arXiv subjects

Michael E. Hoffman

Publications and source records attributed to Michael E. Hoffman.

25 records · Page 2Linked to original sources

arXiv2004

Algebraic Aspects of Multiple Zeta Values

Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves "coding" the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values can then be thought of as defining a map ζ: H^0 -> R, where H^0 is the graded rational vector space generated by the "admissible words" of the noncommutative polynomial algebra Q . Now H^0 admits two (commutative) products making ζa homomorphism: the shuffle product and the "harmonic" product. The latter makes H^0 a subalgebra of the algebra QSym of quasi-symmetric functions. We also discuss some results about multiple zeta values that can be stated in terms of derivations and cyclic derivations of Q , and define an action of the Hopf algebra QSym on Q that appears useful. Finally, we apply the algebraic approach to finite partial sums of multiple zeta value series.

math.QA↗

arXiv2004

The Hopf algebra structure of multiple harmonic sums

Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of their properties that can be exploited in calculations.

math.QA↗

arXiv2004

Updown Categories

A poset can be regarded as a category in which there is at most one morphism between objects, and such that at most one of Hom(c,c') and Hom(c',c) is nonempty for c not equal to c'. If we keep in place the latter axiom but allow for more than one morphism between objects, we can have a sort of generalized poset in which there are multiplicities attached to the covering relations, and possibly nontrivial automorphism groups. We call such a category an "updown category." In this paper we give a precise definition of such categories and develop a theory for them, which incorporates earlier notions of differential posets and weighted-relation posets. We also give a detailed account of ten examples, including the updown categories of integer partitions, integer compositions, planar rooted trees, and rooted trees.

math.CO↗

arXiv2003

Combinatorics of Rooted Trees and Hopf Algebras

We begin by considering the graded vector space with a basis consisting of rooted trees, graded by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices. We define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the multiplicities associated with each operator. The symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this gives the Hopf algebra of Grossman and Larson. We show the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.

math.CO↗

arXiv2000

Relations of multiple zeta values and their algebraic expression

We establish a new class of relations among the multiple zeta values ζ(k_1,k_2,...,k_n), which we call the cyclic sum identities. These identities have an elementary proof, and imply the "sum theorem" for multiple zeta values. They also have a succinct statement in terms of "cyclic derivations" as introduced by Rota, Sagan and Stein. In addition, we discuss the expression of other relations of multiple zeta values via the shuffle and "harmonic" products on the underlying vector space H of the noncommutative polynomial ring Q , and also using an action on Q of the Hopf algebra of quasi-symmetric functions.

math.QA↗

arXiv1999

Periods of mirrors and multiple zeta values

In a recent paper, A. Libgober showed that the multiplicative sequence {Q_i(c_1,...,c_i)} of Chern classes corresponding to the power series Q(z)=1/Gamma(1+z) appears in a relation between the Chern classes of certain Calabi-Yau manifolds and the periods of their mirrors. We show that the polynomials Q_i can be expressed in terms of multiple zeta values.

math.AG↗

arXiv1999

Quasi-shuffle products

Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication * on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product. The resulting commutative algebra can be given the structure of a Hopf algebra (_A_,*,Delta). In the case where A is the set of positive integers and the operation on A is addition, (_A_,*,Delta) is the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, there is a Hopf algebra isomorphism exp from the shuffle Hopf algebra on A onto (_A_,*,Delta). We discuss the dual of (_A_,*,Delta), and define a deformation *_q of * that coincides with * when q = 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples of this construction.

math.QA↗