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arXiv · math/9907211

Versal deformations of a Dirac type differential operator

Anatoliy K. Prykarpatsky·Denis Blackmore
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Abstract

If we are given a smooth differential operator in the variable $x\in {\mathbb R}/2π{\mathbb Z},$ its normal form, as is well known, is the simplest form obtainable by means of the $\mbox{Diff}(S^1)$-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 $\mbox{Diff}(S^1)$-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 $\mbox{Diff}(S^1)$-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.

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Anatoliy K. Prykarpatsky, Denis Blackmore. 1999-07-01. Versal deformations of a Dirac type differential operator. https://doi.org/10.2991/jnmp.1999.6.3.1

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