# On Poisson Brackets of Semi-Invariants

• Hisasi Morikawa
Part of the Progress in Mathematics book series (PM, volume 14)

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

## Keywords

Evolution Equation Poisson Bracket Invariant Theory Formal Power Series Polynomial Algebra
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
I.M. Gelfand, and L.A. Dikii, “The structure of Lie algebras in formal calculus of variations,” Funkts, Analiz Prilozhen., 10, No. 1. 2836 (1976).
2. [2]
H. Morikawa, “Some analytic and geometric application of the invariant theoretic method,” Nagoya Math. J. 80, 1–47 (1980).