what is the difference between $\mathrm{SL}(2)$ and $\mathfrak{sl}(2)$?

Question

Regarding $\mathrm{SL}(2)$ and $\mathfrak{sl}(2)$, may I know what is the difference between them?

I understand that $\mathrm{SL}(2)$ is the set of $2\times 2$ matrices with determinant $1$.

$\mathfrak{sl}(2)$ is refered in the book (Quantum Groups by Kassel) as a Lie algebra.

Sincere thanks for this beginner question.

Answer 1

The two are very different. The first one is the group of all non-singular two by two matrices with determinant 1 with entries in some field. The second one is the set of two by two matrices with vanishing trace, since take the differential of $\det(m)=1$ on $m$ gives $$\det(m)Tr(m^{-1}dm)=0$$ A way of visualizing $SL(2,\mathbb{R})$ can be found at here.

Answer 2

$\mathfrak{sl}(2)$ consists of traceless $2\times 2$ matrices while the other is the standard Special linear group ($\det=1$).

Answer 3

$SL(2)$ is a Lie group (a manifold with a compatible group structure) and $\mathfrak{sl}(2)$ is its corresponding Lie algebra. More details for the general situation can be found here.

For a field $\mathbb{F}$, $SL(n, \mathbb{F})$ is the set of $n\times n$ matrices with values in $\mathbb{F}$ and determinant $1$. The corresponding Lie algebra, $\mathfrak{sl}(n, \mathbb{F})$, is the set of $n\times n$ matrices with values in $\mathbb{F}$ and zero trace.