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Keywords:
symplectic $\mathcal A$-modules; symplectic Gram-Schmidt theorem; symplectic basis; orthosymmetric $\mathcal {A}$-bilinear forms; orthogonal/symplectic geometry; strict integral domain algebra sheaf
symplectic $\mathcal A$-modules; symplectic Gram-Schmidt theorem; symplectic basis; orthosymmetric $\mathcal {A}$-bilinear forms; orthogonal/symplectic geometry; strict integral domain algebra sheaf
Summary:
Like the classical Gram-Schmidt theorem for symplectic vector spaces, the sheaf-theoretic version (in which the coefficient algebra sheaf $\mathcal A$ is appropriately chosen) shows that symplectic $\mathcal A$-morphisms on free $\mathcal A$-modules of finite rank, defined on a topological space $X$, induce canonical bases (Theorem 1.1), called symplectic bases. Moreover (Theorem 2.1), if $(\mathcal {E}, \phi )$ is an $\mathcal A$-module (with respect to a $\mathbb C$-algebra sheaf $\mathcal A$ without zero divisors) equipped with an orthosymmetric $\mathcal A$-morphism, we show, like in the classical situation, that “componentwise” $\phi $ is either symmetric (the (local) geometry is orthogonal) or skew-symmetric (the (local) geometry is symplectic). Theorem 2.1 reduces to the classical case for any free $\mathcal A$-module of finite rank.
References:
[1] Adkins, W. A., Weintraub, S. H.: Algebra. An Approach via Module Theory. Springer New York (1992). MR 1181420 | Zbl 0768.00003
[2] Artin, E.: Geometric Algebra. John Wiley & Sons/Interscience Publishers New York (1988). MR 1009557 | Zbl 0642.51001
[3] Berndt, R.: An Introduction to Symplectic Geometry. American Mathematical Society Providence (2001). MR 1793955 | Zbl 0986.53028
[4] Silva, A. Cannas da: Lectures on Symplectic Geometry. Springer Berlin (2001). MR 1853077
[5] Chambadal, L., Ovaert, J. L.: Algèbre linéaire et algèbre tensorielle. Dunod Paris (1968). MR 0240108 | Zbl 0186.33402
[6] Crumeyrolle, A.: Orthogonal and Symplectic Clifford Algebras. Spinor Structures. Kluwer Dordrecht (1990). MR 1044769 | Zbl 0701.53003
[7] Gosson, M. de: Symplectic Geometry and Quantum Mechanics. Birkhäuser Basel (2006). MR 2241188 | Zbl 1098.81004
[8] Gruenberg, K. W., Weir, A. J.: Linear Geometry, 2nd edition. Springer New York, Heidelberg, Berlin (1977).
[9] Mallios, A.: Geometry of Vector Sheaves. An Axiomatic Approach to Differential Geometry. Volume I: Vector Sheaves. General Theory. Kluwer Dordrecht (1998). Zbl 0904.18001
[10] Mallios, A.: Geometry of Vector Sheaves. An Axiomatic Approach to Differential Geometry. Volume II: Geometry. Examples and Applications. Kluwer Dordrecht (1998). Zbl 0904.18002
[11] Mallios, A.: $\mathcal{A}$-invariance: An axiomatic approach to quantum relativity. Int. J. Theor. Phys. 47 (2008), 1929-1948. DOI 10.1007/s10773-007-9637-2 | MR 2442881
[12] Mallios, A.: Modern Differential Geometry in Gauge Theories. Vol. I: Maxwell Fields. Birkhäuser Boston (2006). MR 2189128
[13] Mallios, A., Ntumba, P. P.: On a sheaf-theoretic version of the Witt's decomposition theorem. A Lagrangian perspective. Rend. Circ. Mat. Palermo 58 (2009), 155-168. DOI 10.1007/s12215-009-0014-2 | MR 2504993 | Zbl 1184.18010
[14] Mallios, A., Ntumba, P. P.: Fundamentals for symplectic $\mathcal{A}$-modules. Affine Darboux theorem. Rend. Circ. Mat. Palermo 58 (2009), 169-198. MR 2533910
[15] Mallios, A., Ntumba, P. P.: Symplectic reduction of sheaves of $\mathcal{A}$-modules. \\arXiv:0802.4224.
[16] Mallios, A., Ntumba, P. P.: Pairings of sheaves of $\mathcal A$-modules. Quaest. Math. 31 (2008), 397-414. DOI 10.2989/QM.2008.31.4.8.612 | MR 2527450
[17] Mostow, M. A.: The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations. J. Diff. Geom. 14 (1979), 255-293. DOI 10.4310/jdg/1214434974 | MR 0587553 | Zbl 0427.58005
[18] Ntumba, P. P.: Cartan-Dieudonné theorem for $\mathcal A$-modules. Mediterr. J. Math. 7 (2010), 445-454. DOI 10.1007/s00009-010-0042-3 | MR 2738570
[19] Sikorski, R.: Differential modules. Colloq. Math. 24 (1971), 45-79. DOI 10.4064/cm-24-1-45-79 | MR 0482794 | Zbl 0226.53004
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