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O que (quem) é symplectic module - definição

Symplectic transformation; Symplectic operator

Symplectic geometry

BRANCH OF DIFFERENTIAL GEOMETRY AND DIFFERENTIAL TOPOLOGY

Symplectic Geometry; Symplectic structure; Symplectic topology

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry was founded by the Russian mathematician Vladimir Arnold and has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.

https://en.wikipedia.org/wiki/Symplectic_geometry

Module (mathematics)

GENERALIZATION OF VECTOR SPACE, WITH SCALARS IN A RING INSTEAD OF A FIELD

Module (algebra); Submodule; Module theory; Submodules; R-module; Module over a ring; Left module; Module Theory; Unital module; Module (ring theory); Right module; Left-module; Module mathematics; Ring action; Z-module; ℤ-module

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers.

https://en.wikipedia.org/wiki/Module_(mathematics)

Dualizing module

In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety. It is used in Grothendieck local duality.

https://en.wikipedia.org/wiki/Dualizing_module

Wikipédia

Symplectic matrix

In mathematics, a symplectic matrix is a $2n\times 2n$ matrix $M$ with real entries that satisfies the condition

where $M^{\text{T}}$ denotes the transpose of $M$ and $\Omega$ is a fixed $2n\times 2n$ nonsingular, skew-symmetric matrix. This definition can be extended to $2n\times 2n$ matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.

Typically $\Omega$ is chosen to be the block matrix

where

I_{n}

is the

n\times n

identity matrix. The matrix

\Omega

has determinant

+1

and its inverse is

\Omega ^{-1}=\Omega ^{\text{T}}=-\Omega

.

https://en.wikipedia.org/wiki/Symplectic matrix