scholarly journals Algebraic Structures on Typed Decorated Rooted Trees

2021 ◽  
Author(s):  
Loïc Foissy ◽  
◽  
Keyword(s):  
Hopf Algebra ◽  
Lie Algebras ◽  
Hopf Algebras ◽  
Rooted Trees ◽  
Stochastic Pdes ◽  
Algebraic Structures ◽  
Contraction Process

Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation process on stochastic PDEs. We here study the algebraic structures on these objects: multiple pre-Lie algebras and related operads (generalizing a result by Chapoton and Livernet), noncommutative and cocommutative Hopf algebras (generalizing Grossman and Larson's construction), commutative and noncocommutative Hopf algebras (generalizing Connes and Kreimer's construction), bialgebras in cointeraction (generalizing Calaque, Ebrahimi-Fard and Manchon's result). We also define families of morphisms and in particular we prove that any Connes-Kreimer Hopf algebra of typed and decorated trees is isomorphic to a Connes-Kreimer Hopf algebra of non-typed and decorated trees (the set of decorations of vertices being bigger), through a contraction process, and finally obtain the Bruned-Hairer-Zambotti construction as a subquotient.

scholarly journals Hopf algebras on planar trees and permutations

2021 ◽  
Author(s):  
Diego Arcis ◽  
Sebastián Márquez
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Rooted Trees ◽  
Labeled Trees ◽  
Different Types ◽  
Structure Lead ◽  
Planar Rooted Trees

We endow the space of rooted planar trees with the structure of a Hopf algebra. We prove that variations of such a structure lead to Hopf algebras on the spaces of labeled trees, [Formula: see text]-trees, increasing planar trees and sorted trees. These structures are used to construct Hopf algebras on different types of permutations. In particular, we obtain new characterizations of the Hopf algebras of Malvenuto–Reutenauer and Loday–Ronco via planar rooted trees.

scholarly journals On the Lie enveloping algebra of a pre-Lie algebra

2008 ◽  
Vol 2 (1) ◽  
pp. 147-167 ◽  
Author(s):  
J.-M. Oudom ◽  
D. Guin
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Rooted Trees ◽  
Enveloping Algebra ◽  
Associative Product ◽  
Planar Rooted Trees ◽  
Symmetric Brace Algebras

AbstractWe construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. It turns S(L) into a Hopf algebra which is isomorphic to the enveloping algebra of LLie. Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees.

scholarly journals Categorification of Hopf algebras of rooted trees

2013 ◽  
Vol 11 (3) ◽  
Author(s):  
Joachim Kock
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Base Change ◽  
Rooted Trees ◽  
Finite Sets ◽  
Polynomial Representation ◽  
Monoidal Structure

AbstractWe exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec ℕ) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ℤ and collapsing H 0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring in the polynomial representation of the free monad on P.

scholarly journals Decomposition Spaces and Restriction Species

2018 ◽  
Vol 2020 (21) ◽  
pp. 7558-7616 ◽  
Author(s):  
Imma Gálvez-Carrillo ◽  
Joachim Kock ◽  
Andrew Tonks
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Directed Graphs ◽  
Rooted Trees ◽  
Independent Interest ◽  
General Construction ◽  
Finite Sets ◽  
Decomposition Spaces ◽  
Segal Spaces ◽  
Finite Posets

Abstract We show that Schmitt’s restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital $2$-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce directed restriction species that subsume Schmitt’s restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and convex maps. We also introduce the notion of monoidal (directed) restriction species, which induce monoidal decomposition spaces and hence bialgebras, most often Hopf algebras. Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. A prominent instance of a resulting incidence bialgebra is the Butcher–Connes–Kreimer Hopf algebra of rooted trees. Both ordinary and directed restriction species are shown to be examples of a construction of decomposition spaces from certain cocartesian fibrations over the category of finite ordinals that are also cartesian over convex maps. The proofs rely on some beautiful simplicial combinatorics, where the notion of convexity plays a key role. The methods developed are of independent interest as techniques for constructing decomposition spaces.

scholarly journals Central invariants and enveloping algebras of braided Hom-Lie algebras

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 3893-3915
Author(s):  
Shengxiang Wang ◽  
Xiaohui Zhang ◽  
Shuangjian Guo
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Hopf Algebras ◽  
Enveloping Algebras ◽  
Definition Of ◽  
Generalized Lie Algebras

Let (H,?) be a monoidal Hom-Hopf algebra and HH HYD the Hom-Yetter-Drinfeld category over (H,?). Then in this paper, we first introduce the definition of braided Hom-Lie algebras and show that each monoidal Hom-algebra in HH HYD gives rise to a braided Hom-Lie algebra. Second, we prove that if (A,?) is a sum of two H-commutative monoidal Hom-subalgebras, then the commutator Hom-ideal [A,A] of A is nilpotent. Also, we study the central invariant of braided Hom-Lie algebras as a generalization of generalized Lie algebras. Finally, we obtain a construction of the enveloping algebras of braided Hom-Lie algebras and show that the enveloping algebras are H-cocommutative Hom-Hopf algebras.

scholarly journals Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky–Novikov Type Symmetry Algebras and Related Hamiltonian Operators

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 601
Author(s):  
Orest Artemovych ◽  
Alexander Balinsky ◽  
Denis Blackmore ◽  
Anatolij Prykarpatski
Keyword(s):  
Dynamical Systems ◽  
Lie Algebras ◽  
Algebraic Structures ◽  
Loop Algebras ◽  
Type Symmetry ◽  
Novikov Algebras ◽  
Algebra Theory ◽  
Hamiltonian Operators ◽  
Adjoint Space ◽  
Algebraic Scheme

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures.

scholarly journals Hopf algebras and skew PBW extensions

2019 ◽  
Vol 10 (2) ◽  
Author(s):  
Luis Alfonso Salcedo Plazas
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Ore Extensions ◽  
Skew Pbw Extensions

In this article we relate some Hopf algebra structures over Ore extensions and over skew PBW extensions ofa Hopf algebra. These relations are illustrated with examples. We also show that Hopf Ore extensions andgeneralized Hopf Ore extensions are Hopf skew PBW extensions.

scholarly journals Antipode Formulas for some Combinatorial Hopf Algebras

10.37236/5949 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Rebecca Patrias
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Explicit Formulas ◽  
Quasisymmetric Functions ◽  
Noncommutative Symmetric Functions ◽  
Combinatorial Hopf Algebras ◽  
Combinatorial Formulas ◽  
K Theory

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.

scholarly journals A combinatorial non-commutative Hopf algebra of graphs

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Adrian Tanasa ◽  
Gerard Duchamp ◽  
Loïc Foissy ◽  
Nguyen Hoang-Nghia ◽  
Dominique Manchon
Keyword(s):  
Hopf Algebra ◽  
Rooted Trees ◽  
Quantum Field ◽  
International Audience ◽  
The One ◽  
Planar Rooted Trees

Combinatorics International audience A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.

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