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In mathematics, a Siegel domain or Piatetski-Shapiro domain is a special open subset of complex affine space generalizing the Siegel upper half plane studied by . They were introduced by in his study of bounded homogeneous domains. ==Definitions== A Siegel domain of the first kind (or first type, or genus 1) is the open subset of C''m'' of elements ''z'' such that : where ''V'' is an open convex cone in R''m''. These are special cases of tube domains. An example is the Siegel upper half plane, where ''V''⊂R''k''(''k'' + 1)/2 is the cone of positive definite quadratic forms in R''k'' and ''m'' = ''k''(''k'' + 1)/2. A Siegel domain of the second kind (or second type, or genus 2), also called a Piatetski-Shapiro domain, is the open subset of C''m''×C''n'' of elements (''z'',''w'') such that : where ''V'' is an open convex cone in R''m'' and ''F'' is a ''V''-valued Hermitian form on C''n''. If ''n'' = 0 this is a Siegel domain of the first kind. A Siegel domain of the third kind (or third type, or genus 3) is the open subset of C''m''×C''n''×C''k'' of elements (''z'',''w'',''t'') such that : and ''t'' lies in some bounded region where ''V'' is an open convex cone in R''m'' and ''L''''t'' is a ''V''-valued semi-Hermitian form on C''n''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Siegel domain」の詳細全文を読む スポンサード リンク
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