# Definition The [[Special Linear Group]] $SL(2,\mathbb{C})$ is the group of all complex $2\times2$ [[matrix|matrices]] with unit [[determinant]]. It is the group of [[Lorentz transformation|Lorentz transformations]] on [[spin-half|spin-1/2]] particles. # Elements The most general element of $SL(2, \mathbb{C})$ can be written as: $ A=\left(\begin{array}{l} a & b \\ c & d \end{array}\right) \quad a, b, c, d \in \mathbb{C}, a d-b c=1 $ This means the group has complex [[dimension]] 3 or real dimension 6. One can show that a boost in $SL(2,\mathbb{C})$ can be written as: $ \tilde{L}=\left(\begin{array}{cc} \cosh \frac{u}{2}+\frac{u_{z}}{u} \sinh \frac{u}{2} & -\frac{1}{u}\left(u_{x}-i u_{y}\right) \sinh \frac{u}{2} \\ -\frac{1}{u}\left(u_{x}+i u_{y}\right) \sinh \frac{u}{2} & \cosh \frac{u}{2}-\frac{u_{z}}{u} \sinh \frac{u}{2} \end{array}\right), \quad \mathbf{u} \in \mathbb{R}^{3} $ Moreover, any $A$ in $SL(2,\mathbb{C})$ can be decomposed as $A = \tilde{L} \tilde{R}$ where $\tilde{L}$ is given above and $\tilde{R} \in SU(2)$. The parameters here are the same used for $SO (3,1)_o$ # Lie Algebra %% The [[Lie Algebra]] of $SL(2,\mathbb{C})$ is $H_2(\mathbb{C})$, the vector space of all [[Hermitian Operator|Hermitian]] $2\times 2$ matrices. %% $\mathfrak{sl}(2, \mathbb{C})_\mathbb{R}$ is defined to be the [[Lie algebra]] of $SL(2, \mathbb{C})$ when viewed as a *real* vector space. As defined above, $SL(2, \mathbb{C})$ is the set of all complex $2\times2$ [[matrix|matrices]] with unit [[determinant]], making $\mathfrak{sl}(2, \mathbb{C})_\mathbb{R}$ the set of all [[trace|traceless]] $2\times2$ complex matrices. This gives us 4-1 complex dimensions, i.e., 6 real dimensions, the same [[dimension|dimensionality]] as $SL(2,\mathbb{C})$ as one would expect. We can take as a basis (i.e., the generators): $ \begin{array}{cc} &S_{1}=\frac{1}{2}\left(\begin{array}{cc} 0 & -i \\ -i & 0 \end{array}\right), \quad &S_{2}=\frac{1}{2}\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right), \quad &S_{3}=\frac{1}{2}\left(\begin{array}{cc} -i & 0 \\ 0 & i \end{array}\right) \\ &\tilde{K}_{1} \equiv \frac{1}{2}\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), \quad &\tilde{K}_{2} \equiv \frac{1}{2}\left(\begin{array}{cc} 0 &-i \\ i & 0 \end{array}\right), \quad &\tilde{K}_{3} \equiv \frac{1}{2}\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right) . \end{array} $ where $\tilde{K}_i = i S_i$. We also have thee commutation relations: $ \begin{aligned} \left[S_{i}, S_{j}\right] &=\sum_{k=1}^{3} \epsilon_{i j k} S_{k} \\ \left[S_{i}, \tilde{K}_{j}\right] &=\sum_{k=1}^{3} \epsilon_{i j k} \tilde{K}_{k} \\ \left[\tilde{K}_{i}, \tilde{K}_{j}\right] &=-\sum_{k=1}^{3} \epsilon_{i j k} S_{k} \end{aligned} $ which are identical to the $\mathfrak{so}(3,1)$ commutation relations.