Representations of sl(2, ℂ) in category O and master symmetries

Abstract

In this paper, we first give a short account on the indecomposable sl(2, C) modules in the Bernstein-Gelfand-Gelfand (BGG) category O. We show these modules naturally arise for homogeneous integrable nonlinear evolutionary systems. We then develop an approach to construct master symmetries for such integrable systems. This method naturally enables us to compute the hierarchy of time-dependent symmetries. We finally illustrate the method using both classical and new examples. We compare our approach to the known existing methods used to construct master symmetries. For the new integrable equations such as a Benjamin-Ono type equation, a new integrable Davey-Stewartson type equation and two different versions of (2+1)-dimensional generalised Volterra Chains, we generate their conserved densities using their master symmetries.

Darboux Transformation for the Vector Sine-Gordon Equation and Integrable Equations on a Sphere

“…Using the above master symmetry, we are able to compute higher order symmetries for (57) sharing with both (34) and (38). Recently, a new method for construction of master symmetries of homogeneous integrable evolution equations (the O-scheme) was proposed in [28]. It would be very useful to extend the O-scheme to the classes of equations studied in this paper.…”

“…Y 12 ν,µ : (x, y, z) → (X(x, y; ν, µ), Y(x, y; ν, µ), z), Y 13 ν,κ : (x, y, z) → (X(x, z; ν, κ), y, Y(x, z; ν, κ)), Y 23 µ,κ : (x, y, z) → (x, X(y, z; µ, κ), Y(y, z; µ, κ)).…”

Master symmetries of non-linear systems

“…Since integrable systems are characterized by having an infinite number of symmetries, master symmetries are crucial for verifying their existence and confirming integrability, as noted in references [8][9][10][11][12]. We employ the O-scheme, introduced by Wang [13][14][15], to calculate these master symmetries. This method relies on the 2, sl( )  representation, initially discussed in [16].…”

“…We call an evolution equation α − homogeneous if [xu x + αu, K] = κK for some constants α and κ. In the case of a homogeneous equation with a scaling h = 2(xu x + αu), the elements e = u x , f = − (x 2 u x + 2αxu) and h form an 2, sl( )  algebra (Lemma 1 in [13]).…”

Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system

“…It appeared in [16] where the authors classified a family of equations with the non-locality of intermediate long wave type. Its infinitely many symmetries and conserved densities are constructed using its master symmetry in [17].…”