Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

http://www.maths.leeds.ac.uk/~marsh/research_articles/pp18.pdf

Tilting theory and cluster combinatorics

Groupes et algébres de lie

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of ‘principal’ coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parameterizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multi-gradings in cluster algebras with principal coefficients. The properties of these parameterizations, some proven and some conjectural, suggest links to duality conjectures of Fock and Goncharov. The coefficient dynamics leads to a natural generalization of Zamolodchikov's -systems, previously known in finite type only, and sharpen the periodicity result from an earlier paper. For cluster algebras of finite type, we identify a canonical ‘universal’ choice of coefficients such that an arbitrary cluster algebra can be obtained from the universal one (of the same type) by an appropriate specialization of coefficients.

/pdf/cluster-algebras-iv-coefficients-2922owyk1s.pdf

Cluster algebras IV: Coefficients

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A_n. We associate to each cluster C of U an abelian category Cat_C such that the indecomposable objects of Cat_C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of Cat_C. Then, we generalize the ``denominator Theorem'' of Fomin and Zelevinsky to any cluster.

/pdf/quivers-with-relations-arising-from-clusters-a-n-case-2cfr4gx9i8.pdf

Quivers with relations arising from clusters $(A_n$ case)

Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding quiver representation category. This also provides some explicit formulas for cluster variables.

/pdf/cluster-algebras-as-hall-algebras-of-quiver-representations-111drjq8as.pdf

Cluster algebras as Hall algebras of quiver representations

Une algebre pre-Lie est un espace vectoriel L muni d'un produit * de L⊗ L dans L qui verifie la relation (x*y)*z-x*(y*z)= (x*z)*y-x*(z*y), pour tous x,y,z dans L. On donne une description combinatoire explicite, en termes d'arbres enracines, de l'operade associee a ce type d'algebres, et on demontre que cette operade est de Koszul.

Pre-Lie algebras and the rooted trees operad

/pdf/quivers-with-relations-arising-from-clusters-a-n-case-35qi7nhgti.pdf

Quivers with relations arising from clusters (A_n case)

Dialgebras.- Dialgebra (co)homology with coefficients.- Un endofoncteur de la categorie des operades.- Un theoreme de Milnor-Moore pour les algebres de Leibniz.

Frédéric Chapoton

Papers

Quivers with relations arising from clusters $(A_n$ case)

Cluster algebras as Hall algebras of quiver representations

Pre-Lie algebras and the rooted trees operad

Quivers with relations arising from clusters (A_n case)

Dialgebras and Related Operads