Generalized $$\widetilde{W}$$ algebras

Abstract Recently, a new generalized family of infinite-dimensional $$ \widetilde{W} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> algebras, each associated with a particular element of a commutative subalgebra of the $$ W_{1+\infty } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:math> algebra, was described. This paper provides a comprehensive account of the aforementioned association, accompanied by the requisite proofs and illustrative examples. This approach allows a derivation of Ward identities for selected WLZZ matrix models and the expansion of corresponding $$ W $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> -operators in terms of an infinite set of variables $$ p_k $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> .