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Shu Oi

Publications and source records attributed to Shu Oi.

8 recordsLinked to original sources
arXiv2015

Fundamental solutions of the Knizhnik-Zamolodchikov equation of one variable and the Riemann-Hilbert problem

In this article, we derive multiple polylogarithms from multiple zeta values by using a recursive Riemann-Hilbert problem of additive type. Furthermore we show that this Riemann-Hilbert problem is regarded as an inverse problem for the connection problem of the KZ equation of one variable, so that the fundamental solutions to the equation are derived from the Drinfel'd associator by using a Riemann-Hilbert problem of multiplicative type. These results say that the duality relation for the Drinfel'd associator can be interpreted as the solvability condition for this inverse problem.

math.CA↗
arXiv2013

The hexagon equations for dilogarithms and the Riemann-Hilbert problem

In this article we present the hexagon equations for dilogarithms which come from the analytic continuation of the dilogarithm $\mathrm{Li}_2(z)$ to ${\mathbf P}^1 \setminus {0,1,\infty}$. The hexagon equations are equivalent to the coboundary relations for a certain 1-cocycle of holomorphic functions on ${\mathbf P}^1$, and are solved by the Riemann-Hilbert problem of additive type. They uniquely characterize the dilogarithm under the normalization condition.

math.CA↗
arXiv2012

The inversion formula of polylogarithms and the Riemann-Hilbert problem

In this article, we set up a method of reconstructing to polylogarithms $\mathrm{Li}_k(z)$ from zeta values $ζ(k)$ via the Riemann-Hilbert problem. This is referred to as "a recursive Riemann-Hilbert problem of additive type." Moreover, we suggest a framework of interpreting the connection problem of the Knizhnik-Zamolodochikov equation of one variable as a Riemann-Hilbert problem.

math.QA↗
arXiv2011

Connection Problem of Knizhnik-Zamolodchikov Equation on Moduli Space ${\mathcal M}_{0,5}$

In this article, we consider the connection problem of the KZ (Knizhnik-Zamolodchikov) equation on the moduli space $\cM_{0,5}$, and show that the connection matrices are expressed in terms of the Drinfel'd associator. As the compatibility condition on the connection problem, we have the pentagon relation for the Drinfeld associators. As an application of the connection problem, we derive the five term relation for dilogarithms.

math.QA↗
arXiv2011

KZ equation on the moduli space ${\mathcal M}_{0,5}$ and the harmonic product of multiple polylogarithms

In this article, we derive a system of functional relations called the generalized harmonic product relations for hyperlogarithms on the moduli space ${\mathcal M}_{0,5}$ and show that the relations contain the harmonic product of multiple polylogarithms. The generalized harmonic product relations are equivalent to the relations which come from two decompositions of the fundamental solution normalized at the origin of the KZ equation on ${\mathcal M}_{0,5}$.

math.QA↗
arXiv2010

Iterated integrals and relations of multiple polylogarithms

This is a summary for the authors' article "The formal KZ equation on the moduli space ${\mathcal M}_{0,5}$ and the harmonic product of multiple zeta values" (prerint (2009) arXiv:0910.0718), including a new result on the five term relation for the dilogarithm. This note will appear in the RIMS Kôkyûroku for the conference on "Representation Theory and Combinatorics" held at Hokkaido University from August 25th to 28th, 2009.

math.QA↗
arXiv2008

Representation of solutions of the Gauss hypergeometric equation by the multiple polylogarithms, functional relations of the multiple polylogarithms and relations of the multiple zeta values

In this article, we express solutions of the Gauss hypergeometric equation as a series of the multiple polylogarithms by using iterated integral. This representation is the most simple case of a semisimple representation of solutions of the formal KZ equation. Moreover, combining this representation with the connection relations of solutions of the Gauss hypergeometric equation, we obtain various relations of the multiple polylogarithms of one variable and the multiple zeta values.

math.QA↗