Geodesic
Galois orders of symmetric differential operators
Algebra and discrete mathematics, Tome 23 (2017) no. 1, pp. 35-46

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In this survey we discuss the theory of Galois rings and orders developed in ([20], [22]) by Sergey Ovsienko and the first author. This concept allows to unify the representation theories of Generalized Weyl Algebras ([4]) and of the universal enveloping algebras of Lie algebras. It also had an impact on the structure theory of algebras. In particular, this abstract framework has provided a new proof of the Gelfand-Kirillov Conjecture ([24]) in the classical and the quantum case for $\mathrm{gl}_n$ and $\mathrm{sl}_n$ in [18] and [21], respectively. We will give a detailed proof of the Gelfand-Kirillov Conjecture in the classical case and show that the algebra of symmetric differential operators has a structure of a Galois order.
Keywords: Weyl algebra, invariant differential operators, filed of fractions.
Mots-clés : Galois order 
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Vyacheslav Futorny; João Schwarz. Galois orders of symmetric differential operators. Algebra and discrete mathematics, Tome 23 (2017) no. 1, pp. 35-46. http://geodesic.mathdoc.fr/item/ADM_2017_23_1_a4/