On the deformation theory of structure constants for associative algebras

Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a deformation driving algebras (DDAs) is discussed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative three-dimensional algebras with the DDA being a three-dimensional Lie algebra and their connection with integrable systems are studied.Comment: minor corrections and references adde

Cohomological, Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of Sato Grassmannian

Cohomological and Poisson structures associated with the special tautological subbundles $TB_{W_{1,2,\dots,n}}$ for the Birkhoff strata of Sato Grassmannian are considered. It is shown that the tangent bundles of $TB_{W_{1,2,\dots,n}}$ are isomorphic to the linear spaces of $2-$coboundaries with vanishing Harrison's cohomology modules. Special class of 2-coboundaries is provided by the systems of integrable quasilinear PDEs. For the big cell it is the dKP hierarchy. It is demonstrated also that the families of ideals for algebraic varieties in $TB_{W_{1,2,\dots,n}}$ can be viewed as the Poisson ideals. This observation establishes a connection between families of algebraic curves in $TB_{W_{\hat{S}}}$ and coisotropic deformations of such curves of zero and nonzero genus described by hierarchies of hydrodynamical type systems like dKP hierarchy. Interrelation between cohomological and Poisson structures is noted.Comment: 15 pages, no figures, accepted in Theoretical and Mathematical Physics. arXiv admin note: text overlap with arXiv:1005.205

Hamiltonian motions of plane curves and formation of singularities and bubbles

A class of Hamiltonian deformations of plane curves is defined and studied. Hamiltonian deformations of conics and cubics are considered as illustrative examples. These deformations are described by systems of hydrodynamical type equations. It is shown that solutions of these systems describe processes of formation of singularities (cusps, nodes), bubbles, and change of genus of a curve.Comment: 15 pages, 12 figure

Birkhoff strata of Sato Grassmannian and algebraic curves

Algebraic and geometric structures associated with Birkhoff strata of Sato Grassmannian are analyzed. It is shown that each Birkhoff stratum $\Sigma_S$ contains a subset $W_{\hat{S}}$ of points for which each fiber of the corresponding tautological subbundle $TB_{W_S}$ is closed with respect to multiplication. Algebraically $TB_{W_S}$ is an infinite family of infinite-dimensional commutative associative algebras and geometrically it is an infinite tower of families of algebraic curves. For the big cell the subbundle $TB_{W_\varnothing}$ represents the tower of families of normal rational (Veronese) curves of all degrees. For $W_1$ such tautological subbundle is the family of coordinate rings for elliptic curves. For higher strata, the subbundles $TB_{W_{1,2,\dots,n}}$ represent families of plane $(n+1,n+2)$ curves (trigonal curves at $n=2$) and space curves of genus $n$. Two methods of regularization of singular curves contained in $TB_{W_{\hat{S}}}$, namely, the standard blowing-up and transition to higher strata with the change of genus are discussed.Comment: 31 pages, no figures, version accepted in Journal of Nonlinear Mathematical Physics. The sections on the integrable systems present in previous versions has been published separatel

Elliptic Euler-Poisson-Darboux equation, critical points and integrable systems

Structure and properties of families of critical points for classes of functions $W(z,\bar{z})$ obeying the elliptic Euler-Poisson-Darboux equation $E(1/2,1/2)$ are studied. General variational and differential equations governing the dependence of critical points in variational (deformation) parameters are found. Explicit examples of the corresponding integrable quasi-linear differential systems and hierarchies are presented There are the extended dispersionless Toda/nonlinear Schr\"{o}dinger hierarchies, the "inverse" hierarchy and equations associated with the real-analytic Eisenstein series $E(\beta,\bar{{\beta}};1/2)$among them. Specific bi-Hamiltonian structure of these equations is also discussed.Comment: 18 pages, no figure