Hopf co-addition for free magma algebras and the non-associative Hausdorff series.
Consider a finite or countable set \(X=\{x_1,x_2,\dots\}\), a field \({\mathbb K}\) of characteristic \(0\), and the free \({\mathbb K}\)-algebra \({\mathbb K}\{X\}\) with unit \(1\) generated by \(X\), also called the free magma algebra over \(X\). Its completion is denoted by \({\mathbb K}\{\{X\}\}\). In the following, let \(A\) be \({\mathbb K}\{X\}\) or \({\mathbb K}\{\{X\}\}\). The co-addition \(\Delta\colon A\to A\otimes A\) is defined by \(x_i\mapsto x_i\otimes 1+1\otimes x_i\). The algebra \((A,\Delta)\) is called the free graded resp. free complete nonassociative Hopf algebra with co-addition. The set of all primitive elements \(f\in A\), i.e., all elements satisfying \(\Delta(f)=f\otimes 1+1\otimes f\) is denoted by \(A_0\). Moreover, one can define the unique \({\mathbb K}\)-linear derivation \(D\) on \(A\) by \(D_i(x_j)=\delta_{ij}\) for all \(i,j\). The authors investigate the co-addition \(\Delta\) and its relation to \(D\). They describe the Taylor expansion in one variable and for several variables. Here, the primitive elements play an important role. Therefore they and \(A_0\) are considered in detail. A number of interesting results and examples are given. The paper ends with an detailed investigation on the nonassociative Hausdorff series. A main result is that the components of this series are primitive elements for the co-addition.
scientific article; zbMATH DE number 1955975