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A [[Differential operator|differential operator]] that does not change its form under certain transformations of the space on which it is defined. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i0522201.png" /> is a partial differential operator written out in some coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i0522202.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i0522203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i0522204.png" />, is some transformation of coordinates inducing a corresponding mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i0522205.png" /> in the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i0522206.png" /> (each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i0522207.png" /> is associated in a natural way with the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i0522208.png" />) and if
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{{MSC|58J70|35Axx}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i0522209.png" /></td> </tr></table>
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An invariant differential operator is a [[Differential operator|differential operator]] that does not change its form under certain transformations of the space on which it is defined. For example, if $L(\partial/\partial x_k)$ is a partial differential operator written out in some coordinate system $(x_1,\ldots,x_n)$, if $x_k=\phi_k(y)$, $y=(y_1,\ldots,y_n)$, is some transformation of coordinates inducing a corresponding mapping $\phi^*$ in the set of functions $u(x)$ (each function $u(x)$ is associated in a natural way with the function $(\phi^* u)(y)$) and if
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\[
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\phi^* L\left( \frac{\partial}{\partial x} \right)u
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= L\left( \frac{\partial}{\partial y} \right) \phi^* u,
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\]
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where the operator $L$ on the right-hand side is expressed in terms of $\partial/\partial y_k$ in the same way as the operator $L$ on the left-hand side is expressed in terms of $\partial/\partial x_k$, then $L$ is said to be invariant under the transformation $\phi$ (or $L$ is said to commute with the operator transformation $\phi^*$). The most important case is when a differential operator is invariant under a family of transformations forming a group. The definition of an invariant differential operator becomes substantially more complicated if one considers a system of functions transformed by some representation of this group of transformations. The invariant differential operators related to the Lorentz group and the orthogonal group (the wave operator, the Klein–Gordon and Laplace operators, etc.) play an important role in mathematical physics. In analysis on differentiable manifolds one extensively uses the operator of exterior differentiation $d$, which is invariant under diffeomorphisms, and the operator $\partial$, metrically dual to it, which is invariant under smooth transformations that preserve the metric tensor. In the theory of Lie groups, the so-called left- and right-invariant operators under the corresponding shifts on the group are of great importance.
  
where the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i05222010.png" /> on the right-hand side is expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i05222011.png" /> in the same way as the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i05222012.png" /> on the left-hand side is expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i05222013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i05222014.png" /> is said to be invariant under the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i05222015.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i05222016.png" /> is said to commute with the operator transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i05222017.png" />). The most important case is when a differential operator is invariant under a family of transformations forming a group. The definition of an invariant differential operator becomes substantially more complicated if one considers a system of functions transformed by some representation of this group of transformations. The invariant differential operators related to the Lorentz group and the orthogonal group (the wave operator, the Klein–Gordon and Laplace operators, etc.) play an important role in mathematical physics. In analysis on differentiable manifolds one extensively uses the operator of exterior differentiation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i05222018.png" />, which is invariant under diffeomorphisms, and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052220/i05222019.png" />, metrically dual to it, which is invariant under smooth transformations that preserve the metric tensor. In the theory of Lie groups, the so-called left- and right-invariant operators under the corresponding shifts on the group are of great importance.
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====References====  
 
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{|
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.G. Petrovskii,   "Partial differential equations" , Saunders  (1967) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"M.A. Naimark,   "Les répresentations linéaires du groupe de Lorentz" , Dunod (1962) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"G. de Rham,   "Differentiable manifolds" , Springer  (1984) (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Helgason,   "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>
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|valign="top"|{{Ref|He}}||valign="top"| S. Helgason, "Differential geometry, Lie groups, and symmetric spaces", Acad. Press (1978)
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|-
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|valign="top"|{{Ref|Na}}||valign="top"| M.A. Naimark, "Les répresentations linéaires du groupe de Lorentz", Dunod (1962) (Translated from Russian)
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|-
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|valign="top"|{{Ref|Pe}}||valign="top"| I.G. Petrovskii, "Partial differential equations", Saunders (1967) (Translated from Russian)
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|-
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|valign="top"|{{Ref|Rh}}||valign="top"| G. de Rham, "Differentiable manifolds", Springer (1984) (Translated from French)
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|-
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|}

Latest revision as of 23:35, 24 July 2012

2010 Mathematics Subject Classification: Primary: 58J70 Secondary: 35Axx [MSN][ZBL]

An invariant differential operator is a differential operator that does not change its form under certain transformations of the space on which it is defined. For example, if $L(\partial/\partial x_k)$ is a partial differential operator written out in some coordinate system $(x_1,\ldots,x_n)$, if $x_k=\phi_k(y)$, $y=(y_1,\ldots,y_n)$, is some transformation of coordinates inducing a corresponding mapping $\phi^*$ in the set of functions $u(x)$ (each function $u(x)$ is associated in a natural way with the function $(\phi^* u)(y)$) and if \[ \phi^* L\left( \frac{\partial}{\partial x} \right)u = L\left( \frac{\partial}{\partial y} \right) \phi^* u, \] where the operator $L$ on the right-hand side is expressed in terms of $\partial/\partial y_k$ in the same way as the operator $L$ on the left-hand side is expressed in terms of $\partial/\partial x_k$, then $L$ is said to be invariant under the transformation $\phi$ (or $L$ is said to commute with the operator transformation $\phi^*$). The most important case is when a differential operator is invariant under a family of transformations forming a group. The definition of an invariant differential operator becomes substantially more complicated if one considers a system of functions transformed by some representation of this group of transformations. The invariant differential operators related to the Lorentz group and the orthogonal group (the wave operator, the Klein–Gordon and Laplace operators, etc.) play an important role in mathematical physics. In analysis on differentiable manifolds one extensively uses the operator of exterior differentiation $d$, which is invariant under diffeomorphisms, and the operator $\partial$, metrically dual to it, which is invariant under smooth transformations that preserve the metric tensor. In the theory of Lie groups, the so-called left- and right-invariant operators under the corresponding shifts on the group are of great importance.

References

[He] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces", Acad. Press (1978)
[Na] M.A. Naimark, "Les répresentations linéaires du groupe de Lorentz", Dunod (1962) (Translated from Russian)
[Pe] I.G. Petrovskii, "Partial differential equations", Saunders (1967) (Translated from Russian)
[Rh] G. de Rham, "Differentiable manifolds", Springer (1984) (Translated from French)
How to Cite This Entry:
Invariant differential operator. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Invariant_differential_operator&oldid=27204
This article was adapted from an original article by A.A. Dezin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article