# Symplectic group

## Contents

## Definition

### In terms of a specific matrix description

For a positive integer and a field , the **symplectic group** is defined as the group, under matrix multiplication, with the following underlying set:

.

where and are respectively the identity and negative identity matrices.

The symplectic group is sometimes termed the symplectic group of degree and sometimes the symplectic group of degree .

### A more general description boiling down to the same thing

Suppose is a field and is a nondegenerate alternating bilinear form on a finite-dimensional vector space over . The **symplectic group** for is defined as the group of all the linear transformations of that preserve , i.e., maps satisfying:

.

Although this definition makes sense for any alternating bilinear form, we typically assume that the alternating bilinear form is *nondegenerate*.

It turns out that, up to change of basis, all nondegenerate alternating bilinear forms are equivalent, and the existence of such forms forces the space to be even-dimensional. Thus, the symplectic groups for any two nondegenerate alternating bilinear forms on vector spaces of the same dimension are isomorphic, and we can use the concrete definition provided above.

## Facts

is a subgroup of the special linear group , and for , it is precisely equal to the special linear group of degree two, i.e., .