Additional symmetries of constrained CKP and BKP hierarchies

Abstract

Abstract. The additional symmetries of the constrained CKP (cCKP) and BKP (cBKP) hierarchies are given by their actions on the Lax operators, and their actions on the eigenfunction and adjoint eigenfunction {Φ i , Ψ i } are presented explicitly. Furthermore, we show that acting on the space of the wave operator, ∂ * k forms new centerless W cC 1+∞ and W cB 1+∞ -subalgebra of centerless W 1+∞ respectively. In order to define above symmetry flows ∂ * k of the cCKP and cBKP hierarchies, two vital operators Y k are introduced to revise the additional symmetry flows of the CKP and BKP hierarchies.

Quantum torus symmetries of the CKP and multi-component CKP hierarchies

“…The Lax operator L c and the Orlov-Schulman's operator M c satisfy the following matrix canonical relation 4) in another, the Orlov-Schulman's operator M c satisfies…”

“…It has perfect structures such as the Virasoro type additional symmetry which has been extensively studied in literature( [1], [2], [3], [4], [5]). Date, Jimbo, Kashiwara and Miwa extended their work on the KP hierarchy to the multi-component KP hierarchy in( [6]).…”

On the sub-KP hierarchy and its constraints, revisited

“…As above, when one considers special cases: Q = 1 and Q = ∂, the additional flows ∂ ζ l reduce to that of the constrained CKP hierarchy and the constrained BKP hierarchy constructed in [18]. To end up this section we want to show that This means that ∂ ζ l commutes with ∂ t2j−1 acting on the space of the dressing operator S, which is our desired result.…”

“…For the KP hierarchy, such kind of symmetries could be represented in Lax equations with operators depending explicitly on the temporal variables as developed by Orlov and Schulman [12], and also in the form of Sato's Bäcklund transformations as the action of vertex operators on tau function of the hierarchy [1,13,14]. But in general, for the constrained KP hierarchy and the (constrained) BKP or CKP hierarchies, one has to revise additional symmetry flows of the KP hierarchy to make them On the sub-KP hierarchy and its constraints, revisited consistent with certain reduction conditions, see [15][16][17][18]20] for details. The aim of this paper is to construct additional symmetry flows and string equations of the BC r -KP hierarchy and its constraints for r ∈ Z ≥0 , and furthermore study their algebraic structures in a unified way.…”

The compatibility of additional symmetry and gauge transformations for the constrained discrete Kadomtsev-Petviashvili hierarchy

“…By now, the gauge transformations of many integrable hierarchies related to KP hierarchy have been derived, for example, the constrained KP (cKP) hierarchy [8,9,10,13,16,20,21], the constrained BKP and CKP hierarchy [15,22] (cBKP and cCKP), the dKP hierarchy [19,18], the cdKP hierarchy [11], the q-KP hierarchy [23,24] and so on. The additional symmetry [25,26,27,28,29,30,31,32,33,34,35,36,37] is a kind of symmetry depending explicitly on the space and time variables, involved in so-called string equation and the generalized Virasoro constraints in matrix models of the 2d quantum gravity (see [6,38] and references therein). Regarding the possible application of the additional symmetry flows of the KP hierarchy in physics, it is natural to ask whether these flows are compatible with the gauge transformation.…”