An extension of U(gln) related to the alternating group and Galois orders

In 2010, V. Futorny and S. Ovsienko gave a realization of $U(\mathfrak{gl}_n)$ as subalgebra the ring invariants certain noncommutative with respect to action $S_1\times S_2\times\cdots\times S_n$, where $S_j$ is symmetric group on $j$ variables. An interesting question what similar algebra would be in invariant product alternating groups. this paper we define such an algebra, denoted $\mathscr{A}(\mathfrak{gl}_n)$, show that it Galois ring. For $n=2$, generalized Weyl for $n=3$ provide generators list verified relations. We also discuss some techniques construct orders from rings. Additionally, study categories finite-dimensional modules generic Gelfand-Tsetlin over $\mathscr{A}(\mathfrak{gl}_n)$. Finally, connections between Gelfand-Kirillov Conjecture, positive solution Noether's problem group.