152 research outputs found
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.Comment: 14 pages, no figure, references update
Renormalisation of q-regularised multiple zeta values
We consider a particular one-parameter family of q-analogues of multiple zeta
values. The intrinsic q-regularisation permits an extension of these q-multiple
zeta values to negative integers. Renormalised multiple zeta values satisfying
the quasi-shuffle product are obtained using an Hopf algebraic Birkhoff
factorisation together with minimal subtraction.Comment: minor correction
Generalized shuffles related to Nijenhuis and TD-algebras
Shuffle and quasi-shuffle products are well-known in the mathematics
literature. They are intimately related to Loday's dendriform algebras, and
were extensively used to give explicit constructions of free commutative
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 product, 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
and show that the free unital commutative Nijenhuis algebra is a TD-algebra. We
relate our construction to Loday's unital commutative dendriform trialgebras,
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.Comment: 20 pages, typos corrected, accepted for publication in Communications
in Algebr
The Hopf algebra of finite topologies and T-partitions
A noncommutative and noncocommutative Hopf algebra on finite topologies H_T
is introduced and studied (freeness, cofreeness, self-duality...). Generalizing
Stanley's definition of P-partitions associated to a special poset, we define
the notion of T-partitions associated to a finite topology, and deduce a Hopf
algebra morphism from H_T to the Hopf algebra of packed words WQSym.
Generalizing Stanley's decomposition by linear extensions, we deduce a
factorization of this morphism, which induces a combinatorial isomorphism from
the shuffle product to the quasi-shuffle product of WQSym. It is strongly
related to a partial order on packed words, here described and studied.Comment: 33 pages. Second version, a few typos correcte
Generalized Matsumoto-Tits sections and quantum quasi-shuffle algebras
In this paper generalized Matsumoto-Tits sections lifting permutations to the
algebra associated to a generalized virtual braid monoid are defined. They are
then applied to study the defining relations of the quantum quasi-shuffle
algebras via the total symmetrization operator.Comment: 18 page
Duality and (q-)multiple zeta values
Following Bachmann's recent work on bi-brackets and multiple Eisenstein
series, Zudilin introduced the notion of multiple q-zeta brackets, which
provides a q-analog of multiple zeta values possessing both shuffle as well as
quasi-shuffle relations. The corresponding products are related in terms of
duality. In this work we study Zudilin's duality construction in the context of
classical multiple zeta values as well as various q-analogs of multiple zeta
values. Regarding the former we identify the derivation relation of order two
with a Hoffman-Ohno type relation. Then we describe relations between the
Ohno-Okuda-Zudilin q-multiple zeta values and the Schlesinger-Zudilin
q-multiple zeta values.Comment: revised version, accepted for publication in Advances in Mathematic
- …