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12
Generalized bialgebras and triples of operads
, 2006
"... Key words and phrases. Bialgebra, generalized bialgebra, Hopf algebra, CartierMilnorMoore, PoincaréBirkhoffWitt, operad, prop, triple of operads, primitive part, dendriform algebra, duplicial algebra, preLie ..."
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Cited by 16 (5 self)
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Key words and phrases. Bialgebra, generalized bialgebra, Hopf algebra, CartierMilnorMoore, PoincaréBirkhoffWitt, operad, prop, triple of operads, primitive part, dendriform algebra, duplicial algebra, preLie
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.
Operads of compatible structures and weighted partitions
 J. Pure Appl. Algebra
"... Abstract. In this paper we describe operads encoding two different kinds of compatibility of algebraic structures. We show that there exist decompositions of these in terms of black and white products and we prove that they are Koszul for a large class of algebraic structures by using the poset meth ..."
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Cited by 6 (1 self)
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Abstract. In this paper we describe operads encoding two different kinds of compatibility of algebraic structures. We show that there exist decompositions of these in terms of black and white products and we prove that they are Koszul for a large class of algebraic structures by using the poset method of B. Vallette. In particular we show that this is true for the operads of compatible Lie, associative and preLie algebras.
Homotopy Batalin–Vilkovisky algebras
"... This paper provides an explicit cofibrant resolution of the operad encoding BatalinVilkovisky algebras. Thus it defines the notion of homotopy BatalinVilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads ..."
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Cited by 6 (3 self)
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This paper provides an explicit cofibrant resolution of the operad encoding BatalinVilkovisky algebras. Thus it defines the notion of homotopy BatalinVilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defined by quadratic and linear relations. The operad encoding Batalin– Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a PoincaréBirkhoffWitt Theorem for such an operad and to give an explicit small quasifree resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BValgebras and of homotopy BValgebras. We show that any topological conformal field theory carries a homotopy BValgebra structure which lifts the BValgebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian–Zuckerman, showing that certain vertex algebras have an explicit homotopy BValgebra structure.
Freeness theorems for operads via Gröbner bases. arXiv:0907.4958
"... Abstract. We show how to use Gröbner bases for operads to prove various freeness theorems: freeness of certain operads as nonsymmetric operads, freeness of an operad Q as a Pmodule for an inclusion P → Q, freeness of a suboperad. This gives new proofs of many known results of this type and helps to ..."
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Cited by 2 (1 self)
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Abstract. We show how to use Gröbner bases for operads to prove various freeness theorems: freeness of certain operads as nonsymmetric operads, freeness of an operad Q as a Pmodule for an inclusion P → Q, freeness of a suboperad. This gives new proofs of many known results of this type and helps to prove some new results. 1.
Theorem (Gerstenhaber).
"... For f ∈ Hom(V ⊗n, V) and g ∈ Hom(V ⊗m, V), binary product f ⋆ g:= n∑ i=1 ..."
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S II
"... Abstract. This paper is the followup of [MV08]. ..."
1 Some problems in operad theory
"... This is a list of some problems and conjectures related to various types of algebras, that is to algebraic operads. Some comments and hints are included. ..."
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This is a list of some problems and conjectures related to various types of algebras, that is to algebraic operads. Some comments and hints are included.
MALCEV DIALGEBRAS
"... Abstract. We apply Kolesnikov’s algorithm to obtain a variety of nonassociative algebras defined by right anticommutativity and a “noncommutative” version of the Malcev identity. We use computational linear algebra to verify that these identities are equivalent to the identities of degree ≤ 4 satisf ..."
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Abstract. We apply Kolesnikov’s algorithm to obtain a variety of nonassociative algebras defined by right anticommutativity and a “noncommutative” version of the Malcev identity. We use computational linear algebra to verify that these identities are equivalent to the identities of degree ≤ 4 satisfied by the dicommutator in every alternative dialgebra. We extend these computations to show that any special identity for Malcev dialgebras must have degree at least 7. Finally, we introduce a trilinear operation which makes any Malcev dialgebra into a Leibniz triple system. 1.