# Symplectic space

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An odd-dimensional projective space over a field endowed with an involutory relation which is a null polarity; it is denoted by .

Let . The absolute null polarity in can always be written in the form , where is a skew-symmetric matrix . In vector form, the absolute null polarity can be written in the form , where is a skew-symmetric operator whose matrix, in a suitable basis, reduces to the form In this case the absolute null polarity takes the canonical form The absolute null polarity induces a bilinear form, written in canonical form as follows: Collineations of that commute with its null polarity are called symplectic transformations; the operators defining these collineations are called symplectic. The above canonical form of defines the square matrix of order of a symplectic operator whose elements satisfy the conditions where is the Kronecker delta. Such a matrix is called symplectic; its determinant is equal to one. The symplectic transformations form a group, which is a Lie group.

Every point of the space lies in its polar hyperplane with respect to the absolute null polarity. One can also define polar subspaces in . The manifold of self-polar -spaces of is called its absolute linear complex. In this context, a symplectic group is also called a (linear) complex group.

Every pair of straight lines, and their polar -spaces in the null polarity, define a unique symplectic invariant in with respect to the group of symplectic transformations of this space. Through every point of each line there passes a transversal of this line and -spaces, which defines a projective quadruple of points. This is the geometrical interpretation of a symplectic invariant, which asserts the equality of the cross ratios of these quadruples of points.

The symplectic -dimensional space admits an interpretation in hyperbolic space and this indicates, among other things, a connection between symplectic and hyperbolic spaces. Thus, the group of symplectic transformations of is isomorphic to the group of motions of the hyperbolic space . In this interpretation, the symplectic invariant is related to the distance between points in hyperbolic space.

How to Cite This Entry:
Symplectic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symplectic_space&oldid=14721
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article