  The purpose of my Ph.D. research is to define and study an analogue of the
classical Bernstein-Gelfand-Gelfand (BGG) category $\mathcal{O}$ for the Lie
algebra $\mathfrak{g}$, where $\mathfrak{g}$ is one of the finitary,
infinite-dimensional Lie algebras $\mathfrak{gl}_\infty(\mathbb{K})$,
$\mathfrak{sl}_\infty(\mathbb{K})$, $\mathfrak{so}_\infty(\mathbb{K})$, and
$\mathfrak{sp}_\infty(\mathbb{K})$. Here, $\mathbb{K}$ is an algebraically
closed field of characteristic $0$. We call these categories "extended
categories $\mathcal{O}$" and use the notation $\bar{\mathcal{O}}$. While the
categories $\bar{\mathcal{O}}$ are defined for all splitting Borel subalgebras
of $\mathfrak{g}$, this research focuses on the categories $\bar{\mathcal{O}}$
for very special Borel subalgebras of $\mathfrak{g}$ which we call Dynkin Borel
subalgebras. Some results concerning block decomposition and Kazhdan-Lusztig
multiplicities carry over from usual categories $\mathcal{O}$ to our categories
$\bar{\mathcal{O}}$. There are differences which we shall explore in detail,
such as the lack of some injective hulls. In this connection, we study
truncated categories $\bar{\mathcal{O}}$ and are able to establish an analogue
of BGG reciprocity in the categories $\bar{\mathcal{O}}$.
