The groups one studies in geometry are usually groups of matrices. The simplest example is the group of inversible matrices. This is just the set of nonzero real numbers with their standard multiplication. The higher dimensional analogue is the group of inversible matrices provided with their usual multiplication.
Before you go on, make sure you can effortlessly solve the following exercises:
If you cannot solve these problems in a very short period of time, then you are not ready for this course. Contact your instructor.
Besides the group of invertible matrices of real numbers, we also have the group of invertible matrices of complex numbers. Notice that is a subgroup of . In fact, almost every interesting group in geometry is a subgroup of . Here are some examples:
The special linear group. The set of real (resp. complex) matrices having determinant one is denoted by (resp. ) and is called the special linear group. If , then the transformation from to itself defined by preserves volume and orientation.
The orthogonal and the special orthogonal group. A square matrix is said to be orthogonal if its transpose is equal to its inverse. The set of orthogonal matrices is denoted by . The subset of consisting of matrices of determinant one is denoted by . They are respectively the orthogonal and the special orthogonal group.
The unitary and the special unitary group. If is a square matrices of complex numbers, the adjoint of , denoted by , is the result of transposing and then conjugating all of its entries. For example
The following exercise was taken from the book of Abraham and Marsden on the foundations of mechanics (see ).
From this exercise we conclude that the group , which we saw earlier identified with the -dimensional sphere, has a natural -to- correspondence with the group of rotations . To this quirk of nature we owe the spin of electrons and a pretty little trick due to Dirac and which we shall describe in chapter 5.