A study of Galois and flag orders

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 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 are Morita equivalent to principal Galois orders and further simplifies their study. The purpose of this dissertation is twofold: (1) To introduce a new example of a Galois order, A(gl_n), which is an extension of the enveloping algebra of gl_n such that the ``Weyl group\u27\u27 of A(gl_n) is the alternating group; (2) To describe some techniques to study such objects including tensor products and morphisms between standard flag orders with conjectured application to the orthogonal Lie algebra