Ricerc@Sapienza

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.