# 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 was previously been called the line complex group.

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, 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, Steen Markvorsen, Maung Min-Oo (editors), Global Riemannian Geometry: Curvature and Topology, Springer, page 72,
In symplectic geometry, there is a notion of fibrations ${\displaystyle \pi :P\rightarrow M}$  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 ${\displaystyle (M,\omega _{M})}$  is also symplectic, there is a weak coupling construction, originally due to Thurston, of defining a symplectic structure on the total space ${\displaystyle P}$ .
6. That moves in the same direction as a system of synchronized waves.

### Noun

symplectic (plural symplectics)

1. (mathematics) A symplectic bilinear form, manifold, geometry, etc.
• 1967, Journal of Mathematics and Mechanics, Volume 16, Issue 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.