  Galois orders, introduced in 2010 by V. Futorny and S. Ovsienko, form a class
of associative algebras that contain many important examples, such as the
enveloping algebra of $\mathfrak{gl}_n$ (as well as its quantum deformation),
generalized Weyl algebras, and shifted Yangians. The main motivation for
introducing Galois orders is they provide a setting for studying certain
infinite dimensional irreducible representations, called Gelfand-Tsetlin
modules. Principal Galois orders, defined by J. Hartwig in 2017, are Galois
orders with an extra property, which makes them easier to study. All of the
mentioned examples are principal Galois orders. In 2019, B. Webster defined
principal flag orders which in most situations are Morita equivalent to
principal Galois orders, and further simplifies their study. This paper
describes some techniques to connect the study pairs of standard flag and
Galois orders with related data. Such techniques include: a sufficient
condition for morphisms between standard flag and Galois orders to exist, a
property of flag orders related to differential operators on affine varieties,
and constructing tensor products of standard and principal flag and Galois
orders.
