symplectic

English

Etymology

A calque of complex, coined by Hermann Weyl in his 1939 book The Classical Groups: Their Invariants and Representations. From Ancient Greek συμπλεκτικός (sumplektikós), from συμ (sum) (variant of σύν (sún)), + πλεκτικός (plektikós) (from πλέκω (plékō)); modelled on complex (from Latin complexus (“braided together”), from com- (“together”) + plectere (“to weave, braid”)).

The symplectic group has previously been called the line complex group.

Pronunciation

• IPA^(key): /sɪmˈplɛktɪk/
• Rhymes: -ɛktɪk

Adjective

symplectic (not comparable)

1. Placed in or among, as if woven together.
2. (group theory, of a group) Whose characteristic abelian subgroups are cyclic.
3. (mathematics, multilinear algebra, of a bilinear form) That is alternating and nondegenerate.
4. (mathematics, multilinear algebra, of a vector space) That is equipped with an alternating nondegenerate bilinear form.
5. (mathematics) Of or pertaining to (the geometry of) a differentiable manifold equipped with a closed nondegenerate bilinear form.
   • 1995, V. I. Arnold, “Some remarks on symplectic monodromy of Milnor fibrations”, in Helmut Hofer, Clifford H. Taubes, Alan Weinstein, Eduard Zehnder, editors, The Floer Memorial Volume, Birkhäuser Verlag, page 99:

     There exist interesting and unexplored relations between symplectic geometry and the theory of critical points of holomorphic functions.

   • 1997, C. H. Cushman-de Vries (translator), Richard H. Cushman, Gijs M. Tuynman (translation editors), Jean-Marie Souriau, Structure of Dynamical Systems: A Symplectic View of Physics, Springer Science & Business Media (Birkhäuser).
   • 2003, Fabrizio Catanese, Gang Tian (editors), Symplectic 4-Manifolds and Algebraic Surfaces: Lectures given at the C.I.M.E Summer School, Springer, Lecture Notes in Mathematics No. 1938.
   • 2003, Yakov Eliashberg, Boris A. Khesin, François Lalonde, editors, Symplectic and Contact Topology: Interactions and Perspectives, American Mathematical Society:
   • 2003, Maung Min-Oo, “The Dirac Operator in Geometry and Physics”, in Steen Markvorsen, Maung Min-Oo, editors, Global Riemannian Geometry: Curvature and Topology, Springer, page 72:

     In symplectic geometry, there is a notion of fibrations $\pi :P\rightarrow M$ with a symplectic manifold F as fiber, where the structure group is the group of (exact) Hamiltonian symplectomorphisms of the fiber. These are called symplectic fibrations. If the base manifold $(M,\omega _{M})$ is also symplectic, there is a weak coupling construction, originally due to Thurston, of defining a symplectic structure on the total space $P$.

6. That moves in the same direction as a system of synchronized waves.
7. (petrology, mineralogy) Of or pertaining to a symplectite; symplectitic.

Antonyms

• antiplectic

Derived terms

• microsymplectic
• symplectic cut
• symplectic form (symplectic bilinear form)
• symplectic group
• symplectic invariant (structure that is invariant under a symplectic form (regarded as a mapping))
• symplectic matrix
• symplectic Clifford algebra

Related terms

• symplectomorphism

Noun

symplectic (plural symplectics)

1. (mathematics) A symplectic bilinear form, manifold, geometry, etc.
   • 1967, Journal of Mathematics and Mechanics, volume 16, number 1, Indiana University, page 339:

     The structure of stable symplectics on finite dimensional spaces has been studied by Krein [8], Gelfand & Lidskii [9], and Moser [10] in work of considerable practical importance.

2. (ichthyology) A bone in the teleostean fishes that forms the lower ossification of the suspensorium, and which articulates below with the quadrate bone by which it is firmly held.
   • 1914, The Philippine Journal of Science, Volume 9, page 27:

     The symplectics (9) consist of a somewhat curved central triangular portion with the base upward, and anteriorly and posteriorly from this extends a wing-like process.

   • 1965, Agra University Journal of Research: Science, Volume 14, page 71:

     The symplectics (Fig. 8, sym) are thin slender bones placed vertically in between the quadrates and the hyomandibulars.

   • 1967, Tyson R. Roberts, Studies on the Osteology and Phylogeny of Characoid Fishes, page 59:

     In many teleosts, on the other hand, including the catfishes, the symplectics have been entirely lost.

References

• The Classical Groups. Their Invariants and Representations, Hermann Weyl; Princeton University Press, 1939 →ISBN, footnote, p. 165
• The Symplectization of Science, Mark J. Gotay and James A. Isenberg, p. 13

Further reading

• symplectic on Wikipedia.Wikipedia
• symplectic bone on Wikipedia.Wikipedia
• Symplectic cut on Wikipedia.Wikipedia
• Symplectic geometry on Wikipedia.Wikipedia
• Symplectic group on Wikipedia.Wikipedia
• Symplectic manifold on Wikipedia.Wikipedia
• Symplectic matrix on Wikipedia.Wikipedia
• Group of symplectic type on Wikipedia.Wikipedia