The elements of
correspond to the
global symmetry currents of the
-model.
There are finitely many global symmetries. We have:
| (49) |
We will now show that
is a
finite-dimensional representation of
,
actually the
adjoint representation of
.
How do we see this?
As a representation of
,
is generated by
the following objects:
| (50) |
Remember 
are defined from
and similarly 
.
The bar over index 
means:
We see that 
is a finite-dimensional space.
Why there are no more elements? For
example, let us consider 
. Modulo 
this is same as
, and using
Eq. (39) this is proportional to 
.
Similarly, all the other covariant derivatives of every element of (50)
can be expressed as a linear sum of other elements. Therefore, they form a closed
representation of 
.