When <em>D</em> is a linear partial differential operator of any order, a <em>direct problem</em> is to look for an operator <em>D</em><sub>1</sub> generating the <em...When <em>D</em> is a linear partial differential operator of any order, a <em>direct problem</em> is to look for an operator <em>D</em><sub>1</sub> generating the <em>compatibility conditions </em>(CC) <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub><em>1</em></sub><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span> =</span><sub></sub> 0 of <em>D</em><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">ξ </span></em></span></span>= <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span>. Conversely, when <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is given, an <em>inverse problem</em> is to look for an operator <span style="white-space:normal;"><em>D</em></span> such that its CC are generated by <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> and we shall say that <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is <em>parametrized</em> by <em>D</em> = <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>0</sub></span>. We may thus construct a differential sequence with successive operators <em>D</em>, <em>D</em><sub>1</sub>, <em>D</em><sub>2</sub>, ..., each operator parametrizing the next one. Introducing the<em> formal adjoint ad</em>() of an operator, we have <img src="Edit_ecbb631c-2896-4dad-8234-cacd5504f138.png" alt="" />but <span style="white-space:nowrap;"><em>ad</em> (<em>D</em><sub><em>i</em>-1</sub>)</span> may not generate <em>all</em> the CC of <em>ad </em>(<em>D</em><sub>i</sub>). When <em>D </em>= <em>K</em> [d<sub>1</sub>, ..., d<sub>n</sub>] = <em>K </em>[<em>d</em>] is the (non-commutative) ring of differential operators with coefficients in a differential field <em>K</em>, then <em>D</em> gives rise by residue to a <em>differential module M</em> over<em> D</em> while <em>a</em><em style="white-space:normal;">d </em><span style="white-space:normal;">(</span><em style="white-space:normal;">D</em><span style="white-space:normal;">)</span> gives rise to a differential module <em>N =ad (M)</em> over <em>D</em>. The <em>differential extension modules</em> <img src="Edit_55629608-629e-4b52-ac8f-52470473af77.png" alt="" /> with <span style="white-space:nowrap;"><em>ext<span style="font-size:10px;"><sup>0</sup></span></em><em>(M) = hom</em><sub><em>D</em></sub><em> (M, D)</em></span> only depend on <em>M</em> and are measuring the above gaps, <em>independently of the previous differential sequence</em>, in such a way that <span style="white-space:nowrap;"><em>ext</em><sup><em>1</em></sup><em> (N) = t (M)</em> </span> is the torsion submodule of <em>M</em>. The purpose of this paper is to compute them for certain Lie operators involved in the theory of Lie pseudogroups in arbitrary dimension <em>n</em> and to prove for the first time that the extension modules highly depend on the Vessiot <em>structure constants c</em>. Comparing the last invited lecture published in 1962 by Lanczos with a commutative diagram that we provided in a recent paper on gravitational waves, we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality. We shall prove that Lanczos was not trying to parametrize the Riemann operator but its formal adjoint <span style="white-space:nowrap;"><em>Beltrami = ad (Riemann)</em></span> which can indeed be parametrized by the operator <span style="white-space:nowrap;"><em>Lanczos = ad (Bianchi) </em></span>in arbitrary dimension, “<em>one step further on to the right</em>” in the Killing sequence. Our purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment, getting closer to the theory of Lie pseudogroups through double differential duality and the construction of finite length differential sequences for Lie operators. In particular, when one is dealing with a Lie group of transformations or, equivalently, when <em>D</em> is a Lie operator of finite type, we shall prove that <img src="Edit_3a20593a-fffe-4a20-a041-2c6bb9738d5d.png" alt="" />. It will follow that the <em>Riemann-Lanczos </em>and <em>Weyl-Lanczos</em> problems just amount to prove such a result for <em>i </em>= 1,2 and arbitrary <em>n</em> when <em>D</em> is the <em>classical or conformal Killing</em> operator. We provide a description of the potentials allowing to parametrize the Riemann and the Weyl operators in arbitrary dimension, both with their adjoint operators. Most of these results are new and have been checked by means of computer algebra.展开更多
文摘When <em>D</em> is a linear partial differential operator of any order, a <em>direct problem</em> is to look for an operator <em>D</em><sub>1</sub> generating the <em>compatibility conditions </em>(CC) <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub><em>1</em></sub><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span> =</span><sub></sub> 0 of <em>D</em><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">ξ </span></em></span></span>= <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span>. Conversely, when <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is given, an <em>inverse problem</em> is to look for an operator <span style="white-space:normal;"><em>D</em></span> such that its CC are generated by <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> and we shall say that <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is <em>parametrized</em> by <em>D</em> = <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>0</sub></span>. We may thus construct a differential sequence with successive operators <em>D</em>, <em>D</em><sub>1</sub>, <em>D</em><sub>2</sub>, ..., each operator parametrizing the next one. Introducing the<em> formal adjoint ad</em>() of an operator, we have <img src="Edit_ecbb631c-2896-4dad-8234-cacd5504f138.png" alt="" />but <span style="white-space:nowrap;"><em>ad</em> (<em>D</em><sub><em>i</em>-1</sub>)</span> may not generate <em>all</em> the CC of <em>ad </em>(<em>D</em><sub>i</sub>). When <em>D </em>= <em>K</em> [d<sub>1</sub>, ..., d<sub>n</sub>] = <em>K </em>[<em>d</em>] is the (non-commutative) ring of differential operators with coefficients in a differential field <em>K</em>, then <em>D</em> gives rise by residue to a <em>differential module M</em> over<em> D</em> while <em>a</em><em style="white-space:normal;">d </em><span style="white-space:normal;">(</span><em style="white-space:normal;">D</em><span style="white-space:normal;">)</span> gives rise to a differential module <em>N =ad (M)</em> over <em>D</em>. The <em>differential extension modules</em> <img src="Edit_55629608-629e-4b52-ac8f-52470473af77.png" alt="" /> with <span style="white-space:nowrap;"><em>ext<span style="font-size:10px;"><sup>0</sup></span></em><em>(M) = hom</em><sub><em>D</em></sub><em> (M, D)</em></span> only depend on <em>M</em> and are measuring the above gaps, <em>independently of the previous differential sequence</em>, in such a way that <span style="white-space:nowrap;"><em>ext</em><sup><em>1</em></sup><em> (N) = t (M)</em> </span> is the torsion submodule of <em>M</em>. The purpose of this paper is to compute them for certain Lie operators involved in the theory of Lie pseudogroups in arbitrary dimension <em>n</em> and to prove for the first time that the extension modules highly depend on the Vessiot <em>structure constants c</em>. Comparing the last invited lecture published in 1962 by Lanczos with a commutative diagram that we provided in a recent paper on gravitational waves, we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality. We shall prove that Lanczos was not trying to parametrize the Riemann operator but its formal adjoint <span style="white-space:nowrap;"><em>Beltrami = ad (Riemann)</em></span> which can indeed be parametrized by the operator <span style="white-space:nowrap;"><em>Lanczos = ad (Bianchi) </em></span>in arbitrary dimension, “<em>one step further on to the right</em>” in the Killing sequence. Our purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment, getting closer to the theory of Lie pseudogroups through double differential duality and the construction of finite length differential sequences for Lie operators. In particular, when one is dealing with a Lie group of transformations or, equivalently, when <em>D</em> is a Lie operator of finite type, we shall prove that <img src="Edit_3a20593a-fffe-4a20-a041-2c6bb9738d5d.png" alt="" />. It will follow that the <em>Riemann-Lanczos </em>and <em>Weyl-Lanczos</em> problems just amount to prove such a result for <em>i </em>= 1,2 and arbitrary <em>n</em> when <em>D</em> is the <em>classical or conformal Killing</em> operator. We provide a description of the potentials allowing to parametrize the Riemann and the Weyl operators in arbitrary dimension, both with their adjoint operators. Most of these results are new and have been checked by means of computer algebra.