Structure of the unitary valuation algebra
Structure of the unitary valuation algebra
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DOI：
10.4310/jdg/1143593748
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发表时间：
2004-10
影响因子：
2.5
通讯作者：
Joseph H. G. Fu
中科院分区：
文献类型：
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作者：
Joseph H. G. Fu
S. Alesker证明了，如果G是O（n）在单位球面S^{n-1}$上传递作用的紧子群，则R^n$上的连续的、平移不变的、G-不变的凸赋值的向量空间瓦尔^G$具有满足Poincare对偶的R$上的有限维分次代数的结构。我们证明了G$的运动学公式是由乘积对决定的。利用这个结果，我们证明了代数瓦尔^{U（n）}$同构于$R[s，t]/（f_{n+1}，f_{n+2}）$，其中s，t$的次数分别为2和1，多项式f_i$是幂级数$\log（1 + s +t）$的次数$i$项.
S. Alesker has shown that if $G$ is a compact subgroup of O(n) acting transitively on the unit sphere $S^{n-1}$ then the vector space $Val^G$ of continuous, translation-invariant, $G$-invariant convex valuations on $R^n$ has the structure of a finite dimensional graded algebra over $R$ satisfying Poincare duality. We show that the kinematic formulas for $G$ are determined by the product pairing. Using this result we then show that the algebra $Val^{U(n) }$ is isomorphic to $R[s,t]/(f_{n+1}, f_{n+2})$, where $s,t$ have degrees 2 and 1 respectively, and the polynomial $f_i$ is the degree $i$ term of the power series $\log(1 + s +t)$.