# Symplectic space

An odd-dimensional projective space $P _ {2n + 1 }$ over a field $k$ endowed with an involutory relation which is a null polarity; it is denoted by $\mathop{\rm Sp} _ {2n + 1 }$.

Let $\mathop{\rm char} k \neq 2$. The absolute null polarity in $\mathop{\rm Sp} _ {2n + 1 }$ can always be written in the form $u _ {i} = a _ {ij} x ^ {j}$, where $\| a _ {ij} \|$ is a skew-symmetric matrix $( a _ {ij} = - a _ {ji} )$. In vector form, the absolute null polarity can be written in the form $\mathbf u = A \mathbf x$, where $A$ is a skew-symmetric operator whose matrix, in a suitable basis, reduces to the form

$$\| A \| = \ \left \| \begin{array}{rrrrrr} 0 &+ 1 &{} &{} &{} &{} \\ - 1 & 0 &{} &{} &{} &{} \\ {} &{} &\cdot &{} &{} &{} \\ {} &{} &{} & 0 &+ 1 &{} \\ {} &{} &{} &- 1 & 0 &+ 1 \\ {} &{} &{} &{} &- 1 & 0 \\ \end{array} \ \right \| .$$

In this case the absolute null polarity takes the canonical form

$$u _ {2i} = x ^ {2i + 1 } ,\ \ u _ {2i + 1 } = - x ^ {2i} .$$

The absolute null polarity induces a bilinear form, written in canonical form as follows:

$$\mathbf x A \mathbf y = \ \sum _ { i } ( x ^ {2i} y ^ {2i + 1 } - x ^ {2i + 1 } y ^ {2i} ).$$

Collineations of $\mathop{\rm Sp} _ {2n + 1 }$ that commute with its null polarity are called symplectic transformations; the operators defining these collineations are called symplectic. The above canonical form of $\| A \|$ defines the square matrix of order $2n + 2$ of a symplectic operator $U$ whose elements satisfy the conditions

$$\sum _ { i } ( U _ {j} ^ {2i} U _ {k} ^ {2i + 1 } - U _ {j} ^ {2i + 1 } U _ {k} ^ {2i} ) = \ \delta _ {j, k - 1 } - \delta _ {j, k + 1 } ,$$

where $\delta _ {a,b}$ 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 $\mathop{\rm Sp} _ {2n + 1 }$ lies in its polar hyperplane with respect to the absolute null polarity. One can also define polar subspaces in $\mathop{\rm Sp} _ {2n + 1 }$. The manifold of self-polar $n$- spaces of $\mathop{\rm Sp} _ {2n + 1 }$ 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 $( 2n - 1)$- spaces in the null polarity, define a unique symplectic invariant in $\mathop{\rm Sp} _ {2n + 1 }$ 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 $( 2n - 1)$- 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 $3$- 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 $\mathop{\rm Sp} _ {3}$ is isomorphic to the group of motions of the hyperbolic space ${} ^ {2} \textrm{ S } _ {4}$. 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=49622
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article