$$\underline{p}$$ p ̲ -reduced Multicomponent KP Hierarchy and Classical $$\mathcal {W}$$ W -algebras $$\mathcal {W}(\mathfrak {gl}_N,\underline{p})$$ W ( gl N , p ̲ )

Abstract

Abstract For each partition

$$\underline{p}$$

p ̲

of an integer

$$N\ge 2$$

N ≥ 2

, consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the

$$\underline{p}$$

p ̲

-reduced r-component KP hierarchy produces a solution of this integrable hierarchy. Along the way we provide an algorithm for the explicit construction of the generators of the corresponding classical

$$\mathcal {W}$$

W

-algebra

$$\mathcal {W}(\mathfrak {gl}_N,\underline{p})$$

W (

gl N

,

p ̲

)

, and write down explicit formulas for evolution of these generators along the commuting Hamiltonian flows.