Commutative Subalgebras in Finite W-algebras

Student: Mitrushchienkova Anna

Supervisor: Leonid Grigoryevich Rybnikov

Educational Programme: Mathematics and Mathematical Physics (Master)

Year of Graduation: 2018

In the paper, we compare two families of commutative subalgebras in the finite $W$-algebra of type $A$ for a nilpotent with $n$ equal Jordan blocks of size $l$. The first family comes from quantum shift argument algebras $\tilde A_{\mu} \subset U(\gg)$, while the second comes from Bethe subalgebras in the Yangian of level $l$ via the Brundan-Kleshchev isomorphism. In the special case that $n = 2$ we prove that these families coincide. The first family of commutative subalgebras generalizes to $W$-algebras of type $A$ assigned to arbitrary nilpotent (but these subalgebras are not maximal in general). Conjecturally they coincide with the images of (not yet known) Bethe subalgebras in the \emph{shifted} Yangian of level $l$ under the Brundan-Kleshchev isomorphism.

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