Bäcklund transformations for new integrable hierarchies related to the polynomial Lie algebra gl∞(n)

The Frobenius–Virasoro algebra and Euler equations

“…(2.20) Furthermore using the formula (2.17), in componentwise forms the Euler equation (2.7) has the following bihamlitonian representation Remark 2.7. Let us remark that when choose F as the Frobenius algebra (Z l , tr l ) ( [3,10,19]), the Frobenius-Virasoro algebra vir F coincides with the polynomial Virasoro algebra introduced by P.Casati and G.Ortenzi in [4]. They also computed Euler equations on vir F * and proved that they admitted a local bihamiltonian structure by using the trace-type map tr l .…”

Integrability of the Frobenius algebra-valued Kadomtsev-Petviashvili hierarchy

“…Proof. As explained in the introduction, the coupled KP hierarchy defined in [4,15] is exactly the Z n -KP hierarchy. According to Example 2.3, the algebra Z n has at least n-"basic" different ways to be realized as the Frobenius algebra.…”

“…With the use of vertex operator representations of polynomial Lie algebras, they obtained a class of coupled KP hierarchy formulated as a "coupled Hirota bilinear equation". Shortly afterwards, Van de Leur in [15], starting from these bilinear equations, recovered the corresponding wave functions and Lax equations with a Z n -valued Lax operator L, where Z n = C[Λ]/(Λ n ) is the maximal commutative subalgebra of gl(n, R) and Λ = (δ i,j+1 ) ∈ gl(m, R). A natural problem then arises as to how to construct Hamiltonian structures for these coupled KP hierarchies.…”

Free-field realizations of the W𝒜 n , N -algebra

“…It is difficult to give such a free-field realization because of the non-local terms except some special cases [3,4]. Recently motivated by the work in [6,13,16,25], Strachan and Zuo began to study the Frobenius algebra-valued integrable systems [21,22,24,26]. In [21] they introduced an F-valued KP hierarchy associated with an F-valued pseudo-differential operator (ΨDO in brief)…”