$${{\,\mathrm{{\mathfrak {L}}}\,}}$$ L -prolongations of graded Lie algebras
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L-prolongations of graded Lie algebras Stefano Marini1
· Costantino Medori1 · Mauro Nacinovich2
Received: 19 April 2019 / Accepted: 2 January 2020 © Springer Nature B.V. 2020
Abstract In this paper we translate the necessary and sufficient conditions of Tanaka’s theorem on the finiteness of effective prolongations of a fundamental graded Lie algebras into computationally effective criteria, involving the rank of some matrices that can be explicitly constructed. Our results would apply to geometries, which are defined by assigning a structure algebra on the contact distribution. Keywords G-structure · Fundamental graded Lie algebra · Tanaka’s prolongation Mathematics Subject Classification (2000) Primary 17B70 · 53C10; Secondary 53C15 · 70G65
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1 Fundamental graded Lie algebras and prolongations . . 2 Tanaka’s finiteness criterion . . . . . . . . . . . . . . . 2.1 Left and right V -modules . . . . . . . . . . . . . . 2.2 Right m-modules . . . . . . . . . . . . . . . . . . 2.3 Reduction to first kind . . . . . . . . . . . . . . . . 2.4 Comparing maximal effective prolongations . . . . 3 L-prolongations of graded Lie algebras of the first kind 3.1 Prolongation of irreducible representations . . . . . 4 glK (V )-prolongations of FGLA’s of the second kind . . 5 glK (V )-prolongations of FGLA’s of higher kind . . . . 6 L-prolongations of FGLA’s of general kind . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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Stefano Marini [email protected] Costantino Medori [email protected] Mauro Nacinovich [email protected]
1
Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Unità di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A (Campus), 43124 Parma, Italy
2
Dipartimento di Matematica, II Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy
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Geometriae Dedicata
Introduction The concept of G-structure was introduced to treat various interesting differential geometrical structures in a unified manner (see e.g. [16–22]). At a chosen point p0 of a manifold M, a G-strucure can be described by the datum of a Lie algebra L = Lie(G) of infinitesimal transformations, acting as linear maps on the tangent space V =Tp0 M. It is convenient to envision V
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