The theory of representation of a group in group theory is very important and thus have huge contribution to the not only physics community but also other diverse science world.
To start with, a representation can be understood as a linear action of a group on a vector space, 
(may either the real or complex vector space). So when we understand the term of linear action, we automatically recall for the word mapping (see terminology) for which we denoted as 
to be the notation of a representation.
In short we have the following mapping
with homomorphism that follows 
for all 
for all 
and 
Also for a Lie group, 
representations there is an associated Lie algebra, 
representations. This is follows by
.
The fact that for every Lie group there is an associated Lie algebra is related by a mapping
.
To be more explicit, we need below definition.
Definition:
Let be a Lie group and its Lie algebra, then for a set of matrices 
there is 
for all 
. This is the important properties of what is known as the exponential of a matrix.
to be continued…
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Filed under: Group Theory
