The Multiple Zeta Value Algebra and the Stable Derivation Algebra
The Multiple Zeta Value Algebra and the Stable Derivation Algebra
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多重Zeta值代数和稳定导数代数
DOI：
10.2977/prims/1145476044
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发表时间：
2000
影响因子：
1.2
通讯作者：
H. Furusho
中科院分区：
文献类型：
--
作者：
H. Furusho
MZV 代数是由所有多个 zeta 值生成的 Q 上的分级代数。稳定导数代数是 GrothendieckTeichmuller 群的分级李代数版本。我们将证明，从 Q 上的稳定推导代数的分级对偶向量空间到新 zeta 空间，存在一个规范满射 Q 线性映射，即 MZV 代数的子向量空间的商空间，其等级大于 2 的最大理想的平方。作为推论，我们根据稳定推导代数的分级部分的相应维数得到每个权重的 MZV 代数分级部分的维数上限。如果Y. Ihara和P. Deligne关于稳定导数代数结构的一些标准猜想成立，这将成为Zagier在第一届欧洲数学大会演讲中提出的界限猜想。通过稳定导数代数，我们可以将新 zeta 空间与 l 进伽罗瓦图像李代数进行比较，后者与 P Q − {0, 1,∞} 的 pro-l 基本群上的伽罗瓦表示相关。
The MZV algebra is the graded algebra over Q generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the GrothendieckTeichmuller group. We shall show that there is a canonical surjective Q-linear map from the graded dual vector space of the stable derivation algebra over Q to the new-zeta space, the quotient space of the sub-vector space of the MZV algebra whose grade is greater than 2 by the square of the maximal ideal. As a corollary, we get an upper-bound for the dimension of the graded piece of the MZV algebra at each weight in terms of the corresponding dimension of the graded piece of the stable derivation algebra. If some standard conjectures by Y. Ihara and P. Deligne concerning the structure of the stable derivation algebra hold, this will become a bound conjectured in Zagier’s talk at 1st European Congress of Mathematics. Via the stable derivation algebra, we can compare the new-zeta space with the l-adic Galois image Lie algebra which is associated with the Galois representation on the pro-l fundamental group of P Q − {0, 1,∞}.