On some commutative subalgebras of the universal enveloping algebra of the Lie algebra \({\mathfrak {gl}} (n,\mathbb{C})\) (Q2710728)

Let \(\mathfrak g\) be a Lie algebra, and \(U(\mathfrak g)\) its universal enveloping algebra. One has a natural filtration \(U^k=U^k(\mathfrak g)\), \(k\geq 0\); the associated graded algebra \(P(\mathfrak g)=\operatorname {gr} U(\mathfrak g)=\bigoplus_{k=0}^\infty P_k\) is the symmetric algebra of space \(\mathfrak g\). Now \(P(\mathfrak g)\) has the structure of a Lie algebra (Poisson bracket) by NEWLINE\[NEWLINE\{u+U^{k-1},v+U^{l-1}\}=[u,v]+U^{k+l-2},NEWLINE\]NEWLINE where \(u\in U^k\), \(v\in U^l\). NEWLINENEWLINENEWLINEIn the case \(\mathfrak g=\mathfrak g\mathfrak l(n,\mathbb C)\), the authors consider shifts of maximal commutative subalgebras of the Poisson algebra \(P(\mathfrak g)\) and prove that this commutative subalgebras can be lifted into the universal enveloping algebra \(U(\mathfrak g)\). Moreover, this lifting is obtained by the symmetrization.