Finite W-algebras for glN

We study the quantum finite W-algebras W(glN,f), associated to the Lie algebra glN, and its arbitrary nilpotent element f. We construct for such an algebra an r1×r1 matrix L(z) of Yangian type, where r1 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(glN,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.