Re: the concept of a representation of a group

m1ngleg02@xxxxxxxxxxxxxx wrote:

> Thank you for your detailed answer Igor. About the dimension of the
> group, I restrict to the case of Lie groups which can be represented by
> matrix groups. With the dimension of the group I mean the number of
> parameters of the matrix group.

More precisely, it is the dimension of its Lie algebra.
There are many books on "linear groups" which go into this in detail.
My favourite is Adams, "Lectures on Lie Groups",
which takes a very concrete approach.

> What I do not understand is exactly the
> relation between the dimension of the group and the dimension of their
> representations.

There is none.

> After all my question remains unanswered. Is the dimension of a
> representation the dimension of H? Can the quotient group G/ker(f) be
> used for something? Maybe you could explain it with the help of a small
> simple group, SU(2).

The dimension - more often called the degree - of a representation
is the dimension of the vector space in which the representation is defined.

Presumably, if the group is simple then ker(f) = 1 (or G),
ie the representation is faithful (or trivial).


--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
.

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