Shuffle products for multiple zeta values and partial fraction decompositions of zeta-functions of root systems

The shuffle product plays an important role in the study of multiple zeta values (MZVs). This is expressed in terms of multiple integrals, and also as a product in a certain non-commutative polynomial algebra over the rationals in two indeterminates. In this paper, we give a new interpretation of the shuffle product. In fact, we prove that the procedure of shuffle products essentially coincides with that of partial fraction decompositions of MZVs of root systems. As an application, we give a proof of extended double shuffle relations without using Drinfel??d integral expressions for MZVs. Furthermore, our argument enables us to give some functional relations which include double shuffle relations.     

The shuffle relation of fractions from multiple zeta values

Partial fraction methods play an important role in the study of multiple zeta values. One class of such fractions is related to the integral representations of MZVs. We show that this class of fractions has a natural shuffle algebra structure. This finding conceptualizes the connections among the various methods of stuffle, shuffle and partial fractions in the study of MZVs. This approach also gives an explicit product formula for the fractions.     

On the q-ampleness of the tensor product of two line bundles

Multiple zeta values (MZVs) are generalizations of Riemann zeta values at positive integers to multiple variable setting. These values can be further generalized to level N multiple polylog values by evaluating multiple polylogs at Nth roots of unity. In this paper, we consider another level N generalization by restricting the indices in the iterated sums defining MZVs to congruence classes modulo N, which we call the MZVs at level N. The goals of this paper are twofold. First, we shall lay down the theoretical foundations of these values such as their regularizations and double shuffle relations. Second, we will generalize the bracket functions related to multiple divisor sums defined by Bachmann and Kühn to arbitrary level N and study their relations to MZVs at level N. The brackets are all q-series and similar to MZVs, they have both weight and depth filtrations. But unlike that of MZVs, the product of brackets usually has mixed weights; however, after projecting to the highest weight we can obtain an algebra homomorphism from brackets to MZVs. Moreover, the image of the derivation \({\mathfrak{D}=q\frac{d}{dq}}\) on brackets vanishes on the MZV side, which gives rise to many nontrivial \({\mathbb{Q}}\)-linear relations.     

Polylogarithmes multiples uniformes en une variable

Various single-valued versions of ordinary polylogarithms Li_n(z) have been constructed by Ramakrishnan, Wojtkowiak, Zagier, and others. These single-valued functions are generalisations of the Bloch–Wigner dilogarithm and have many applications in mathematics. In this Note we show how to construct explicit single-valued versions of multiple polylogarithms in one variable. We prove the functions thus constructed are linearly independent, that they satisfy the shuffle relations, and that every possible single-valued version of multiple polylogarithms in one variable can be obtained in this way. To cite this article: F.C.S. Brown, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Structure Theorems of Mixable Shuffle Algebras

Mixable shuffle algebras are generalizations of the well-known shuffle algebra and quasi-shuffle algebra with broad applications. In this article we study the ring theoretic structures of mixable shuffle algebras with coefficients in a field motivated by the well-known work of Radford that a shuffle algebra with rational coefficients is a polynomial algebra in Lyndon words. To consider coefficients in a field of positive characteristic p, we carefully study the Lyndon words and their p-variations. As a result, we determine the structures of a quite large class of mixable shuffle algebras by providing explicit sets of generators and relations.     

Renormalization of multiple zeta values

Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at non-positive integers since the values are usually undefined. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of Ihara–Kaneko–Zagier on renormalization of MZVs with positive arguments. We further show that the important quasi-shuffle (stuffle) relation for usual MZVs remains true for the renormalized MZVs.     

Mixable shuffles, quasi-shuffles and Hopf algebras

The quasi-shuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasi-shuffle product algebras as subalgebras of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free Rota–Baxter algebras.     

Cohomological properties of the quantum shuffle product and application to the construction of quasi-Hopf algebras

For a commutative algebra the shuffle product is a morphism of complexes. We generalize this result to the quantum shuffle product, associated to a class of non-commutative algebras (for example all the Hopf algebras). As a first application we show that the Hochschild-Serre identity is the dual statement of our result. In particular, we extend this identity to Hopf algebras. Secondly, we clarify the construction of a class of quasi-Hopf algebras.     

Generalized Shuffles Related to Nijenhuis and TD-Algebras

Shuffle type products are well known in mathematics and physics. They are intimately related to Loday's dendriform algebras and were extensively used to give explicit constructions of free Rota–Baxter algebras. In the literature there exist at least two other Rota–Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators. Our construction is related to Loday's unital commutative tridendriform algebra, including the involutive case. The concept of Rota–Baxter, Nijenhuis and TD-bialgebras is introduced at the end, and we show that any commutative bialgebra provides such objects.     

Permutation Groups, a Related Algebra and a Conjecture of Cameron

We consider the permutation group algebra defined by Cameron and show that if the permutation group has no finite orbits, then no homogeneous element of degree one is a zero-divisor of the algebra. We proceed to make a conjecture which would show that the algebra is an integral domain if, in addition, the group is oligomorphic. We go on to show that this conjecture is true in certain special cases, including those of the form H Wr S and H Wr A, and show that in the oligormorphic case, the algebras corresponding to these special groups are polynomial algebras. In the H Wr A case, the algebra is related to the shuffle algebra of free Lie algebra theory.     

Polynomial Poisson Algebras with Regular Structure of Symplectic Leaves

We study polynomial Poisson algebras with some regularity conditions. Linear (Lie–Berezin–Kirillov) structures on dual spaces of semisimple Lie algebras, quadratic Sklyanin elliptic algebras, and the polynomial algebras recently described by Bondal, Dubrovin, and Ugaglia belong to this class. We establish some simple determinant relations between the brackets and Casimir functions of these algebras. In particular, these relations imply that the sum of degrees of the Casimir functions coincides with the dimension of the algebra in the Sklyanin elliptic algebras. We present some interesting examples of these algebras and show that some of them arise naturally in the Hamiltonian integrable systems. A new class of two-body integrable systems admitting an elliptic dependence on both coordinates and momenta is among these examples.     

On generators and relations for certain involutory subalgebras of kac-moody lie algebras ∗

We find generators and relations for those subalgebras of Kac-Moody Lie algebras that are the fixed point algebras of certain involutions. Specifically the involution must involve the Cartan involution which interchanges the positive and negative generators. We go on to apply these results to the G.I.M. algebras, which were introduced as natural generalizations of Kac-Moody algebras by P. Slodowy. We show such algebras are isomorphic to subalgebras of Kac-Moody algebras. From this we are able to derive someinteresting interrelations between certain Kac-Moody algebras.     

Combinatorial study of colored Hurwitz polyzêtas

A combinatorial study discloses two surjective morphisms between generalized shuffle algebras and algebras generated by the colored Hurwitz polyzêtas. The combinatorial aspects of the products and co-products involved in these algebras will be examined.     

Hilbert-Carleman and regularized determinants for linear operators

A general theory of regularized and Hilbert-Carleman determinants in normed algebras of operators acting in Banach spaces is proposed. In this approach regularized determinants are defined as continuous extensions of the corresponding determinants of finite dimensional operators. We characterize the algebras for which such extensions exist, describe the main properties of the extended determinants, obtain Cramer's rule and the formulas for the resolvent which are expressed via the extended tracestr(A ^k ) of iterations and regularized determinants.This paper is a continuation of the paper [GGKr].     

Lattice subordinations and Priestley duality

There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. We show that the correspondence between Heyting algebras and S4-algebras extends naturally to distributive lattices and Boolean algebras with a lattice subordination. We also introduce Heyting lattice subordinations and prove that the category of Boolean algebras with a Heyting lattice subordination is isomorphic to the category of S4-algebras, thus obtaining the correspondence between Heyting algebras and S4-algebras as a particular case of our approach. In addition, we provide a uniform approach to dualities for these classes of algebras. Namely, we generalize Priestley spaces to quasi-ordered Priestley spaces and show that lattice subordinations on a Boolean algebra B correspond to Priestley quasiorders on the Stone space of B. This results in a duality between the category of Boolean algebras with a lattice subordination and the category of quasi-ordered Priestley spaces that restricts to Priestley duality for distributive lattices. We also prove that Heyting lattice subordinations on B correspond to Esakia quasi-orders on the Stone space of B. This yields Esakia duality for S4-algebras, which restricts to Esakia duality for Heyting algebras.     

Pluriassociative algebras I: The pluriassociative operad

Diassociative algebras form a category of algebras recently introduced by Loday. A diassociative algebra is a vector space endowed with two associative binary operations satisfying some very natural relations. Any diassociative algebra is an algebra over the diassociative operad, and, among its most notable properties, this operad is the Koszul dual of the dendriform operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of diassociative algebras, called γ-pluriassociative algebras, so that 1-pluriassociative algebras are diassociative algebras. Pluriassociative algebras are vector spaces endowed with 2γ associative binary operations satisfying some relations. We provide a complete study of the γ-pluriassociative operads, the underlying operads of the category of γ-pluriassociative algebras. We exhibit a realization of these operads, establish several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct the free objects in the corresponding categories. We also study several notions of units in γ-pluriassociative algebras and propose a general way to construct such algebras. This paper ends with the introduction of an analogous generalization of the triassociative operad of Loday and Ronco.     

Cocommutative Hopf Algebras of Permutations and Trees

Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980's by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra. Aguiar supported in part by NSF grant DMS-0302423. Sottile supported in part by NSF CAREER grant DMS-0134860, the Clay Mathematics Institute, and MSRI.     

Based algebras and standard bases for quasi-hereditary algebras

Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly based algebra, which is a generalized version of a cellular algebra introduced by Graham and Lehrer, and discuss their representation theory. The main result is that an algebra over a commutative local noetherian ring with finite rank is split quasi-hereditary if and only if it is standardly full-based. As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.     

On an algebraic version of the Knizhnik–Zamolodchikov equation

A difference equation analogue of the Knizhnik?CZamolodchikov equation is exhibited by developing a theory of the generating function H(z) of Hurwitz polyzeta functions to parallel that of the polylogarithms. By emulating the role of the KZ equation as a connection on a suitable bundle, a difference equation version of the notion of connection is developed for which H(z) is a flat section. Solving a family of difference equations satisfied by the Hurwitz polyzetas leads to the normalized multiple Bernoulli polynomials (NMBPs) as the counterpart to the Hurwitz polyzeta functions, at tuples of non-positive integers. A generating function for these polynomials satisfies a similar difference equation to that of H(z), but in contrast to the fact that said polynomials have rational coefficients, the algebraic independence of the usual Hurwitz zeta functions is proven, and the Hurwitz polyzeta functions are shown to satisfy no algebraic relations other than those arising from the shuffle relations. The values of the NMBPs at z=1 provide a regularization of the multiple zeta values at tuples of negative integers, which is shown to agree with the regularization given in Akiyama et al. (Acta Arith. 98:107?C116, 2001). Various elementary properties of these values are proven.