Involution on a quotient space of multiple zeta values in positive characteristic

Abstract

In this paper, we introduce $\ast$-inverse multiple zeta values and $\ast$-inverse Carlitz multiple polylogarithms in positive characteristic and study their algebraic structures and relations. Using special values of Carlitz multiple polylogarithms, we show that a natural quotient of the space of multiple zeta values in positive characteristic admits a Hopf algebra structure and that the original space admits a compatible comodule structure over this quotient. These structures may be regarded as function field analogues of the Hopf algebra and comodule structures arising from motivic multiple zeta values in characteristic zero. In particular, the antipode on this quotient gives a non-trivial involution corresponding to the $\ast$-inverse operation. We also establish, in certain cases, the $q$-shuffle product formula and linear relations for $\ast$-inverse multiple zeta values, providing evidence for our conjectures on their relations. Finally, we consider another characteristic-zero analogue of our construction, based on the Hopf algebra structure given by the harmonic product and the deconcatenation coproduct, and formulate a related problem.

20 pages