The loop of formal power series with noncommutative coefficients under substitution

Abstract

The set of formal power series with coefficients in an associative but noncommutative algebra becomes a loop with the substitution product. We study this loop by describing certain Lie and Sabinin algebras related to it. Some examples of Lie algebras satisfying the standard identities of degrees 5 and 6 appear naturally.

Appendix: One-sided local loops

We observe that the formula $\alpha _m * \beta _n := (m+1) \alpha _m \beta _n$ has not right unit element in case we assume that 1 is of degree 0, which leads to a left loop rather than a two-sided loop.

In this section we would like to briefly discuss how to adapt the approach of Mikheev and Sabinin to the study of one-side local loops. Let $(Q,x*y,e)$ be a right (local) loop, i.e. e is the right unit element and the Jacobian of the left and right multiplication operators at e is non-zero, and define a two-sided loop by

$$\begin{aligned} xy = x*(L^{*}_e)^{-1}y. \end{aligned}$$

This loop is classified by its Sabinin algebra $(T_e Q, \langle - ; -,- \rangle , \varPhi (-;-))$. Thus, if we include a new family of totally symmetric multilinear operation, lets say $[x_1,\ldots , x_n]$ ($n \ge 1$), corresponding to the Taylor series of the map $L^{*}_e$ in normal coordinates to classify $L^{*}_e$ (see [22]), then the algebraic structure

$$\begin{aligned} (T_e Q, \langle - ; -,- \rangle , \varPhi (-;-),[-]) \end{aligned}$$

(26)

classifies the right local loop $(Q,x*y,e)$. The integration of these structures to left loops only requires the usual convergence conditions (see [22]).

From a geometrical point of view, given a right loop we define the parallel transport as

$$\begin{aligned} \tau ^e_y := dL^*_y(L^*_e)^{-1}\vert _e. \end{aligned}$$

so that

$$\begin{aligned} \tau ^x_y := (dL^*_y\vert e) (dL^*_x\vert _e)^{-1}. \end{aligned}$$

This parallel transport defines a right monoalternative geodesic loop $x \times y := \exp _x \tau ^e_x \exp ^{-1}_e(y)$. For this loop we have

$$\begin{aligned} dL^{\times }_y\vert _e = \tau ^e_y = dL^*_y(L^*_e)^{-1}\vert _e. \end{aligned}$$

We can consider the map $\varPsi _x$ defined by $x*y = x \times \varPsi _x(y)$. The only restrictions on $\varPsi$ are

$$\begin{aligned} \varPsi _x(e) = e, \quad \varPsi _e(y) = e*y \quad \text {and} \quad d\varPsi _x\vert _e = dL^*_e\vert _e. \end{aligned}$$

and $(Q,*)$ is classified by $x \times y$ and $\varPsi _x$. Thus, to locally classify $x*y$, we need the same structure as in (26) since we require $\langle - ; -,- \rangle$ to recover $x \times y$; the operations $[-]$ that encode the coefficients of the Taylor expansion in normal coordinates for $L^*_e$ to classify $\varPsi _e$; and, to recover $\varPsi _x$, we need $\varPhi (-;-)$ that collects the coefficients of the Taylor expansion of $\varPsi _x(y)$ on degrees $\ge 1$ and $\ge 2$ of x and y respectively (notice that the coefficients of monomials in the Taylor expansion of $\varPsi _x(y)$ of degree 0 on the coordinates of x and of degree $\ge 1$ on the coordinates of y are determined by $\varPsi _e = L^*_e$, and those of degree $\ge 1$ on the coordinates of x and of degree 0 or 1 on the coordinates of y are determined by $d\varPsi _x\vert _e = dL^*_e\vert _e$).

In fact, if we define $\varPhi '_x:=\varPsi _x(L^*_e)^{-1}$ then $xy = x *(L^{*}_e)^{-1}y = x \times \varPhi '_x(y)$ with

$$\begin{aligned} \varPhi '_x(e) = e, \quad \varPhi '_e(y) = y \quad \text {and} \quad d\varPhi '_x\vert _e = \mathrm {Id}. \end{aligned}$$

This proves that the loop $x \times y$ is also the monoalternative perturbation of the loop xy and that $\varPsi$ is recovered with the help of $\varPhi$ and $L^*_e$. Moreover, if the right local loop $(Q,*)$ is right monoalternative then xy is a monoalternative loop. Thus $xy = x \times y$ and $x \times y = x* (L^{*}_e)^{-1}(y)$.

Therefore, the classification of local one-sided loops only requires the usual structure of a Sabinin algebra and an extra family of totally symmetric multilinear operations $[x_1,\ldots , x_n]$ ($n\ge 1$).