John Stillwell - Naive Lie Theory.pdf - Index of

54 3 Generalized rotation groups
the exercises below to show that path-connectedness has interesting algebraic
implications.
Exercises
The following exercises study the identity component in a matrix group G,thatis,
the set of matrices A ∈ G for which there is a continuous path from 1 to A that lies
inside G.
3.2.1 Bearing in mind that matrix multiplication is a continuous operation, show
that if there are continuous paths in G from 1 to A ∈ G and to B ∈ G then
there is a continuous path in G from A to AB.
3.2.2 Similarly, show that if there is a continuous path in G from 1 to A, then
there is also a continuous path from A −1 to 1.
3.2.3 Deduce from Exercises 3.2.1 and 3.2.2 that the identity component of G is
a subgroup of G.
3.3 The unitary groups
The unitary groups U(n) and SU(n) are the analogues of the orthogonal
groups O(n) and SO(n) for the complex vector space C n , which consists
of the ordered n-tuples (z 1 ,z 2 ,...,z n ) of complex numbers. The sum operation
on C n is the usual vector addition:
(u 1 ,u 2 ,...,u n )+(v 1 ,v 2 ,...,v n )=(u 1 + v 1 ,u 2 + v 2 ,...,u n + v n ).
And the multiple of (z 1 ,z 2 ,...,z n ) ∈ C n by a scalar c ∈ C is naturally
(cz 1 ,cz 2 ,...,cz n ). The twist comes with the inner product, because we
would like the inner product of a vector v with itself to be a real number—
the squared distance |v| 2 from the zero matrix 0 to v. We ensure this by
the definition
(u 1 ,u 2 ,...,u n ) · (v 1 ,v 2 ,...,v n )=u 1 v 1 + u 2 v 2 + ···+ u n v n . (*)
With this definition of u · v we have
v · v = v 1 v 1 + v 2 v 2 + ···+ v n v n = |v 1 | 2 + |v 2 | 2 + ···+ |v n | 2 = |v| 2 ,
and |v| 2 is indeed the squared distance of v =(v 1 ,v 2 ,...,v n ) from 0 in the
space R 2n that equals C n when we interpret each copy of C as R 2 .