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  "body": "\n## Example\n\nThe group $SL(2, \\C)$ is the [[special linear group|0V9PMN0RBH]] of complex $2 \\times 2$ matrices with determinant $1$:\n$$SL(2, \\C) = \\set{\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix} : a, b, c, d \\in \\C, \\ ad - bc = 1}.$$\nIt is cut out of $M_2(\\C) = \\C^4$ by the single polynomial equation $ad - bc = 1$, whose differential is nonzero at every solution.\nSo it is a complex [[Lie group|7JBRGCVJRB]] of complex dimension $3$, that is, of real dimension $6$.\n\nIt acts on the [[Riemann sphere|0J78KBPFZB]] $\\C \\cup \\set{\\infty}$ by [[Möbius transformations|0SG74WBFA2]] $z \\mapsto (az + b)/(cz + d)$.\nCheck: the matrix $g$ acts as the identity map exactly when $(az + b)/(cz + d) = z$ for all $z$, which forces $b = c = 0$ and $a = d$, and then $ad = 1$ gives $g = \\pm I$.\nSo the group of Möbius transformations is $PSL(2, \\C) = SL(2, \\C)/\\set{\\pm I}$.\n\n## Remarks\n\nEvery $g \\in SL(2, \\C)$ has a unique polar decomposition $g = u p$ with $u \\in$ [[SU(2)|E5P8AG6CK7]] and $p$ a positive definite [[Hermitian matrix|0P17NGQQMQ]] of determinant $1$.\nThese $p$ form a space homeomorphic to $\\R^3$, so $SL(2, \\C)$ is homeomorphic to $S^3 \\times \\R^3$.\nHence it is connected and simply connected, and $SU(2)$ is a maximal compact subgroup.\nIts Lie algebra $\\mathfrak{sl}(2, \\C)$ is the complex simple Lie algebra of type $A_1$.\n$SL(2, \\C)$ acts on the Hermitian $2 \\times 2$ matrices by $X \\mapsto g X g^*$, preserving $\\det X$, which is a quadratic form of signature $(1, 3)$.\nThis gives a double cover of the identity component of the Lorentz group, so $SL(2, \\C)$ is the spin group $\\mathrm{Spin}(1, 3)$.\n\n## Explanations\n\n### L1 human\n\n$SL(2, \\C)$ is the complex version of the group of $2 \\times 2$ matrices with determinant $1$.\nEach of its elements moves the points of the extended complex plane by a fractional linear map, and two matrices $g$ and $-g$ give the same map.\nThese maps send circles to circles, and they are exactly the orientation-preserving conformal symmetries of the sphere.\nIn physics the same group describes how spinors change under the rotations and boosts of special relativity.\n",
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