A Lax type operator for quantum finite W-algebras

Abstract

© 2018, Springer Nature Switzerland AG. For a reductive Lie algebra g, its nilpotent element f and its faithful finite dimensional representation, we construct a Lax operator L(z) with coefficients in the quantum finite W-algebra W(g, f). We show that for the classical linear Lie algebras glN, slN, soN and spN, the operator L(z) satisfies a generalized Yangian identity. The operator L(z) is a quantum finite analogue of the operator of generalized Adler type which we recently introduced in the classical affine setup. As in the latter case, L(z) is obtained as a generalized quasideterminant.