In Blumenhagen's book "Introduction to Conformal Field Theory", I found the statement
The algebra of infinitesimal conformal transformations in an Euclidean 2-dimensional space is infinite dimensional.
He concludes this after finding that the generators of the infinitesimal conformal transformations (i.e. a basis for its Lie algebra) are $l_n=-z^{n+1}\partial$ and $\bar l_n= -\bar z^{n+1}\bar\partial$ for $n\in\mathbb N$.
My problem with this is: shouldn't the elements of the Lie algebra be linear combinations (with real coefficients) of the basis elements $\partial,\bar\partial$ of $T_eG$? (Instead of a roduct of the derivatives with polynomials). Furthermore (and related to the previous point), if our Lie group is 2-dimensional (as a manifold), that implies that its Lie algebra is 2-dimensinal (as vector space), so definitely not infinitely dimensional. What is going on here?