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  "body": "\n## Statement\n\n1. Every [[bounded symmetric domain|Y7Q2SNKG0C]] is [[biholomorphic|FX92PP3HZR]] to a product of irreducible bounded symmetric domains, and the factors are unique up to order and biholomorphism.\n2. Every irreducible bounded symmetric domain is biholomorphic to one of the following domains.\n   Here $Z^*$ is the conjugate transpose of $Z$, and $M \u003e 0$ means that the Hermitian matrix $M$ is positive definite.\n   - type $I_{m,n}$: $\\set{Z \\in M_{m \\times n}(\\C) : I_n - Z^* Z \u003e 0}$;\n   - type $II_n$: $\\set{Z \\in M_{n \\times n}(\\C) : Z^T = -Z,\\ I_n - Z^* Z \u003e 0}$;\n   - type $III_n$: $\\set{Z \\in M_{n \\times n}(\\C) : Z^T = Z,\\ I_n - Z^* Z \u003e 0}$;\n   - type $IV_n$: $\\set{z \\in \\C^n : \\abs{z^T z} \u003c 1,\\ 1 - 2 z^* z + \\abs{z^T z}^2 \u003e 0}$;\n   - two exceptional domains, of complex dimensions $16$ and $27$.\n\n## Remarks\n\nÉlie Cartan obtained the classification in 1935.\nThe proof passes to Lie theory.\nA bounded symmetric domain is $G/K$, where $G$ is the identity component of its automorphism group, a semisimple Lie group, and $K$ is a maximal compact subgroup.\nIrreducible domains correspond to simple $G$ for which $K$ has a nontrivial centre.\nReading these off from the [[classification of symmetric spaces|9P8VK3H4YZ]] gives the list: $G$ is $SU(m, n)$, $SO^*(2n)$, $Sp(2n, \\R)$, $SO_0(n, 2)$, or real forms of $E_6$ and $E_7$.\n\nExample: $I_{m,1}$ is the unit ball of $\\C^m$, and $I_{1,1} = III_1$ is the unit disc.\nThe type $III_n$ domain is the bounded model of the [[Siegel upper half-space|5PTYE97PF6]] of degree $n$.\n\nFor small indices some types coincide, so the list is without repetitions only for suitable ranges of the indices.\n\n## Explanations\n\n### L1 human\n\nBounded symmetric domains are the most symmetric bounded regions in $\\C^n$.\nThe theorem says that there are only four infinite families of them, made of matrices of small norm, plus two exceptional ones, and products of these.\nSo a seemingly general class of domains is described by an explicit list.\n\n### L2 human\n\nThrough the list, analysis on bounded symmetric domains reduces to matrix computations.\nThe domains are the [[Hermitian symmetric spaces|NT4797Z9H8]] of noncompact type, and each has a compact dual, such as a Grassmannian for type $I$.\nQuotients of these domains by arithmetic groups include the Shimura varieties and the moduli spaces of abelian varieties, so the list organises much of the theory of automorphic forms.\n",
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