On Witten multiple zeta-functions associated with semisimple Lie algebras I

Kohji Matsumoto[1]; Hirofumi Tsumura[2]

  • [1] Nagoya University Graduate School of Mathematics Chikusa-ku, Nagoya 464-8602 (Japan)
  • [2] Tokyo Metropolitan University Department of Mathematics 1-1, Minami-Ohsawa Hachioji-shi, Tokyo 192-0397 (Japan)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 5, page 1457-1504
  • ISSN: 0373-0956

Abstract

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We define Witten multiple zeta-functions associated with semisimple Lie algebras 𝔰𝔩 ( n ) , ( n = 2 , 3 , ... ) of several complex variables, and prove the analytic continuation of them. These can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case 𝔰𝔩 ( 4 ) , we determine the singularities of this function. Furthermore we prove certain functional relations among this function, the Mordell-Tornheim double zeta-functions and the Riemann zeta-function. Using these relations, we prove new and non-trivial evaluation formulas for special values of this function at positive integers.

How to cite

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Matsumoto, Kohji, and Tsumura, Hirofumi. "On Witten multiple zeta-functions associated with semisimple Lie algebras I." Annales de l’institut Fourier 56.5 (2006): 1457-1504. <http://eudml.org/doc/10182>.

@article{Matsumoto2006,
abstract = {We define Witten multiple zeta-functions associated with semisimple Lie algebras $\{\mathfrak\{sl\}\}(n)$, $(n=2,3,\ldots )$ of several complex variables, and prove the analytic continuation of them. These can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case $\{\mathfrak\{sl\}\}(4)$, we determine the singularities of this function. Furthermore we prove certain functional relations among this function, the Mordell-Tornheim double zeta-functions and the Riemann zeta-function. Using these relations, we prove new and non-trivial evaluation formulas for special values of this function at positive integers.},
affiliation = {Nagoya University Graduate School of Mathematics Chikusa-ku, Nagoya 464-8602 (Japan); Tokyo Metropolitan University Department of Mathematics 1-1, Minami-Ohsawa Hachioji-shi, Tokyo 192-0397 (Japan)},
author = {Matsumoto, Kohji, Tsumura, Hirofumi},
journal = {Annales de l’institut Fourier},
keywords = {Witten multiple zeta-functions; Mordell-Tornheim zeta-functions; Riemann zeta-function; analytic continuation; semisimple Lie algebra},
language = {eng},
number = {5},
pages = {1457-1504},
publisher = {Association des Annales de l’institut Fourier},
title = {On Witten multiple zeta-functions associated with semisimple Lie algebras I},
url = {http://eudml.org/doc/10182},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Matsumoto, Kohji
AU - Tsumura, Hirofumi
TI - On Witten multiple zeta-functions associated with semisimple Lie algebras I
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 5
SP - 1457
EP - 1504
AB - We define Witten multiple zeta-functions associated with semisimple Lie algebras ${\mathfrak{sl}}(n)$, $(n=2,3,\ldots )$ of several complex variables, and prove the analytic continuation of them. These can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case ${\mathfrak{sl}}(4)$, we determine the singularities of this function. Furthermore we prove certain functional relations among this function, the Mordell-Tornheim double zeta-functions and the Riemann zeta-function. Using these relations, we prove new and non-trivial evaluation formulas for special values of this function at positive integers.
LA - eng
KW - Witten multiple zeta-functions; Mordell-Tornheim zeta-functions; Riemann zeta-function; analytic continuation; semisimple Lie algebra
UR - http://eudml.org/doc/10182
ER -

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