Arthemy Kiselev (Johann Bernoulli Institute)
Non-Abelian Lie algebroids over jet spaces
The Inverse Scattering Transform (IST) is a powerful method for solving Cauchy problems for many nonlinear partial differential equations of mathematical physics (e.g., the Korteweg - de Vries equation). A Lie-algebra valued zero-curvature representation for an equation under study is the initial ingredient for finding solutions via IST. Geometrically, such representations are flat connection one-forms in principal fibre bundles over jet spaces where the equations live. If the equation's unknowns depend on two (but not three of more) independent variables (x,t), the zero-curvature representation must contain a spectral parameter which can not be removed under gauge transformations by the respective Lie group (for three or more variables, having such parameter is not obligatory). Recently, M.Marvan developed a very convenient cohomological technique which checks whether the parameter in a given family of representations is or is not removable; that theory's differential is explicitly constructed for every zero-curvature representation and is then used in the verification procedure.
I shall relate Marvan's idea to a natural class of variational Lie algebroids in which his operators are the anchors. By following Manin's heuristic principle ("If there is a Lie algebra, look around for its neighbours, i.e., its dual and the parity-opposites of the two"), I enlarge the geometry of IST bundles and introduce another, very natural cohomological theory which captures the initial setup. Naturally, this is the realisation of variational Lie algebroid at hand by the homological vector field --- or by the master-functional S which satisfies the classical master-equation [ [S,S] ]=0 with respect to the variational Schouten bracket (e.g., all Poisson bi-vectors satisfy the same equation). This approach, not being Reshetikhin's immediate BFV-quantisation of the zero-curvature model (see 1311.2490) but linking it to Drinfeld's geometry around Yang-Baxter equation and quantum IST, opens a way for deformation quantisation of the initial differential equation and IST problem for it.
The objective of the talk is to render a basic geometric idea by following the joint paper arXiv:1305.4598 [math.DG] by the speaker and A.Krutov.