The Algebra of Multiple Harmonic Series
The Algebra of Multiple Harmonic Series
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DOI：
10.1006/jabr.1997.7127
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发表时间：
1997-08
影响因子：
0.9
通讯作者：
Michael E. Hoffman
中科院分区：
文献类型：
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作者：
Michael E. Hoffman
近年来，多重调和级数ζ(I1，I2，…)引起了人们的极大兴趣，ik)=∑n 1>；n2>；···>；n k≥11 n 1 i 1 n 2 i 2···n k i k(当指数为正整数且1>；1时收敛)，也称为多重zeta值或欧拉/zagier和。从非交换多项式代数Q<x，y>出发，定义了第二个交换结合乘法，称其为调和代数h。作为一个分次交换代数，h是一个自由多项式代数，它的生成元数是由两个生成元上的自由李代数的度基元个数N(N)的Witt公式给出的。多重调和级数可以看作是映射ζ：h0→R下的像，它是关于交换乘法的同态，其中h0是h的适当子代数。如果我们把序列ζ(i1，…)的权称为1+···+，ik)，则(Forn>；1)至多存在N(N)个不可约的权重级数，因为它们不是较低权重的多重调和级数乘积的有理倍数之和。事实上，我们的方法给出了一个“代数”不可约重调和级数的显式集合。我们还证明了有一个子代数h0⊂h1⊂h与包含对称函数的代数的混洗代数有关：实际上，∩将Sζh0的元素映射为整数≥2的zeta值的代数组合ζ(I)。映射ζ不是内射的；我们证明了关于重调和级数的几个结果如何被重塑为关于ζ核的陈述，并对代数h0/Ker的结构提出了一些猜想。
Abstract Recently there has been much interest in multiple harmonic series ζ(i 1 , i 2 ,…,i k ) = ∑ n 1 > n 2 > ··· > n k ≥ 1 1 n 1 i 1 n 2 i 2 ···n k i k (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 : h 0 → Rwhich is a homomorphism with respect to the commutative multiplication, where h 0is 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 h 0 ⊂ h 1 ⊂ h related to the shuffle algebra which contains the algebra S of symmetric functions: in fact, ζ maps the elements of S ∩ h 0to 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 h 0/ker ζ.