2.1 General DGLAs

For simplicity, we will work over $k=\mathbb {Q}$ , though the discussion also holds for any field of characteristic zero. Recall that a DGLA over k is a vector space A over $k$ with $\mathbb {Z}$ -grading $A=\oplus _{n\in \mathbb {Z}}A_n$ along with a bilinear map $[\cdot ,\cdot ]{\colon }{}A\times {}A\rightarrow {}A$ (bracket, respecting the grading) and a linear map ${\partial }{\colon }{}A\rightarrow {}A$ (differential, grading shift $-1$ ) for which ${\partial }^2=0,$ while

1. symmetry of bracket $[b,a]=-(-1)^{|a||b|}[a,b];$

2. Jacobi identity $(-1)^{|a||b|}[[b,c],a]+(-1)^{|b||c|}[[c,a],b]+(-1)^{|c||a|}[[a,b],c]=0;$

3. Leibniz rule ${\partial }[a,b]=[{\partial }{a},b]+(-1)^{|a|}[a,{\partial }{b}],$

for all homogeneous $a,b,c\in {}A$ . Defining the adjoint action of A on itself by ${\mathrm {ad}}_e(a)=[e,a]$ , the operator ${\text {ad}}_e{\colon } A\rightarrow {}A$ has grading shift $|e|$ , for homogeneous $e\in {}A$ . The Jacobi identity and Leibniz rule can now be reformulated as operator equalities:

1. Jacobi identity ${\text {ad}}_{[a,b]}=[{\text {ad}}_a,{\text {ad}}_b]$ ;

2. Leibniz rule ${\text {ad}}_{{\partial }{a}}=[{\partial },{\text {ad}}_a]$ ;

in terms of the graded operator commutator, $[A,B]\equiv {}A\circ {}B-(-1)^{|A||B|}B\circ {}A$ . Since the relations all preserve the number of brackets, it is meaningful to define an additional grading by the number of (Lie) brackets; in particular, for $x\in {}A$ , let $x^{[m]}$ denote the part of x containing precisely $m$ brackets.

2.2 Points and Localisation

An element $a\in {}A_{-1}$ is called a point (or is said to be flat) in the model if it satisfies the Maurer–Cartan equation ${\partial }{a}+{\tfrac 12}[a,a]=0$ . For any point $a\in {}A_{-1}$ , define the twisted differential ${\partial }_a$ by ${\partial }_a\equiv {\partial }+{\text {ad}}_a$ ; the fact that ${\partial }_a^2=0$ is guaranteed by the Maurer–Cartan condition. By the localisation of A to a point $a$ , denoted $A(a)$ , we will mean the DGLA that as a graded Lie algebra is

$$\begin{align*}(\ker{\partial}_a|_{A_0})\oplus\bigoplus_{n>0}A_n\end{align*}$$

with the induced bracket from A and the differential ${\partial }_a$ . This contains only non-negative gradings. Leibniz guarantees that $\ker {\partial }_a|_{A_0}$ is closed under Lie bracket.

2.3 Edges and Flows

Any element $e\in {}A_0$ defines a flow on A by

(2.1) $$ \begin{align} \frac{dx}{dt}={\partial}{e}-{\text{ad}}_e(x)\quad\text{on}\quad A_{-1}\>,\qquad \frac{dx}{dt}=-{\text{ad}}_e(x)\quad\text{on}\quad A_{\not=-1}. \end{align} $$

This flow is called the flow by $e$ and preserves the grading. (To define this rigorously, one can work in a space quotiented by all expressions involving $N+1$ Lie brackets, as in [Reference Lawrence and Sullivan10], effectively truncating to the space of linear combinations of terms involving at most $N$ Lie brackets, whose coefficients are polynomials in $t$ with rational coefficients. Then one considers the tower of spaces as $N$ increases. Equivalently, one can choose a basis for the finite-dimensional space of expressions involving exactly $N$ Lie brackets, and then allowed expressions are formal combinations of these basis elements, over all $N$ , with coefficients that are polynomials in $t$ . While we talk about functions of $t$ and their derivatives, these are well defined for rational $t$ , with derivatives being well-defined, since all the coefficients are polynomial functions of $t$ .)

Proof As in the proof of [Reference Lawrence and Sullivan10, Theorem 1], consider the curvature $f(t)\in {}A_{-2}$ defined by $f\equiv {}{\partial }{x}+{\frac 12}[x,x]$ . It satisfies

$$ \begin{align*} \frac{df}{dt}&={\partial}\frac{dx}{dt}+\Big[x,\frac{dx}{dt}\Big]=-{\partial}({\text{ad}}_ex)+[x,{\partial}{e}]-[x,{\text{ad}}_e(x)]\\ &=-{\partial}\circ{\text{ad}}_e(x)+{\text{ad}}_{{\partial}{e}}(x)+({\text{ad}}_x)^2e=-{\text{ad}}_e\circ{\partial}(x)+{\tfrac12}{\text{ad}}_{[x,x]}e=-{\text{ad}}_ef\>, \end{align*} $$

a first order homogeneous linear ode for $f(t)$ with initial condition $f(0)=0$ , since $x(0)$ satisfies the Maurer–Cartan condition. Thus, $f(t)=0$ for all $t$ , as required. ▪

Linearity of the differential equations (2.1) in e ensures that flowing by $e$ for time t is equivalent to flowing by $te$ for a unit time. Denote the result of flowing by $e$ from $a\in {}A_{-1}$ for unit time, by $u_e(a)$ , so that the solution of the first equation in (2.1) is $x(t)=u_{te}(x(0))$ . Explicitly,

$$\begin{align*}u_e(a)=e^{-{\text{ad}}_e}a+\frac{1-e^{-{\text{ad}}_e}}{{\text{ad}}_e}{\partial}{e} =a+{\partial}{e}-[e,a+\tfrac12{\partial}{e}]+(\geq2\ \text{brackets}),\end{align*}$$

where the meaning of the operator quotient is the series $\sum _{n=0}^\infty \frac {(-1)^n}{(n+1)!}({\text {ad}}_e)^n({\partial }{e})$ .

Example 2.4 The unique DGLA model, $A(I)$ , of an interval has three generators; $a,\;b$ of grading $-1$ (the endpoints), and $e$ of grading $0$ (the 1-cell). The differential is given by the condition $u_e(a)=b$ (see [Reference Lawrence and Sullivan10]). Explicitly,

$$\begin{align*}{\partial}{e}=({\text{ad}}_e)b+\sum\limits_{i=0}^\infty{\frac{B_i}{i!}}({\text{ad}}_e)^i(b-a)=\frac{E}{1-e^E}a+\frac{E}{1-e^{-E}}b=b-a+\tfrac{E}2(a+b)+\cdots,\end{align*}$$

where $B_i$ denotes the $i$ -th Bernoulli number defined as coefficients in the expansion $\frac {x}{e^x-1}=\sum _{n=0}^\infty {}B_n \frac {x^n}{n!}$ , $E\equiv {\text {ad}}_e,$ and the expressions in $E$ are considered as formal power series.

Denote by ${\text {BCH}}(x,y)$ the Baker–Campbell–Hausdorff formula for the element of the free Lie algebra (over $\mathbb {Q}$ ) in two variables $x,\;y$ such that as formal series $\exp (x).\exp (y)=\exp {\text {BCH}}(x,y)$ (see [Reference Eichler4] for a short proof of existence and [Reference Dynkin3] for a computational formula). Here, we collect some elementary properties that follow from the definition, Jacobi, and uniqueness of $\mathrm{BCH}$ as a free Lie algebra element.

Proof By [Reference Lawrence and Sullivan10, Lemma 3], and the explicit formula given for $u_e(a)$ above, it follows that

$$\begin{align*}u_{e_2}\big(u_{e_1}(a)\big)=u_{{\text{BCH}}(e_1,e_2)}(a)\>,\end{align*}$$

for any $a\in {}A_{-1}$ . Thus, on elements of grading $-1$ , a flow by $e_1$ for unit time followed by a flow by $e_2$ for unit time is equivalent to a flow by ${\text {BCH}}(e_1,e_2)$ for unit time. Note that the flow for unit time by e acting on $A_n$ for $n\not =-1$ , is just the exponential operator $\exp (-{\text {ad}}_e)$ for which it is immediate that $\exp (-{\text {ad}}_{e_2})\circ \exp (-{\text {ad}}_{e_1})=\exp (-{\text {ad}}_{\mathrm{BCH}(e_1,e_2)})$ .▪

Lemma 2.9 (see [Reference Buijs, Félix, Murillo and Tanré2])

If $X$ has $c$ connected components and $\{a_1,\dots ,a_c\}$ is a choice of basepoints, one in each connected component, then the set of points in $A(X)$ is

$$\begin{align*}\bigcup\limits_{i=1}^c\big\{u_e(a_i)\bigm|e\in{}A_0\big\}\cup \big\{u_e(0)\bigm|e\in{}A_0\big\}\>.\end{align*}$$

For each $i$ , the map $\pi _i{\colon }{}e\mapsto {}u_e(a_i)$ is a ‘fibration’, with fibre $\pi _i^{-1}(a_i)$ generated as a vector space by $\{{\text {BCH}}(\gamma )|\gamma \in \pi _1(X,a_i)\}$ , while the map $\pi _0{\colon }{}e\mapsto {}u_e(0)$ is injective.

2.4 Localisation of Models

As noted in Section 1, $A(X)$ is not unique, but is well defined up to (exact) DGLA isomorphism.

By Lemma 2.9, equivalently, a point is local to the cell $f$ if it can be written as $u_e(a_0)$ , where $a_0$ is a 0-cell in $\overline {f}$ and $e$ is a zero-graded element in the (free) Lie algebra generated by cells in $\overline {f}$ .

Here, by abuse of notation, we have used the same symbol for the geometric $k$ -cell $f$ and the element $f\in {}A_{k-1}$ in its model. By locality of the model and using freeness of the Lie algebra, we see that the point of localisation of a cell $f$ in a model (if it exists) is unique and must be local to the cell (in the sense of Definition 2.10).

Lemma 2.12 If A is a model of a regular cell complex $X$ in which the $k$ -cell $f$ ( $k>1$ ) is localised at the point $a\in {}A_{-1}$ , then for any $e\in {}A_0$ in the subalgebra generated by the 1-skeleton of $\overline {f}$ , there is a variation $A'$ of the model in which the generator for f is replaced by

$$\begin{align*}f'=\exp(-{\text{ad}}_e)\cdot{}f\end{align*}$$

and the cell $f$ is localised at the point $a'=u_e(a)$ .

The above lemma means that we can ‘twist’ any model in which a cell is localised, so that the cell is localised at any point we please (which is local to the cell). This technique was used in [Reference Gadish, Griniasty and Lawrence6, Reference Griniasty and Lawrence7] to generate symmetric models, by writing first a (non-symmetric) model of the relevant $2$ -cell localised at a point on its boundary and then twisting it so that it is localised at a symmetric point. All that remained was to verify that the model obtained was indeed symmetric.

2.5 Universal Averages

In [Reference Lawrence and Siboni9], we were shown how to construct a universal expression $\mu _n(x_1,\dots ,x_n)$ in the completed free Lie algebra generated by $x_1,\dots ,x_n$ such that

1. (i) $\mu _n$ is totally symmetric in its arguments;

2. (ii) in any DGLA model in which $a,b$ are points with $u_{x_i}(a)=b$ for all $i=1,\dots ,n$ , also $u_{\mu _n(x_1,\dots ,x_n)}(a)=b$ .

It was also shown that the expansion of $\mu _n$ up to three Lie brackets is

$$\begin{align*}\mu_n(x_1,\dots,x_n)=\tfrac1n\sum\limits_ix_i-\tfrac1{12n^2}\sum\limits_{i,j,\ i\not=j}[x_i,[x_i,x_j]]+\cdots ,\end{align*}$$

and we call $\mu _n$ the universal average. There is a closed formula for $\mu _2$ , found in [Reference Gadish, Griniasty and Lawrence6],

$$\begin{align*}\mu_2(x,y)={\text{BCH}}\big(x,\tfrac12{\text{BCH}}(-x,y)\big).\end{align*}$$

3.1 Symmetries

The geometric symmetry group of the banana is $D_n\times {}\mathbb {Z}_2$ . The dihedral group is generated by the rotation by $\frac {2\pi }n$ around the axis $ab$ ,

$$\begin{align*}\tau{\colon} e_i\longmapsto{}e_{i+1},\quad{}f_i\longmapsto{}f_{i+1},\quad{}a,b,h\ \text{fixed,}\end{align*}$$

and the reflection in the plane containing $e_n$ and the central axis of the banana,

$$\begin{align*}\sigma{\colon} a,b\ \text{fixed},\quad{}e_i\longmapsto{}e_{n-i}, \quad{}f_i\longmapsto{}-f_{n-i-1},\quad{}h\longmapsto-h,\end{align*}$$

while the further $\mathbb {Z}_2$ factor is generated by reflection in the plane equidistant from $a$ and $b$ ,

$$\begin{align*}\iota: a\longleftrightarrow{}b,\quad{}e_i\longmapsto-e_i,\quad{} f_i\longmapsto-f_i,\quad{}h\longmapsto-h,\end{align*}$$

which commutes with both $\sigma $ and $\tau $ . For some purposes we will restrict to the subgroup $D_n=\langle \sigma ,\tau \rangle $ of the symmetry group fixing the vertices.

3.2 $0$ -, $1$ -, $2$ -cells

By the Leibniz rule, the differential ${\partial }$ is determined by its values on generators. On vertices, ${\partial }$ is fixed by the Maurer–Cartan condition, namely,

(3.1) $$ \begin{align} {\partial}{a}=-{\frac{1}{2}}[a,a],\qquad {\partial}{b}=-{\frac{1}{2}}[b,b]\>. \end{align} $$

On $1$ -cells, ${\partial }$ is also unique (see Example 2.4):

(3.2) $$ \begin{align} {\partial}{e_i}=\frac{E_i}{1-e^{E_i}}a+\frac{E_i}{1-e^{-E_i}}b =b-a+\tfrac12[a+b,e_i]+(\geq2\ \text{brackets}), \end{align} $$

where $E_i={\text {ad}}_{e_i}$ . The faces $f_i$ are bi-gons, and we use the symmetric model of the bi-gon from [Reference Gadish, Griniasty and Lawrence6] in which

(3.3) $$ \begin{align} {\partial}{}f_i={\text{BCH}}(-\tfrac12v_i,e_i,-e_{i+1},\tfrac12v_i)-[x_i,f_i] =e_i-e_{i+1}-\tfrac12[a+b,f_i]+\cdots \end{align} $$

so that $f_i$ is localised at its centre

$$\begin{align*}x_i=u_{\frac12v_i}(a) =\tfrac12(a+b)+\frac1{16}[e_i+e_{i+1},b-a]+(\geq2\ \text{brackets}), \end{align*}$$

where $v_i$ is the centreline from $a$ to $b$ given by the average

$$\begin{align*}v_i=\mu_2(e_i,e_{i+1})={\text{BCH}}(e_i,\tfrac12{\text{BCH}}(-e_i,e_{i+1})) =\tfrac12(e_i+e_{i+1})+(\geq2\ \text{brackets}) \end{align*}$$

in terms of the universal average $\mu _2$ of [Reference Lawrence and Siboni9].

3.3 Central Point of Banana

A totally symmetric point should be at the centre of the banana,

$$\begin{align*}x=u_{\frac12v}(a) =\tfrac12(a+b)+\tfrac1{8n}\sum\limits_{i=1}^nE_i(b-a)+\cdots,\end{align*}$$

which is half way along a central diagonal of the banana from $a$ to $b$ , a path in the direction $v=\mu _n(e_1,\dots ,e_n)$ , the universal average of $e_1,\dots ,e_n$ . The fact that $x$ is invariant under $\sigma $ , $\tau ,$ and $\iota $ follows from the total symmetry of $\mu _n$ along with the fact that ([Reference Lawrence and Siboni9, Lemma 4.3])

$$\begin{align*}\mu_n(-e_1,\dots,-e_n)=-\mu_n(e_1,\dots,e_n).\end{align*}$$

The only freedom remaining in the model is the boundary of the $3$ -cell, ${\partial }{h}\in {}A_1$ , which, to give a valid model of $X_n$ must be such that ${\partial }^2(h)=0$ , while $({\partial }{h})^{[0]}$ must coincide with the topological boundary ${\partial }_0h=f_1+\cdots +f_n$ . The purpose of this section is to give a formula for ${\partial }{h}$ that is invariant under the $D_n$ action of the symmetries of the banana fixing the vertices and in which $h$ is localised at a central point. In order to do this, we will first construct a model in which all the $2$ -cells and the $3$ -cell are localised at $a$ , and then twist using Lemma 2.12 to localise at a symmetric point.

3.4 Model Localised at a

Since $f_i$ is localised at $x_i$ , $f^{\prime }_i=\exp (\frac 12{\text {ad}}_{v_i})\cdot {}f_i$ is localised at $a$ ; in particular,

(3.4) $$ \begin{align} {\partial}_a{}f^{\prime}_i={\text{BCH}}(e_i,-e_{i+1}). \end{align} $$

Similarly, set $h'=\exp (\frac 12{\text {ad}}_{v})\cdot {}h$ ; by Lemma 2.12, a model in which $h$ is localised at $x$ will have $h'$ localised at $a$ .

Lemma 3.1 A DGLA model of $X_n$ is defined by generators $a,b,e_i,f^{\prime }_i,h^{\prime }$ , with the differential defined by (3.1), (3.2), and (3.4) along with

(3.5) $$ \begin{align} {\partial}_ah^{\prime}=\sum\limits_{i=1}^nP_i\big({\text{BCH}}(E_1,-E_2),\dots,{\text{BCH}}(E_{n-1},-E_n)\big)\cdot{}f^{\prime}_i \end{align} $$

for any polynomials $P_i$ in $(n-1)$ non-commuting variables whose initial term is $1$ and satisfy the identity

(3.6) $$ \begin{align} \sum\limits_{i=1}^{n-1}P_i({\text{ad}}_{x_1},\dots,{\text{ad}}_{x_{n-1}})&\cdot{}x_i \nonumber\\ &\quad=P_n({\text{ad}}_{x_1},\dots,{\text{ad}}_{x_{n-1}})\cdot{\text{BCH}}(x_1,\dots,x_{n-1}). \end{align} $$

In this model, the $3$ -cell and $2$ -cells are all localised at $a$ .

Proof It is apparent that the initial term of ${\partial }_ah'$ as defined by (3.5) is $\sum _{i=1}^nf^{\prime }_i,$ as required. It remains only to verify that ${\partial }_a^2h'=0$ , that is, that the RHS of (3.5) defines an element of $\ker {\partial }_a$ . For this, observe that since $u_{e_i}(a)=b$ , by Lemma 2.7, $u_{{\text {BCH}}(e_i,-e_{i+1})}(a)=a,$ and so by Lemma 2.2, ${\text {BCH}}(e_i,-e_{i+1})\in \ker {\partial }_a$ . However, for $y_1,\dots ,y_r\in \ker {\partial }_a$ of degree 0,

$$\begin{align*}{\partial}_a[y_1,[y_2,\dots,[y_r,w]\cdots]]=[y_1,[y_2,\dots,[y_r,{\partial}{}w]\cdots]],\end{align*}$$

and so for any non-commuting polynomial $P$ , $P({\text {ad}}_{y_1},\dots ,{\text {ad}}_{y_r})$ commutes with ${\partial }_a$ . By Lemma 2.6(ii), ${\text {BCH}}({\text {ad}}_x,{\text {ad}}_y)={\text {ad}}_{{\text {BCH}}(x,y),}$ and it follows that

$$ \begin{align*} &\partial_a \sum\limits_{i=1}^nP_i \big({\text{BCH}}(E_1,-E_2),\dots,{\text{BCH}}(E_{n-1},-E_n)\big)\cdot{}f^{\prime}_i\\ &\quad=\sum\limits_{i=1}^nP_i\big({\text{BCH}}(E_1,-E_2),\dots,{\text{BCH}}(E_{n-1},-E_n)\big)\cdot{}{\partial}_af^{\prime}_i\\ &\quad=\sum\limits_{i=1}^nP_i\big({\text{BCH}}(E_1,-E_2),\dots,{\text{BCH}}(E_{n-1},-E_n)\big)\cdot{}{\text{BCH}}(e_i,-e_{i+1})\\ &\quad=0, \end{align*} $$

where the second step follows from (3.4) and the third from (3.6) applied to $x_i={\text {BCH}}(e_i,-e_{i+1}),$ since $-x_n={\text {BCH}}(x_1,\cdots ,x_{n-1})$ .▪

To see that the previous lemma’s constructions actually lead to a model of the $n$ -faceted banana, it remains only to show that $P_1,\dots ,P_n$ exist with initial term 1 and satisfying identity (3.6). This follows immediately from Lemma 2.6(iii), even with $P_n\equiv 1$ ; see the next subsection and (3.12). Indeed, there are many possible choices for $P_1,\dots ,P_n$ .

3.5 Model Localised at Central Point x

Twisting the model of Lemma 3.1 back so that the $2$ -cells are localised at their centres, and the $3$ -cell is localised at its centre $x$ , we find the differential is given by (3.1), (3.2), and (3.3) along with

(3.7) $$ \begin{align} {\partial}_xh&=\nonumber\\&e^{-\frac12{\text{ad}}_v} \Big(\sum\limits_{i=1}^nP_i({\text{BCH}}(E_1,-E_2),\dots,{\text{BCH}}(E_{n-1},-E_n))\cdot{} e^{\frac12{\text{ad}}{v_i}}f_i\Big). \end{align} $$

The final requirement is that the model is symmetric under the action of $D_n$ ; that is, it is invariant under $\sigma $ , $\tau $ . By the geometry and total symmetry of the bi-gon model, (3.1), (3.2), and (3.3) will be invariant under $\sigma $ , $\tau ,$ and $\iota $ . Under $\tau $ , $x,h,v$ are fixed, while $e_i\mapsto {}e_{i+1}$ , $f_i\mapsto {}f_{i+1}$ , $v_i\mapsto {}v_{i+1,}$ so (3.7) is invariant under $\tau $ so long as

$$ \begin{align*} P_i \big({\text{BCH}}(E_2,-E_3),\dots,{\text{BCH}}&(E_n,-E_1)\big)=\\ &P_{i+1}\big({\text{BCH}}(E_1,-E_2),\dots,{\text{BCH}}(E_{n-1},-E_n)\big), \end{align*} $$

which is ensured by

(3.8) $$ \begin{align} P_{i+1}(x_1,\dots,x_{n-1})=P_i\big(x_2,\dots,x_{n-1},-{\text{BCH}}(x_1,\dots,x_{n-1})\big). \end{align} $$

Under $\sigma $ , $v,x$ are fixed, $e_i\mapsto {}e_{n-i}$ , $f_i\mapsto {}-f_{n-i-1}$ , $v_i\mapsto {}v_{n-i-1,}$ while $h$ changes sign. In order for (3.7) to be invariant under $\sigma $ , it is required that

$$ \begin{align*} P_i\big({\text{BCH}}(E_{n-1},-E_{n-2}),&\dots,{\text{BCH}}(E_1,-E_n)\big)=\\ &P_{n-i-1}\big({\text{BCH}}(E_1,-E_2),\dots,{\text{BCH}}(E_{n-1},-E_n)\big), \end{align*} $$

which is ensured by (3.8) along with

(3.9) $$ \begin{align} P_{n-i}(x_1,\dots,x_{n-1})=P_i(-x_{n-1},\dots,-x_1). \end{align} $$

Under $\iota $ , the quantities $e_i$ , $f_i$ , $v_i$ , $h,$ and $v$ all change sign, while $x$ is invariant, and so (3.7) is invariant under $\iota $ so long as

(3.10) $$ \begin{align} e^{\frac12{\text{ad}}_v}&P_i({\text{BCH}}(-E_1,E_2),\dots,{\text{BCH}}(-E_{n-1},E_n))e^{-\frac12{\text{ad}}_{v_i}}=\nonumber\\ &\qquad e^{-\frac12{\text{ad}}_v}P_{i}({\text{BCH}}(E_1,-E_2),\dots,{\text{BCH}}(E_{n-1},-E_n))e^{\frac12{\text{ad}}_{v_i}}, \end{align} $$

where $v_i=\mu _2(e_i,e_{i+1})$ and $v=\mu _n(e_1,\dots ,e_n)$ .

In conclusion, we have the following lemma.

By Lemma 2.6(i), we can write

(3.11) $$ \begin{align} BCH(x,y)=x+y+Q(X,Y)y, \end{align} $$

where $Q$ is a polynomial in two non-commuting variables whose lowest order terms are $\tfrac 12X+\tfrac 1{12}(X^2-YX)+\cdots $ . By Lemma 2.6(iii), iterating (3.11) gives

$$\begin{align*}{\text{BCH}}(x_1,\dots,x_n)=\sum\limits_{i=1}^nx_i+\sum\limits_{i=2}^nQ({\text{BCH}}(X_1,\dots,X_{i-1}),X_i)x_i\end{align*}$$

so that

(3.12) $$ \begin{align}P_1=P_n=1,\quad{}P_i=1+Q \big({\text{BCH}}(X_1,\dots,X_{i-1}),X_i\big)\ \text{for } 1<i<n \end{align} $$

is a solution of (3.6). Note that (3.6) is a linear condition on $(P_1,\dots ,P_n)$ that is invariant under the action of cyclic permutation of the $P_i,$ while cyclically permuting the $X_i$ (with $X_n=-{\text {BCH}}(X_1,\dots ,X_{n-1})$ ), as well as reversing the order of the $P_i$ while reversing the order and signs of the $x_i$ . That is, (3.6) is invariant under a dihedral group action, and so there exists an invariant solution (that will satisfy (3.8) and (3.9)) given by averaging the orbit of the solution (3.12) under the just described dihedral action. The result is that

$$ \begin{align*} P_i&=1+\tfrac1{2n}\!\!\!\!\sum\limits_{j=1+\delta_{in}}^{i-1}\!\!\!\!Q\big({\text{BCH}}(X_j,\dots,X_{i-1}),X_i\big) \\&\quad+\tfrac1{2n}\sum\limits_{j=i+1}^{n-1}\!\!\!Q\big(-{\text{BCH}}(X_i,\dots,X_j),X_i\big)\\ &\quad+\tfrac1{2n}\!\!\!\!\sum\limits_{j=1+\delta_{in}}^{i-1}\!\!\!\!Q\big({\text{BCH}}(X_j,\dots,X_i),-X_i\big) \\&\quad+\tfrac1{2n}\sum\limits_{j=i+1}^{n-1}\!\!\!Q\big(-{\text{BCH}}(X_{i+1},\dots,X_j),-X_i\big) \end{align*} $$

satisfies (3.6), (3.8), and (3.9). The coefficient of $f^{\prime }_i$ in ${\partial }_ah'$ from (3.5) is now

$$ \begin{align*} P_i&\big({\text{BCH}}(E_1,-E_2),\dots,{\text{BCH}}(E_{n-1},-E_n)\big)\\ &=1+\tfrac1{2n}\sum\limits_{\substack{{j=1}\\{j\not=i,i+1}}}^{n} \Big(Q\big({\text{BCH}}(E_j,-E_i),{\text{BCH}}(E_i,-E_{i+1})\big)\\[-18pt] &\qquad\qquad\qquad\qquad+Q\big({\text{BCH}}(E_j,-E_{i+1}),{\text{BCH}}(E_{i+1},-E_i)\big )\Big). \end{align*} $$

Theorem 3.3 The completed DGLA with (free) Lie algebra generators $a,b,e_i,f_i,h$ and differential defined by (3.1), (3.2), (3.3) and

$$ \begin{align*} e^{\frac12{\text{ad}}_v}{\partial}_xh &= \sum\limits_{i=1}^nf^{\prime}_i+\tfrac1{2n} \sum\limits_{\substack{{i,j=1}\\{j\not=i,i+1}}}^n \Big(Q\big({\text{BCH}}(E_j,-E_i),{\text{BCH}}(E_i,-E_{i+1})\big)\\ &\quad+Q\big({\text{BCH}}(E_j,-E_{i+1}),{\text{BCH}}(E_{i+1},-E_i)\big )\Big)f^{\prime}_i \end{align*} $$

defines a model for the $n$ -faceted banana that is symmetric under the geometric symmetries of the cell fixing the vertices, where $Q$ is defined by (3.11), $f^{\prime }_i=e^{\frac 12{\text {ad}}_{v_i}}f_i$ , $v_i=\mu _2(e_i,e_{i+1})$ , $v=\mu _n(x_1,\dots ,x_n),$ and $x=u_{\frac 12v}(a)$ . The $3$ -cell is localised at $x$ in this model and the bi-gon $2$ -cells are localised at their centres $x_i=u_{\frac 12v_i}(a)$ .

Up to the first two non-trivial orders (in Lie brackets),

$$ \begin{align*} v_i&=\tfrac12(e_i+e_{i+1})-\tfrac1{48}(E_i^2e_{i+1}+E_{i+1}^2e_i)+\cdots,\\ v&=\tfrac1n\sum\limits_ie_i-\tfrac1{12n^2}\sum\limits_{i\not-j}E_i^2e_j+\cdots,\\ x&=\tfrac12(a+b)+\tfrac1{8n}\sum\limits_i[e_i,b-a]+\cdots,\\ x_i&=\tfrac12(a+b)+\tfrac1{16}(E_i+E_{i+1})(b-a)+\cdots. \end{align*} $$

Meanwhile, up to second order (in Lie brackets) the differential is given by

$$ \begin{align*} {\partial}{a}&=-\tfrac12[a,a],\qquad {\partial}{b}=-\tfrac12[b,b],\\[3pt] {\partial}{e_i}&=b-a+\tfrac12E_i(a+b)+\tfrac1{12}E_i^2(b-a)+\cdots,\\[3pt] {\partial}{}f_i&=e_i-e_{i+1}-\tfrac12[a+b,f_i]+\tfrac1{48}(E_{i+1}^2e_i-E_i^2e_{i+1}) +\tfrac1{16}[(E_i+E_{i+1})(a-b),f_i]+\cdots,\\[3pt] {\partial}{}h&=\sum\limits_if_i-\tfrac12[a+b,h]+\tfrac1{8n}\sum\limits_i[E_i(a-b),h]+\sum\limits_i \text{(quadratic in } E_j\text{'s)}f_i+\cdots. \end{align*} $$