> $N$ consists of the strictly lower triangular matrices, and
$$
N^i=\operatorname{span}\{e_{rs}:r-s\geq i\}.
$$
The $n$ simple left modules $S_r$ are one-dimensional, with a matrix acting through its $r$th diagonal entry. For the left regular module,
$$
N^i/N^{i+1}\cong S_{i+1}\oplus\cdots\oplus S_n.
$$