On the Representation Theory of Mbius Groups in R~n

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作者
方爱农
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[1] Department of Mathematics
[2] Hunan
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<正> We will solve several fundamental problems of M(?)bius groups M(R~n)which havebeen matters of interest such as the conjugate classification,the establishment of a standardform without finding the fixed points and a simple discrimination method.Let g=[abcd]be a Clifford matrix of dimension n,c≠0.We give a complete conjugateclassification and prove the following necessary and sufficient conditions:g is f.p.f.(fixed pointsfree)iff g~[αβcα'],丨α丨<1 and丨E-AE1丨≠0;g is elliptic iff g~[αβcα'],丨α丨<1 and丨E-AE1丨=0;g is parabolic iff g~[α 0 c α'],丨α丨=1;and g is loxodromic iff g~[αβcα'],丨α丨>1 or rank(E-AE1)≠rank(E-AE1,αc-1+c-1d),where α is represented by the solutions ofcertain linear algebraic equations and satisfies丨c-1α'丨=丨(E-AE1)-1(αc-1+c-1α')丨.
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