Is there a standard notation for the set of lower triangular matrices?

Question

Is there a standard notation for the set of lower triangular matrices, and if so what is it. Additionally, is there a notation for the set of strictly lower triangular matrices, and if so what is it? Thanks.

Answer 1

Very often the vector space of $m\times m$ lower triangular matrices with coefficients in a field $K$, or a ring $R$, is denoted by $\mathfrak{b}_m(K)$, and the subspace of strictly lower-triangualr ones by $\mathfrak{n}_m(K)$. Here the $\mathfrak{b}$ stands for the Lie algebra of the Borel subgroup $B$ of $GL_m(K)$, of lower-triangular invertible matrices, and the $\mathfrak{n}$ for its nilradical $N$. In this sense, $\mathfrak{b}_m(K)$ and $\mathfrak{n}_m(K)$ are Lie subalgebras of $\mathfrak{gl}_m(K)$, where $\mathfrak{b}_m(K)$ is solvable, and $$\mathfrak{n}_m(K)=[\mathfrak{b}_m(K),\mathfrak{b}_m(K)]$$ is nilpotent. The commutator of two matrices here is given by $[A,B]:=AB-BA$. So the notation $B$ comes from "Borel", and $N$ from Nilradical, or nilpotent.

Without the context of Lie algebras, see here.