The Theory of Zeta-Functions of Root Systems / Yasushi Komori, Kohji Matsumoto, and Hirofumi Tsumura.

Format:

Book

Author/Creator:

Komori, Yasushi, author.

Matsumoto, Kohji, author.

Tsumura, Hirofumi, author.

Series:

Springer monographs in mathematics.

Springer Monographs in Mathematics Series

Language:

English

Subjects (All):

Functions, Zeta.

Physical Description:

1 online resource (IX, 414 p. 13 illus.)

Edition:

First edition.

Place of Publication:

Singapore : Springer, [2023]

Summary:

The contents of this book was created by the authors as a simultaneous generalization of Witten zeta-functions, Mordell–Tornheim multiple zeta-functions, and Euler–Zagier multiple zeta-functions. Zeta-functions of root systems are defined by certain multiple series, given in terms of root systems. Therefore, they intrinsically have the action of associated Weyl groups. The exposition begins with a brief introduction to the theory of Lie algebras and root systems and then provides the definition of zeta-functions of root systems, explicit examples associated with various simple Lie algebras, meromorphic continuation and recursive analytic structure described by Dynkin diagrams, special values at integer points, functional relations, and the background given by the action of Weyl groups. In particular, an explicit form of Witten’s volume formula is provided. It is shown that various relations among special values of Euler–Zagier multiple zeta-functions—which usually are called multiple zeta values (MZVs) and are quite important in connection with Zagier’s conjecture—are just special cases of various functional relations among zeta-functions of root systems. The authors further provide other applications to the theory of MZVs and also introduce generalizations with Dirichlet characters, and with certain congruence conditions. The book concludes with a brief description of other relevant topics.

Contents:

Introduction

Fundamentals of the theory of Lie algebras and root systems

Definitions and examples

Values at positive even integer points

Convex polytopes and the rationality

The recursive structure

The meromorphic continuation

Functional relations (I)

Functional relations (II)

Poincar´e polynomials and values at integer points

The case of the exceptional algebra G2

Applications to multiple zeta values (I)

Applications to multiple zeta values (II)

L-functions

Zeta-functions of Lie groups

Lattice sums of hyperplane arrangements

Miscellaneous results.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

981-9909-10-4