The automorphism group of a composition of quadratic forms
The automorphism group of a composition of quadratic forms
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二次形式组合的自同构群
DOI：
10.1090/s0002-9947-1982-0637698-2
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发表时间：
1982
影响因子：
1.3
通讯作者：
C. Riehm
中科院分区：
文献类型：
--
作者：
C. Riehm
。设U X X-？X是特征=2的域F上两个二次空间U和X的(双线性)复合(u，x)h？UX，并假定U中有一个向量，它通过这个合成导出X上的单位映射.定义G是O(U)x O(X)的子群，它由满足(j>；(U)IP(X)=Ik？x)的那些对(<；/>；，p)组成，并定义Gx是G在0(X)上的投影。研究了群G，特别地，证明了它的连通分支作为一个代数群同于两个或三个经典群的乘积，因此是约化的。给出了当U和X是欧氏空间时，GX在X的单位球面上传递的充要条件。
. Let U X X -» X be a (bilinear) composition (u, x) h» ux of two quadratic spaces U and X over a field F of characteristic =¡t 2 and assume there is a vector in U which induces the identity map on X via this composition. Define G to be the subgroup of O(U) x O(X) consisting of those pairs (</>, \p) satisfying (j>(u)ip(x) = iK«x) identically and define Gx to be the projection of G on 0(X). The group G is investigated and in particular it is shown that its connected component, as an algebraic group, is isogenous to a product of two or three classical groups and so is reductive. Necessary and sufficient conditions are given for Gx to be transitive on the unit sphere of X when U and X are Euclidean spaces.