234 research outputs found

### 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

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