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The quantum toroidal algebra of $gl_1$ provides many deformed W-algebrasassociated with (super) Lie algebras of type A. The recent work by Gaiotto andRapcak suggests that a wider class of deformed W-algebras includingnon-principal cases are obtained by gluing the quantum toroidal algebras of$gl_1$. These algebras are expected to be related with 5d AGT correspondence.In this paper, we discuss quantum deformation of the W-algebras obtained from$\widehat{su}(N)$ by the quantum Drinfeld-Sokolov reduction with su(2)embedding [N-1,1]. They were studied by Feigin and Semikhatov and we refer tothem as Feigin-Semikhatov's W-algebras. We construct free field realization andfind several quadratic relations. We also compare the norm of the Whittakerstates with the instanton partition function under the presence of a simplesurface operator in the N=3 case.

Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator