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29
RotaBaxter algebras, dendriform algebras and PoincaréBirkhoffWitt theorem
, 2004
"... Abstract. RotaBaxter algebras appeared in both the physics and mathematics literature. It is of great interest to have a simple construction of the free object of this algebraic structure. For example, free commutative RotaBaxter algebras relate to double shuffle relations for multiple zeta values ..."
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Cited by 18 (11 self)
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Abstract. RotaBaxter algebras appeared in both the physics and mathematics literature. It is of great interest to have a simple construction of the free object of this algebraic structure. For example, free commutative RotaBaxter algebras relate to double shuffle relations for multiple zeta values. The interest in the noncommutative setting arised in connection with the work of Connes and Kreimer on the Birkhoff decomposition in renormalization theory in perturbative quantum field theory. We construct free noncommutative RotaBaxter algebras and apply the construction to obtain universal enveloping RotaBaxter algebras of dendriform dialgebras and trialgebras. We also prove an analog of the PoincaréBirkhoffWitt theorem for universal enveloping algebra in the context of dendriform trialgebras. In particular, every dendriform dialgebra and trialgebra is a subalgebra of a RotaBaxter algebra. We explicitly show that the free dendriform dialgebras and trialgebras, as represented by
Cocommutative Hopf algebras of permutations and trees
 Journal Algebraic Combinatorics
, 2004
"... Abstract. Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the la ..."
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Cited by 13 (2 self)
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Abstract. Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980’s by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heapordered trees, respectively. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra.
On the algebra of quasishuffles
 Manuscripta Mathematica
"... Abstract. For any commutative algebra R the shuffle product on the tensor module T (R) can be deformed to a new product. It is called the quasishuffle algebra, or stuffle algebra, and denoted T q (R). We show that if R is the polynomial algebra, then T q (R) is free for some algebraic structure cal ..."
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Cited by 10 (1 self)
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Abstract. For any commutative algebra R the shuffle product on the tensor module T (R) can be deformed to a new product. It is called the quasishuffle algebra, or stuffle algebra, and denoted T q (R). We show that if R is the polynomial algebra, then T q (R) is free for some algebraic structure called Commutative TriDendriform (CTDalgebras). This result is part of a structure theorem for CTDbialgebras which are associative as coalgebras and whose primitive part is commutative. In other words, there is a good triple of operads (As, CT D, Com) analogous to (Com, As, Lie). In the last part we give a similar interpretation of the quasishuffle algebra in the noncommutative setting.
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S I
"... Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCarta ..."
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Cited by 9 (4 self)
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Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results.
Coassociative magmatic bialgebras and the fine numbers. arXiv:math.RA/0609125
"... Abstract. We prove a structure theorem for the connected coassociative magmatic bialgebras. The space of primitive elements is an algebra over an operad called the primitive operad. We prove that the primitive operad is magmatic generated by n − 2 operations of arity n. The dimension of the space of ..."
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Cited by 8 (2 self)
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Abstract. We prove a structure theorem for the connected coassociative magmatic bialgebras. The space of primitive elements is an algebra over an operad called the primitive operad. We prove that the primitive operad is magmatic generated by n − 2 operations of arity n. The dimension of the space of all the nary operations of this primitive operad turns out to be the Fine number Fn−1. In short, the triple of operads (As, Mag,MagFine) is good.
FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES
, 2008
"... Abstract. A Rota–Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota–Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota–Baxter algebras have been for commutative algebras. Two constru ..."
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Cited by 8 (1 self)
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Abstract. A Rota–Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota–Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota–Baxter algebras have been for commutative algebras. Two constructions of free commutative Rota–Baxter algebras were obtained by Rota and Cartier in the 1970s and a third one by Keigher and one of the authors in the 1990s in terms of mixable shuffles. Recently, noncommutative Rota–Baxter algebras have appeared both in physics in connection with the work of Connes and Kreimer on renormalization in perturbative quantum field theory, and in mathematics related to the work of Loday and Ronco on dendriform dialgebras and trialgebras. This paper uses rooted trees and forests to give explicit constructions of free noncommutative Rota–Baxter algebras on modules and sets. This highlights the combinatorial nature of Rota–Baxter algebras and facilitates their further study. As an application, we obtain the unitarization of Rota–Baxter algebras. 1.
From left modules to algebras over an operad: application to combinatorial Hopf algebras
, 2006
"... The purpose of this paper is two fold: we study the behaviour of the forgetful functor from Smodules to graded vector spaces in the context of algebras over an operad and derive from this theory the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness resu ..."
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Cited by 7 (2 self)
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The purpose of this paper is two fold: we study the behaviour of the forgetful functor from Smodules to graded vector spaces in the context of algebras over an operad and derive from this theory the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for these Hopf algebras. Let O denote the forgetful functor from Smodules to graded vector spaces. Left modules over an operad P are treated as Palgebras in the category of Smodules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad P: the functor O sends Palgebras to Palgebras. If P is a Hopf operad the functor O sends Hopf Palgebras to Hopf Palgebras. If the operad P is regular one gets two different structures of Hopf Palgebras in the category of graded vector spaces. We develop the notion of unital infinitesimal Pbialgebra and prove freeness and cofreeness results for Hopf algebras built from Hopf operads. Finally, we prove that many combinatorial Hopf algebras arise from our theory, as Hopf algebras on the faces of the permutohedra and associahedra.
PBW deformations of braided symmetric algebras and a MilnorMoore type theorem
"... Abstract. Braided bialgebras were defined by mimicking the definition of bialgebras in a braided category; see [Ta]. In this paper we are interested in those braided bialgebras that are connected as a coalgebra, and such that, up to multiplication by a certain scalar, their braiding restricted to th ..."
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Cited by 4 (3 self)
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Abstract. Braided bialgebras were defined by mimicking the definition of bialgebras in a braided category; see [Ta]. In this paper we are interested in those braided bialgebras that are connected as a coalgebra, and such that, up to multiplication by a certain scalar, their braiding restricted to the primitive part is a Hecke operator. To every braided bialgebra as above we associate a braided Lie algebra. Conversely, for each braided Lie algebra we construct a braided bialgebra, namely its enveloping algebra. Braided symmetric and exterior algebras are examples of enveloping algebras (they correspond to a trivial Lie bracket). We show many of the properties of ordinary enveloping algebras still hold in the braided case: the graded associated (with respect to the standard filtration) is a braided symmetric algebra; the coalgebra counterpart is isomorphic to the coalgebra structure of a braided symmetric algebra; the coradical filtration and the standard filtration are identical. As in the classical case, braided symmetric and exterior algebras are Koszul algebras. The proof of this fact is based on a new characterization of Koszul algebras. These properties are used to prove a MilnorMoore type theorem for infinitesimally cocommutative connected braided bialgebras (see Theorem 5.5). We apply our result to three different classes of braided bialgebras: connected bialgebras in the category of comodules over a coquasitriangular cosemisimple Hopf algebra (e.g. superbialgebras and enveloping algebras of