Baxter algebras and number theory
by
Li Guo
Rutgers University at Newark
Coauthors: Kurusch Ebrahimi-Fard (IHES)

A Baxter algebra is an algebra with a linear operator that satisfies the Baxter identity

P(x)P(y) = P(xP(y)+P(x)+lambda xy)

for a constant lambda. It is also known as Rota-Baxter algebras to the physicists. Studied in early years by Baxter, Cartier and Rota, Baxter algebras have recently found applications in Hopf algebras, operad theory, combinatorics and math physics. We will describe two relations between Baxter algebras and number theory.

The first relation is with generating functions in number theory. We will explain how, like power series, elements in Baxter algebras serve as generating functions of number sequences, such as Stirling numbers and multinomial coefficients. This allows us to generalize the concept of generating functions to represent more number sequences.

The second relation is with multiple zeta values. Relations among multiple zeta values were already studied by Euler in the 17th century and have been studied very actively in the last decade in the analysic, algebraic, combinatorial and motivic aspects. Many more such relations were found and general forms of such relations were conjectured that have profound consequences. Central to these developments are the so-called shuffle and quasi-shuffle (stuffle) products. We show that these products are special cases of mixable shuffle products in free Baxter algebras. We will explain the role play by Baxter algebras in multiple zeta values and q-multiple zeta values and the relation with recent work of Bowman, Bradley and Hoffman in this area.

For further details, visit http://newark.rutgers.edu/ liguo/lgpapers.html

Date received: March 8, 2004