  Diagram algebras (e.g. graded braid groups, Hecke algebras, Brauer algebras)
arise as tensor power centralizer algebras, algebras of commuting operators for
a Lie algebra action on a tensor space. This work explores centralizers of the
action of a complex reductive Lie algebra $\mathfrak{g}$ on tensor space of the
form $M \otimes N \otimes V^{\otimes k}$. We define the degenerate two-boundary
braid algebra $\mathcal{G}_k$ and show that centralizer algebras contain
quotients of this algebra in a general setting. As an example, we study in
detail the combinatorics of special cases corresponding to Lie algebras
$\mathfrak{gl}_n$ and $\mathfrak{sl}_n$ and modules $M$ and $N$ indexed by
rectangular partitions. For this setting, we define the degenerate extended
two-boundary Hecke algebra $\mathcal{H}_k^{\mathrm{ext}}$ as a quotient of
$\mathcal{G}_k$, and show that a quotient of $\mathcal{H}_k^{\mathrm{ext}}$ is
isomorphic to a large subalgebra of the centralizer. We further study the
representation theory of $\mathcal{H}_k^{\mathrm{ext}}$ to find that the
seminormal representations are indexed by a known family of partitions. The
bases for the resulting modules are given by paths in a lattice of partitions,
and the action of $\mathcal{H}_k^{\mathrm{ext}}$ is given by combinatorial
formulas.
