I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as :
$$ \left( \begin{array}{ccc} x+a \\ y+a \end{array} \right) = \left( \begin{array}{ccc} x \\ y \end{array} \right) + \left( \begin{array}{ccc} a \\ b \end{array} \right)$$
Now the set of all translations $ T = \left( \begin{array}{ccc} a \\ b \end{array} \right) $ form a two parameter lie group (I presume) with addition of column as the composition rule.
If that is so, how do I go about finding the generators of this transformation. I know the generators of translation are linear momenta in the corresponding directions. But I am not able to see this here.
PS: In my course I have been taught that the generators are found by calculating the Taylor expansion of the group element about the Identity of the group. For instance, $\operatorname{SO}(2)$ group $$ M = \left( \begin{array}{cc} \cos \:\phi & -\sin \:\phi \\ \sin \:\phi & \cos \:\phi \end{array} \right) $$ I obtain the generator by taking $$ \frac{\partial M}{\partial \phi}\Bigg|_{\phi=0} = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) $$
Now if I exponentiate this, I can obtain back the group element. My question how do I do this for Translation group.
EDIT :This edit is to summarise and get a view of the answers obtained.
Firstly, the vector representation of the translation group (for 2D) would in general have the form : $$ \begin{pmatrix} 1 & 0 & a_x\\ 0 & 1 & a_y \\ 0 & 0 & 1 \end{pmatrix}\ $$ with generators (elements of Lie algebra) $$ T_x =\begin{pmatrix} 0 & 0 & i\\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\ , \;\; T_y = \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & i \\ 0 & 0 & 0 \end{pmatrix}\ $$
Secondly, the scalar-field representation of the same is given by the differential operators $$ exp^{ i(a_x\frac{\partial}{\partial x}+ a_y\frac{\partial}{\partial y} )} $$ with generators $$ T_x^s = i\frac{\partial}{\partial x},\;\;T_y^s = i\frac{\partial}{\partial y} $$
The Lie algebra is two-dimensional and abelian : $ [T_x,T_y] = 0$