  For a quiver with weighted arrows we define gauge-theory K-theoretic
W-algebra generalizing the definition of Shiraishi et al., and Frenkel and
Reshetikhin. In particular, we show that the qq-character construction of gauge
theory presented by Nekrasov is isomorphic to the definition of the W-algebra
in the operator formalism as a commutant of screening charges in the free field
representation. Besides, we allow arbitrary quiver and expect interesting
applications to representation theory of generalized Borcherds-Kac-Moody Lie
algebras, their quantum affinizations and associated W-algebras.
