  Let $k$ be an algebraically closed field. Let $B$ be the Borel subgroup of
$\mGL_n(k)$ consisting of nonsingular upper triangular matrices. Let $\frb =
\mLie B$ be the Lie algebra of upper triangular $n \times n$ matrices and
$\fru$ the Lie subalgebra of $\frb$ consisting of strictly upper triangular
matrices. We classify all Lie ideals $\frn$ of $\frb$, satisfying $\fru'
\subseteq \frn \subseteq \fru$, such that $B$ acts (by conjugation) on $\frn$
with a dense orbit. Further, in case $B$ does not act with a dense orbit, we
give the minimal codimension of a $B$--orbit in $\frn$. This can be viewed as a
first step towards the difficult open problem of classifying of all ideals
$\frn \subseteq \fru$ such that $B$ acts on $\frn$ with a dense orbit.
  The proofs of our main results require a translation into the representation
theory of a certain quasi-hereditary algebra $\cA_{t,1}$. In this setting we
find the minimal dimension of $\mExt^1_{\cA_{t,1}}(M,M)$ for a $\Delta$-good
$\cA_{t,1}$--module of certain fixed $\Delta$-dimension vectors.
