The Algebra of Multiple Harmonic Series

Abstract

Recently there has been much interest in multiple harmonic seriesζ(i1,i2,…,ik)=∑n1>n2>···>nk≥11n1i1n2i2···nkik(which converge when the exponentsijare positive integers andi1>1), also known as multiple zeta values or Euler/Zagier sums. Starting with the noncommutative polynomial algebraQ〈x,y〉, we define a second multiplication which is commutative and associative, and call the resulting structure the harmonic algebra h. As a graded commutative algebra, h turns out to be a free polynomial algebra with the number of generators in degreengiven by the Witt formula for the numberN(n) of basis elements of degreenin the free Lie algebra on two generators. Multiple harmonic series can be thought of as images under a map ζ:h0→Rwhich is a homomorphism with respect to the commutative multiplication, where h0is an appropriate subalgebra of h. If we calli1+···+ikthe weight of the series ζ(i1,…,ik), then (forn>1) there are at mostN(n) series of weightnthat are irreducible in the sense that they are not sums of rational multiples of products of multiple harmonic series of lower weight. In fact, our approach gives an explicit set of “algebraically” irreducible multiple harmonic series. We also show that there is a subalgebra h0⊂h1⊂h related to the shuffle algebra which contains the algebra S of symmetric functions: in fact, ζ maps the elements of S∩h0to algebraic combinations of zeta values ζ(i), for integeri≥2. The map ζ is not injective; we show how several results about multiple harmonic series can be recast as statements about the kernel of ζ, and propose some conjectures on the structure of the algebra h0/kerζ.