We give a simple proof of the formula for the coproduct ∆ in the Hopf algebra of motivic iterated integrals on the affine line. We show that it encodes the group law in the group of automorphisms of certain non commutative variety. We relate the coproduct ∆ with the coproduct in the Hopf algebra of decorated rooted planar trivalent trees defined in chapter 4 – a version of the one defined by Co...

Sustainable use of biomass as a renewable source of energy can be an alternative solution to the cost of fossilbased energy and global warming. Production of biofuel from plant biomass results not only in bio-based energy, but also in coproducts containing lignin, modified lignin, and lignin derivatives. This paper discusses the moisture susceptibility of subgrade soil stabilized by bio-based e...

We formulate the Hopf algebra underlying the su(2|2) worldsheet S-matrix of the AdS 5 × S 5 string in the AdS/CFT correspondence. For this we extend the previous construction in the su(1|2) subsector due to Janik to the full algebra by specifying the action of the coproduct and the antipode on the remaining generators. The nontriviality of the coproduct is determined by length-changing effects ...

We study how the comultiplication on a Hopf algebra can be modified in such a way that the new comultiplication together with the original multiplication and a suitable antipode gives a new Hopf algebra. To this end, we have to study Harrison type cocycles, and it turns out that there is a relation with the Yang-Baxter equation. The construction is applied to deform the coalgebra structure on t...

We introduce a new class of combinatorial objects, which we call “bipieri tableaux,” which arise in a natural way from the evaluation of products consisting of commutative or noncommutative complete homogeneous symmetric functions and elementary symmetric functions through repeated applications of Pieri rules. We prove using sign-reversing involutions on bipieri tableaux an elegant coproduct fo...

We explain our previous results about Hochschild actions (2007/2008) pertaining, in particular, to the coproduct, which appeared a different form work by Goresky and Hingston (2009), provide fresh look at results. recall general action, specialize aforementioned coproduct prove that assumption of commutativity, made for convenience article (2008), is not needed. give detailed background materia...

We extend our work in [1] to the case of Hopf comodule coalgebras. We introduce the cocylindrical module C♮H, where H is a Hopf algebra with bijective antipode and C is a Hopf comodule coalgebra over H. We show that there exists an isomorphism between the cocyclic module of the crossed coproduct coalgebra C>◭H and ∆(C♮H), the cocyclic module related to the diagonal of C♮H. We approximate HC(C >...

In [1], we associate an oriented graph with each coalgebra and define the notion of (Markov) L-coalgebra, i.e. a k-vector space equipped with a left coproduct, ∆̃ and a right coproduct ∆, verifying the coassociativity breaking equation (∆̃⊗id)∆ = (id⊗∆)∆̃. The aim of this work is twofold. Firstly, we show that some coassociative coalgebras are closely related to Markov co-dialgebras, (i.e. Markov ...

Extension of the braid relations to the multiple braided tensor product of algebras that can be used for quantization of nonultralocal models is presented. The Yang– Baxter–type consistency conditions as well as conditions for the existence of the multiple coproduct (monodromy matrix), which can be used for construction of the commuting subalgebra, are given. Homogeneous and local algebras are ...