General and Special Linear Groups

Special Linear Groups

Recall the special linear groups:

Claim that the set of all linear operators such that determinant is $1$ is a group. Write it as ${\mathrm{SL}}_n(\mathbb{F})$where $n$ denotes the dimension of the corresponding square matrix.

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e.g.

• The Lie algebra ${\mathfrak{sl}}_2(\mathbb{R})$ consists of all $2\times 2$ real matrices with trace zero. A standard basis is given by $n^{+}=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),n^{-}=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),h=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$with Lie brackets $[h,n^{+}]=2n^{+},[h,n^{-}]=-2n^{-},[n^{+},n^{-}]=h.$As ${\text{sl}}_2(\mathbb{R})$ is so important in physics, people give special names to these basis elements: $n^{+}$ is called the raising operator, $n^{-}$ the lowering operator, and $h$ the Cartan operator.
• The complexification of ${\text{sl}}_2(\mathbb{R})$ is ${\text{sl}}_2(\text{C})$. i.e., ${\text{sl}}_2(\mathbb{R}){\otimes }_{\mathbb{R}}\text{C}\cong {\text{sl}}_2(\text{C})$. So the above basis also works for ${\text{sl}}_2(\text{C})$. It is also sometimes useful to use Pauli matrices as a basis of it. Suppose $\{H,E,F\}$ is a basis of ${\text{sl}}_2(\text{C})$ satisfying $[H,E]=2E,[H,F]=-2F,[E,F]=H.$Then the Casimir operator of ${\text{sl}}_2(\text{C})$ is defined as $\Omega =\frac{1}{2}{H}^2+EF+FE=\frac{1}{2}{H}^2+H+2FE.$A notable result is that $\Omega$ lies in the center of ${\text{sl}}_2(\text{C})$, for which one can verify that $[\Omega ,E]=[\Omega ,F]=[\Omega ,H]=0.$