Finite W-algebras for gl_N

TL;DR

This paper constructs a matrix of Yangian type for quantum finite W-algebras associated with gl_N and arbitrary nilpotent elements, providing explicit formulas and detailed examples for key cases.

Contribution

It introduces a new matrix L(z) of Yangian type for W(gl_N,f), generalizing classical operators and encoding the algebra's structure with explicit generators and relations.

Findings

Constructed L(z) matrix of Yangian type for W(gl_N,f)

Provided explicit formulas for generators and relations

Detailed analysis of principal, rectangular, and minimal nilpotent cases

Abstract

We study the quantum finite W-algebras W(gl_N,f), associated to the Lie algebra gl_N, and its arbitrary nilpotent element f. We construct for such an algebra an r_1 x r_1 matrix L(z) of Yangian type, where r_1 is the number of maximal parts of the partition corresponding to f. The matrix L(z) is the quantum finite analogue of the operator of Adler type which we introduced in the classical affine setup. As in the latter case, the matrix L(z) is obtained as a generalized quasideterminant. It should encode the whole structure of W(gl_N,f), including explicit formulas for generators and the commutation relations among them. We describe in all detail the examples of principal, rectangular and minimal nilpotent elements.