Higher symmetries of symplectic Dirac operator
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Higher symmetries of symplectic Dirac operator Petr Somberg1 · Josef Šilhan2 Received: 15 February 2019 / Accepted: 31 March 2020 © Springer Nature B.V. 2020
Abstract We construct in projective differential geometry of the real dimension 2 higher symmetry / s acting on symplectic spinors. The higher symmealgebra of the symplectic Dirac operator D try differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of sl(3, R). Keywords Symplectic Dirac operator · Higher symmetry algebra · Projective differential geometry · Minimal nilpotent orbit · sl(3, R) Mathematics Subject Classification 53D05 · 35Q41 · 58D19 · 17B08 · 53A20
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Symplectic spinors and projective geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Symplectic spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Projective geometry and tractor calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Projective connection on symplectic spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /s . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Symbols and construction of higher symmetries of D /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Algebra of higher symmetries of D 6 Comments and open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Josef Šilhan [email protected] Petr Somberg [email protected]
1
Mathematical Institute, Charles University, Sokolovská 83, Prague, Czech Republic
2
Institute of Mathematics and Statistics, Masaryk University, Building 08, Kotláˇrská 2, 611 37 Brno, Czech Republic
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Geometriae Dedicata
1 Introduction It is always desirable to convert a purely algebraic construct into its geometrical realization with the hope to gain its better understanding, as well as a potential generalization of the former algebraic structure. The present article is an example of this phenomenon: the algebraic / s (cf. the structure is the algebra of higher symmetries of the symplectic Dirac operator D seminal work [13]) realized in projective differential geometry of the real dimension two. This algebra corresponds to a completely prime primitive ideal which has as its associated variety the minimal nilpotent orbit of the complexification of sl(3, R), while the geometric realization pursued in our article relies on the use of certain class of projectively invariant
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