On extensive amenability of algebras

Amenability of groups developed into a thriving and fundamental notion in part because of the variety of equivalent definitions of the notion: almost-invariant sets, invariant functionals, subexponential decay of return probabilities, triviality of a suitable Poisson boundary, etc., see [2]. The equivalences between the definitions lead both to useful applications and a powerful method of proof of (non)amenability for diverse examples and constructions. For example, an extension of groups is amenable if and only if both subgroup and quotient are amenable, and the implications are of varying difficulty depending on the definition used.

It turned out that it is much more convenient to study more generally the amenability of $G$-sets; most notions and results extend, and one may define amenability of a group either as amenability of the regular $G$-set $G$ or equivalently as amenability of all non-empty $G$-sets. Remarkably however, amenability does not propagate through extensions: if $f:X\twoheadrightarrow Y$ is a map of $G$-sets, then it is not true that $G X$ is amenable if and only if $G Y$ and all the fibre actions $G_y(f^{-1}(y))$ are amenable.

This led Juschenko, Matte Bon, Monod and de la Salle to introduce extensive amenability, a stronger property of $G$-sets, which also characterizes amenable groups and does propagate through extensions [6 Proposition 2.4].

To define amenability of associative algebras, we have essentially only one definition at our disposal:

Definition (Elek [4]). Let $\mathcal A$ be an algebra over a field $\mathbb{k}$. The left $\mathcal A$-module $M$ is amenable if for every $\epsilon>0$ and every finite $S\subseteq\mathcal A$ there is a finite $F\subseteq M$ with

(Here and throughout, “finite” means “finite-dimensional” when applied to vector spaces. The strict inclusion ‘$\subset$’ prevents $F=0$.)

The case in which $\mathcal A$ is a cocommutative Hopf algebra is of particular interest, covering both group rings and universal enveloppes of Lie algebras, and we restrict ourselves to this case in this article. Note that the $0$ module is never amenable. The regular module $\mathcal{A}$ is amenable if and only if every nonzero $\mathcal{A}$-module is amenable, by Corollary 1.2 below; in this case, $\mathcal{A}$ itself is called amenable. In a previous article, I prove that amenability is preserved by Hopf quotients, Hopf subalgebras, and cleft extensions, see [3 Theorems 1.9, 2.2 and 2.5].

One soon discovers that there is little structure on amenable modules over non-amenable Hopf algebras; for example, a direct sum of an amenable and a non-amenable module is amenable, so amenability need not pass to submodules or quotients. I propose the following definition, to recover preservation under exact sequences (see Theorem E):

Here $\operatorname{Sym}(M)$ is a Hopf algebra with each $m \in M$ primitive, and $\operatorname{Sym}(M)\#\mathcal{A}$, which from now on will be written $M \wr\mathcal{A}$, is $\operatorname{Sym}(M) \otimes\mathcal{A}$ qua coalgebra, with multiplication and action

The induced action makes $\operatorname{Sym}(M)$ an $\mathcal{A}$-module algebra and coalgebra, and $M \wr\mathcal{A}$ is a cocommutative Hopf algebra.

The zero module is extensively amenable, since $\operatorname{Sym}(0)=k$, although it is not amenable. More generally,

Proof. For given finite $S \leq M \wr\mathcal{A}$, choose a finite $\mathcal{A}$-submodule $V \leq M$ containing all variables in $S$’s polynomial coefficients. Set $q=\dim V$, and bound the degrees of those coefficients by $d$. The $\mathcal{A}$-action preserves polynomial degree. For $E_n=\operatorname{Sym}^{\leq n}(V)$, the expanded space is contained in $\operatorname{Sym}^{\leq n+d}(V)$, so its expansion ratio is at most

The main results of this article are:

The following proves, in the case of extensive amenability, the corresponding result [1 Theorem 1.4] for usual amenability:

In fact, there is quite a broad flexibility in the definition of extensive amenability of $G$-sets. One may replace the functor $(\mathbb{Z}/2)^{(X)}$ by $B^{(X)}$ for any nontrivial amenable group $B$, or by $\operatorname{Sym}_{\mathrm{fin}}(X)$, the group of finitely supported permutations of $X$, see [6 Theorem 1.3 and Examples 3.1–3.2].

Other functorial constructions VectorSpace $\to$ AmenableAlgebra may be used in place of the symmetric algebra, provided they carry a compatible induced $\mathcal{A}$-action and Hopf structure. I have not gone into the general theory which is beyond my non-AI brain, but contented myself with the exterior algebra, interpreted as a Hopf superalgebra, and call $M$ $\wedge$-extensively amenable when $\bigwedge M$ is amenable under multiplication and the induced $\mathcal{A}$-action. We regard $\mathcal{A}$ as purely even and $M$ as purely odd in the Hopf superalgebra $\bigwedge M \mathrel{\#} \mathcal{A}$.

I have managed to prove that all the above main results remain valid for $\wedge$-extensive amenability, but not that $M$ is extensively amenable if and only if it is $\wedge$-extensively amenable, in contrast with the group case.

Acknowledgments to humans and AI. I am grateful to Mikael de la Salle for numerous stimulating discussions on this topic. My definition of extensive amenability for algebras is directly inspired from the group case.

ChatGPT 5.6 Sol was used to assist with drafting, editing, checking parts of this manuscript, and exploratory formal verification. All mathematical content was reviewed, checked, and substantially revised by the author, who is responsible for the final text.

The main results have been formalized in Lean/Mathlib and are available at

https://github.com/laurentbartholdi/hopfamenability

Two transfer lemmas

In equations “dim$((F+SF)/F)<\epsilon\dim F$”, we refer to $S$ as a testing space and $F$ as a Følner space. By the fundamental theorem of coalgebras, finite testing spaces in a Hopf algebra may be enlarged to finite subcoalgebras containing $1$. We use this reduction below.

The following is the tensor-space form of the argument proving [3 Theorem 1.9]. We record the slight strengthening needed below.

Proof. Put $E^+=FE$ and choose a finite subspace $W\leq V$ containing all second tensor coefficients of $E$ and $E^+$. For $Z=E,E^+$ put $r_Z(L)=\dim(Z\cap(U\otimes L))$. The functions $r_Z$ are monotone and supermodular on the subspace lattice of $W$. For $t>0$, let $C_t(Z)$ be the largest maximizer of $r_Z(L)-t\dim L$, which exists by [3 Lemma 1.2]. These are also maximizers over all finite subspaces of $V$: replacing $L$ by $L\cap W$ leaves its rank unchanged and cannot increase its dimension.

First suppose that $F$ has a complete subcoalgebra flag $0=F_0<\cdots<F_s=F$. We claim that $FC_t(E)\subseteq C_t(E^+)$. Write $C=C_t(E)$, $C^+=C_t(E^+)$, and suppose inductively that $F_{i-1}C\subseteq C^+$. For $a\in F_i\setminus F_{i-1}$, there is a group-like $g\in F_i$, by the codimension-one argument in [3 Proposition 1.8], with $\Delta(a)-g\otimes a\in F_i\otimes F_{i-1}$. Set $K=\{c\in C:ac\in C^+\}$, let $q:V\to V/C^+$ be the quotient map, and put $\bar a:C\to V/C^+$, $\bar a(c)=q(ac)$. Write $\delta_a$ for the diagonal action of $a$ on $U\otimes V$, and $\alpha_g$ for the action of $g$ on $U$. The relation for $\Delta(a)$ and the inductive inclusion $F_{i-1}C\subseteq C^+$ give

Since $g_U$ is invertible with inverse $S(g)_U$ and $\ker\bar a=K$, exactness of the tensor product gives

Moreover, $\delta_a(E)\leq E^+$, because $a\in F$, and the same triangular relation gives $\delta_a(U\otimes C)\leq U\otimes(C^++aC)$. Thus the restriction of $\delta_a$ to $E\cap(U\otimes C)$, followed by quotienting by $E^+\cap(U\otimes C^+)$, has kernel exactly $E\cap(U\otimes K)$. It therefore induces an injection

Maximality of $C$ bounds the dimension on the left below by $t(\dim C-\dim K)$; rank-nullity identifies this with $t(\dim(C^++aC)-\dim C^+)$. Hence $C^++aC$ is also a maximizer for $E^+$, and the choice of the largest maximizer gives $aC\subseteq C^+$.

The density/coarea identity [3 Lemma 1.2] now gives

Some nonzero $C_t(E)$ has the asserted ratio.

For the general case, split a subcoalgebra flag after a finite field extension and descend the ratio, using [3 Lemmas 1.6 and 1.7].

Proof. The reverse implication follows by taking the regular module. Conversely, suppose that $\mathcal A$ is amenable and let $M\ne0$. The linear isomorphism

with inverse $a\otimes m\mapsto\sum a_{(1)}\otimes S(a_{(2)})m$, intertwines the action on the first factor with the diagonal action. For any fixed $0\ne m\in M$, the diagonal module therefore contains the amenable submodule $T(\mathcal A\otimes km)\simeq\mathcal A\mathcal A$. Lemma 1.1 makes $M$ amenable.

For commutative Hopf algebras $\mathcal H\leq\mathcal B$, let $Q=\mathcal B/\mathcal B\mathcal K$ be the quotient left $\mathcal B$-module coalgebra, with quotient map $q$. The left $Q$-coaction on $\mathcal B$ is $\rho=(q\otimes\operatorname{id})\Delta_{\mathcal B}:\mathcal B\to Q\otimes\mathcal B$.

Proof. Let $F\leq\mathcal B$ be a finite subcoalgebra containing $1$, and fix $0<\delta$. By [3 Theorem 1.1], choose a nonzero finite subcoalgebra $C\leq Q$ with $\dim FC<(1+\delta)\dim C$, and put $D=FC$. Collinearity and the module-coalgebra identity give

Applying $\operatorname{id}_{Q} \otimes\Theta^{-1}$ and using $Q$-colinearity of $\Theta$, we obtain

and applying $\Theta(\operatorname{id}_{Q}\otimes\varepsilon_{Q}\otimes\operatorname{id}_{\mathcal{K}})$ yields $F\sigma(C) \le\Theta(D\otimes\mathcal{K})$. Consequently there is a finite $T\le\mathcal{K}$, containing $1$, such that $F\sigma(C)\le\sigma(D)T$. Choose a finite $E\le V$ with $\dim TE<(1+\delta)\dim E$. The induced vector-space identification $\mathcal{B}\otimes_{\mathcal{K}}V\cong Q\otimes V$ then gives, for $W=C\otimes E$,

Let $\delta$ tend to zero. $\square$

Amenability and extensively amenable modules

Proof. Let $M$ be a nonzero extensively amenable $\mathcal{A}$-module and let

be the universal derivation. It is $\mathcal{A}$-linear for the diagonal action and by the Leibniz identity satisfies

Fix a finite $F\le\mathcal{A}$ containing $1$, a nonzero $m\in M$, and $\eta>0$. Extensive amenability gives a finite $E\le\operatorname{Sym}(M)$ such that

Put $E'=E+mE$ and $V=d(E')$. The map

is injective: if both coordinates vanish, the Leibniz identity gives $u\otimes m=0$, hence $u=0$. Thus $\dim E\le2\dim V$. Since $d$ is $\mathcal{A}$-linear, $d(FE')=FV$, so $d$ induces a surjection $FE'/E'\twoheadrightarrow FV/V$. As $FE'=Z$ and $E\le E'$, we have

Hence $\operatorname{Sym}(M)\otimes M$ is amenable. Lemma 1.1 extracts Følner spaces in $M$. $\square$

Proof. The commutative Hopf algebra $\operatorname{Sym}(M)$ is amenable [3 Theorem 2.4]. The split cleft extension

therefore makes $M\wr\mathcal{A}$ amenable by [3 Theorem 2.5]. Its nonzero module $\operatorname{Sym}(M)$ is amenable by Corollary 1.2. $\square$

Exact sequences

We first recall an elementary filtered-space observation. For compatible increasing exhaustive filtrations $(R_n)_{n\geq-1}$ and $(V_n)_{n\geq-1}$ on an algebra $R$ and an $R$-module $V$, with $R_{-1}=V_{-1}=0$, amenability of $V$ implies amenability of $\operatorname{gr} V$ over $\operatorname{gr} R$. Indeed, lift a finite homogeneous subspace $\overline{F}\leq\operatorname{gr} R$ to $F\leq R$, take a Følner space $E$, and put $E'=E+FE$. Then

This is the leading-symbol argument of [3 Lemma 7.2]; the exhaustive nonnegative filtration ensures that the induced filtrations on finite spaces preserve dimension. The same argument applies to decreasing filtrations $V=V_0\geq V_1\geq\cdots$ with $\bigcap_n V_n=0$: their intersections with any finite-dimensional space eventually vanish. In particular, for an already graded algebra and module one may take homogeneous Følner spaces, after enlarging the finite testing space to a homogeneous one.

Proof. Suppose first that $N$ is extensively amenable. Give $N$ the two-step filtration $0\leq M\leq N$, giving $M$ weight zero and $N/M\cong P$ in degree 1. Then $\operatorname{gr}N\cong M\oplus P$, and the induced multiplicative filtrations satisfy

The preceding observation proves that $M\oplus P$ is extensively amenable. The two projections of $M\oplus P$ induce surjective counital module-coalgebra maps to $\operatorname{Sym}(M)$ and $\operatorname{Sym}(P)$. The quotient theorem [3 Theorem 1.9], with the actions factored through $M\wr\mathcal{A}$ and $P\wr\mathcal{A}$, proves both conclusions.

Conversely, take $\mathcal{B}=N\wr\mathcal{A}$, $\mathcal{K}=M\wr\mathcal{A}$, and $\mathcal{Q}=\mathcal{B}/\mathcal{B}\ker\varepsilon_{\mathcal{K}}\cong\operatorname{Sym}(P)$, with quotient map $q$ and coaction $\rho=(q\otimes\operatorname{id})\Delta_{\mathcal{B}}$. A vector-space section $P\to N$ induces an algebra and coalgebra section $\sigma:\operatorname{Sym}(P)\to\operatorname{Sym}(N)\leq\mathcal{B}$. Multiplication gives

It is a right $\mathcal{K}$-module isomorphism by the polynomial normal form, and is colinear because $\rho(\sigma(p))=\sum\sigma(p_{(1)})\otimes\sigma(p_{(2)})$ and $q(bk)=q(b)\varepsilon(k)$. The $\mathcal{B}$-action on $\operatorname{Sym}(P)$ factors through $P\wr\mathcal{A}$, so $\mathcal{B}(\operatorname{Sym}(P))$ is amenable. Apply Lemma 1.3 to the amenable $\mathcal{K}$-module $\operatorname{Sym}(M)$. The action map

identifies these modules, proving that $N$ is extensively amenable. □

Proof. The first assertions and the finite direct-sum assertion follow from Theorem 3.1. If $M=\bigcup_i M_i$ is directed and every $M_i$ is extensively amenable, the coefficients of any finite subspace of $M\wr\mathcal{A}$ lie in some $\operatorname{Sym}(M_i)$; a Følner space there works in $\operatorname{Sym}(M)$. The converse follows by submodule permanence. Arbitrary direct sums are directed unions of finite ones. □

Group actions and permutation modules

Juschenko, Matte Bon, Monod and de la Salle call a $G$-set $X$ extensively amenable if there is a $G$-invariant mean on the collection $\mathcal{P}_{f}(X)$ of finite subsets of $X$ giving mass one to $\{B : T \subseteq B\}$ for every finite $T \subseteq X$, see [6 Definition 1.1] or [2 Definition 9.1]. We immediately pass to an equivalent definition, see [6 Theorem 1.3]: with $L = \mathbb{Z}^{(X)}$, $X$ is extensively amenable if and only if the affine action $L \rtimes G \curvearrowright L$ is amenable.

In case $\mathcal{A}$ is a group ring $k[G]$, our wreath product $M \wr\mathcal{A} = \operatorname{Sym}(M) \# \mathcal{A}$ is also known as the crossed product algebra $\operatorname{Sym}(M) \rtimes G$.

Proof. Set $L = \mathbb{Z}^{(X)}$ and $R = k[L]$, and write $u_x$ for the standard generators of $L$ and $z_x = u_x - 1$. Let $\varpi= \ker(\epsilon: R \to k)$ be the augmentation ideal, and filter $R$ by $\varpi^n$. The classical description of the augmentation filtration of a free abelian group gives

see [7 Chapter VI and Corollary VIII.8.2]. The isomorphism sends $x$ to the initial form of $u_x - 1$, so is $G$-equivariant. Since $G$ preserves $\varpi$, the compatible filtration $(R \# k[G])_n = \varpi^n \# k[G]$ gives

Here $R \# k[G] \simeq k[L \rtimes G]$, and $R$ is the permutation module of the affine action $L \rtimes G \curvearrowright L$. By the preceding criterion and the permutation-module equivalence [3 §3], it suffices to prove that $\operatorname{Sym}(k[X])$ is amenable under $\operatorname{Sym}(k[X]) \# k[G]$ if and only if $R$ is amenable under $R \# k[G]$.

Suppose first that $\operatorname{Sym}(k[X])$ is amenable under $k[X] \wr k[G]$. The $G$-equivariant map

identifies the symmetric algebra with the polynomial subalgebra $k[z_x : x \in X]$; these are also the canonical lifts of the generators in $\operatorname{gr}_{\varpi} R \simeq\operatorname{Sym}(k[X])$. If $E$ is a Følner space in $\operatorname{Sym}(k[X])$, let $\widetilde{E}$ be its image in $R$. For $g \in G$ the expansion ratios of $E$ and $\widetilde{E}$ agree, while $\widetilde{E} + u_x\widetilde{E} = \widetilde{E} + z_x\widetilde{E}$ shows that multiplication by $u_x$ has the same boundary as multiplication by $z_x$.

For any finite-dimensional subspace $W$ and invertible linear operators $a,b$ on its ambient space, put $b_W(a) = \dim(W + aW) - \dim W$. Then

The first inequality follows by enlarging $W + abW$ to $W + aW + abW$; the additional dimension beyond $W + aW$ is at most $\dim((aW + abW)/aW) = \dim((W + bW)/W) = b_W(b)$; the second follows by applying $a$ to $W + a^{-1}W$. Given a finite subset of $L \rtimes G$, express its elements as words in finitely many $u_x^{\pm1}$ and elements of $G$. Choose $E$ simultaneously Følner for the corresponding variables $x$ and group operators, with error small compared with the sum of the word lengths. By (4.2) and

the image $\widetilde{E}$ is Følner for the prescribed finite subset. Thus $R$ is amenable under $R \# k[G]$, so the affine action is amenable and $G \curvearrowright X$ is extensively amenable.

Conversely, suppose that $R$ is amenable under $R \#_{\mathbb{k}}[G]$. The separated-filtration argument preceding Theorem 3.1 proves that amenability is preserved on the associated graded, so $\mathbb{k}[X]$ is extensively amenable by (4.1). □

By the lamp-functor theorem [6 Theorem 1.3], or its action-valued form [2 Proposition 9.5], these conditions are also equivalent to amenability of the affine actions

for $L = \mathbb{Z}/2$, $L = \mathbb{Z}$, or $L = (\mathbb{k},+)$. We used $L = \mathbb{Z}$ in the proof above, and will consider $L = \mathbb{Z}/2$ in § 5.

The exterior algebra version

We consider a variant of the definition of extensive amenability: let us call the $\mathcal{A}$-module $M$ $\Lambda$-extensively amenable if the exterior algebra $\bigwedge M$ is amenable under multiplication and the induced $\mathcal{A}$-action. The results above have exterior analogues, but outside characteristic two one must retain the supercoalgebra signs. We give the required modifications.

Superalgebras and the coalgebra arguments

Regard $\mathcal{A}$ as purely even and $M$ as purely odd. We work in supervector spaces, with the conventions

Here as usual $|u|$ denotes degree. The exterior algebra $\bigwedge M$ is a supercommutative, supercocommutative Hopf superalgebra with $\Delta(m) = m \otimes1 + 1 \otimes m$ and $\varepsilon(m) = 0$. Every $m \in M$ satisfies $m^2 = 0$ in $\bigwedge M$. Put

a Hopf superalgebra admitting $\bigwedge M$ as a supermodule coalgebra. Both $\bigwedge M$ and $M \wr_{\wedge} \mathcal{A}$ are $\mathbb{N}$-graded objects. Following the observation preceding Theorem 3.1, we may restrict our considerations to degree-graded Følner spaces.

We unfortunately need to reprove some of the previous results in the setting of Hopf superalgebras. We gather the required conclusions in the following lemma.

Proof. Every finite subspace of a graded coalgebra is contained in a finite graded subcoalgebra: take a finite subcoalgebra containing it and then its homogeneous hull. Indeed, if $p_n$ denotes projection to degree $n$, then

Thus arbitrary finite testing spaces may be enlarged to homogeneous subcoalgebras containing $1$; the filtered-space observation preceding Theorem 3.1 then supplies homogeneous Følner spaces.

Complete homogeneous subcoalgebra flags exist by coalgebra duality: let $0 \ne F = \bigoplus_{n \ge0} F_n \leq\mathcal{H}$ be a finite graded subcoalgebra, put $\Omega= \overline{\mathbb{k}}$, and consider the ordinary convolution algebra $\mathcal{R} = (\Omega\otimes F)^*$, with product

Give $R$ the grading $R_n=(\Omega\otimes F_n)^*$, extending functionals by zero on the other degrees. Then $R_iR_j\subseteq R_{i+j}$, so $R_{>0}$ is a nilpotent two-sided ideal. Moreover, $R/R_{>0}\simeq R_0$ is commutative, since $F_0$ is commutative. Thus, writing $J_0$ for the radical of $R_0$, we have

for some $r$. In particular, $J$ is homogeneous and nilpotent, while $R/J$ is concentrated in degree zero.

For every $m\geq0$, the radical layer $J^m/J^{m+1}$ is a graded bimodule over $R/J$. Its left and right actions commute and preserve each degree; equivalently, it is a graded module over

This algebra is split semisimple and concentrated in degree zero, so each layer is a direct sum of homogeneous one-dimensional sub-bimodules. Refining each radical layer by these summands and taking inverse images gives a complete flag of homogeneous two-sided ideals of $R$. Their ordinary annihilators give, in reverse order, a complete homogeneous subcoalgebra flag in $\Omega\otimes F$. Indeed, if $I$ is a two-sided ideal and $c\in I^\perp$, then $\Delta(c)$ annihilates $I\otimes R+R\otimes I$, whence $\Delta(c)\in I^\perp\otimes I^\perp$.

Finally, a homogeneous basis adapted to the flag involves only finitely many coefficients in $\Omega$. They generate a finite extension of $k$, over which the entire homogeneous subcoalgebra flag is defined.

For a homogeneous codimension-one step $F'\subset F'+k a$, write

where $g,h$ are even group-like elements. The super module-coalgebra identity is

More explicitly, let $C$ be a finite graded subcoalgebra of an $\mathcal{H}$-module coalgebra, not necessarily $\mathcal{H}$-stable, and let $D\subseteq(F'+ka)C$ be a graded subcoalgebra. Put $V=D+F'C$ and $\phi:C\to(F'+ka)C/V$, $c\mapsto ac+V$, as in [3 Proposition 1.8]. On underlying vector spaces, using ordinary tensor products of linear operators, the first triangular relation gives $(\phi\otimes g)\Delta(c)=0$ on $\ker\phi$; the second gives $(h\Pi^{[a]}\otimes\phi)\Delta(c)=0$, where $\Pi(c)=(-1)^{|c|}c$. Since $g$ and $h\Pi^{[a]}$ are invertible on the ambient module, their restrictions to $C$ are injective, and the two tensor-kernel arguments are unchanged. Their intersection shows that $\ker\phi$ is a graded subcoalgebra. The dimension inequalities in [3 Proposition 1.8] therefore telescope unchanged along the homogeneous flag. To remove the splitting assumption, extend scalars to a finite splitting field. The tensor-amplification and base-change arguments of [3 Lemmas 1.4–1.5] apply with a purely even field-extension factor and graded subspaces, so preserve semistability on the lattice of graded subcoalgebras. Apply the split transfer inequality to scalar extensions of the original graded data and descend it by preservation of dimensions and intersections. The density and comparison arguments of [3 Lemmas 1.2–1.3] and the proof of Theorem 1.1 then give rounding on the lattice of graded subcoalgebras over the original field; no descent of the rounded subcoalgebra is needed.

Likewise, in Lemma 1.1, the leading action on $U\otimes V$ is $h\Pi^{[a]}_{U}\otimes a$; its first factor is invertible. Use graded subspaces in the density argument, and descend the extracted subspace’s expansion ratio by [3 Lemma 1.7]; its coefficient filtration preserves homogeneity. This ratio descent is used only for subspaces, not for subcoalgebras. For an even counital coalgebra quotient $q:C\to Q$, the map $(\operatorname{id}\otimes q)\Delta_C:C\to C\otimes Q remains an injective $\mathcal{H}$-module map with the super diagonal action, proving quotient permanence. Finally, the normal-basis proof uses only rounding, colinearity and dimensions, and is unchanged with graded spaces.

Permanence and ordinary amenability.

Proof. For a locally finite module, $\bigwedge V$, with $V$ a sufficiently large finite invariant subspace, is already a finite invariant space for the given test.

For an exact sequence, the weight filtration used in Theorem 3.1 again has associated graded $\bigwedge(M\oplus P)$. (In applying Lemma 5.1, the nonnegative grading is still total exterior degree.) Quotient permanence from Lemma 5.1 gives descent to $M$ and $P$. For ascent put $\mathcal{B}=N\rtimes_{\wedge}\mathcal{A}$, $\mathcal{K}=M\rtimes_{\wedge}\mathcal{A}$, and $Q=\bigwedge P$. A linear splitting $P\to N$ induces a graded Hopf-superalgebra section $\sigma:\bigwedge P\to\bigwedge N\subseteq\mathcal{B}$. Exterior normal form gives the right-$\mathcal{K}$ isomorphism $\Theta(p\otimes k)=\sigma(p)k$, and

Thus $\Theta$ is also a left-$Q$ comodule map. The super version of Lemma 1.3 proves ascent exactly as before. The direct-sum and directed-union arguments are unchanged.

If $\mathcal{A}$ is amenable, put $\mathcal{B}=M\rtimes_{\wedge}\mathcal{A}$ and $\mathcal{K}=\bigwedge M$. The algebra $\mathcal{K}$ is locally finite-dimensional and therefore amenable. The quotient $\mathcal{B}\otimes_{\mathcal{K}}k=\mathcal{A}$, with its regular action, is amenable. The Hopf section $\mathcal{A}\to\mathcal{B}$ gives the right-$\mathcal{K}$ normal basis $\mathcal{A}\otimes\mathcal{K}\to\mathcal{B}$; bijectivity follows from the smash-product relations and the bijective antipode of $\mathcal{A}$. The super induction lemma, applied to $V=\mathcal{K}$, makes $\mathcal{B}$ amenable. Its quotient module coalgebra $\mathcal{B}\otimes_{\mathcal{A}}k\cong\bigwedge M$ is amenable by Lemma 5.1.

It remains to pass from $\bigwedge$-extensive amenability of $M$, for $M\ne0$, to ordinary amenability. Put $J=\bigwedge^{>0}M$, and let $\pi:\bigwedge M\to J$ be the $\mathcal{A}$-linear projection. Fix $0\ne m\in M$, a finite subspace $F\le\mathcal{A}$ containing $1$, and $\eta>0$. Choose a finite $E\le\bigwedge M$ with $\dim(FE+F(mE))<(1+\eta)\dim E$, and put $E'=E+mE$ and $W=\pi(E')$. As in Theorem 2.1,

is injective: $\pi(u)=0$ gives $u\in k1$, and then $\pi(mu)=mu=0$ forces $u=0$. Thus $\dim E\le2\dim W$. The quotient-surjection argument of that theorem, applied to $\pi$, gives

Letting $\eta$ tend to zero proves amenability of $J$. The coproduct components of bidegree $(n-1,1)$ give an $\mathcal{A}$-linear injection

It is injective, since distinct wedge basis monomials have disjoint nonempty image supports. Apply Lemma 1.1.

Permutation modules and comparison.

Proof. Put $L=(\mathbb{Z}/2)^{(X)}$, identified with $\mathcal{P}_{f}(X)$ by support. By [6 Theorem 1.3], the action $G\curvearrowright X$ is extensively amenable if and only if the affine action $L\rtimes G\curvearrowright L$ is amenable.

Suppose first that this affine action is amenable. Choose an order on $X$ and the resulting wedge basis

of $\bigwedge k[X]$. A finite testing space $S$ in $\bigwedge k[X]\# k[G]$ is contained in the span of finitely many elements $r_i\# g_i$, where each $r_i$ is an exterior monomial. Write $T_i=\operatorname{supp}(r_i)\in L$, and put $h_i=(T_i,g_i)\in L\rtimes G$.

Choose a finite Følner set $F\subseteq L$ for the finitely many $h_i$, and set

For $B\in F$, the vector $(r_i\#g_i)e_B$ is either $0$, or a nonzero scalar multiple of $e_{T_i\triangle g_iB}=e_{h_iB}$; indeed, it is nonzero precisely when $T_i\cap g_iB=\varnothing$. Hence

Choosing $F$ sufficiently Følner therefore makes $E$ Følner for the original testing space. Thus $k[X]$ is $\bigwedge$-extensively amenable.

Conversely, suppose that $k[X]$ is $\bigwedge$-extensively amenable. For $x\in X$, define

let $p_1:\bigwedge k[X]\to k[X]$ be projection onto degree one, and let $x^*:k[X]\to k$ be the coordinate functional of $x$. Define contractions by

and put

On the wedge basis, $c_x(e_B)=\pm e_{B\triangle\{x\}}$, so $c_x$ is a signed permutation whose underlying permutation is the toggle $B\mapsto B\triangle\{x\}$. Likewise $G$ acts by signed permutations, with underlying permutation $B\mapsto gB$. Let $\Gamma\leq\operatorname{GL}(\bigwedge k[X])$ be the group generated by all $c_x$ and the operators induced by $G$. Its underlying permutation action on $\mathcal{P}_{f}(X)$ has the same image as the affine action $L\rtimes G\curvearrowright L$.

Let $T\subseteq X$ and $S\subseteq G$ be finite. By Lemma 5.1, there is a nonzero finite graded subcoalgebra $C\leq\bigwedge k[X]$ which is arbitrarily Følner for multiplication by the $x\in T$ and for the $g\in S$. Since $C$ is a subcoalgebra, every coefficient operator of its coproduct preserves $C$; in particular,

Consequently

so the same $C$ is arbitrarily Følner for the signed permutation operators $c_x$, $x\in T$, and for $g\in S$. The boundary estimates (3), applied to words in these operators, show that this monomial representation of $\Gamma$ is amenable.

Gromov’s ordering argument for linear actions [5 §3.6] applies unchanged to monomial actions: the nonzero scalar coefficients do not affect the leading coordinates. Therefore amenability of this monomial representation implies amenability of its underlying permutation action on $\mathcal{P}_{f}(X)$. Since this action has the same image as $L\rtimes G\curvearrowright L$, the affine action is amenable. By [6 Theorem 1.3], $G\curvearrowright X$ is extensively amenable.

In characteristic two the binary-lamp correspondence is particularly explicit:

This is a $G$-equivariant algebra isomorphism, not a Hopf isomorphism for the primitive exterior coproduct: $\Delta(u_x-1)$ has the additional term $(u_x-1)\otimes(u_x-1)$.

Proof. The quotient $\operatorname{Sym}(M)\longrightarrow\operatorname{Sym}(M)/(m^2:m\in M)=\bigwedge M$ is an $\mathcal{A}$-equivariant Hopf map: in characteristic two, cocommutativity makes the ideal generated by squares $\mathcal{A}$-stable. The module-coalgebra quotient theorem [3 Theorem 1.9] gives the first implication.

For the converse put $R=\operatorname{Sym}(M)$, let $R^{[2]}$ be the subalgebra generated by squares, and put $\mathcal{B}=R\mathbin{\#}\mathcal{A}$, $\mathcal{K}=R^{[2]}\mathbin{\#}\mathcal{A}$. Primitives act as derivations and annihilate every square, so $\mathcal{A}$ acts on $R^{[2]}$ through its counit. The commutative Hopf algebra $R^{[2]}$ is amenable by [3 Theorem 2.4]; since the $\mathcal{K}$-action factors through its regular action, it is an amenable $\mathcal{K}$-module. Moreover $\mathcal{B}\otimes_{\mathcal{K}} k=\bigwedge M$, which is amenable by hypothesis. Choose a basis of $M$. Sending a wedge basis monomial to its squarefree polynomial gives a coalgebra section; separating even exponents from squarefree exponents gives the colinear right-$\mathcal{K}$ normal basis. Lemma 1.3 applied to $V=R^{[2]}$ proves amenability of $\mathcal{B}\otimes_{\mathcal{K}}R^{[2]}\cong R$. □

Remark 5.5. Replacing squares by $p$-th powers gives the analogous equivalence between the symmetric and truncated symmetric constructions in characteristic $p$, for primitively generated $\mathcal{A}$.

I still do not know whether the symmetric and exterior definitions are equivalent.

Institut Camille Jordan, Université de Lyon 1 and Section de Mathématiques, Université de Genève

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