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

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

We prove a conjecture proposed in [A. De Sole, V. G. Kac and D. Valeri, Finite W-algebras for N, Adv. Math. 327 (2018) 173-224.] describing the Lax type operator L(z) for the quantum finite W-algebras of N in terms of a PBW generating system for the W-algebra. In doing so, we extend this result to an arbitrary good grading and an arbitrary isotropic subspace of [1 2].

We prove that the singularities of the -matrix () of the minimal quantization of the adjoint representation of the Yangian () of a finite dimensional simple Lie algebra are the opposite of the roots of the monic polynomial ()
entering in the OPE expansions of quantum fields of conformal weight 3/2 of the universal minimal affine -algebra at level attached to