7,880 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
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
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.Comment: 23 pages, 7 Postscript figures (uses epsfig.sty
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
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
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