## Abstract

If we are given a smooth differential operator in the variable x ∈ R/2πZ, its normal form, as is well known, is the simplest form obtainable by means of the Diff(S ^{1})-group action on the space of all such operators. A versal deformation of this operator is a normal form for some parametric infinitesimal family including the operator. Our study is devoted to analysis of versal deformations of a Dirac type differential operator using the theory of induced Diff(S ^{1})-actions endowed with centrally extended Lie-Poisson brackets. After constructing a general expression for tranversal deformations of a Dirac type differential operator, we interpret it via the Lie-algebraic theory of induced Diff(S ^{1})-actions on a special Poisson manifold and determine its generic moment mapping. Using a Marsden-Weinstein reduction with respect to certain Casimir generated distributions, we describe a wide class of versally deformed Dirac type differential operators depending on complex parameters.

Original language | English (US) |
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Pages (from-to) | 246-254 |

Number of pages | 9 |

Journal | Journal of Nonlinear Mathematical Physics |

Volume | 6 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1999 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics