酉赋值代数的结构
Structure of the unitary valuation algebra
摘要：
S. Alesker证明了，如果$G$是O（n）的紧致子群，作用于单位球面$S^{n-1}$上传递，则$R^n$上连续的、平移不变的、$G$不变的凸赋值的向量空间$Val^G$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配对决定的。利用这个结果，我们证明了代数$Val^{U(n) }$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.
英文摘要：
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 $\log(1 + s +t)$.
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