On Poisson Brackets of Semi-Invariants

Abstract

In the present article we shall treat Gelfand-Dikii’s theory of formal calculus of variations [1] from a new point of view, invariant theory of formal power series

$${f_\lambda }(\xi x) = \sum\limits_\ell {\frac{{{{(\lambda )}_\ell }}}{{\ell !}}} {\xi ^{(\ell )}}{x^\ell }$$

, and we shall give natural expllicit expressions of Poisson brackets. In formal calculus of variations Poisson brackets are defined on the quotient module

$${\raise0.7ex\hbox{${K\left[ {...,{{\left( {\frac{\partial }{{\partial x}}} \right)}^\ell }y,...} \right]}$} \!\mathord{\left/{\vphantom {{K\left[ {...,{{\left( {\frac{\partial }{{\partial x}}} \right)}^\ell }y,...} \right]} {K + \frac{\partial }{{\partial x}}K\left[ {...,{{\left( {\frac{\partial }{{\partial x}}} \right)}^\ell }y,...} \right]}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${K + \frac{\partial }{{\partial x}}K\left[ {...,{{\left( {\frac{\partial }{{\partial x}}} \right)}^\ell }y,...} \right]}$}}$$

, in our case, however, they are defined on ring of semi-invaiants

$$G = \left\{ {\varphi \in K\left[ \xi \right]D\varphi = \sum\limits_\ell {\ell {\xi ^{(\ell - 1)}}} \frac{{\partial \varphi }}{{\partial {\xi ^{(\ell )}}}} = 0} \right\}$$

.