We give a new presentation of the Lie cooperad as a quotient of the graph cooperad, a presentation which is not linearly dual to any of the standard presentations of the Lie operad. We use this presentation to explicitly compute duality between Lie algebras and coalgebras, to give a new presentation of Harrison homology, and to show that Lyndon words yield a canonical basis for cofree Lie coalgebras.
We give a new solution of the "homotopy periods" problem, as highlighted by Sullivan [25], which places explicit geometrically meaningful formulae first dating back to Whitehead [28] in the context of Quillen's formalism for rational homotopy theory and Koszul-Moore duality [20].We build on [24], which uses graph coalgebras to breathe combinatorial life into the category of differential graded Lie coalgebras. We use that framework to construct a new isomorphism of Lie coalgebras η : H * −1 (E(A * (X))) → Hom(π * (X), Q) for X simply connected. Here A * (X) denotes a model for commutative rational-valued cochains on X, and E is isomorphic to the Harrison complex. While the existence of such an isomorphism follows from Quillen's seminal work in rational homotopy theory, giving a direct, explicit isomorphism has benefits in both theory and applications.On the calculational side, we are able to evaluate Hopf invariants on iterated Whitehead products in terms of the "configuration pairing." We can use this, for example, to take the well-known calculation of the rational homotopy groups of a wedge of spheres as a free graded Lie algebra and give a geometric algorithm to determine which element of that algebra a given map would correspond to. On the formal side, we are able to understand the naturality of these maps in the long exact sequence of a fibration. For applications, we can show for example that the rational homotopy groups of homogeneous spaces are detected by classical linking numbers. Ultimately all of these Hopf invariants are essentially generalized linking invariants, as we explain in Section 1.2.We proceed in two steps, first using the classical bar complex to define integer-valued homotopy functionals which coincide with evaluation of the cohomology of ΩX on the looping of a map from S n to X. This was also the starting point of Hain's work [12] using Chen integrals, but our definition of functionals is clearly distinct from his. We establish basic properties and give examples using the classical bar complex. In the second part, we use the Harrison complex on commutative cochains, and thus must switch to rational coefficients. Using our graph coalgebraic presentation, we show that a product-coproduct formula established geometrically in the bar complex descends to the duality predicted by Koszul-Moore theory.Our basic, apparently new, observation is that calculations in bar complexes yield the Hopf invariant formula of Whitehead [28], as well as those of Haefliger, Novikov and Sullivan. This observation could have been made fifty years ago. Our approach incorporates a modern viewpoint by directly using HarrisonAndré-Quillen homology, the standard algebraic bridge from commutative algebras to Lie coalgebras, with the new graphical presentation essential for a self-contained development. One direction we plan to pursue further is the use of Hopf invariants to realize Koszul-Moore duality isomorphisms in general. A second direction we plan to pursue is that of spaces which are not simply connected, where...
We give an algebraic construction of the topological graph-tree configuration pairing of Sinha and Walter beginning with the classical presentation of Lie coalgebras via coefficients of words in the associative Lie polynomial. Our work moves from associative algebras to preLie algebras to the graph complexes of Sinha and Walter, justifying the use of graph generators for Lie coalgebras by iteratively expanding the set of generators until the set of relations collapses to two simple local expressions. Our focus is on new computational methods allowed by this framework and the efficiency of the graph presentation in proofs and calculus involving free Lie algebras and coalgebras. This outlines a new way of understanding and calculating with Lie algebras arising from the graph presentation of Lie coalgebras.
Abstract. We construct a basis for free Lie algebras via a "left-greedy" bracketing algorithm on Lyndon-Shirshov words. We use a new tool -the configuration pairing between Lie brackets and graphs of Sinha-Walter -to show that the left-greedy brackets form a basis. Our constructions further equip the left-greedy brackets with a dual monomial Lie coalgebra basis of "star" graphs. We end with a brief example using the dual basis of star graphs in a Lie algebra computation.
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