TY - JOUR

T1 - Versal deformations of a dirac type differential operator

AU - Prykarpatsky, Anatoliy K.

AU - Blackmore, Denis

N1 - Funding Information:
A. Prykarpatsky is greatful to the Dept. of Applied Mathematics at AGH for its support of this work through an AGH research grant. D. Blackmore would like to express his gratitude to the Courant Institute of Mathematical Sciences for the hospitality extended to him as a visiting member during the time when this research was conducted.

PY - 1999

Y1 - 1999

N2 - 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.

AB - 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.

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U2 - 10.2991/jnmp.1999.6.3.1

DO - 10.2991/jnmp.1999.6.3.1

M3 - Article

AN - SCOPUS:33846089934

SN - 1402-9251

VL - 6

SP - 246

EP - 254

JO - Journal of Nonlinear Mathematical Physics

JF - Journal of Nonlinear Mathematical Physics

IS - 3

ER -