85 research outputs found
Derived bracket construction and Manin products
We will extend the classical derived bracket construction to any algebra over
a binary quadratic operad. We will show that the derived product construction
is a functor given by the Manin white product with the operad of permutation
algebras. As an application, we will show that the operad of prePoisson
algebras is isomorphic to Manin black product of the Poisson operad with the
preLie operad. We will show that differential operators and Rota-Baxter
operators are, in a sense, Koszul dual to each other.Comment: This is the final versio
Post-Lie Algebras, Factorization Theorems and Isospectral-Flows
In these notes we review and further explore the Lie enveloping algebra of a
post-Lie algebra. From a Hopf algebra point of view, one of the central
results, which will be recalled in detail, is the existence of a second Hopf
algebra structure. By comparing group-like elements in suitable completions of
these two Hopf algebras, we derive a particular map which we dub post-Lie
Magnus expansion. These results are then considered in the case of
Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined
in terms of solutions of modified classical Yang-Baxter equation. In this
context, we prove a factorization theorem for group-like elements. An explicit
exponential solution of the corresponding Lie bracket flow is presented, which
is based on the aforementioned post-Lie Magnus expansion.Comment: 49 pages, no-figures, review articl
Combinatorial Hopf algebras from renormalization
In this paper we describe the right-sided combinatorial Hopf structure of
three Hopf algebras appearing in the context of renormalization in quantum
field theory: the non-commutative version of the Fa\`a di Bruno Hopf algebra,
the non-commutative version of the charge renormalization Hopf algebra on
planar binary trees for quantum electrodynamics, and the non-commutative
version of the Pinter renormalization Hopf algebra on any bosonic field. We
also describe two general ways to define the associative product in such Hopf
algebras, the first one by recursion, and the second one by grafting and
shuffling some decorated rooted trees.Comment: 16 page
Rota-Baxter algebras and new combinatorial identities
The word problem for an arbitrary associative Rota-Baxter algebra is solved.
This leads to a noncommutative generalization of the classical Spitzer
identities. Links to other combinatorial aspects, particularly of interest in
physics, are indicated.Comment: 8 pages, improved versio
Geometric combinatorial algebras: cyclohedron and simplex
In this paper we report on results of our investigation into the algebraic
structure supported by the combinatorial geometry of the cyclohedron. Our new
graded algebra structures lie between two well known Hopf algebras: the
Malvenuto-Reutenauer algebra of permutations and the Loday-Ronco algebra of
binary trees. Connecting algebra maps arise from a new generalization of the
Tonks projection from the permutohedron to the associahedron, which we discover
via the viewpoint of the graph associahedra of Carr and Devadoss. At the same
time that viewpoint allows exciting geometrical insights into the
multiplicative structure of the algebras involved. Extending the Tonks
projection also reveals a new graded algebra structure on the simplices.
Finally this latter is extended to a new graded Hopf algebra (one-sided) with
basis all the faces of the simplices.Comment: 23 figures, new expanded section about Hopf algebra of simplices,
with journal correction
Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element
When W is a finite reflection group, the noncrossing partition lattice NCP_W
of type W is a rich combinatorial object, extending the notion of noncrossing
partitions of an n-gon. A formula (for which the only known proofs are
case-by-case) expresses the number of multichains of a given length in NCP_W as
a generalised Fuss-Catalan number, depending on the invariant degrees of W. We
describe how to understand some specifications of this formula in a case-free
way, using an interpretation of the chains of NCP_W as fibers of a
Lyashko-Looijenga covering (LL), constructed from the geometry of the
discriminant hypersurface of W. We study algebraically the map LL, describing
the factorisations of its discriminant and its Jacobian. As byproducts, we
generalise a formula stated by K. Saito for real reflection groups, and we
deduce new enumeration formulas for certain factorisations of a Coxeter element
of W.Comment: 18 pages. Version 2 : corrected typos and improved presentation.
Version 3 : corrected typos, added illustrated example. To appear in Journal
of Algebraic Combinatoric
A Unified Algebraic Approach to Classical Yang-Baxter Equation
In this paper, the different operator forms of classical Yang-Baxter equation
are given in the tensor expression through a unified algebraic method. It is
closely related to left-symmetric algebras which play an important role in many
fields in mathematics and mathematical physics. By studying the relations
between left-symmetric algebras and classical Yang-Baxter equation, we can
construct left-symmetric algebras from certain classical r-matrices and
conversely, there is a natural classical r-matrix constructed from a
left-symmetric algebra which corresponds to a parak\"ahler structure in
geometry. Moreover, the former in a special case gives an algebraic
interpretation of the ``left-symmetry'' as a Lie bracket ``left-twisted'' by a
classical r-matrix.Comment: To appear in Journal of Physics A: Mathematical and Theoretica
Polyhedral models for generalized associahedra via Coxeter elements
Motivated by the theory of cluster algebras, F. Chapoton, S. Fomin and A.
Zelevinsky associated to each finite type root system a simple convex polytope
called \emph{generalized associahedron}. They provided an explicit realization
of this polytope associated with a bipartite orientation of the corresponding
Dynkin diagram.
In the first part of this paper, using the parametrization of cluster
variables by their -vectors explicitly computed by S.-W. Yang and A.
Zelevinsky, we generalize the original construction to any orientation. In the
second part we show that our construction agrees with the one given by C.
Hohlweg, C. Lange, and H. Thomas in the setup of Cambrian fans developed by N.
Reading and D. Speyer.Comment: 31 pages, 2 figures. Changelog: 20111106: initial version 20120403:
fixed errors in figures 20120827: revised versio
Categorification of skew-symmetrizable cluster algebras
We propose a new framework for categorifying skew-symmetrizable cluster
algebras. Starting from an exact stably 2-Calabi-Yau category C endowed with
the action of a finite group G, we construct a G-equivariant mutation on the
set of maximal rigid G-invariant objects of C. Using an appropriate cluster
character, we can then attach to these data an explicit skew-symmetrizable
cluster algebra. As an application we prove the linear independence of the
cluster monomials in this setting. Finally, we illustrate our construction with
examples associated with partial flag varieties and unipotent subgroups of
Kac-Moody groups, generalizing to the non simply-laced case several results of
Gei\ss-Leclerc-Schr\"oer.Comment: 64 page
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