Symmetry and Integrability of Equations of Mathematical Physics − 2013
Arthemy Kiselev (Johann Bernoulli Institute, Royal University of Groningen, the Netherlands)
Non-Abelian Lie algebroids over jet spaces
In the paper arXiv:1305.4598 [math.DG] (joint with A. Krutov) we relate Marvan's idea to a natural class of variational Lie algebroids in which his (that is, Marvan's not Lie's) operators are the anchors. Extending the fibres in vector bundles for Lie algebra-valued ZCRs by the dual of that algebra and by the opposite-parity neighbours of the two vector spaces, we enlarge the IST geometry and construct another complex. Of course, it stems from the realisation of variational Lie algebroid at hand via homological evolutionary vector field - or by the master-functional S satisfying the classical master-equation [[S,S]]=0 with respect to the variational Schouten bracket. This reveals an interesting approach to quantisation of the setup: because the cohomological realisation is well defined at any number of independent variables, we argue that its quantum extension does not reduce to the standard Batalin-Vilkovisky quantisation of Chern-Simons models over threefolds but that it should be obtained via deformation quantisation of the IST problem for the PDE under study.
This is joint work with A.O. Krutov (Department of Higher Mathematics, Ivanovo State Power University, Russia).