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.
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Notes
We identify the formal power series \(t + \alpha _1 t^2 + \alpha _2 t^3 + \cdots \in t + S_1 t^2 + S_2 t^3 + \cdots \) with \(1 + \alpha _1 + \alpha _2 + \cdots \).
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I would like to thank the referee for detailed comments.
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Dedicated with admiration, respect and affection to Ivan Shestakov on the occasion of his 70th birthday. Thanks for making your home our mathematical home.
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Appendix: One-sided local loops
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
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
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
so that
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
We can consider the map \(\varPsi _x\) defined by \(x*y = x \times \varPsi _x(y)\). The only restrictions on \(\varPsi \) are
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
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 \([x_1,\ldots , x_n]\) (\(n\ge 1\)).
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Pérez-Izquierdo, J.M. The loop of formal power series with noncommutative coefficients under substitution. São Paulo J. Math. Sci. 13, 133–157 (2019). https://doi.org/10.1007/s40863-019-00131-w
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DOI: https://doi.org/10.1007/s40863-019-00131-w