A Lax type operator for quantum finite W-algebras

Abstract

For a reductive Lie algebra $$\mathfrak {g}$$ , 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 $$W(\mathfrak {g},f)$$ . We show that for the classical linear Lie algebras $$\mathfrak {gl}_N$$ , $$\mathfrak {sl}_N$$ , $$\mathfrak {so}_N$$ and $$\mathfrak {sp}_N$$ , 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.