Shuffle Product Research Articles

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.

An algebraic scheme associated with the non-commutative KP hierarchy and some of its extensions

A well-known ansatz (`trace method') for soliton solutions turns the equations of the (noncommutative) KP hierarchy, and those of certain extensions, into families of algebraic sum identities. We develop an algebraic formalism, in particular involving a (mixable) shuffle product, to explore their structure. More precisely, we show that the equations of the noncommutative KP hierarchy and its extension (xncKP) in the case of a Moyal-deformed product, as derived in previous work, correspond to identities in this algebra. Furthermore, the Moyal product is replaced by a more general associative product. This leads to a new even more general extension of the noncommutative KP hierarchy. Relations with Rota-Baxter algebras are established.

Itô calculus and quantisation of Lie bialgebras

Motivated by classical and quantum stochastic flows, particularly the distinction between inner and outer flows in the quantum case, we develop analogous notions which live in the Itô Hopf algebra, which is got by replacing the shuffle product in a shuffle Hopf algebra by a noncommutative multiplication which abstracts the product rule for iterated quantum stochastic integrals. Inner flows got by conjugation by ordered double product integrals ∏ → ← ( 1 + d r [ h ] ) are used to quantise Lie bialgebras.

Double product integrals and Enriquez quantization of Lie bialgebras I: The quasitriangular identities

Let 𝒯(ℒ) be the space of all tensors over a Lie algebra ℒ in which the Lie bracket is obtained by taking commutators in an associative algebra. We show that 𝒯(ℒ) becomes a Hopf algebra when equipped with a noncommutative modification of the shuffle product together with the standard coproduct. A definition is given of directed double product integrals as iterated single product integrals driven by formal power series with coefficients in the tensor product of ℒ with an appropriate associative algebra. For the Hopf algebra T(L)[[h]] of formal power series we show that elements R[h] of (T(L)⊗T(L))[[h]] satisfying (Δ⊗id)R[h]=R[h]13R[h]23, (id⊗Δ)R[h]=R[h]13R[h]12,and which are unitalized by the counit in either copy of 𝒯(ℒ), can be characterized as such directed double product integrals ∏∏(1+d⃗⊗d↙r[h]) where r[h] is a formal power series with coefficients in L⊗L and vanishing constant term.

The Hochschild homology of fatpoints

This paper calculates the Hochschild homology of fatpoints—rings of the formk[x 1,x 2,…,x d ]/m a wherek is a field of characteristic zero and m=(x 1 x 2,…,x d ). The calculation includes the multiplicative structure induced by the shuffle product. The answer is given in terms of the homology of tori relative to theira-diagonal. Since the rings in question are monoidal, this calculation also determines their topological Hochschild homology. By a theorem of Goodwillie, the dimensions of the cyclic homology groups of these rings are determined as well.

Where do the tedious products of ζ's come from?

Lamentably, the full analytical content of the ε-expansion of the master two-loop two-point function, with arbitrary self-energy insertions in 4−2ε dimensions, is still unknown. Here we show that multiple zeta values (MZVs) of weights up to 12 suffice through O( ε 9). Products of primitive MZVs are generated by a processes of “pseudo-exponentiation” whose combinatorics faithfully accord with expectations based on Kreimer's modified shuffle product and on the Drinfeld-Deligne conjecture. The existence of such a mechanism, relating thousands of complicated rational numbers, enables us to identify precise and simple combinations of MZVs specific to quantum field theories in even numbers of spacetime dimensions.

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.

Lyndon words, polylogarithms and the Riemann [formula omitted] function

The algebra of polylogarithms (iterated integrals over two differential forms ω 0= dz/z and ω 1= dz/(1−z) ) is isomorphic to the shuffle algebra of polynomials on non-commutative variables x 0 and x 1 . The multiple zeta values (MZVs) are obtained by evaluating the polylogarithms at z=1. From a second shuffle product, we compute a Gröbner basis of the kernel of this evaluation morphism. The completeness of this Gröbner basis up to order 12 is equivalent to the classical conjecture about MZVs. We also show that certain known relations on MZVs hold for polylogarithms.

Finite generation of Hochschild homology algebras

We prove converses of the Hochschild-Kostant-Rosenberg Theorem, in particular: If a commutative algebra S is flat and essentially of finite type over a noetherian ring \(\mathbb{k}\), and the Hochschild homology HH*(S|\(\mathbb{k}\)) is a finitely generated S-algebra for shuffle products, then S is smooth over \(\mathbb{k}\).

Baxter Algebras and Shuffle Products

In this paper we generalize the well-known construction of shuffle product algebras by using mixable shuffles, and prove that any free Baxter algebra is isomorphic to a mixable shuffle product algebra. This gives an explicit construction of the free Baxter algebra, extending the work of Rota and Cartier.

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 a on the set of noncommutative polynomials in A which we call a quasi-shuffle products it can be viewed as a generalization of the shuffle product III. We extend this commutative algebra structure to a Hopf algebra (U, a, Δ)s in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, the quasi-shuffle product is in fact no more general than the shuffle products we give an isomorphism exp of the shuffle Hopf algebra (U, III, Δ) onto (U, a, Δ) the set L of Lyndon words on A and their images l exp(w) m w e Lr freely generate the algebra (U, a). We also consider the graded dual of (U, a, Δ). We define a deformation a_q of a that coincides with a when q e 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples, particularly the algebra of quasi-symmetric functions (dual to the noncommutative symmetric functions) and the algebra of Euler sums.

On free p-superalgebras

In this article we consider several aspects of algebraic combinatorics and combinatorial algebra over fields of prime characteristics. P-super-Radford theorem gives the structure of the free associative algebra over a field of prime characteristic with the new multiplication given by the super shuffle product, we show that this algebra is isomorphic to the reduced free super commutative algebra on s-regular words. We prove the elimination theorem for free partially commutative color Lie p-superalgebras and obtain a Schreier type formula for free Lie p-superalgebras using formal power series techniques.

Almost Sure Convergence of the Numerical Discretization of Stochastic Jump Diffusions

Based on the shuffle product expansion of exponential Lie series in terms of a Philip Hall basis for the stochastic differential equations of jump-diffusion type, we can establish Stratonovich–Taylor–Hall (STH) schemes. However, the STHr scheme converges only at order r in the mean-square sense. In order to have the almost sure Stratonovich–Taylor–Hall (ASTH) schemes, we have to include all the terms related to multiple Poissonian integrals as the moments of multiple Poissonian integrals always have lower orders of magnitudes as compared with those of multiple Brownian integrals.

Inverse for the shuffle for algebraic series

Let K be a commutative field of characteristic 0, and a an algebraic series of one variable with coefficients in K, which is invertible for the shuffle product, i.e. it exists an algebraic series b over K such that a̫ b==1. Then a is a rational series. The same result is true when a and b are an algebraic series of many non-commutative variables. One application of this result is: If M is a linear system which is invertible, and with an output function ) and and N is a linear system with an output function ) suc such that N and the gen and the generating series H for M and G for N are both algebraic, then the systems M and N are bilinear.

Bi-catenation and shuffle product of languages

The shuffle product of two words consists of all words obtained by inserting one word into another word sparsely. The shuffle product of two languages is the union of all the shuffle products of two words taken one from each of these two languages. The bi-catenation of two languages A andB is the set \(A\star B = AB\cup BA\). A non-empty word which is not a power of any other word is called a primitive word. A language is a prefix code if no word in this language is a prefix of any other word in this language. This paper is devoted to the investigation of the elementary properties of bi-catenation and shuffle product of languages. The families of prefix codes, disjunctive languages and languages consisting of primitive words with respective to these two operations are studied. We characterize languages of which the bi-catenation or the shuffle product with any non-empty word are prefix codes. We also derive that for any bifix code A, both \(A\star Q\) and \(A\star Q^{(n)}\), \(n\geq 2\), are disjunctive languages, where Q is the set of all primitive words over an alphabet X with more than one letter and \(Q^{(n)} = \{ f^n\mid f\in Q\}\). For the shuffle product case, surprisingly \(a\diamond Q\) is a regular language, where a is a letter of the alphabet X.

Combinatorial Aspects of Multiple Zeta Values

Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle product rule allows the possibility of a combinatorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with certain repeated arguments. We also prove a similar cyclic sum identity. Finally, we present extensive computational evidence supporting an infinite family of conjectured MZV identities that simultaneously generalize the Zagier identity.

Nonfinite axiomatizability of the equational theory of shuffle

We consider language structures ${\bf L}_\Sigma = (P_\Sigma,\cdot,\otimes,+,1,0)$ , where $P_\Sigma$ consists of all subsets of the free monoid $\Sigma^*$ ; the binary operations $\cdot$ , $\otimes$ and $+$ are concatenation, shuffle product and union, respectively, and where the constant 0 is the empty set and the constant 1 is the singleton set containing the empty word. We show that the variety Lang generated by the structures ${\bf L}_\Sigma$ has no finite axiomatization. In fact we establish a stronger result: The variety Lang has no finite axiomatization over the variety of ordered algebras ${\bf Lg}_\leq$ generated by the structures $(P_\Sigma,\cdot,\otimes,1,\subseteq)$ , where $\subseteq$ denotes set inclusion.

Axiomatizing Shuffle and Concatenation in Languages

We consider the varietyLanggenerated by all language structures (PΣ, ·, ⊗ +, 0, 1), and the varietyLg⩽of ordered algebras generated by the structures (PΣ, ·, ⊗ 0, 1, ⊆), wherePΣis the powerset ofΣ*, and whereB·Cis the complex concatenation of the languagesB, C⊆Σ*,B⊗Cis their shuffle product, andB+Cis their union. We prove that for each finite setEof equations valid inLangthere is a (finite) modelSEofEin which some inequation valid inLg⩽fails. It follows that neither variety is finitely axiomatizable.

Adams operations for projective modules over group rings

Let R be a commutative ring, Γ a finite group acting on R, and let k∈ℕ be invertible in R. Generalizing a definition of Kervaire, we construct an Adams operation ψk on the Grothendieck group and on the higher K-theory of projective modules over the twisted group ring R#Γ. For this, we generalize Atiyah's cyclic power operations and use shuffle products in higher K-theory. For the Grothendieck group, we show that ψk is multiplicative and that it commutes with base change, with the Cartan homomorphism, and with ψl for any other l which is invertible in R.

Noncommutative symmetric functions III : Deformations of Cauchy and convolution algebras

[in "Special Issue : Lie Computations", G. Jacob, V. Koseleff, Eds.] This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple q-analogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in mathematical physics (the quon algebra, generalized Brownian motion).