多重zeta值上的代數關係式

Algebraic Relations of Multiple Zeta Values

陳暐捷(Wei-Chieh Chen)

指導教授 ： 廖文欽

國立中正大學/理學院/數學系研究所/碩士(2014年)

摘要

在這篇論文，我們將介紹多重zeta值。
這些值在幾何學、弦論、物理數學以及代數幾何上有其一席之地。
在此，我們主要關切多重zeta值之間的有理線性關係式。

我們將藉由非交換代數 Q 來研究多重zeta值，
並引入洗牌乘積與調和乘積來推導有限雙洗牌關係式。

接著，將介紹數個Q上的微分算子，
並將其取指數後得到自同型映射，以達推導出
諸如Hoffman關係式和大 野關係式。

最終，我們探討環狀和公式以及川島關係式。

關鍵字

多重zeta值

並列摘要

In this thesis we will introduce multiple zeta values.
These numbers have arisen in various contexts in geometry,
knot theory, mathematical physics, and arithmetical algebraic geometry.
As number theorists, we only concern about the relations on the multiple zeta values.
Here we especially study the linear relations over Q among the multiple zeta values.

We study multiple zeta values through a non-commutative algebra Q.
Also, we will define the shuffle products on this algebra,
to get the finite double shuffle relations for multiple zeta values.
Next, we will take a number of derivations on some subalgebras of Q
and the exponentiate them to obtain automorphisms,
in order to consider some identities of multiple zeta values,
such as Hoffman's relations and Ohno's relations.
Finally, we introduce the cyclic sum formula and Kawashima's relations.

並列關鍵字

multiple zeta values

參考文獻

[1] M. E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (2) (1992), 275-290.

[2] K. Ihara, M. Kaneko, and D. Zagier, Derivation and double shuffle relations for multiple zeta values,

Compositio Math. 142 (2) (2006), 307-338.

[3] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed.,

[4] G. Kawashima, A class of relations among multiple zeta values,