Nonassociative Solomon's descent algebras

TL;DR

This paper extends Solomon's descent algebra to a nonassociative setting using Sabinin algebras, introducing new algebraic structures and combinatorial frameworks related to nonassociative symmetric functions.

Contribution

It establishes a nonassociative analogue of Solomon's descent algebra via Sabinin algebras and develops a combinatorial nonassociative Hopf algebra framework.

Findings

Proves isomorphism between universal enveloping algebras of Sabinin algebras and nonassociative symmetric functions.

Introduces a nonassociative lifting of the Malvenuto-Reutenauer Hopf algebra.

Shows the nonassociative algebra is free and cofree with a Mackey type formula.

Abstract

Descent algebras of graded bialgebras were introduced by F. Patras as a generalization of Solomon's descent algebras for Coxeter groups of type $A$, i.e. symmetric groups. The universal enveloping algebra of the free Lie algebra on a countable number of generators, its descent algebra and Solomon's descent algebra, with its outer product, for symmetric groups are isomorphic to the Hopf algebra of noncommutative symmetric functions, a free associative algebra on a countable number of generators. In this paper we prove a similar result for universal enveloping algebras of relatively free Sabinin algebras, which makes them nonassociative analogues of the Solomon's descent algebra and of the algebra of noncommutative symmetric functions. In the absolutely free case a combinatorial setting is also presented by introducing a nonassociative lifting of the Malvenuto-Reutenauer Hopf algebra of permutations and of the dual of the Hopf algebra of noncommutative quasi-symmetric functions. We prove that this lifting is a free nonassociative algebra as well as a cofree graded coassociative coalgebra which also admits an associative inner product that satisfies a Mackey type formula when restricted to its nonassociative Solomon's descent algebra. Pseudo-compositions of sets instead of pseudo-compositions of numbers is the natural language for the underlying combinatorics.

Equations

$$\bigg{\{}D_{\subseteq S}:=\sum_{\operatorname{Des}(\sigma)\subseteq S}\mathbf{F}_{\sigma}\,\bigg{|}\,S\subseteq\{1,\dots,m-1\}\bigg{\}}$$

$$1=\cdot,\quad x=\begin{minipage}{8.0pt} \begin{picture}(8.0,24.0) \put(4.0,20.0){\makebox(0.0,0.0)[b]{$x$}} \put(4.0,16.0){\line(0,-1){8.0}} \put(4.0,8.0){\line(0,-1){8.0}} \end{picture} {\vskip 10.76385pt} \end{minipage},\quad xx=\begin{minipage}{16.0pt} \begin{picture}(16.0,24.0) \put(4.0,20.0){\makebox(0.0,0.0)[b]{$x$}} \put(12.0,20.0){\makebox(0.0,0.0)[b]{$x$}} \put(8.0,16.0){\oval(8.0,12.0)[b]} \put(8.0,10.0){\line(0,-1){2.0}} \end{picture} {\vskip 10.76385pt} \end{minipage},\quad(xx)x=\begin{minipage}{24.0pt} \begin{picture}(24.0,32.0) \put(4.0,28.0){\makebox(0.0,0.0)[b]{$x$}} \put(12.0,28.0){\makebox(0.0,0.0)[b]{$x$}} \put(20.0,28.0){\makebox(0.0,0.0)[b]{$x$}} \put(8.0,24.0){\oval(8.0,12.0)[b]} \put(8.0,18.0){\line(0,-1){2.0}} \put(20.0,24.0){\line(0,-1){8.0}} \qbezier(8.0,16.0)(8.0,14.0)(10.0,12.0)\qbezier(10.0,12.0)(12.0,12.0)(12.0,8.0) \put(20.0,16.0){\line(0,-1){8.0}} \put(16.0,8.0){\oval(8.0,12.0)[b]} \put(16.0,2.0){\line(0,-1){2.0}} \end{picture} {\vskip 10.76385pt} \end{minipage},\quad x(xx)=\begin{minipage}{24.0pt} \begin{picture}(24.0,32.0) \put(4.0,28.0){\makebox(0.0,0.0)[b]{$x$}} \put(12.0,28.0){\makebox(0.0,0.0)[b]{$x$}} \put(20.0,28.0){\makebox(0.0,0.0)[b]{$x$}} \put(4.0,24.0){\line(0,-1){8.0}} \put(16.0,24.0){\oval(8.0,12.0)[b]} \put(16.0,18.0){\line(0,-1){2.0}} \put(4.0,16.0){\line(0,-1){8.0}} \qbezier(16.0,16.0)(16.0,14.0)(14.0,12.0)\qbezier(14.0,12.0)(12.0,12.0)(12.0,8.0) \put(8.0,8.0){\oval(8.0,12.0)[b]} \put(8.0,2.0){\line(0,-1){2.0}} \end{picture} {\vskip 10.76385pt} \end{minipage},\quad\dots$$

$$⟨x_1,\dots ,x_m;y,z⟩,m\ge 0,\text{ and }$$

$$\Phi (x_1,\dots ,x_m;y_1,\dots ,y_n),m\ge 1,n\ge 2$$

$$⟨x_1,\dots ,x_m;y,z⟩=-⟨x_1,\dots ,x_m;z,y⟩,$$

$$⟨x_1,\dots ,x_r,a,b,x_{r+1},\dots ,x_m;y,z⟩-⟨x_1,\dots ,x_r,b,a,x_{r+1},\dots ,x_m;y,z⟩$$

$$+\sum _{k=0}^r\sum _{\alpha }⟨x_{\alpha (1)},\dots ,x_{\alpha (k)},⟨x_{\alpha (k+1)},\dots ,x_{\alpha (r)};a,b⟩,\dots ,x_m;y,z⟩=0,$$

$${\sigma }_{x,y,z}\left(⟨x_1,\dots ,x_r;y,z⟩+\sum _{k=0}^r\sum _{\alpha }⟨x_{\alpha (1)},\dots ,x_{\alpha (k)};⟨x_{\alpha (k+1)},\dots ,x_{\alpha (r)};y,z⟩,x⟩\right)=0$$

$$\Phi (x_1,\dots ,x_m;y_1,\dots ,y_n)=\Phi (x_{\tau (1)},\dots ,x_{\tau (m)};y_{\delta (1)},\dots ,y_{\delta (n)}),$$

$$\sum u_{(1)}\backslash (u_{(2)}v)$$

$$\sum (u{v}_{(1)})/v_{(2)}$$

$$f\ast h:u$$

$$f\backslash h:u$$

$$f/h:u$$

$$F\circ \Delta =\Delta \circ f.$$

$$\sum _{i+j=n}{I}_i\otimes I_j\text{ is a pseudo-coproduct for }{I}_n,$$

$$t(I_I)\circ t^{\prime }(I_J)=\sum _{M}t(t^{\prime })(I_M)$$

$$t(I_{i_1},\dots ,I_{i_r})\circ t^{\prime }(I_{j_1},\dots ,I_{j_s})(u)$$

$$=\sum t(I_{i_1},\dots ,I_{i_r})(t^{\prime }(I_{j_1}(u_{(1)}),\dots ,I_{j_s}(u_{(s)})))$$

$$=\sum _{\begin{array}{c}|u_{(k)}|=j_k\\ k=1,\dots ,s\end{array}}t(I_{i_1},\dots ,I_{i_r})(t^{\prime }(u_{(1)},\dots ,u_{(s)}))$$

$$=\sum _{\begin{array}{c}{\sum }_{l=1}^r|u_{(k)(l)}|=j_k\\ {\sum }_{k=1}^s|u_{(k)(l)}|=i_l\end{array}}t(t^{\prime }(u_{(1)(1)},\dots ,u_{(s)(1)}),\dots ,t^{\prime }(u_{(1)(r)},\dots ,u_{(s)(r)}))$$

$$=\sum _{\begin{array}{c}{\sum }_{l=1}^r|u_{(l)(k)}|=j_k\\ {\sum }_{k=1}^s|u_{(l)(k)}|=i_l\end{array}}t(t^{\prime }(u_{(1)(1)},\dots ,u_{(1)(s)}),\dots ,t^{\prime }(u_{(r)(1)},\dots ,u_{(r)(s)}))$$

$$=\sum _{M}t(t^{\prime })(I_M)$$

$$e^x:={\exp }_l(x):=\sum _{n\ge 0}\frac{1}{n!}\underset{n}{\underbrace{(((xx)\cdots )x)x}}$$

$${\log }_g((1+y)\otimes (1+z))={\log }_g(1+y)\otimes 1+1\otimes {\log }_g(1+z).$$

$$g(y\otimes 1+1\otimes z)$$

$${\log }_g^{\ast }(I):=\sum _{n\ge 1}{l}_n^{\ast }(I_{+})\in {\mathrm{End}}_K(U),$$

$$\Delta ({\log }_g^{\ast }(I)(u))$$

$$e^n:=g_n^{\ast }({\log }_g^{\ast }(I))\in {\mathrm{End}}_K(U).$$

$$e^m\circ e^n={\delta }_{m,n}{e}^m\text{and}\sum _{n=0}^{\infty }{e}^n=I.$$

$$e^m\circ g^{\ast }(\lambda L)(u)$$

$$e_n^1:=I_n\circ e^1$$

$$I_n\circ (f\ast h)=\sum _{i+j=n}(I_i\circ f)\ast (I_j\circ h)$$

$$e_i^1(w(g(x_1),\dots ,g(x_n)))\in K\{X\}.$$

$$\partial \Omega :=\{e_i^1(w(g(x_1),\dots ,g(x_n)))\sim 0\mid w(z_1,\dots ,z_n)\in \Omega ,i=1,2,\dots \}.$$

$${\mathrm{PBW}}_g(U(s);B):=\{g_n(b_{i_1},\dots ,b_{i_n})\mid n\in N,b_{i_1},\dots ,b_{i_n}\in B\text{ and }{b}_{i_1}\le \cdots \le b_{i_n}\}$$

Full text

1. Introduction

A descent of a permutation $\sigma=(\sigma(1),\dots,\sigma(m))$ is an index $1\leq i\leq m-1$ such that $\sigma(i)>\sigma(i+1)$. In [MR0444756] Solomon proved, in the more general context of finite Coxeter groups, that

[TABLE]

spans a $2^{m-1}$-dimensional subalgebra $(\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}_{m}),\operatorname{\circ\,})$ of the group algebra $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}_{m}]$ of the symmetric group $\operatorname{\mathcal{S}}_{m}$, where $\operatorname{Des}(\sigma)$ denotes the set of descents of $\sigma$ and $\{\mathbf{F}_{\sigma}\mid\sigma\in\operatorname{\mathcal{S}}_{m}\}$ is the usual basis of $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}_{m}]$, i.e. $\mathbf{F}_{\sigma}\operatorname{\circ\,}\mathbf{F}_{\tau}=\mathbf{F}_{\sigma\operatorname{\circ\,}\tau}$. The structure of $(\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}_{m}),\operatorname{\circ\,})$ was studied by Garsia and Reutenauer in [MR1020585]. The product of basic elements (1.1) obeys a Mackey type formula, thus leading to an algebra homomorphism $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}_{m})\rightarrow\operatorname{Cl}_{\operatorname{{\mathbb{K}}}}(\operatorname{\mathcal{S}}_{m})$ that maps $D_{\subseteq S}$ to the Young character $\xi^{I(S)}$, where $I(S):=(m_{1},m_{2}-m_{1},\dots,m_{r}-m_{r-1},m-m_{r})$ if $S=\{m_{1}<\cdots<m_{r}\}$ and $\operatorname{Cl}_{\operatorname{{\mathbb{K}}}}(\operatorname{\mathcal{S}}_{m})$ is the algebra of class functions on $\operatorname{\mathcal{S}}_{m}$. Motivated by the induction and restriction of characters, Geissinger [MR0506405] introduced a graded Hopf algebra structure on $\operatorname{Cl}_{\operatorname{{\mathbb{K}}}}(\operatorname{\mathcal{S}}):=\oplus_{m\geq 0}\operatorname{Cl}_{\operatorname{{\mathbb{K}}}}(\operatorname{\mathcal{S}}_{m})$, as well as a scalar product, that encoded the representation theory of the symmetric group. This structure was lifted to $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}):=\oplus_{m\geq 0}\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}_{m})$ making the new graded Hopf algebra $(\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}),*,\Delta)$ a tempting noncommutative setting for an approach to the character theory of symmetric groups where noncommutativity clarifies the underlying combinatorics, and results like the theorems of Murnaghan-Nakayama and Littewood-Richardson become simple computations [Joellenbeck1999]. In practice, larger Hopf algebras such as the Jöllenbeck algebra [Joellenbeck1999] or the coplactic algebra [MR1334836] by Poirier and Reutenauer are required [MR2424338]. All these algebras are subalgebras of a larger Hopf algebra $(\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}]:=\oplus_{m\geq 0}\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}_{m}],*,\Delta)$ defined by Malvenuto and Reutenauer in [MR1358493] and studied by Aguiar and Solttile in [MR2103213], and recently by Duchamp, Hivert, Novelli and Thibon in [MR2854786]. The Malvenuto-Reutenauer Hopf algebra of permutations is a self-dual Hopf algebra that also appears as the graded dual of the free quasi-symmetric functions introduced in [MR1935570].

Solomon’s descent Hopf algebra $(\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}),*,\Delta)$ is isomorphic to the graded dual of the Hopf algebra of quasi-symmetric functions used by Stanley in connection with plane partitions [MR0332509] and studied by Gessel [MR777705]. Thus, Solomon’s descent algebra is isomorphic to the Hopf algebra of noncommutative symmetric functions introduced by Gelfand et al in [MR1327096], i.e. it is a graded free associative algebra with a generator in each degree. The study of quasi-symmetric functions is an active area of research too rich to be presented here, so we will just remark that “combinatorial objects like to be counted by quasi-symmetric functions” [MR3617217]. This observation also holds true in the context of Hopf algebras since the Hopf algebra of quasi-symmetric functions is a terminal object in the category of combinatorial Hopf algebras [MR2196760], which makes these functions ubiquitous. See [1810.07148, MR1975559] and references therein for more information. Noncommutative analogues of quasi-symmetric functions were introduced in [hivert_2004] and studied by Bergeron and Zabrocki in [MR2555523]. The Hopf algebra of noncommutative quasi-symmetric functions has a monomial basis labeled by set compositions and it is a free and cofree Hopf algebra [MR2555523]. The graded dual of this Hopf algebra also appears when passing from the combinatorics of compositions to the combinatorics of set compositions by means of twisted descent algebras [PaSch08].

Solomon’s descent algebra $(\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}),*,\Delta,\circ)$ also appears as a convolution algebra. The free associative algebra on a countable set of generators, one in each degree, is the universal enveloping algebra $U(\operatorname{Lie}\{\operatorname{\mathbf{X}}\})$ of the free Lie algebra $\operatorname{Lie}\{\operatorname{\mathbf{X}}\}$ on a countable number of generators $\operatorname{\mathbf{X}}$. $U(\operatorname{Lie}\{\operatorname{\mathbf{X}}\})$ is a graded Hopf algebra with a right action of $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}_{m}]$ by permuting the factors of each monomial of degree $m$ while killing the other homogeneous components. This action identifies the opposite of $(\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}_{m}],\operatorname{\circ\,})$ as a subalgebra of $\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U(\operatorname{Lie}\{\operatorname{\mathbf{X}}\}))$. Under this identification, $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}]=\oplus_{m\geq 0}\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}_{m}]$ also inherits a new product $*$ from the convolution product of $\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U(\operatorname{Lie}\{\operatorname{\mathbf{X}}\}))$, as well as a comultiplication, thus recovering the Malvenuto-Reutenauer Hopf algebra $(\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}],*,\Delta)$ of permutations. The subalgebra of $(\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}],*,\Delta)$ generated by $\{(1,2,\dots,m)\in\operatorname{\mathcal{S}}_{m}\mid m\geq 0\}$ coincides with $(\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}),*,\Delta)$. In other words, $(\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}),*)$ can be identified with the convolution algebra generated by the projections of $U(\operatorname{Lie}\{\operatorname{\mathbf{X}}\})$ onto its homogeneous components. Since these projections are free generators, $(\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}),*,\Delta)$ is isomorphic to the Hopf algebra $U(\operatorname{Lie}\{\operatorname{\mathbf{X}}\})$ [MR1231799]. This approach to Solomon’s descent algebras allowed Patras to associate descent algebras to graded bialgebras [MR1302855].

Lie algebras are particular instances of Sabinin algebras. In the same way as Lie algebras are tangent algebras of Lie groups, Sabinin algebras are tangent algebras of local analytic loops. These algebras were introduced in [MR924255] as the basis of a nonassociative Lie theory. Motivated by the work of Shestakov and Umirbaev [MR1899864], in [MR2304338] the Poincaré-Birkhoff-Witt theorem was proved in a nonassociative setting by extending the construction of universal enveloping algebras for Lie algebras to Sabinin algebras. Any Sabinin algebra admits a universal enveloping algebra isomorphic to the bialgebra of distributions with support at the identity of any local analytic loop having the Sabinin algebra as its tangent algebra [MR2718940]. A survey on this topic appeared in [MR3174282]. Any variety $\operatorname{\mathcal{V}}_{\Omega}$ of loops induces a variety $\operatorname{\mathcal{V}}_{\partial\Omega}$ of Sabinin algebras. For instance, the variety of associative loops, i.e. groups, induces the variety of Lie algebras; the variety of Moufang loops induces the variety of Malcev algebras; the variety of Bruck loops induces the variety of Lie triple systems, etc. For each of these varieties $\operatorname{\mathcal{V}}_{\Omega}$ we can consider the relatively free Sabinin algebra $\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\}$ generated by a countable set of generators $\operatorname{\mathbf{X}}$ in $\operatorname{\mathcal{V}}_{\partial\Omega}$ and its universal enveloping algebra $(U(\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\}),\cdot,\Delta)$, which is a graded nonassociative bialgebra. The convolution product on $\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U(\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\}))$ can be defined as usual but it is no longer associative. The approach to Solomon’s descent algebra for the symmetric group adopted by Patras naturally moves to this context making the convolution algebra generated by the projections of $U(\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\})$ onto its homogeneous components an interesting nonassociative object. The main result in the first part of this paper is that this convolution algebra is isomorphic to $U(\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\})$. As a consequence, $(U(\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\}),\cdot,\Delta)$ naturally inherits an associative inner product $\operatorname{\bullet}$, induced by the composition of linear maps, that satisfies a Mackey type formula. The splitting formula for $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}})$ [MR1327096, MR1433196] relating the inner and outer products and the comultiplication also holds in this new context. This suggests that $U(\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\})$ might be an adequate nonassociative generalization of the Hopf algebra $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}})$ and of the Hopf algebra of noncommutative symmetric functions.

Once we accept the new role of $U(\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\})$, it is natural to look for a nonassociative lifting of the Malvenuto-Reutenauer algebra of permutations to accommodate $U(\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\})$ so that the name “descent algebra” makes some combinatorial sense. In the second part of this paper we present a candidate in the absolutely free case, i.e. when $U(\operatorname{Sab}\{\operatorname{\mathbf{X}}\})$ is a free nonassociative algebra. While permutations are the fundamental basis for the Malvenuto-Reutenauer algebra, when dealing with nonassociativity planar binary rooted trees must appear. Surprisingly enough, pseudo-compositions of sets are more natural than permutations, and the fundamental basis of our nonassociative lifting $(\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,],*,\Delta)$ of the Malvenuto-Reutenauer algebra, which is also a nonassociative lifting of the dual of the algebra of noncommutative quasi-symmetric functions studied in [MR2555523], will consist of equivalence classes of symbols $t\pi$, where $t$ is a planar binary rooted tree and $\pi$ is a pseudo-composition. $U(\operatorname{Sab}\{\operatorname{\mathbf{X}}\})$ is identified with a subalgebra $\operatorname{\mathrm{Sol}}(\operatorname{\hbox{\cyrillic X}}\,)$ of $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]$ spanned by elements $D^{t}_{\subseteq S}=\sum_{\operatorname{Des}(\sigma^{\pi})\subseteq S}t\pi$, where $\sigma^{\pi}$ denotes the underlying permutation of $\pi$. The set $\operatorname{\hbox{\cyrillic X}}\,$ of all the equivalence classes $t\pi$, that we call compermutations, is a monoid with respect to certain associative operation $\circ$. The induced inner product $\circ$ on $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]$ extends the inner product of $\operatorname{\mathrm{Sol}}(\operatorname{\hbox{\cyrillic X}}\,)$ inherited from $U(\operatorname{Sab}\{\operatorname{\mathbf{X}}\})$ in the same way as the inner product of $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}]$ extends the inner product of $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}})$. Finally we will discuss the structure of $(\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,],*,\Delta)$ by proving that it is a free nonassociative algebra as well as a graded free coassociative coalgebra. A more detailed study of the inner product of $\operatorname{\mathrm{Sol}}(\operatorname{\hbox{\cyrillic X}}\,)$ and $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]$ will be the topic of a forthcoming paper.

This work originated when exploring the context to extend to a nonassociative setting the beautiful techniques on transformation of alphabets available for noncommutative symmetric functions [MR1433196]. Transformation of alphabets is essential in the study of Lie idempotents, and it will probably play a similar role in the study of projections of nonassociative universal enveloping algebras onto their Sabinin algebras of primitive elements, such as the Dynkin idempotent used in [MR3717941] to obtain a nonassociative Baker-Campbell-Hausdorff formula.

1.1. Notation and conventions.

In this paper the base field $\operatorname{{\mathbb{K}}}$ is assumed to be of characteristic zero. The set of natural numbers is denoted by $\operatorname{{\mathbb{N}}}:=\{0,1,2,\dots\}$. For any $m\in\operatorname{{\mathbb{N}}}$, $\mathbf{m}$ stands for $\{1,\dots,m\}$. The cardinality of a set $A$ is denoted by $|A|$ while the identity map of $A$ is $\operatorname{\mathrm{I}}$ or $\operatorname{\mathrm{I}}_{A}$. The symmetric group of degree $m$ is $\operatorname{\mathcal{S}}_{m}$. Permutations $\sigma\in\operatorname{\mathcal{S}}_{m}$ are written as tuples $\sigma=(\sigma_{1},\dots,\sigma_{m})$ where $\sigma_{i}:=\sigma(i)$.

A pseudo-composition of $m\geq 0$ is an $r$-tuple $I=(i_{1},\dots,i_{r})$ of natural numbers whose weight $|I|:=i_{1}+\dots+i_{r}$ is equal to $m$. The length $l(I)$ of the pseudo-composition $I$ is the number $r$ of components of $I$. The tuple $I$ is composition of $m$, denoted by $I\models m$, if it is a pseudo-composition of $m$ without zero components. The composition associated to a pseudo-composition $I$ is obtained by removing all the zero components in $I$. The set $S(I):=\{i_{1},i_{1}+i_{2},\dots,i_{1}+\cdots+i_{r-1})$ is associated to the composition $I=(i_{1},\dots,i_{r})\models m$ and the mapping $I\mapsto S(I)$ defines a bijection between the compositions of $m$ and the subsets of $\{1,\dots,m-1\}$.

A pseudo-composition $\pi$ of a set $A$ is a tuple $\pi=(\pi_{1},\dots,\pi_{k})$ of (possibly empty) subsets of $A$ so that $A$ is the disjoint union of $\pi_{1},\dots,\pi_{r}$. The type of $\pi$ is the pseudo-composition of $|A|$ defined by $\operatorname{\mathrm{type}}(\pi):=(|\pi_{1}|,\dots,|\pi_{r}|)$. The length $l(\pi)$ of $\pi$ is the number $r$ of components of $\pi$, while the weight of $\pi$ is $|\pi|:=|\pi_{1}|+\cdots+|\pi_{r}|$, the cardinality of $A$. A composition of a set $A$ is a pseudo-composition of $A$ that has no empty components. The composition associated to a pseudo-composition $\pi$ is obtained by removing all the empty sets in $\pi$. We will use the notation $\pi\models A$ to indicate that $\pi$ is a composition of $A$. The type of any composition of $A$ is a composition of $|A|$. The set of pseudo-compositions of $A$ will be denoted by $\operatorname{\mathcal{P}}(A)$ or by $\operatorname{\mathcal{P}}_{m}$ if $A=\mathbf{m}$.

$\operatorname{{\mathbb{K}}}\langle\operatorname{\mathbf{X}}\rangle$ is the unital associative algebra freely generated by a set $\operatorname{\mathbf{X}}$, $\operatorname{{\mathbb{K}}}\{\operatorname{\mathbf{X}}\}$ is the unital nonassociative algebra freely generated by $\operatorname{\mathbf{X}}$ and $\operatorname{{\mathbb{K}}}\{\{\operatorname{\mathbf{X}}\}\}$ is the unital algebra of formal power series on the noncommutative and nonassociative generators $\operatorname{\mathbf{X}}$.

The projection of $\oplus_{m\geq 0}U_{m}$ onto $U_{m}$ will be denoted by $\operatorname{\mathbf{I}}_{m}$ and the degree $m$ of $u\in U_{m}$ is denoted by $|u|$. The graded dual of $\oplus_{m\geq 0}U_{m}$ is $\oplus_{m\geq 0}U^{*}_{m}$, where $U^{*}_{m}$ is the dual space of $U_{m}$. For linear maps defined on tensor products we will write $\varphi(u\otimes v\otimes w\otimes\cdots)$ or $\varphi(u,v,w,\dots)$ at our convenience.

The set of planar binary rooted trees with $m$ leaves is $\mathcal{T}_{m}$. It can be identified with the set of monomials in a non-associative and non-commutative generator $x$ by

[TABLE]

The grafting $t\vee t^{\prime}$ of trees corresponds to the product of monomials. Trees in $\mathcal{T}_{m}$ also describe products of $m$ elements $a_{1},\dots,a_{m}$ in any nonassociative algebra (change from left to right the labels $x$ of the leaves in the previous trees by $a_{1},a_{2},\dots$). Given $t\in\mathcal{T}_{m}$ we will write $t(a_{1},\dots,a_{m})$ for the resulting element. If the nonassociative monomial of degree $m$ associated to $t$ is $g_{m}(x)$, sometimes we will write $g_{m}(a_{1},\dots,a_{m})$ instead of $t(a_{1},\dots,a_{m})$. If $a_{1}=a_{2}=\cdots=a$ we will just write $g_{m}(a)$ or $t(a)$ for the corresponding element.

2. Descent algebras of nonassociative bialgebras

2.1. Nonassociative bialgebras

We will follow [MR0252485, MR594432] for basic definitions on coalgebras, bialgebras and Hopf algebras. A coalgebra $(U,\Delta,\epsilon)$ is a $\operatorname{{\mathbb{K}}}$-vector space $U$ equipped with two linear maps $\Delta\colon U\rightarrow U\otimes U$ (coproduct or comultiplication) and $\epsilon\colon U\rightarrow\operatorname{{\mathbb{K}}}$ (counit) such that, if we denote the image $\Delta(u)$ of $u$ by $\sum u_{(1)}\otimes u_{(2)}$ then $\sum\epsilon(u_{(1)})u_{(2)}=u=\sum u_{(1)}\epsilon(u_{(2)})$. Subcoalgebras are defined in the natural way. A cocommutative (or c.c.) coalgebra is a coalgebra for which $\sum u_{(1)}\otimes u_{(2)}=\sum u_{(2)}\otimes u_{(1)}$. If $(\Delta\otimes\operatorname{\mathrm{I}})\Delta=(\operatorname{\mathrm{I}}\otimes\Delta)\Delta$ then we say that the coalgebra is coassociative (or c.a.). For coassociative coalgebras the iterated application of $\Delta$, lets say $m$ times, to $u$ does not depend on the chosen factors and it is denoted by $\sum u_{(1)}\otimes u_{(2)}\otimes\cdots\otimes u_{(m+1)}$. A group-like element is an element $g\in U$ such that $\epsilon(g)=1$ and $\Delta(g)=g\otimes g$. Clearly $\operatorname{{\mathbb{K}}}g$ is a subcoalgebra of $U$. The set of group-like elements of $U$ is denoted by $\operatorname{G}(U)$. A simple coalgebra is a nonzero coalgebra $U$ with no other subcoalgebras than $\{0\}$ and $U$. A coalgebra with only one simple subcoalgebra is called irreducible. A graded coalgebra is a coalgebra with a decomposition $U=\oplus_{n\geq 0}U_{n}$ such that $\Delta(U_{n})\subseteq\sum_{i+j=n}U_{i}\otimes U_{j}$ and $\epsilon(U_{n})=0$ if $n\geq 1$. Given two coalgebras $(U,\Delta,\epsilon)$ and $(U^{\prime},\Delta^{\prime},\epsilon^{\prime})$, a linear map $\varphi\colon U\rightarrow U^{\prime}$ is a coalgebra morphism if $\Delta^{\prime}\varphi=(\varphi\otimes\varphi)\Delta$ and $\epsilon^{\prime}\varphi=\epsilon$. The tensor product $U\otimes U^{\prime}$ is a coalgebra with $\Delta(u\otimes u^{\prime}):=\sum(u_{(1)}\otimes u^{\prime}_{(1)})\otimes(u_{(2)}\otimes u^{\prime}_{(2)})$ and $\epsilon(u\otimes u^{\prime}):=\epsilon(u)\epsilon^{\prime}(u^{\prime})$. In case that $U$ and $U^{\prime}$ are graded coalgebras then $U\otimes U^{\prime}$ is also a graded coalgebra with homogeneous component of degree $n$ given by $\sum_{i+j=n}U_{i}\otimes U^{\prime}_{j}$. The base field $\operatorname{{\mathbb{K}}}$ is a graded coalgebra condensed in degree [math] with comultiplication $\Delta(\lambda):=\lambda 1\otimes 1$ and $\epsilon(\lambda):=\lambda$.

A unital $\operatorname{{\mathbb{K}}}$-algebra $(U,\mu,\nu)$ is a $\operatorname{{\mathbb{K}}}$-vector space $U$ equipped with two linear maps $\mu\colon U\otimes U\rightarrow U$ (the product or multiplication) and $\nu\colon\operatorname{{\mathbb{K}}}\rightarrow U$ (the unit) such that $\nu(\lambda)u=\lambda u=u\nu(\lambda)$ where $uv:=\mu(u\otimes v)$. The image $1:=1_{U}:=\nu(1_{\operatorname{{\mathbb{K}}}})$ is the unit or identity element of $U$. Commutative and associative algebras are defined in the usual way. A unital graded algebra is a unital algebra with a decomposition $U=\oplus_{n\geq 0}U_{n}$ satisfying $U_{i}U_{j}\subseteq U_{i+j}$. Clearly $1\in U_{0}$.

A unital $\operatorname{{\mathbb{K}}}$-bialgebra $(U,\mu,\nu,\Delta,\epsilon)$ is a coalgebra $(U,\Delta,\epsilon)$ and a unital algebra $(U,\mu,\nu)$ so that $\mu$ and $\nu$ are coalgebra morphisms. Depending on the properties of the underlying coalgebra and algebra structure we will a bialgebra can be associative, commutative, coassociative, cocommutative or irreducible. We say that $U$ is graded unital bialgebra if it is a graded coalgebra and a unital graded algebra with the same gradation. The unit element $1$ of any unital bialgebra is a group-like element, so for any irreducible bialgebra the simple subcoalgebra must be $\operatorname{{\mathbb{K}}}1$. A primitive element of a unital bialgebra is an element $u$ such that $\Delta(u)=u\otimes 1+1\otimes u$ and $\epsilon(u)=0$. The subspace of primitive elements of $U$ is denoted by $\operatorname{Prim}(U)$. This space is closed under the commutator product $[u,v]:=uv-vu$, so it is a Lie algebra if $U$ is associative. An associative Hopf algebra is an associative unital bialgebra $(U,\mu,\nu,\Delta,\epsilon)$ equipped with an extra linear map, the antipode, $S\colon U\rightarrow U$ satisfying $\sum S(u_{(1)})u_{(2)}=\epsilon(u)1=\sum u_{(1)}S(u_{(2)})$. For short, we will omit the unit, counit and the antipode from the notation for algebras, bialgebras and Hopf algebras, and sometime we will omit all the operations. So we will say that $(U,\mu,\Delta)$ or just $U$ is a (unital) bialgebra or a Hopf algebra, for instance.

Group-like elements play an important role in the following. However, these elements are scarce in our graded bialgebras $U=\oplus_{n\geq 0}U_{n}$ but they naturally appear in the completion $\hat{U}:=\prod_{n=0}^{\infty}U_{n}$ of $U$ relative to the grading when we endow $\hat{U}$ with the continuous extension of the operations of $U$ and we change $\otimes$ by the completed tensor product $\hat{\otimes}$. When required the reader must assume that the operations take place in these completed bialgebras. For short, $\oplus_{n\geq 1}U_{n}$ and $\prod_{n=1}^{\infty}U_{n}$ will be denoted by $U_{+}$ and $\hat{U}_{+}$ respectively.

Irreducible c.a. and c.c. unital bialgebras over fields of characteristic zero are well understood [MR2304338, MR3174282]. In the associative case, the Milnor-Moore theorem [MR0252485, MR594432] ensures that any such bialgebra is isomorphic to the universal enveloping algebra $U(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}=\operatorname{Prim}(U)$ of primitive elements. In the nonassociative setting $\operatorname{Prim}(U)$ is closed under many more multilinear operations other than the commutator and it becomes a Sabinin algebra [MR1899864], i.e. a vector space endowed with two families of multilinear operations

[TABLE]

which satisfy the identities

[TABLE]

and

[TABLE]

where $\alpha$ runs the set of all permutations in the symmetric group $\operatorname{\mathcal{S}}_{r}$ with $\alpha(1)<\cdots<\alpha(k)$, $\alpha(k+1)<\cdots<\alpha(r)$, $k=0,1,\dots,r$, $r\geq 0$, $\sigma_{x,y,z}$ denotes the cyclic sum on $x,y$ and $z$, $\tau\in\operatorname{\mathcal{S}}_{m}$ and $\delta\in\operatorname{\mathcal{S}}_{n}$.

Any irreducible c.a. and c.c. unital bialgebra $U$ is isomorphic to the universal enveloping algebra $U(\operatorname{\mathfrak{s}})$ of the Sabinin algebra $\operatorname{\mathfrak{s}}=\operatorname{Prim}(U)$ of primitive elements [MR2304338], and the Poincaré-Birkhoff-Witt for Sabinin algebras establishes that any Sabinin algebra $\operatorname{\mathfrak{s}}$ is isomorphic to the Sabinin algebra of primitive elements of its universal enveloping algebra $U(\operatorname{\mathfrak{s}})$, which is an irreducible c.a. and c.c. unital bialgebra. For instance, the universal enveloping algebra of the Sabinin algebra $\operatorname{Sab}\{\operatorname{\mathbf{X}}\}$ freely generated by $\operatorname{\mathbf{X}}$ is the free nonassociative unital algebra $\operatorname{{\mathbb{K}}}\{\operatorname{\mathbf{X}}\}$, which is an irreducible c.a. and c.c. unital bialgebra with the comultiplication determined by making the generators in $\operatorname{\mathbf{X}}$ primitive and $\Delta(uv)=\Delta(u)\Delta(v)$. $\operatorname{Sab}\{\operatorname{\mathbf{X}}\}$ coincides with the primitive elements of $\operatorname{{\mathbb{K}}}\{\operatorname{\mathbf{X}}\}$. In particular, any algebra can be seen as a Sabinin algebra with the operations induced by the evaluation of the nonassociative polynomials $\langle x_{1},\dots,x_{m};x_{m+1},x_{m+2}\rangle$ and $\Phi(x_{1},\dots,x_{m};x_{m+1},\dots,x_{m+n})$ of $\operatorname{{\mathbb{K}}}\{\operatorname{\mathbf{X}}\}$, so we can always consider the Sabinin subalgebra generated by a set of elements in any algebra [MR1899864]. A unified approach to Poincaré-Birkhoff-Witt type theorems in terms of monads has recently appeared in [DoTa18]. Geometrically, $U(\operatorname{\mathfrak{s}})$ can be identified with the bialgebra of distributions with support at the identity of any analytic loop with Sabinin tangent algebra $\operatorname{\mathfrak{s}}$ [MR924255, MR2718940].

Recall that a loop is a nonempty set with three operations $xy$, $x\backslash y$ and $x/y$ that satisfy $x\backslash(xy)=y=x(x\backslash y)$, $(xy)/y=y=(x/y)y$ and $x\backslash x=y/y$. Groups are associative loops with divisions $x\backslash y=x^{-1}y$, $x/y=xy^{-1}$ and $x\backslash x=e=y/y$, where $e$ denotes the identity element of the group. The antipode for Hopf algebras is related to the inverse map for groups, and this map is unnatural for loops. Instead of antipodes, nonassociative (c.a. and c.c. irreducible unital) bialgebras have two extra bilinear maps, the left and right divisions, $\backslash$ and $/$ that satisfy

[TABLE]

Irreducible c.a. and c.c. unital bialgebras, as for instance $\operatorname{{\mathbb{K}}}\{\operatorname{\mathbf{X}}\}$ or any other universal enveloping algebra $U(\operatorname{\mathfrak{s}})$, always have these divisions.

2.2. Descent algebras of nonassociative graded bialgebras

The descent algebra of a graded bialgebra $U$ was introduced in [MR1302855] by Patras as a subalgebra of $\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U)$ for the convolution product. In a nonassociative setting it is natural to consider $\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U)$ endowed not only with the convolution $*$ but with the left and right divisions as well:

[TABLE]

In contrast to the composition $f\operatorname{\circ\,}h$ of linear maps, due to the lack of associativity of the product $uv$ on $U$, the convolution product $f*h$ on $\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U)$ is nonassociative in general.

Definition 2.1**.**

Given a graded bialgebra $U=\oplus_{n\geq 0}U_{n}$, let $\operatorname{\mathbf{I}}_{n}$ be the projection of $U$ onto $U_{n}$. The descent algebra of $U$ is defined as the subalgebra $\Sigma^{U}$ of $(\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U),*)$ generated by $\{\operatorname{\mathbf{I}}_{n}\mid n\in\operatorname{{\mathbb{N}}}\}$. It is a graded algebra $\Sigma^{U}=\oplus_{n\geq 0}\Sigma^{U}_{n}$ with $\Sigma^{U}_{n}:=\{f\in\Sigma^{U}\mid f(U_{n})\subseteq U_{n},f(U_{i})=0,\,i\neq n\}$.

Following [MR1900008] we say that $f\in\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U)$ admits $F\in\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U)\otimes\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U)$ as a pseudo-coproduct if

[TABLE]

In case that $F=f\otimes\epsilon 1_{U}+\epsilon 1_{U}\otimes f$ we say that $f$ is pseudo-primitive. Pseudo-coproducts appear since, as proved by Patras and Reutenauer [MR1900008]*Appendix, in general there is not a well-defined coproduct $f\mapsto\Delta(f)$ on $\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U)$ satisfying $\Delta(f)\operatorname{\circ\,}\Delta=\Delta\operatorname{\circ\,}f$. The following result [MR1900008]*Theorem 2 shows that pseudo-primitive elements are useful to study projections of $U$ onto $\operatorname{Prim}(U)$. The original proof does not require of the associativity.

Theorem 2.2**.**

Let $U$ be a c.a. and c.c. unital bialgebra and $f,g\in\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U)$. We have that:

• •

If $f,g$ admit the pseudo-coproducts $F,G$ and $\alpha\in\operatorname{{\mathbb{K}}}$, then $f+g,\alpha f,f*g,f\operatorname{\circ\,}g$ admit respectively the pseudo-coproducts $F+G,\alpha F,F*G$ and $F\operatorname{\circ\,}G$.

• •

An element $f\in\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U)$ takes values in $\operatorname{Prim}(U)$ if and only if it is pseudo-primitive.

A rather trivial but important observation in this context is that

[TABLE]

thus, by Theorem 2.2, all the elements in the descent algebra $\Sigma^{U}$ have at least a pseudo-coproduct in $\Sigma^{U}\otimes\Sigma^{U}$. Looking at projections of $U$ onto $\operatorname{Prim}(U)$ as pseudo-primitive maps has provided to be very useful. For instance, the identity map $\operatorname{\mathbf{I}}$ admits $\operatorname{\mathbf{I}}\otimes\operatorname{\mathbf{I}}$ as a pseudo-coproduct. By analogy with the behavior of group-like elements in Hopf algebras this suggested that the logarithm of $\operatorname{\mathbf{I}}$ should be pseudo-primitive, which led to a better understanding of some remarkable projections [MR1274784, MR1231799] of $U$ on $\operatorname{Prim}(U)$.

Given an $r\times s$ matrix $M=(m_{ij})_{i,j}$ with entries in $\operatorname{{\mathbb{N}}}:=\{0,1,2,\dots\}$, $\operatorname{\mathbf{I}}_{M}$ will denote the element $\operatorname{\mathbf{I}}_{m_{11}}\otimes\cdots\otimes\operatorname{\mathbf{I}}_{m_{1s}}\otimes\cdots\otimes\operatorname{\mathbf{I}}_{m_{r1}}\otimes\cdots\otimes\operatorname{\mathbf{I}}_{m_{rs}}$. For any planar binary rooted tree $t\in\mathcal{T}_{rs}$, $t(\operatorname{\mathbf{I}}_{M})$ stands for the element in $\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U)$ obtained by performing the multilinear operation indicated by $t$ with respect to the convolution product $*$ to $\operatorname{\mathbf{I}}_{M}$. Pseudo-compositions are examples of the previous matrices $M$, so the notation $t(\operatorname{\mathbf{I}}_{I})$ makes sense for pseudo-compositions $I$. The row sum of $M$ is $(\sum_{j}m_{1,j},\dots,\sum_{j}m_{r,j})$ while the column sum of $M$ is $(\sum_{i}m_{i,1},\dots,\sum_{i}m_{i,s})$. The following result is well-known for associative bialgebras [MR1231799]*Theorem 9.2 or [MR1302855]*Théorème II,7.

Theorem 2.3** (Mackey formula).**

Let $U$ be a c.a. and c.c. graded unital bialgebra. Given two pseudo-compositions $I$, $J$ of $m$ and trees $t\in\mathcal{T}_{l(I)}$ and $t^{\prime}\in\mathcal{T}_{l(J)}$, we have

[TABLE]

where $\operatorname{\circ\,}$ denotes the composition of linear maps and $M$ runs the set of all matrices with entries in $\operatorname{{\mathbb{N}}}$ whose row sum and column sum are respectively $I$ and $J$, and $t(t^{\prime})$ is the tree obtained after placing a copy of $t^{\prime}$ in each leaf of $t$ (the root of $t^{\prime}$ is identified with the leaf). In particular, the descent algebra $\Sigma^{U}$ is closed under composition.

Proof.

Let $I=(i_{1},\dots,i_{r})$ and $J=(j_{1},\dots,j_{s})$ be pseudo-compositions. Given $u\in U$ homogeneous,

[TABLE]

where $M$ runs the set of all matrices with entries in $\operatorname{{\mathbb{N}}}$ whose row sum and column sum are respectively $I$ and $J$. ∎

Definition 2.4**.**

The composition $\operatorname{\circ\,}$ of linear maps is called the inner product of $\Sigma^{U}$ while the convolution product $*$ that is called the outer product.

2.3. Eulerian idempotents

Our goal is to understand $\Sigma^{U}$ in a rather general nonassociative setting by proving that it is ‘relatively free’. However, first we have to explain what we mean by this and which varieties are adequate for our purposes, which requires of eulerian idempotents.

Recall that there are several ways of extending the exponential when associativity is not assumed [MR2029036, MR2074402, MR1984911, MR2718940]. As discussed in [MR3717941], each one leads to a different definition of logarithm, but many times there are no algebraic reasons to make a choice. This situation is similar to selecting a base for a logarithm of positive real numbers. A group-like element $g(x)=\sum_{n}g_{n}(x)\in\operatorname{{\mathbb{K}}}\{\{x\}\}$ with $|g_{n}(x)|=n$ –here $\operatorname{{\mathbb{K}}}\{\{x\}\}$ stands for the bialgebra of nonassociative and noncommutative formal power series on the primitive element $x$– is called a base for logarithms if $g_{1}(x)\neq 0$. For any such $g(x)$ we clearly have $\Delta(g_{n}(x))=\sum_{i+j=n}g_{i}(x)\otimes g_{j}(x)$. A natural choice for $g(x)$ could be

[TABLE]

but it is only one of the infinitely many possible bases, and we will not adhere to this particular choice.

Lemma 2.5**.**

For any base for logarithms $g(x)\in\operatorname{{\mathbb{K}}}\{\{x\}\}$ and any graded c.a. and c.c. unital bialgebra $U$, the map $g\colon\hat{U}_{+}\rightarrow 1+\hat{U}_{+}$ given by $u\mapsto g(u)$ is bijective.

The bijective map in the lemma is called the exponentiation on $U$ with base $g(x)$, while its inverse $\log_{g}$ is called logarithm to the base $g(x)$.

Lemma 2.6**.**

In $\operatorname{{\mathbb{K}}}\{\{y\}\}\hat{\otimes}\operatorname{{\mathbb{K}}}\{\{z\}\}$ we have

[TABLE]

Proof.

Since

[TABLE]

the statement follows by taking inverses. ∎

Given a fixed base $g(x)$, we expand $\log_{g}(1+x)$ as $\log_{g}(1+x)=\sum_{n}l_{n}(x)\in\operatorname{{\mathbb{K}}}\{\{x\}\}$ with $|l_{n}(x)|=n$. For any irreducible c.a. and c.c. unital bialgebra $U$, the map

[TABLE]

where $\operatorname{\mathbf{I}}_{+}(u)=u-\epsilon(u)1$ is well-defined. By including $*$ in the notation we want to emphasize that the computation of $l^{*}_{n}(\operatorname{\mathbf{I}}_{+})$ depends on the convolution product rather than on the usual composition $\circ$.

Proposition 2.7**.**

Let $U$ be an irreducible graded c.a. and c.c. unital bialgebra, then $\log_{g}^{*}(\operatorname{\mathbf{I}})$ is pseudo-primitive.

Proof.

The result follows from the equalities

[TABLE]

for any $u\in U$, where the last equality is consequence of Lemma 2.6. ∎

Let $g(x)=\sum_{n\geq 0}g_{n}(x)$ be a base for logarithms and $U$ be an irreducible graded c.a. and c.c. unital bialgebra. For any $n\geq 0$ consider linear maps

[TABLE]

We again include $*$ in the notation to emphasize that the operations indicated by $g_{n}(x)$ are performed with respect to the convolution product. In the associative setting the elements $\operatorname{\mathbf{I}}_{n}\operatorname{\circ\,}e^{m}$ correspond to the eulerian idempotents of $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}]$ [MR1274784, MR1302855].

Proposition 2.8**.**

We have

[TABLE]

Proof.

Take $\lambda\in\operatorname{{\mathbb{K}}}$ and $L:=\log_{g}^{*}(\operatorname{\mathbf{I}})$. Since $g(x)$ is group-like and $L$ is pseudo-primitive, then the linear map $g^{*}(\lambda L)$ admits $g^{*}(\lambda L)\otimes g^{*}(\lambda L)$ as a pseudo-coproduct. Thus, for any $u\in U$ we have

[TABLE]

Since $\operatorname{{\mathbb{K}}}$ is infinite then $e^{m}\operatorname{\circ\,}e^{n}=\delta_{m,n}e^{m}$ as desired. The second claim in the statement is obvious. ∎

The map $e^{1}$ is a projection of $U$ onto $\operatorname{Prim}(U)$. The elements

[TABLE]

belong to the descent algebra $\Sigma^{U}$ because

[TABLE]

for any $f,h\in\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U)$. Since $e^{1}_{n}$ takes values in $\operatorname{Prim}(U)$, by Theorem 2.2 these maps are pseudo-primitive, so any other map in the Sabinin algebra generated by $\{e^{1}_{n}\mid n\geq 1\}$ is.

Proposition 2.9**.**

Let $U$ be an irreducible graded c.a. and c.c. unital bialgebra. The descent algebra $\Sigma^{U}$ is generated by $\{e^{1}_{n}\}_{n\geq 1}$ as a unital algebra.

Proof.

It follows from $\operatorname{\mathbf{I}}_{n}=\operatorname{\mathbf{I}}_{n}\operatorname{\circ\,}\operatorname{\mathbf{I}}=\operatorname{\mathbf{I}}_{n}\operatorname{\circ\,}g^{*}(\log_{g}^{*}(\operatorname{\mathbf{I}}))$ and (2.6). ∎

2.4. Varieties of Sabinin algebras determined by varieties of loops.

Let $\Omega$ be a subset of a free loop on generators $z_{1},z_{2},\dots$. We may think of $\Omega$ as the identities that define a variety $\operatorname{\mathcal{V}}_{\Omega}$ of loops. We will restrict to varieties that contains the abelian groups, so in the following we will assume that any abelian group belongs to $\operatorname{\mathcal{V}}_{\Omega}$.

Once we have eulerian idempotents at our disposal, we can easily define the variety $\operatorname{\mathcal{V}}_{\partial\Omega}$ of Sabinin algebras associated to $\Omega$. First fix a base for logarithms $g(x)=\sum_{n\geq 0}g_{n}(x)$. Let $e^{1}=g^{*}_{1}(\log_{g}^{*}(\operatorname{\mathbf{I}}))$ and $e^{1}_{n}=\operatorname{\mathbf{I}}_{n}\operatorname{\circ\,}e^{1}$ be as in (2.5). Given $w(z_{1},\dots,z_{n})\in\Omega$, consider

[TABLE]

Since $e^{1}$ is a projection onto the primitives, these elements belong to the free Sabinin algebra $\operatorname{Sab}\{\operatorname{\mathbf{X}}\}$.

Definition 2.10**.**

The variety $\operatorname{\mathcal{V}}_{\partial\Omega}$ of Sabinin algebras determined by $\Omega$ is the variety of Sabinin algebras that satisfy the identities

[TABLE]

As we will show, $\operatorname{\mathcal{V}}_{\partial\Omega}$ does not depend on the choice of the base for logarithms $g(x)$.

Lemma 2.11**.**

Let $g(x)=\sum_{n\geq 0}g_{n}(x)$ be a base for logarithms, $\operatorname{\mathfrak{s}}$ a Sabinin algebra and $B$ a totally ordered basis of $\operatorname{\mathfrak{s}}$. The set

[TABLE]

is a basis of $U(\operatorname{\mathfrak{s}})$.

Proof.

The graded algebra of $U(\operatorname{\mathfrak{s}})$ associated to the coradical filtration of $U(\operatorname{\mathfrak{s}})$ is isomorphic to the symmetric algebra $\operatorname{{\mathbb{K}}}[\operatorname{\mathfrak{s}}]$ [MR2304338] and, up to scalars, when associativity is assumed $g_{n}(x)$ is $x^{n}$ since the group-like elements in the completion of $\operatorname{{\mathbb{K}}}\langle x\rangle$ are of the form $e^{\lambda x}=\sum_{n\geq 0}(\lambda x)^{n}/n!$ with $\lambda\in\operatorname{{\mathbb{K}}}$. Thus, the image $\operatorname{PBW}_{g}(U(\operatorname{\mathfrak{s}});B)$ in the graded algebra of $U(\operatorname{\mathfrak{s}})$ is a basis. This proves the statement. ∎

2.5. Relative freeness of the descent algebra of $U_{\Omega}$.

Let us denote by $\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\}$ the relatively free Sabinin algebra on $\operatorname{\mathbf{X}}:=\{x_{1},x_{2},\dots\}$ in the variety $\operatorname{\mathcal{V}}_{\partial\Omega}$ and $U_{\Omega}:=U(\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\})$ its universal enveloping algebra. Since abelian group belong to $\operatorname{\mathcal{V}}_{\Omega}$, by the universal property of $U_{\Omega}$ [MR2304338] there exists a homomorphism of algebras determined by

[TABLE]

for all $i\geq 1$. As it is customary in dealing with noncommutative symmetric functions the grading on $U_{\Omega}$ and $\operatorname{Sab}_{\Omega}(\operatorname{\mathbf{X}})$ are induced by

[TABLE]

for all $i\geq 1$. This is rather important in the following since $\operatorname{\mathbf{I}}_{i}$ is the projection of $U_{\Omega}$ with respect to this grading.

Theorem 2.12**.**

Let $\Sigma^{U_{\Omega}}$ be the descent algebra of $U_{\Omega}$. There exists an isomorphism of unital algebras $\varphi\colon U_{\Omega}\rightarrow\Sigma^{U_{\Omega}}$ determined by $\varphi\colon x_{i}\mapsto e^{1}_{i}$ $(i\geq 1)$.

Proof.

First we will check that $U(\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\})$ satisfies the linearization of the identities in $\Omega$. Recall that the linearization of $w(z_{1},\dots,z_{n})\sim 1$ is obtained by substituting repeated occurrences of $z_{i}$ in $w(z_{1},\dots,z_{n})$ with $(x_{i})_{(1)},(x_{i})_{(2)},\dots$ –include a summatory symbol if needed– and the unit element $1$ of the right-hand side with $\epsilon(x_{1})\cdots\epsilon(x_{n})1$ in case that all $z_{1},\dots,z_{n}$ appear in $w(z_{1},\dots,z_{n})$ (see [MR2304338] for details and examples). Linearization converts identities $w(z_{1},\dots,z_{n})\sim 1$ on loops in multilinear identities $w^{\prime}(x_{1},\dots,x_{n})\sim\epsilon(x_{1})\cdots\epsilon(x_{n})1$ for nonassociative c.a. and c.c. unital bialgebras with left and right divisions such as $U(\operatorname{\mathfrak{s}})$.

Given $w(z_{1},\dots,z_{n})\in\Omega$, let $w^{\prime}$ be its linearization. For any $a_{1},\dots,a_{n}\in\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\}$, $\lambda_{1},\dots,\lambda_{n}\in\operatorname{{\mathbb{K}}}$ and $m\geq 1$, in the completed algebra $\widehat{U_{\Omega}}$ we have

[TABLE]

Thus, since $g(\lambda_{i}a_{i})$ is group-like,

[TABLE]

which implies $\sum_{i_{1},\dots,i_{n}}\lambda^{i_{1}}_{1}\cdots\lambda^{i_{n}}_{n}w^{\prime}(g_{i_{1}}(a_{1}),\dots,g_{i_{n}}(a_{n}))=1$ and, since $\operatorname{{\mathbb{K}}}$ is infinite,

[TABLE]

By Lemma 2.11 we can conclude that

[TABLE]

for all $u_{1},\dots,u_{n}\in U_{\Omega}$.

The set of coalgebra morphisms $\operatorname{Coalg}(U_{\Omega})$ of $U_{\Omega}$ is a loop with the operations $*$, $\backslash$ and $/$ in (2.1), (2.2) and (2.3) (see [MR2304338] for more details). The unit element is the map $\epsilon 1$. Identity (2.7) in $U_{\Omega}$ implies that the loop $\operatorname{Coalg}(U_{\Omega})$ satisfies the identity $w(z_{1},\dots,z_{n})\sim 1$. Thus, $\operatorname{Coalg}(U_{\Omega})$ is a loop in $\operatorname{\mathcal{V}}_{\Omega}$.

By Theorem 2.2, for any pseudo-primitive $f\in\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U_{\Omega})$, the map $g^{*}(f)$ has $g^{*}(f)\otimes g^{*}(f)$ as pseudo-product, so $g^{*}(f)\in\operatorname{Coalg}(U_{\Omega})$. Since $\{e^{1}_{i}\mid i\geq 1\}$ are pseudo-primitive, elements $f_{1},\dots,f_{n}$ in the Sabinin algebra generated by $\{e^{1}_{i}\mid i\geq 1\}$ inside $\operatorname{End}_{\operatorname{{\mathbb{K}}}}(U_{\Omega})$ are pseudo-primitive too, so $w(g^{*}(f_{1}),\dots,g^{*}(f_{n}))=\epsilon 1$. This proves that $\log_{g}^{*}(w(g^{*}(f_{1}),\dots,g^{*}(f_{n})))=0$ for any $w\in\Omega$. In particular, the Sabinin algebra generated by $\{e^{1}_{i}\mid i\geq 1\}$ inside $\Sigma^{U_{\Omega}}$ satisfies the identities in $\partial\Omega$. By the universal property of $U_{\Omega}$ and Proposition 2.9, there exists a well defined epimorphisms $\varphi\colon U_{\Omega}\rightarrow(\Sigma^{U_{\Omega}},*)$ induced by $x_{i}\mapsto e^{1}_{i}$.

Let us prove that $\varphi$ is an isomorphism. Consider

[TABLE]

in $\Sigma^{U_{\Omega}}$ for some scalars $\alpha_{w}\in\operatorname{{\mathbb{K}}}$, where $\operatorname{{\mathbb{K}}}\{y_{1},\dots,y_{n}\}$ denotes the nonassociative unital algebra freely generated by $\{y_{1},\dots,y_{n}\}$. Evaluating (2.8) on the group-like element $g(x_{1}+\cdots+x_{n})\in\widehat{U_{\Omega}}$ we get

[TABLE]

Since $g_{1}(x)$ is a scalar multiple of $x$ and $e^{1}_{i}(g(x_{1}+\cdots+x_{n}))=\operatorname{\mathbf{I}}_{i}\operatorname{\circ\,}g_{1}(x_{1}+\cdots+x_{n})=g_{1}(x_{i})$ –recall that $|x_{i}|=i$– then we obtain, after re-scaling, that $\sum_{w\in\operatorname{{\mathbb{K}}}\{y_{1},\dots,y_{n}\}}\alpha_{w}w(x_{1},\dots,x_{n})=0$, i.e. $\alpha_{w}=0$ for all $w$. This proves the statement. ∎

Remark 2.13*.*

In the proof of the theorem we got that $U(\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\})$ satisfies the linearization of the identities in $\Omega$. Thus, given another base for logarithms $g^{\prime}(x)$ and $a_{1},\dots,a_{n}\in\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\}$ we have $w(g^{\prime}(a_{1}),\dots,g^{\prime}(a_{n}))=1$ for any $w(z_{1},\dots,z_{n})\in\Omega$. In particular, $\log_{g^{\prime}}(w(g^{\prime}(a_{1}),\dots,g^{\prime}(a_{n})))=0$. This proves that $\operatorname{Sab}_{\Omega}\{\operatorname{\mathbf{X}}\}$ satisfies the identities in $\partial\Omega$ when this set is computed with respect to another base. In other words, $U_{\Omega}$ only depends on $\Omega$.

The inner product of $\Sigma^{U_{\Omega}}$ induces an inner product on $U_{\Omega}$ by

[TABLE]

while the comultiplication on $U_{\Omega}$ induces a comultiplication on $\Sigma^{U_{\Omega}}$ for which $e^{1}_{n}$ are primitive rather than pseudo-primitive. For this comultiplication, $\Delta(\operatorname{\mathbf{I}}_{m})=\sum_{i=0}^{m}\operatorname{\mathbf{I}}_{i}\otimes\operatorname{\mathbf{I}}_{j}$.

Given a nonassociative word, $w(y_{1},\dots,y_{n})\in\operatorname{{\mathbb{K}}}\{y_{1},\dots,y_{n}\}$, let us denote by $w^{*}(f_{1},\dots,f_{n})$ the evaluation of $w(y_{1},\dots,y_{n})$ on $f_{1},\dots,f_{n}$. The following formula holds on $(U_{\Omega},*,\Delta,\operatorname{\circ\,})$.

Proposition 2.14** (Splitting formula).**

For any maps $f_{1},\dots,f_{n},g\in\Sigma^{U_{\Omega}}$ and any $w(y_{1},\dots,y_{n})\in\operatorname{{\mathbb{K}}}\{y_{1},\dots,y_{n}\}$we have

[TABLE]

Proof.

Any $g\in\Sigma^{U_{\Omega}}$ has a pseudo-product an it is given by $\Delta(g)$. Thus

[TABLE]

This proves the statement. ∎

3. A nonassociative lifting of the Malvenuto-Reutenauer algebra

The associative Malvenuto-Reutenauer algebra [MR1358493] of permutations is the Hopf algebra structure determined on $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}]$ by

[TABLE]

for $\sigma\in\operatorname{\mathcal{S}}_{m},\tau\in\operatorname{\mathcal{S}}_{n}$. Here $\operatorname{Sh}(m,n)$ is the set of all $(m,n)$-shuffles, i.e. permutations $\gamma=(\gamma_{1},\dots,\gamma_{m+n})\in\operatorname{\mathcal{S}}_{m+n}$ such that $\gamma_{1}<\cdots<\gamma_{m}$ and $\gamma_{m+1}<\cdots<\gamma_{m+n}$, $\sigma\times\tau[m]$ is the image of $(\sigma,\tau)$ under the usual embedding of $\operatorname{\mathcal{S}}_{m}\times\operatorname{\mathcal{S}}_{n}$ in $\operatorname{\mathcal{S}}_{m+n}$, $\sigma|_{A}$ means that we remove from $\sigma$ all the components not in $A$, and $\operatorname{st}(\sigma|_{\mathbf{m}\setminus\mathbf{l}})$ means that we subtract $l$ to the components of $\sigma|_{\mathbf{m}\setminus\mathbf{l}}$ to get an element in $\operatorname{\mathcal{S}}_{m-l}$. This is a self-dual Hopf algebra and the reader should be aware that sometimes it is the algebra with the dual operations which is referred as the Malvenuto-Reutenauer algebra.

3.1. A monoid of pseudo-compositions.

Two pseudo-compositions are equivalent if their associated compositions agree. Pseudo-compositions of type $(1,1,\dots,1)$ correspond to permutations in an obvious way: the permutation $\sigma=(\sigma_{1},\dots,\sigma_{m})$ is identified with the pseudo-composition $\pi^{\sigma}:=(\{\sigma_{1}\},\dots,\{\sigma_{m}\})$. In fact, any $\pi=(\pi_{1},\dots,\pi_{r})\in\operatorname{\mathcal{P}}_{m}$ determines an underlying permutation $\sigma^{\pi}$ by placing the elements of $\pi_{1}$ in increasing order, followed by those of $\pi_{2}$, also in increasing order, etc. For instance, if $\pi=(\{3,4\},\emptyset,\{5,2\},\{1\})$ then $\sigma^{\pi}=(3,4,2,5,1)$. We will use the notation $\pi(i)$ for the image of $i$ by $\sigma^{\pi}$ to alleviate the notation. The symmetric group $\operatorname{\mathcal{S}}_{m}$ acts on $\operatorname{\mathcal{P}}_{m}$ by $\sigma(\pi):=(\sigma(\pi_{1}),\dots,\sigma(\pi_{r}))$. In particular, given $\pi,\bar{\pi}\in\operatorname{\mathcal{P}}_{m}$ we can define $\bar{\pi}(\pi)$ as $\sigma^{\bar{\pi}}(\pi)$. Thus, we have the following product on $\operatorname{\mathcal{P}}_{m}$:

[TABLE]

This formula also defines a product for compositions after removing all empty sets. The opposite product of $\circ$ is denoted by $\operatorname{\bullet}$, i.e. $\pi\operatorname{\bullet}\bar{\pi}:=\bar{\pi}\operatorname{\circ\,}\pi$.

Proposition 3.1**.**

$(\operatorname{\mathcal{P}}_{m},\circ)$* is a monoid with identity element $(\mathbf{m})=(\{1,\dots,m\})$ and the map $\operatorname{\mathcal{P}}_{m}\rightarrow\operatorname{\mathcal{S}}_{m}$ given by $\pi\mapsto\sigma^{\pi}$ is an epimorphism of monoids.*

Proof.

First notice that the pseudo-composition associated to $\sigma^{\pi}\operatorname{\circ\,}\sigma^{\bar{\pi}}$ is equivalent to $\sigma^{\pi}(\bar{\pi}_{1}\cap\{1\},\dots,\bar{\pi}_{1}\cap\{m\},\dots,\bar{\pi}_{s}\cap\{1\},\dots,\bar{\pi}_{s}\cap\{m\})=(\pi(\bar{\pi}_{1})\cap\{\bar{\pi}(1)\},\dots,\pi(\bar{\pi}_{s})\cap\{\bar{\pi}(m)\})$ which is equivalent to $\sigma^{\pi\operatorname{\circ\,}\bar{\pi}}$. Therefore, $\sigma^{\pi}\operatorname{\circ\,}\sigma^{\bar{\pi}}=\sigma^{\pi\operatorname{\circ\,}\bar{\pi}}$. To prove the associativity of $(\operatorname{\mathcal{P}}_{m},\circ)$ we observe that

[TABLE]

while, since $\sigma^{\pi\operatorname{\circ\,}\bar{\bar{\pi}}}=\sigma^{\pi}\operatorname{\circ\,}\sigma^{\bar{\pi}}$,

[TABLE]

Finally, $\pi\operatorname{\circ\,}(\mathbf{m})=(\dots,\pi(\mathbf{m})\cap\pi_{j},\dots)=\pi=(\dots,\operatorname{\mathrm{I}}(\pi_{i})\cap\mathbf{m},\dots)=(\mathbf{m})\operatorname{\circ\,}\pi$. ∎

Given $A=\{p+1,\dots,m\}\subseteq\mathbf{m}$, any $\pi\in\operatorname{\mathcal{P}}(A)$ has a standard form $\operatorname{st}(\pi)\in\operatorname{\mathcal{P}}_{\mathbf{m-p}}$ that it is obtained by subtracting $p$ to the elements that appear in $\pi$. For any set $A$ and any $B\subseteq A$, we can restrict $\pi=(\pi_{1},\dots,\pi_{r})\in\operatorname{\mathcal{P}}(A)$ to get a pseudo-composition in $\operatorname{\mathcal{P}}(B)$ by $\pi|_{B}:=(\pi_{1}\cap B,\dots,\pi_{r}\cap B)$. The shift of $\pi\in\operatorname{\mathcal{P}}_{m}$ by $n$ is the pseudo-composition of $n+A$ defined by $\pi[n]:=(n+\pi_{1},\dots,n+\pi_{r})$. Given another $\bar{\pi}=(\bar{\pi}_{1},\dots,\bar{\pi}_{s})\in\operatorname{\mathcal{P}}_{n}$, we define

[TABLE]

The previous definitions also apply to permutations $\sigma$ and $\tau$, leading to $\operatorname{st}(\sigma)$, $\sigma|_{A}$, $\sigma[n]$, $\sigma\operatorname{\underline{\times}}\tau$ and $\sigma\times\tau[m]$ respectively, but in these cases the result has to be interpreted as a permutation by taking the underlying permutation, in agreement with the notation used to define the Malvenuto-Reutenauer algebra.

3.2. A monoid of compermutations.

Consider the set of all planar binary rooted trees with $r$ leaves $\mathcal{T}_{r}$ and symbols $t\pi$ with $\pi=(\pi_{1},\dots,\pi_{r})\in\operatorname{\mathcal{P}}_{m}$ and $t\in\mathcal{T}_{r}$. Declare two symbols $t\pi$ and $\bar{t}\bar{\pi}$ to be equivalent in case that $\pi$ and $\bar{\pi}$ are equivalent pseudo-compositions and $t$ and $\bar{t}$ give the same tree after removing the leaves corresponding to empty set entries in $\pi$ and $\bar{\pi}$ respectively. The equivalence classes of these symbols will be called compermutations 111The prefix ‘com’ in the name comes from the interpretation of trees as ways of composing nonassociative monomials, while the suffix ‘permutation’ comes from the interpretation of pseudo-compositions as generalized permutations. $\operatorname{\hbox{\cyrillic X}}\,$ is a Cyrillic character related to the Greek letter $\Sigma$. of $\mathbf{m}$ and the set of all these compermutations will be denoted by $\operatorname{\hbox{\cyrillic X}}_{m}$. Each $t\pi$ is equivalent to a unique $\operatorname{\mathfrak{t}}(t\pi)\pi^{*}$, its reduced form, where $\pi^{*}$ is a composition of $\mathbf{m}$. The vector space with basis $\{\mathbf{F}_{t\pi}\mid t\pi\in\operatorname{\hbox{\cyrillic X}}_{m}\}$ will be denoted by $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]_{m}$ or $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}_{m}]$, and $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]:=\oplus_{m=0}^{\infty}\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]_{m}$. When needed we will write $t\cdot\pi$ to better distinguish between the tree and the pseudo-composition. Given $t\pi\in\operatorname{\hbox{\cyrillic X}}_{m},\bar{t}\bar{\pi}\in\operatorname{\hbox{\cyrillic X}}_{n}$, $t\pi\operatorname{\underline{\times}}\bar{t}\bar{\pi}$ denotes $t\vee\bar{t}\cdot\pi\times\bar{\pi}[m]$. For $t\pi$ and $\bar{t}\bar{\pi}$ in $\operatorname{\hbox{\cyrillic X}}_{m}$, define

[TABLE]

where $t(\bar{t})$ means that we place a copy of $\bar{t}$ in each leaf of $t$ by identifying the root of $\bar{t}$ and the leaf of $t$.

Lemma 3.2**.**

$(\operatorname{\hbox{\cyrillic X}}_{m},\operatorname{\bullet})$* is a monoid with unit element $|(\mathbf{m})$, and the map $\operatorname{\hbox{\cyrillic X}}_{m}\times\mathcal{T}_{m}\rightarrow\mathcal{T}_{m}$ given by*

[TABLE]

where $\iota:=(\{1\},\dots,\{m\})$, defines an action of $\operatorname{\hbox{\cyrillic X}}_{m}$ on $\mathcal{T}_{m}$.

Proof.

That $(\operatorname{\hbox{\cyrillic X}}_{m},\operatorname{\bullet})$ is a monoid is a consequence of Proposition 3.1. Since $|(\mathbf{m})$ is the identity for the product $\operatorname{\bullet}$, then it acts as the identity map on $\mathcal{T}_{m}$. Also, by the very definitions we have

[TABLE]

While $\pi\operatorname{\bullet}\iota$ and $\pi\operatorname{\bullet}(\bar{\pi}\operatorname{\bullet}\iota)^{*}$ might differ, we only need to check that the empty sets in these pseudo-compositions occur at the same positions, being the other components irrelevant in our considerations; but, since $(\bar{\pi}\operatorname{\bullet}\iota)^{*}$ is the underlying permutation of $\bar{\pi}$, this immediately follows from $\pi\operatorname{\bullet}(\bar{\pi}\operatorname{\bullet}\iota)^{*}=(\dots,\bar{\pi}(\pi_{i})\cap(\bar{\pi}\operatorname{\bullet}\iota)^{*}_{j},\dots)=\bar{\pi}(\dots,\pi_{i}\cap\{j\},\dots)=\bar{\pi}(\pi\operatorname{\bullet}\iota)$. ∎

The opposite product of $\bullet$ will be denoted by $\circ$. For $\operatorname{\mathbf{X}}:=\{x_{1},x_{2},\dots\}$, the set $\operatorname{\mathbf{X}}^{m}\subseteq\operatorname{{\mathbb{K}}}\{\operatorname{\mathbf{X}}\}$ denotes the set of all nonassociative words of length $m$ in the alphabet $\operatorname{\mathbf{X}}$. The following result shows an action of $(\operatorname{\hbox{\cyrillic X}}_{m},\circ)$ on $\operatorname{\mathbf{X}}^{m}$ that extends the usual action of $(\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}_{m}],\circ)$ on associative words of length $m$ by permuting the factors.

Lemma 3.3**.**

The monoid $(\operatorname{\hbox{\cyrillic X}}_{m},\circ)$ acts on $\operatorname{\mathbf{X}}^{m}$ by

[TABLE]

3.3. Bialgebra structure of $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]$.

We define the outer product of $t\pi\in\operatorname{\hbox{\cyrillic X}}_{m}$ and $\bar{t}\bar{\pi}\in\operatorname{\hbox{\cyrillic X}}_{n}$ as

[TABLE]

Because of the grafting $t\vee\bar{t}$ this product is nonassociative, although it is obviously related to the associative outer product of the Malvenuto-Reutenauer algebra of permutations. The comultiplication is defined by

[TABLE]

for any $\mathbf{F}_{t\pi}\in\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]_{m}$.

Remark 3.4*.*

Beware, compermutations are equivalence classes, thus the reduced forms of $t\pi|_{\mathbf{p}}$ and $t\operatorname{st}(\pi|_{\mathbf{m}\setminus\mathbf{p}})$ might involve trees quite different from $t$.

The inner product is defined for $\mathbf{F}_{t\pi}\in\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]_{m},\mathbf{F}_{\bar{t}\bar{\pi}}\in\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]_{n}$ by

[TABLE]

where $\delta_{m,n}:=1$ if $m=n$ and [math] otherwise.

Proposition 3.5**.**

$(\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,],*,\Delta)$* is a coassociative unital bialgebra.*

Proof.

First we have to check that

[TABLE]

and

[TABLE]

are equal for any $t\pi\in\operatorname{\hbox{\cyrillic X}}_{m}$ and $\bar{t}\bar{\pi}\in\operatorname{\hbox{\cyrillic X}}_{n}$. This amounts to proving that the summands in both expressions are the same. Clearly, as long as we do not remove empty sets, we can safely forget about the trees since they are the same in both cases. Thus, we are concerned with the equality of the following two sets, where the use of $\otimes$ is as a separator for clarity:

[TABLE]

and

[TABLE]

Let $K^{\prime}\otimes K^{\prime\prime}=\gamma(\pi\operatorname{\underline{\times}}\bar{\pi})|_{\mathbf{l}}\otimes\operatorname{st}(\gamma(\pi\operatorname{\underline{\times}}\bar{\pi})|_{\mathbf{m+n}\setminus\mathbf{l}})\in\mathcal{C}$ with

[TABLE]

where $r=l(\pi)$ and $s=l(\bar{\pi})$. The value of $l$ is determined by $l=|K^{\prime}|$. We obviously have

[TABLE]

and

[TABLE]

Define $p:=|K^{\prime}_{1}|+\cdots+|K^{\prime}_{r}|$ and $q:=|\bar{K}^{\prime}_{1}|+\cdots+|\bar{K}^{\prime}_{s}|$, thus $p+q=l$. Since $\gamma\in\operatorname{Sh}(m,n)$, equation (3.2) implies

[TABLE]

Therefore the types of $\pi|_{\mathbf{p}}\operatorname{\underline{\times}}\bar{\pi}|_{\mathbf{q}}$ and $K^{\prime}$ agree, so the types of $\operatorname{st}(\pi|_{\mathbf{m}\setminus\mathbf{p}})\operatorname{\underline{\times}}\operatorname{st}(\bar{\pi}|_{\mathbf{n}\setminus\mathbf{q}})$ and $K^{\prime\prime}$ do. Once we know that these types coincide, we can ensure that there exist permutations $\gamma^{\prime}$, $\gamma^{\prime\prime}$ such that

[TABLE]

These permutations satisfy

[TABLE]

which, since $\gamma\in\operatorname{Sh}(m,n)$, allows us to choose $\gamma^{\prime}$ and $\gamma^{\prime\prime}$ in $\operatorname{Sh}(p,q)$ and $\operatorname{Sh}(m-p,n-q)$ respectively. This proves that $K^{\prime}\otimes K^{\prime\prime}\in\mathcal{C}^{\prime}$, i.e. $\mathcal{C}\subseteq\mathcal{C}^{\prime}$. Conversely, given $K^{\prime}\otimes K^{\prime\prime}\in\mathcal{C}^{\prime}$

[TABLE]

the type of $K^{\prime}$ determines $l:=p+q$ and we obtain, by (3.3) and (3.4), a well-defined permutation $\gamma\in\operatorname{\mathcal{S}}_{m+n}$. In fact, (3.3) and (3.4) easily imply that $\gamma\in\operatorname{Sh}(m,n)$. Therefore, $K^{\prime}\otimes K^{\prime\prime}\in\mathcal{C}$, i.e. $\mathcal{C}^{\prime}\subseteq\mathcal{C}$.

With respect to the coassociativity, it is easy to observe that both $(\Delta\otimes\operatorname{\mathbf{I}})\Delta(\mathbf{F}_{t\pi})$ and $(\operatorname{\mathbf{I}}\otimes\Delta)\Delta(\mathbf{F}_{t\pi})$ coincide with

[TABLE]

∎

The usual product $\operatorname{\circ\,}$ of $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}_{m}]$ extends to an inner product $\operatorname{\circ\,}$ on $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}]$ by declaring $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}_{m}]\circ\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}_{m}]=\{0\}$ if $m\neq n$.

Proposition 3.6**.**

The map

[TABLE]

is an epimorphism of bialgebras and it is also a homomorphism with respect to the inner products.

Proof.

It is rather obvious that

[TABLE]

∎

The algebra of noncommutative quasi-symmetric functions $\operatorname{\mathbf{NCQSym}}$ appeared in [MR2555523]. Its graded dual $\operatorname{\mathbf{NCQSym}}^{*}$ has a basis $\{\mathbf{W}_{I}\mid I\models\mathbf{m},\,m\geq 0\}$ labeled by compositions of sets. The product in this basis is defined as

[TABLE]

for $I\models\mathbf{m}$ and $J\models\mathbf{n}$. The comultiplication is given by

[TABLE]

It is surprisingly obvious that $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]$ is a nonassociative lifting of $\operatorname{\mathbf{NCQSym}}^{*}$, although it was not intended to be since when defining $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]$ we only looked for the natural combinatorial place generalizing $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}]$ for the descent algebra of $\operatorname{{\mathbb{K}}}\{\operatorname{\mathbf{X}}\}$.

3.4. Nonassociative Solomon’s descent algebra.

Given a composition $I=(i_{1},\dots,i_{r})\models m$ and $t\in\mathcal{T}_{r}$ consider the element

[TABLE]

in analogy with the elements $D_{\leq I}:=D_{\subseteq S(I)}$ in $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}})$.

Remark 3.7*.*

We can expand the notation for compermutations by adding a third symbol: instead of $t\pi$ write $t\pi\sigma^{\pi}$. Any $\pi\in\operatorname{\mathcal{P}}_{m}$ with $\operatorname{\mathrm{type}}(\pi)=I$ has an underlying permutation $\sigma^{\pi}$ with $\operatorname{Des}(\sigma^{\pi})\subseteq S(I)$, and conversely, once we fix $I$, any permutation $\sigma\in\operatorname{\mathcal{S}}_{m}$ with $\operatorname{Des}(\sigma)\subseteq S(I)$ determines a unique partition $\pi^{\sigma}\in\operatorname{\mathcal{P}}_{m}$ of type $I$ and underlying permutation $\sigma$. So, with this notation, $D^{t}_{\leq I}$ will be identified with

[TABLE]

Thus, these elements resemble the elements $D_{\leq I}$ in $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}})$. In fact

[TABLE]

With the notation $I\|J:=(i_{1},\dots,i_{r},j_{1},\dots,j_{s})$ for $I=(i_{1},\dots,i_{r})$ and $J=(j_{1},\dots,j_{s})$, the operations of $\operatorname{{\mathbb{K}}}[\operatorname{\mathcal{S}}]$ give for $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}})$:

[TABLE]

where $M=(m_{i,j})$ runs the set of all matrices $r\times s$ with entries in $\operatorname{{\mathbb{N}}}$ whose row sum and column sum are respectively $J$ and $I$ –beware that the role of $I$ and $J$ has changed with respect to Theorem 2.3– and $I(M)$ is the composition associated to $(m_{1,1},\dots,m_{1,s},m_{2,1},\dots,m_{r,1},\dots,m_{r,s})$.

Proposition 3.8**.**

In $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]$ we have

[TABLE]

Proof.

In $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}})$ one has $D_{\leq I}*D_{\leq\bar{I}}=D_{I\|\bar{I}}$ [MR1358493], which, in view of the formula for the outer product, proves the statement. ∎

Proposition 3.9**.**

[TABLE]

With regard to the inner product, given two compositions $I,\bar{I}\models m$ of lengths $r$ and $s$ respectively, consider the sets

[TABLE]

and

[TABLE]

Lemma 3.10**.**

The map $M\mapsto(M_{11},M_{12},\dots,M_{21},M_{22},\dots)$ defines a bijection between $\mathcal{M}(I,\bar{I})$ and $\mathcal{M}^{\prime}(I,\bar{I})$.

Proof.

Given $M\in\mathcal{M}$ consider $\bar{\pi}_{l}:=M_{1l}\cup\cdots\cup M_{rl}$, $\bar{\pi}:=(\bar{\pi}_{1},\dots,\bar{\pi}_{s})$, $\pi_{k}:=(\sigma^{\bar{\pi}})^{-1}(M_{k1}\cup\cdots\cup M_{ks})$ and $\pi:=(\pi_{1},\dots,\pi_{s})$. Clearly $\pi\operatorname{\bullet}\bar{\pi}=(\dots,\bar{\pi}(\pi_{k})\cap\bar{\pi}_{l},\dots)=(\dots,M_{kl},\dots)$. This proves the surjectivity. The injectivity is obvious. ∎

This leads to an analogue of the usual Mackey formula for multiplying basic elements $D_{\leq I}$ in $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}})$ that we enunciate for the opposite product $\operatorname{\bullet}$. The type of $M\in\mathcal{M}(I,\bar{I})$ has to be understood as the type of its image in $\mathcal{M}^{\prime}(I,\bar{I})$.

Proposition 3.11**.**

In $\operatorname{\mathrm{Sol}}(\operatorname{\hbox{\cyrillic X}}\,)$ we have

[TABLE]

Remark 3.12*.*

While in $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}})$ we can collect, for a fixed $J$, the contributions of all $M\in\mathcal{M}^{\prime}(I,\bar{I})$ that give $J$ after removing empty sets, in $\operatorname{\mathrm{Sol}}(\operatorname{\hbox{\cyrillic X}}\,)$ this reduction runs parallel to the pruning of $t(\bar{t})$, and the resulting coefficient will strongly depend on $t$ and $\bar{t}$.

Definition 3.13**.**

The space

[TABLE]

will be called nonassociative Solomon’s descent algebra. It is a subbialgebra of $(\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,],*,\Delta)$ and a subalgebra of $(\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,],\circ)$ whose projection by $\zeta$ is $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}})$.

Proposition 3.14**.**

We have

[TABLE]

Moreover, $(\operatorname{\mathrm{Sol}}(\operatorname{\hbox{\cyrillic X}}),\operatorname{\bullet})\cong(\Sigma^{\operatorname{{\mathbb{K}}}\{\operatorname{\mathbf{X}}\}},\circ)$.

Remark 3.15*.*

The inner product for pseudo-compositions we use is rather different from the $\wedge$ consider for instance by Bidigare [Bid97] (see also [MR2424338]*Appendix B). We sketch Bidigare’s approach to Solomon’s descent algebra for completeness. The monoid $\Pi_{m}$ of compositions of $\mathbf{m}$ is endowed with the product

[TABLE]

Let $\sigma\pi=(\tau(\pi_{1}),\dots,\tau(\pi_{r}))$ be the natural action of $\operatorname{\mathcal{S}}_{m}$ on $\Pi_{m}$. The $\mathbb{Z}$-module $\mathbb{Z}[\Pi_{m}^{(1,\dots,1)}]$ spanned by $\Pi_{m}^{(1,\dots,1)}=\{\pi\in\Pi_{m}\mid\operatorname{\mathrm{type}}(\pi)=(1,\dots,1)\}$ is isomorphic to $\mathbb{Z}[\operatorname{\mathcal{S}}_{m}]$ by $\pi\mapsto\sigma^{\pi}$ and it is a two sided ideal with respect to $\wedge$. Under this isomorphism, $\sigma^{\tau(\pi)}=\tau\sigma^{\pi}$. Since $\tau(\pi\wedge\bar{\pi})=\tau(\pi)\wedge\tau(\bar{\pi})$, the left multiplication by elements of $\mathcal{B}_{m}:=\{X\in\mathbb{Z}[\Pi_{m}]\mid\tau(X)=X\,\,\forall\sigma\in\operatorname{\mathcal{S}}_{m}\}$ commutes with the action of $\operatorname{\mathcal{S}}_{m}$. Under the isomorphism $\mathbb{Z}[\Pi_{m}^{(1,\dots,1)}]\cong\mathbb{Z}[\operatorname{\mathcal{S}}_{m}]$, the centralizer of this action is the algebra generated by the right multiplication operators by elements of $\operatorname{\mathcal{S}}_{m}$, which is isomorphic to $(\mathbb{Z}[\operatorname{\mathcal{S}}_{m}],\operatorname{\bullet})$ –the isomorphism is given by evaluation the operators at $(1,\dots,m)\in\operatorname{\mathcal{S}}_{m}$. The image of $\sum_{\operatorname{\mathrm{type}}(\pi)=I}\pi\in\mathcal{B}_{m}$ in the centralizer algebra is the right multiplication operator by $\sum_{\operatorname{\mathrm{type}}(\pi)=I}\sigma^{\pi}=\sum_{\operatorname{Des}(\sigma)\leq I}\sigma$, and after evaluating at $(1,\dots,m)$ we get $\sum_{\operatorname{Des}(\sigma)\leq I}\sigma$. Thus, Bidigare’s algebra $\mathcal{B}_{m}$ is isomorphic to $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}_{m})$. The product $\wedge$ has the rather trivial but surprisingly important property $\pi\wedge\pi=\pi$ [Br00] that the product $\operatorname{\bullet}$ lacks. In fact, for any $\pi\in\Pi_{m}$ and $\bar{\pi}\in\Pi^{(1,\dots,1)}_{m}$,

[TABLE]

which proves that the left multiplication $\bar{\pi}\mapsto\pi\operatorname{\bullet}\bar{\pi}$ corresponds to the right multiplication operator by $\pi\operatorname{\bullet}(\{1\},\dots,\{m\})=\sigma^{\pi}$. While this suffices, by means of $\zeta$, to prove that $\operatorname{\mathrm{Sol}}(\operatorname{\mathcal{S}}_{m})$ is closed under $\operatorname{\bullet}$, it is not enough to prove that $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}_{m}]$ is.

4. Structure of the bialgebra of compermutations

Consider the dual basis $\{\mathbf{G}_{t\pi}\mid t\pi\in\operatorname{\hbox{\cyrillic X}}\,\}$ of $\{\mathbf{F}_{t\pi}\mid t\pi\in\operatorname{\hbox{\cyrillic X}}\,\}$ and the vector space $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]^{\prime}$ spanned by it. The operations $*$ and $\Delta$ have dual maps $*^{\prime}$ and $\Delta^{\prime}$ given by

[TABLE]

and

[TABLE]

Call a compermutation $t\pi\in\operatorname{\hbox{\cyrillic X}}_{m}$ atomic in case that $t\pi\neq t^{\prime}\pi^{\prime}\operatorname{\underline{\times}}t^{\prime\prime}\pi^{\prime\prime}$ for compermutations $t^{\prime}\pi^{\prime}\in\operatorname{\hbox{\cyrillic X}}_{i},t^{\prime\prime}\pi^{\prime\prime}\in\operatorname{\hbox{\cyrillic X}}_{m-i}$ and some $1\leq i\leq m-1$. If $\pi=\pi^{\prime}\operatorname{\underline{\times}}\pi^{\prime\prime}$ with $|\pi^{\prime}|,|\pi^{\prime\prime}|\geq 1$ we say that $\pi$ has a splitting at $|\pi^{\prime}|$. Let $w(x)$ be a nonassociative monomial in $x$ that we will use to codify products with parentheses. If $t\pi=w^{\operatorname{\underline{\times}}}(t^{\prime}\pi^{\prime},\dots,t^{\prime\prime}\pi^{\prime\prime}\operatorname{\underline{\times}}\bar{t}\bar{\pi},\dots,\bar{\bar{t}}\bar{\bar{\pi}})$ and $|\pi^{\prime}|+\cdots+|\pi^{\prime\prime}|$ is a splitting of $\pi$ we say that $t\pi$ has a local splitting at $|\pi^{\prime}|+\cdots+|\pi^{\prime\prime}|$. Recall that $w^{\operatorname{\underline{\times}}},w^{*},\dots$ means that the operations indicated by $w$ has been computed wit respect to the operation $\operatorname{\underline{\times}},*,\dots$.

Theorem 4.1**.**

$(\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,],*)$* is a nonassociative unital algebra freely generated by the set $\{\mathbf{F}_{t\pi}\mid t\pi\in\operatorname{\hbox{\cyrillic X}}\,\text{ is atomic}\}$.*

Proof.

We introduce a partial order on $\operatorname{\hbox{\cyrillic X}}_{m}$. We will say that $t^{\prime}\pi^{\prime}\leq\bar{t}\bar{\pi}$ if the local splittings of $t^{\prime}\pi^{\prime}$ are local splittings of $\bar{t}\bar{\pi}$.

Consider a summand $t^{\prime}\vee t^{\prime\prime}\cdot\gamma\pi^{\prime}\operatorname{\underline{\times}}\pi^{\prime\prime}$ in the expansion $\sum_{\gamma\in\operatorname{Sh}(m,n)}t^{\prime}\vee t^{\prime\prime}\cdot\gamma\pi^{\prime}\operatorname{\underline{\times}}\pi^{\prime\prime}$ of $t^{\prime}\pi^{\prime}*t^{\prime\prime}\pi^{\prime\prime}$ and a local spitting $p$ of it. If $p<|\pi^{\prime}|$ (resp. $p>|\pi^{\prime}|$) then $p$ is a local splitting of $t^{\prime}\pi^{\prime}$ (resp. $t^{\prime\prime}\pi^{\prime\prime}$), while if $p=|\pi^{\prime}|$ then $\gamma=(1,\dots,m+n)$ and $t^{\prime}\vee t^{\prime\prime}\cdot\gamma\pi^{\prime}\operatorname{\underline{\times}}\pi^{\prime\prime}=t^{\prime}\vee t^{\prime\prime}\cdot\pi^{\prime}\operatorname{\underline{\times}}\pi^{\prime\prime}$. In any case $t^{\prime}\vee t^{\prime\prime}\gamma\pi^{\prime}\operatorname{\underline{\times}}\pi^{\prime\prime}\leq t^{\prime}\pi^{\prime}\operatorname{\underline{\times}}t^{\prime\prime}\pi^{\prime\prime}$.

Any compermutation $t\pi$ can be uniquely written as $t\pi=w^{\operatorname{\underline{\times}}}(t^{\prime}\pi^{\prime},\dots,t^{\prime\prime}\pi^{\prime\prime})$ for a nonassociative word $w(x)$ and atomic compermutations $t^{\prime}\pi^{\prime},\dots,t^{\prime\prime}\pi^{\prime\prime}$. The element $w^{\operatorname{\underline{\times}}}(t^{\prime}\pi^{\prime},\dots,t^{\prime\prime}\pi^{\prime\prime})-w^{*}(t^{\prime}\pi^{\prime},\dots,t^{\prime\prime}\pi^{\prime\prime})$ is a linear combination of compermutations $\leq w^{\operatorname{\underline{\times}}}(t^{\prime}\pi^{\prime},\dots,t^{\prime\prime}\pi^{\prime\prime})$, so atomic compermutations generate $(\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,],*)$. In case that $\sum_{i}\lambda_{i}w_{i}^{*}(t^{\prime}\pi^{\prime},\dots,t^{\prime\prime}\pi^{\prime\prime})=0$ for nonzero scalars $\lambda_{i}\in\operatorname{{\mathbb{K}}}$ and nonassociative words $w_{i}(x)$, then comparing the highest compermutations we obtain a nontrivial linear combination of certain elements $w_{i}^{\operatorname{\underline{\times}}}(t^{\prime}\pi^{\prime},\dots,t^{\prime\prime}\pi^{\prime\prime})$, which contradicts the uniqueness of the factorization in terms of atomic compermutations. This proves the freeness of the generators. ∎

Given $t\pi\in\operatorname{\hbox{\cyrillic X}}\,$ there exists a unique factorization $t\pi=t^{\prime\prime\prime}\pi^{\prime\prime\prime}\operatorname{\underline{\times}}(\cdots\operatorname{\underline{\times}}(t^{\prime\prime}\pi^{\prime\prime}\operatorname{\underline{\times}}t^{\prime}\pi^{\prime}))$ with maximum number of factors. We say that $t\pi$ is connected in case that this decomposition involves an odd number of factors, otherwise we say that $t\pi$ is not connected. It is clear that either $t\pi$ is connected or $t\pi=t^{\prime}\pi^{\prime}\operatorname{\underline{\times}}t^{\prime\prime}\pi^{\prime\prime}$ with $t^{\prime\prime}\pi^{\prime\prime}$ connected. Thus, we have a factorization $t\pi=((t^{\prime}\pi^{\prime}\operatorname{\underline{\times}}t^{\prime\prime}\pi^{\prime\prime})\operatorname{\underline{\times}}\cdots)\operatorname{\underline{\times}}t^{\prime\prime\prime}\pi^{\prime\prime\prime}$ with $t^{\prime}\pi^{\prime},t^{\prime\prime}\pi^{\prime\prime},\dots$ connected. In fact, this decomposition is unique. Consider two different factorizations

[TABLE]

with connected factors. If the right-hand side only involves a factor, lets say $\bar{t}\bar{\pi}$ then $t\pi$ is connected. However, since $t^{\prime\prime\prime}\pi^{\prime\prime\prime}$ is connected too then the compermutation $t\pi=((t^{\prime}\pi^{\prime}\operatorname{\underline{\times}}t^{\prime\prime}\pi^{\prime\prime})\operatorname{\underline{\times}}\cdots)\operatorname{\underline{\times}}t^{\prime\prime\prime}\pi^{\prime\prime\prime}$ is not, a contradiction. Thus, both sides involve at least two factors, which implies that $t^{\prime\prime\prime}\pi^{\prime\prime\prime}=\bar{\bar{\bar{t}}}\bar{\bar{\bar{\pi}}}$. Iterating we get the uniqueness. From the factorization in connected compermutations we define the connected type of $t\pi$ as $\operatorname{\mathrm{ctype}}(t\pi):=(|\pi^{\prime}|,|\pi^{\prime\prime}|,\dots)$.

Theorem 4.2**.**

$(\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]^{\prime},*^{\prime})$* is a unital associative algebra freely generated by the set $\{\mathbf{G}_{t\pi}\mid t\pi\in\operatorname{\hbox{\cyrillic X}}\,\text{ connected}\}$.*

Proof.

Let us define a partial order on $\operatorname{\hbox{\cyrillic X}}_{m}$ as follows: we say that $t\pi<\bar{t}\bar{\pi}$ if

• i)

the set of local splittings of $t\pi$ is strictly contained in the set of local splittings of $\bar{t}\bar{\pi}$, or

• ii)

$t\pi$ and $\bar{t}\bar{\pi}$ share the same local splittings and, reading from right to left, the first component in which differ $\operatorname{\mathrm{ctype}}(t\pi)$ and $\operatorname{\mathrm{ctype}}(\bar{t}\bar{\pi})$ is greater in $\operatorname{\mathrm{ctype}}(t\pi)$.

Given a product $((t^{\prime}\pi^{\prime}\operatorname{\underline{\times}}t^{\prime\prime}\pi^{\prime\prime})\operatorname{\underline{\times}}\cdots)\operatorname{\underline{\times}}t^{\prime\prime\prime}\pi^{\prime\prime\prime}$ of connected compermutations, the summands in $((t^{\prime}\pi^{\prime}*^{\prime}t^{\prime\prime}\pi^{\prime\prime})*^{\prime}\cdots)*^{\prime}t^{\prime\prime\prime}\pi^{\prime\prime\prime}$ are compermutations $t\pi$ such that its restrictions $t\operatorname{st}(\pi|_{S})$ to adequate sets $S$ give $t^{\prime}\pi^{\prime},t^{\prime\prime}\pi^{\prime\prime},\dots$. Each local splitting of $t\pi$ either induces a local splitting of a factor or it belongs to $\{|\pi^{\prime}|,|\pi^{\prime}|+|\pi^{\prime\prime}|,\dots\}$. Thus, either $t\pi<((t^{\prime}\pi^{\prime}\operatorname{\underline{\times}}t^{\prime\prime}\pi^{\prime\prime})\operatorname{\underline{\times}}\cdots)\operatorname{\underline{\times}}t^{\prime\prime\prime}\pi^{\prime\prime\prime}$ or both compermutations share the same local splittings. In the latter case we must have $\pi=\pi^{\prime}\operatorname{\underline{\times}}\pi^{\prime\prime}\operatorname{\underline{\times}}\cdots\operatorname{\underline{\times}}\pi^{\prime\prime\prime}$.

Now let us consider the factorization $t\pi=((\bar{t}\bar{\pi}\operatorname{\underline{\times}}\bar{\bar{t}}\bar{\bar{\pi}})\cdots)\operatorname{\underline{\times}}\bar{\bar{\bar{t}}}\bar{\bar{\bar{\pi}}}$ in connected compermutations:

• i)

If $|\bar{\bar{\bar{\pi}}}|>|\pi^{\prime\prime\prime}|$ then $t\pi<((t^{\prime}\pi^{\prime}\operatorname{\underline{\times}}t^{\prime\prime}\pi^{\prime\prime})\operatorname{\underline{\times}}\cdots)\operatorname{\underline{\times}}t^{\prime\prime\prime}\pi^{\prime\prime\prime}$.

• ii)

If $|\bar{\bar{\bar{\pi}}}|<|\pi^{\prime\prime\prime}|$ then when we restrict to an adequate set $S$ we get that $\bar{\bar{\bar{t}}}\bar{\bar{\bar{\pi}}}$ is of the form $\hat{t}\hat{\pi}\operatorname{\underline{\times}}t^{\prime\prime\prime}\pi^{\prime\prime\prime}$ for some $\hat{t}\hat{\pi}$, which is not possible since both $\bar{\bar{\bar{t}}}\bar{\bar{\bar{\pi}}}$ and $t^{\prime\prime\prime}\pi^{\prime\prime\prime}$ are connected.

• iii)

If $|\pi^{\prime\prime\prime}|=|\bar{\bar{\bar{\pi}}}|$ then taking an adequate restriction we obtain $t^{\prime\prime\prime}\pi^{\prime\prime\prime}=\bar{\bar{\bar{t}}}\bar{\bar{\bar{\pi}}}$ and iterating we get that either $t\pi<((t^{\prime}\pi^{\prime}\operatorname{\underline{\times}}t^{\prime\prime}\pi^{\prime\prime})\operatorname{\underline{\times}}\cdots)\operatorname{\underline{\times}}t^{\prime\prime\prime}\pi^{\prime\prime\prime}$ or they are equal.

Thus, the summands in $((t^{\prime}\pi^{\prime}*^{\prime}t^{\prime\prime}\pi^{\prime\prime})*^{\prime}\cdots)*^{\prime}t^{\prime\prime\prime}\pi^{\prime\prime\prime}$ are either the compermutation $((t^{\prime}\pi^{\prime}\operatorname{\underline{\times}}t^{\prime\prime}\pi^{\prime\prime})\operatorname{\underline{\times}}\cdots)\operatorname{\underline{\times}}t^{\prime\prime\prime}\pi^{\prime\prime\prime}$ or compermutations less than this. An argument as in the proof of Theorem 4.1 concludes the proof. ∎

A graded c.a. coalgebra $C=\oplus_{n\geq 0}C_{n}$ is said to be cofree if for any graded c.a. coalgebra $D=\oplus_{n\geq 0}D_{n}$ and any linear map $f\colon D\rightarrow C_{1}$ such that $f(D_{i})=0$ if $i\neq 1$ there exists a unique morphism of graded coalgebras $\hat{f}\colon D\rightarrow C$ such that $p_{1}\hat{f}=f$ where here $p_{1}\colon C\rightarrow C_{1}$ is the usual projection. By dualizing Theorem 4.2 we get

Corollary 4.3**.**

$(\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,],\Delta,\epsilon)$* is cofree as a graded coassociative coalgebra.*

5. Appendix: examples of primitive elements

We will compute a basis for the space of primitives of $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]_{3}$. Clearly

[TABLE]

where $S_{m,k}$ denotes the Stirling numbers of the second kind and $C_{k-1}$ is the Catalan number that counts the number of nonassociative monomials $w(x)$ of degree $k$. Thus, for $m=0,1,2,3$ we get $\dim\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]_{m}=1,1,3,19$. The number of connected compermutations in $\operatorname{\hbox{\cyrillic X}}_{2}$ (resp. $\operatorname{\hbox{\cyrillic X}}_{3}$) is $3-1=2$ (resp. $19-2\cdot 2-1=14$). Compermutations in $\operatorname{\hbox{\cyrillic X}}_{3}$ are given by the following table:

[TABLE]

where, for short, we write $(12,3)$ instead of $(\{1,2\},\{3\})$, etc. The non-connected compermutation in $\operatorname{\hbox{\cyrillic X}}_{2}$ is \begin{minipage}{16.0pt} \begin{picture}(16.0,16.0) \put(8.0,16.0){\oval(8.0,12.0)[b]} \put(8.0,10.0){\line(0,-1){2.0}} \end{picture} {\vskip 10.76385pt} \end{minipage}(1,2), while the non-connected compermutations in $\operatorname{\hbox{\cyrillic X}}_{3}$ are

[TABLE]

For instance the last compermutation in (5.1) is of the form \begin{minipage}{8.0pt} \begin{picture}(8.0,24.0) \put(4.0,24.0){\line(0,-1){8.0}} \put(4.0,16.0){\line(0,-1){8.0}} \end{picture} {\vskip 10.76385pt} \end{minipage}(1)\operatorname{\underline{\times}}\begin{minipage}{16.0pt} \begin{picture}(16.0,24.0) \put(4.0,24.0){\line(0,-1){8.0}} \put(12.0,24.0){\line(0,-1){8.0}} \put(8.0,16.0){\oval(8.0,12.0)[b]} \put(8.0,10.0){\line(0,-1){2.0}} \end{picture} {\vskip 10.76385pt} \end{minipage}(2,1) with both factors connected, so it is not connected.

To compute a basis for the space of primitive elements of $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]$, we first consider the monomials in $\{\mathbf{G}_{t\pi}\mid t\pi\text{ connected}\}$ with respect to the product $*^{\prime}$, which form a basis of $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]^{\prime}$; then we take the dual of this basis to obtain a basis of $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]$ and finally we choose the elements of this basis that are the dual of $\{\mathbf{G}_{t\pi}\mid t\pi\text{ connected}\}$. In particular, the dimension of the space of primitive elements in $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]_{m}$ coincides with the number of connected compermutations in $\operatorname{\hbox{\cyrillic X}}_{m}$. After some computations we get the following basis for $\operatorname{{\mathbb{K}}}[\operatorname{\hbox{\cyrillic X}}\,]_{3}$:

1. (1)

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\begin{minipage}{16.0pt} \begin{picture}(16.0,24.0) \put(4.0,24.0){\line(0,-1){8.0}} \put(12.0,24.0){\line(0,-1){8.0}} \put(8.0,16.0){\oval(8.0,12.0)[b]} \put(8.0,10.0){\line(0,-1){2.0}} \end{picture} {\vskip 10.76385pt} \end{minipage}(13,2)-\begin{minipage}{24.0pt} \begin{picture}(24.0,32.0) \put(4.0,32.0){\line(0,-1){8.0}} \put(16.0,32.0){\oval(8.0,12.0)[b]} \put(16.0,26.0){\line(0,-1){2.0}} \put(4.0,24.0){\line(0,-1){8.0}} \qbezier(16.0,24.0)(16.0,22.0)(14.0,20.0)\qbezier(14.0,20.0)(12.0,20.0)(12.0,16.0) \put(8.0,16.0){\oval(8.0,12.0)[b]} \put(8.0,10.0){\line(0,-1){2.0}} \end{picture} {\vskip 10.76385pt} \end{minipage}(1,3,2) 3. (3)

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