Co-addition for Free Non-associative Algebras and the Hausdorff Series

Generalizations of the series exp and log to noncommutative nonassociative and other types of algebras were regarded by M. Lazard, and recently by V. Drensky and L. Gerritzen. There is a unique power series exp(x) in one non-associative variable x such that exp(x) exp(x) = exp(2x), exp(0) = 1. We call the unique series H = H(x, y) in two non-associative variables satisfying exp(H) = exp(x) exp(y) the non-associative Hausdorff series, and we show that the homogeneous components Hn of H are primitive elements with respect to the coaddition for non-associative variables. We describe the space of primitive elements for the co-addition in non-associative variables using Taylor expansion and a projector onto the algebra A0 of constants for the partial derivations. By a theorem of Kurosh, A0 is a free algebra. We describe a procedure to construct a free algebra basis consisting of primitive elements. In this article we are studying the co-addition ∆ : K{X} → K{X} ⊗ K{X}, where K{X} denotes the non-associative noncommutative algebra with unit freely generated by a set X, and where K is a field of characteristic 0. The space Prim(K{X}) = {f ∈ K{X} : ∆(f) = f⊗1+1⊗f} of primitive elements for the co-addition is considered. The first Lazard-cohomology group of the ⊗-Kurosh analyzer is given by the primitive elements for the co-addition, see [Ho1], and they are also called pseudo-linear, following [La]. The smallest space that contains the variables and is closed under commutators [f1, f2] := f1f2 − f2f1 and associators (f1, f2, f3) := (f1f2)f3− f1(f2f3) is contained in Prim(K{X}), and is strictly smaller than Prim(K{X}). It does not contain the primitive element xx−2x(xx)+x(xx), for example. The algebra (K{X})0 of constants relative to all partial derivations d dxi , xi ∈ X, contains all primitive elements of order ≥ 2. It is expected that (K{X})0 is freely generated by homogeneous primitive polynomials. For the algebra K{x} in one variable x, we describe a construction of such an algebra basis for the algebra of constants, in which Lazard-cohomology is used. The first element of the free algebra basis consisting of primitive elements is xx − xx, which is the only generator in degree 3. In degree n ≥ 4, there are 3(cn−1 − cn−2) generators, with Catalan number cn := (2(n−1))! n!(n−1)! . Mathematics Subject Classification: 17A50, 16W30, 16W60. 1 2 LOTHAR GERRITZEN, RALF HOLTKAMP The basic method is based on the concept of Taylor expansion in K{X}, which provides a projector onto the algebra (K{X})0 of constants. The classical Hausdorff series log(ee) = H(x, y) = ∑∞ n=1Hn, Hn homogeneous of degree n, is a series in associative variables with rational coefficients. The famous Campbell-Baker-Hausdorff formula states that all Hn are Lie polynomials. By a Theorem of Friedrichs, Lie polynomials are characterized as the primitive elements of the free associative co-addition Hopf algebra, cf. [R], p. 20. Generalizations of the series exp and log to noncommutative non-associative and other types of algebras were regarded by M. Lazard, see [La], and recently by V. Drensky and L. Gerritzen, see [DG]. There is a unique power series exp(x) in one non-associative variable x such that exp(x) exp(x) = exp(2x), exp(0) = 1, and it holds that exp(x) = exp(x). There exists a unique series H = H(x, y) without constant term in two non-associative variables satisfying exp(H) = exp(x) exp(y). We suggest to call H the non-associative Hausdorff series. We show that the homogeneous components Hn of H are primitive elements with respect to the co-addition for non-associative variables. We obtain a recursive formula for the coefficients c(τ) of H , see section 6. Each non-associative monomial τ is a word with parenthesis and can be identified with an X-labeled planar binary rooted tree. Each coefficient c̄(w), w an associative word in x and y, of the classical Hausdorff series is obtained as a sum ∑ c(τ) over non-associative monomials τ for which the foliage is equal to w (see [R], p.84). Several formulas for the coefficients in the classical case are given in [R] §3.3. There is no analogue of the given formula, though. In section 1 we show that the co-addition is cocommutative, coassociative, and that there are left and right antipodes, which however are not anti-homomorphisms. We get a Hopf algebra structure on the free non-associative noncommutative algebra. If we identify the monomials with X-labeled planar binary trees, the grafting of trees occurs as the multiplication, see also [Ho2], Remark (3.7). The Hopf algebras on planar binary trees described in [LR] or [BF] are Hopf algebra structures on free associative algebras. The comultiplications can be described by cuts that are given by subsets of the set of vertices. For the co-addition Hopf algebra of section 1, the image ∆(τ) of a non-associative monomial under ∆ is described in terms of contractions of the tree τ onto subsets of the set of leaves, see Lemma (1.9). Polynomials in one variable x are considered in section 2. The space of constants of degree n homogeneous elements has dimension cn−cn−1. We study Taylor expansions