Dialgebras

By J.-L. Loday. 2001

311 citations recorded by the source corpus.

There is a notion of non-commutative Lie algebra called"Leibniz algebra", which is characterized by the condition: left bracketing is a derivation. The purpose of this article is to introduce and study a new notion of algebra, called"associative dialgebra", which is, to the Leibniz algebras, what associative algebras are to Lie algebras. An associative dialgebra is a vector space equipped with two associative operations satisfying three more conditions. For instance, any differential associative algebra gives rise to a dialgebra. In this article we construct and study a (co)homology theory for associative dialgebras. The surprizing fact, in the construction of the chain complex, is the appearance of the combinatorics of planar binary trees (grafting and nesting). The principal result about this homology theory is its vanishing on free associative dialgebras. The Koszul dual (in the sense of Ginzburg and Kapranov) of the operad of associative dialgebras is the operad of"dendriform algebras". The dendriform algebras are characterized by two operations satisfying three linear conditions. The sum of these two operations defines a new operation which is associative. The free dendriform algebra can be described in terms of planar binary trees. As a consequence we give an explicitly description of strong homotopy associative dialgebras. This paper is part of a long-standing project whose ultimate aim is to study periodicity phenomenons in algebraic K-theory, as explained in ``Overview on Leibniz algebras, dialgebras and their homology". Fields Inst. Commun. 17 (1997), 91--102.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/25f913c128a108a5a2ad840fda28f0139c3fd4d8

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Citations in the corpus

Spitzer's identity and the algebraic Birkhoff decomposition in pQFT (2004, background): “On the mathematics side we would like to underline its intimate link to Loday’s dendriform algebra structures [1, 12, 14, 23, 24].”
On products and duality of binary, quadratic, regular operads (2004, background): “The generator-relation construction of ABQR operads allows a concrete description which is quite simple and can be found in the existing literature [33].”
Operated semigroups, Motzkin paths and rooted trees (2007, background): “Since the 1990s, algebraic structures on rooted trees have been studied in the work of Connes and Kreimer [10, 11] on the renormalization of quantum field theory, Grossman and Larson [24] on data structures and of Loday and Ronco [36, 37] on operads.”
Rota–Baxter algebras and dendriform algebras (2005, methodology): “AT/0205043) [58] J.-L. Loday and M. Ronco, Algèbre de Hopf colibres, C. R. Acad.”
Nonabelian Generalized Lax Pairs, the Classical Yang-Baxter Equation and PostLie Algebras (2009, unknown): “Dendriform dialgebras [35] and trialgebras [36] are introduced with moti vation from algebraicK-theory and topology.”
Generalized bialgebras and triples of operads (2006, methodology): “The free Dup-algebra is easy to describe (analogous to the free diassociative algebra, see [47]).”
Deformations of associative Rota-Baxter operators (2019, methodology): “[24] J.-L. Loday, Dialgebras, Dialgebras and related operads, 7-66, Lecture Notes in Math., 1763, Springer, Berlin, 2001.”
Combinatorial Hopf algebras (2008, methodology): “To prove that it is a dipterous morphism we proceed by induction (on the degree of the trees) like in the proof of the dendriform case performed in [35] Proposition 5.”
Homology of generalized partition posets (2004, background): “Loday [L1] and in the appendix B of [L2].”
Splitting of operations, Manin products and Rota-Baxter operators (2011, methodology): “Introduction Since the late 1990s, several algebraic structures with mul tiple binary operations have emerged: first the dendriform dialgebra of Loday [30] and then the dend riform trialgebra of Loday and Ronco [33], discovered from studying algebraic K-theory, o perads and algebraic topology.”
Polynomial realizations of some trialgebras (2006, background): “Since then, Loday and Ronco have introduced the operads of dendriform trialgebras and of tricubical algebras [15], in which the free algebras on one generator are respectively based on faces of associahedras and hypercubes, and are isomorphic (as Hopf algebras) to the corresponding algebras of Chapo”
On the operad of associative algebras with derivation (2009, background): “Other examples include magmatic, dendriform [9], cubical [11], duplicial [10], compatible-two-associative [4].”
On the structure of cofree Hopf algebras (2004, unknown): “or [12] Appendix B, for a brief introduction), the dual of the operad 2as is the operad 2as whose algebras have two associative operations ∗ and · verifying: (x ∗ y) · z = 0 = x ∗ (y · z) (x · y) ∗ z = 0 = x · (y ∗ z) It is clear that the free 2as-algebra over V is V in dimension 1 and the sum of tw”
Curved Koszul duality theory (2010, unknown): “We follow Appendix B of [Lod01] defining P ! := F(V )/(R), where V ∨ := V ∗ ⊗ (sgn) with the signature representation (sgn) and R is the orthogonal space for the natural pairing 〈−,−〉 : V ∨ ⊗ V → K.”
FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES (2005, methodology): “· · · This is used, for example in the Hopf algebra of planar rooted trees of Loday and Ronco [47, 49] and noncommutative variation of the Connes-Kreimer Hopf algebra [41, 29].”
Varieties of dialgebras and conformal algebras (2006, background): “By the definition [2, 3], dialgebras are linear spaces with two bilinear products ⊣, ⊢.”