POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON
NILPOTENT GROUPS

arXiv:1503.04068v2 [] 5 Nov 2017

DAVID KYED AND HENRIK DENSING PETERSEN
Abstract. We introduce a refined version of group cohomology for locally compact, second
countable groups which is naturally related to the space of polynomials on the group in
question. In the case of trivial coefficients it is proven that the polynomial cohomology
admits a description in terms of ordinary cohomology with polynomial coefficients and that
the first polynomial cohomology admits a description given directly in terms polynomials. We
furthermore give a complete description of the polynomials on a connected, simply connected
nilpotent Lie group identifying these with the classical algebra of regular functions.

1. Introduction
Group cohomology is by now a standard tool with a wide range of applications spanning
from finite to locally compact groups and across a variety of disciplines, including differential
geometry, ergodic theory, topology and operator algebras. The aim of the present paper is
to introduce a refined version of group cohomology, dubbed polynomial cohomology, which
consists of a family of functors Hn(d) (G, −) for which the case d = 1 corresponds to the ordinary
cohomology Hn (G, −) of the group G in question. As the name suggests, this cohomology
theory is intimately linked with the notion of polynomials (see Section 6) on groups, and
in the case of trivial coefficients this relationship is made precise by means of the following
theorem.
Theorem A (see Proposition 5.1 and 6.6). Let G be a locally compact, second countable group
and denote by Pold (G) the space of polynomials on G of degree at most d ∈ N0 . Then there
exists an isomorphism of topological vector spaces
Hn(d) (G, R) ≃ Hn (G, Pold−1 (G)) , for all n ∈ N0 .
Moreover, in degree 1 there exists an isomorphism of topological vector spaces
H1(d) (G, R) ≃ Pold (G)/Pol d−1 (G).
Often the polynomial cohomology captures no new information about the group (this is for
instance the case if the group has compact abelianization; cf. Remark 3.7) but for nilpotent
groups we show that the situation is quite different. More precisely, for a connected, simply
connected nilpotent Lie group, Theorem 6.18 provides a complete description of its space of
polynomials in terms of a choice of Mal’cev basis, and shows, in particular, that the space of
polynomials in this case is nothing but the classical algebra of regular functions on the Lie
group (Remark 7.1).
Theorem B (see Theorem 6.18). For a connected, simply connected Lie group G the algebra
of polynomials is equal to the algebra of regular functions on G.
2010 Mathematics Subject Classification. 57M07, 22E41, 22E25.
Key words and phrases. Locally compact groups, nilpotent groups, group cohomology, polynomials.
1

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DAVID KYED AND HENRIK DENSING PETERSEN

Theorem A and Theorem B therefore show, that adorned with the additional data coming
from the degree of the polynomials, the classical algebra of regular functions can be described
naturally via the cohomological functors H1(−) (−, R). Using Theorem B, we furthermore provide a new simple proof of the classical fact that the Hopf algebra of regular functions is a
complete invariant for connected, simply connected Lie groups.
Corollary C (See Theorem 7.9). Let G and H be connected, simply connected, nilpotent Lie
groups and suppose that Ψ : Pol(G) → Pol(H) is a Hopf algebra homomorphism. Then there
is a unique continuous group homomorphism ϕ : H → G such that Ψ is induced by ϕ and ϕ is
an isomorphism if and only if Ψ is.
Acknowledgements. We are grateful to Nicolas Monod and Ryszard Nest for several useful
discussions, and for their hospitality and generosity. We would also like to thank Yves Cornulier and Peter Schlicht for a number of useful comments. The first named author gratefully
acknowledges the financial support from the Lundbeck foundation (grant R69-A7717), the
Villum foundation (grant 7423) and DFF (grant 7014-00145B).

2. Notation and conventions
Topological vector spaces. Unless explicitly stated otherwise, all generic topological vector
spaces are implicitly assumed to be Hausdorff. Our primary need for treating non-Hausdorff
topological vector spaces stems from the fact that the cohomology of a topological group is
(generally) a non-Hausdorff topological vector space, but this will not lead to any confusion,
as it will always be clear from the context whether or not the space in question is (assumed)
Hausdorff or not.
A morphism ϕ : E → F between (not necessarily Hausdorff) topological vector spaces E and
F is a continuous linear map, and an isomorphism is a morphism such that there exists an
inverse morphism ϕ−1 : F → E.
Topological groups. The term ‘group’ G will always mean simply an abstract group without
any topology. We will abbreviate ‘locally compact second countable (unimodular)’ by ‘lcsc’
(respectively ‘lcscu’), and whenever G is a lcscu group we will denote by λG a choice of (biinvariant) Haar measure on it. We denote the identity element in G by 1G , leaving out the
subscript whenever this does not lead to ambiguity. Lastly, we will denote the inversion map
g 7→ g −1 by inv : G → G whenever notationally convenient, and the center of G by Z(G). The
product map is occasionally denoted by m : G × G → G and the map (g, h) 7→ gh−1 by m̃.
Topological G-modules. Let G be a lcsc group. By a topological (or continuous) G-module we
shall mean a (Hausdorff) topological vector space E over either R or C together with an action
of G by invertible linear maps such that the action map G × E → E is continuous. Note
that when E is a Hilbert space this, a priori quite strong continuity requirement, coincides
with the more familiar notion of a strongly continuous G-action [8, Lemme D8]. A morphism
of topological G-modules is a morphism of the underlying topological vector spaces which
intertwines their respective G-actions.

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

3

Extended natural numbers. We denote by N0 the set N ∪ {0}. We furthermore denote the set
{−∞} ∪ N0 by Z∗ and define

x + y,
x, y ∈ N0 ⊆ Z∗
x+̇y :=
−∞, if either of x, y = −∞

x − y,
x > y ∈ N0 ⊆ Z∗
x−̇y :=
−∞, x = −∞ or x < y ∈ N0
We leave x−̇y undefined if both x, y = −∞.
Multi-index notation. A multi-index over a finite set I (which will always be assumed endowed
with a fixed well-ordering) is an element d = (di )i∈I ∈ NI0 . For a Mal’cev group G (see Section
4 for the definition of Mal’cev groups; in particular this includes connected, simply connected
cl(G)
Lie groups) we denote by cl(G) the multi-index (1, . . . , cl(G)) ∈ N0 , by rk(G) the multicl(G)
index (dimR g[i] /g[i+1] )i=1,...,cl(G) ∈ N0 . For any multi-index k over I we denote by Nk0
the disjoint union ∪˙ i∈I Nk0i . Finally we denote by dim(G) the multi-index (where we write
m := rk(G))
dim(G) := ((1)j=1,...,m1 , . . . , (cl(G))j=1,...,mcl(G) ) ∈ Nm
0 .
For any d ∈ N0 and any multi-index k over I we define
(
)
X
I
Dd,k := d ∈ N0 |
ki di 6 d ,
i∈I

and denote by

D=
d,k

the subset for which equality holds. Finally, we set
Bk := ∪˙ i∈I {1, 2, . . . , ki },

and consider on this set the lexicographic ordering.
Product
Q notation. For a group G, a finite, well-ordered set I and a map
Q I → G : i 7→ gi we
write i∈I gi for the element g in G defined recursively by g = gi0 · i∈I\{i0 } gi , where i0 is
the
Q smallest element in I. In other words, if I = {i1 , . . . , in } with i1 < i2 < · · · < in then
i∈I gi := gi1 gi2 · · · gin .
3. Polynomial cohomology of lcsc groups

In this section we recall the definition of continuous cohomology for locally compact groups,
and define, more generally, a notion of polynomial cohomology, for which the “linear” (or degree
one) case coincides with the usual cohomology.
Definition 3.1 (strengthened morphism). A morphism v : E → F between topological Gmodules is said to be strengthened if there exists a morphism of topological vector spaces
η : F → E such that v ◦ η ◦ v = v.
We emphasize that it is not part of the definition that the map η be G-equivariant. The
definition of a strengthened morphism just given might not be completely standard, but is
easily seen to be equivalent with the one used e.g. in [8, Chapter III & Appendix D]; in
particular, for injective morphisms, being strengthened is the same as being left invertible in
the category of topological vector spaces.

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DAVID KYED AND HENRIK DENSING PETERSEN

Definition 3.2 (relative injectivity). A continuous G-module E is said to be relatively injective
if, given any diagram
/ F1 u / F2
0
⑤
v

 ~⑤

⑤
⑤

∃w

E
where u : F1 → F2 is a strengthened injective morphism, there exists a morphism w : F2 → E
such that the augmented diagram commutes.
For G a lcsc group and X a locally compact space on which G acts continuously by homeomorphisms, the space of continuous functions C(X, E) is a continuous G-module for every
continuous G-module E, when endowed the standard action (g.f )(x) = g.f (g −1 .x). Recall
that the topology on C(X, E) is the projective topology generated by the restriction maps
C(X, E) → C(K, E) over all compact subsets K of X, that is, the topology of uniform convergence on compact sets. In particular, note that if X is second countable and E is a Fréchet
space (as will often be the case in this paper), then C(X, E) is a Fréchet space as well.
Lemma 3.3 ([8, Chapter III]). Let G be a lcsc group and E be a continuous G-module. Then
C(G, E) is relatively injective. In particular, the category of continuous G-modules contains
sufficiently many relatively injectives. Further, there is a strengthened injective resolution
0

/E

ε

/ C(G, E)

d0

/ C(G × G, E)

d1

/ ···

dn−1 /

C(Gn+1 , E)

dn

/ ··· .

where the coboundary maps are given by ε(ξ)(g) := ξ and
dn (f )(g0 , . . . , gn+1 ) :=

n+1
X

(−1)i f (g0 , . . . , ĝi , . . . , gn+1 ).

(3.1)

i=0

Here, as usual, the symbol ĝi denotes omission of the element gi .
Lemma 3.4 ([8, Chapter III]). Let G be a lcsc group and let E, F be continuous G-modules.
Then for any morphism ϕ : E → F and any two relatively injective strengthened resolutions
/E
/ (E• ) and 0
/F
/ (F• ) there is a lift ϕ• : E• → F• to a (continuous)
0
morphism of complexes of G-modules, which is unique up to G-equivariant continuous cochain
homotopy.
Definition 3.5 (differential notation and higher order invariants). Let G be a lcsc group and
E be a continuous G-module. For g ∈ G, we denote by ∂ g : E → E the continuous, linear map
ξ 7→ g.ξ − ξ, and for d ∈ N we define the d’th order invariants in E as
EG(d) := {ξ ∈ E | ∀g1 , . . . , gd ∈ G : ∂ g1 ◦ · · · ◦ ∂ gd (ξ) = 0}.
Note that EG(d) is the pre-image under the quotient map E → E/EG(d−1) of the subspace
G
E/EG(d−1) — an observation we will be using frequently (without reference) in the sequel.
The relation (g1 − 1) · · · (gd − 1)(gx) = g(g −1 g1 g − 1) · · · (g−1 gd g − 1)x shows that EG(d)
is a (closed) G-invariant subspace in E and hence (−)G(d) defines an endo-functor on the
category of topological G-modules, which recovers the classical invariants functor when d = 1.
Furthermore it is easy to see that (−)G(d) is left exact, and thus has well defined right-derived
functors, and these are the object of study in this section. We spell out this construction by
means of the following:

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

5

Definition 3.6 (continuous polynomial cohomology). Let G be a lcsc group and let E be a
continuous G-module. For d ∈ N, we define the d-th order continuous polynomial cohomology
of G with coefficients in E as


ker dn |EG(d)
n

 , n ∈ N0 ,
Hn(d) (G, E) :=
n−1
im d
|EG(d)
n−1

where 0

/E

/ (E• , d• ) is any strengthened, relatively injective resolution of E. The

space ker(dn |EG(d) ) is denoted by Z1(d) (G, E) and referred to as the space of homogeneous (degree
n

d) polynomial n-cocycles and the space im(dn−1 |EG(d) ) is denoted by B1(d) (G, E) and referred
n−1

to as the space of homogeneous (degree d) polynomial n-coboundaries.
By standard arguments [8], using the left-exactness of (−)G(d) and Lemma 3.4, the polynomial cohomology Hn(d) (G, E) is indeed well defined as a (generally non-Hausdorff) topological
vector space; that is, using different relatively injective resolutions to compute Hn(d) (G, E) yields
bijective, bi-continuous, linear maps between the resulting polynomial cohomology spaces.
Remark 3.7. A direct computation shows that, given any topological G-module E and any
ξ ∈ EG(2) , the map g 7→ ∂ g .ξ is a continuous homomorphism from G to E. Thus, if G
has compact abelianization we conclude that EG(2) = EG for every topological G-module
E, and inductively that EG(d) = EG for all d. Hence, for such G, the continuous polynomial
cohomology coincides with the ordinary continuous cohomology in the sense that Hn(d) (G, E) =
Hn (G, E) for all d ∈ N.
4. Nilpotent groups and their cohomology
In this section we collect the necessary prerequisites concerning nilpotent groups; for general
background we refer to [6, 11]. Let G be a group and recall that the commutator of two
elements g, h ∈ G is defined as the element [g, h] := g −1 h−1 gh ∈ G. For subgroups H, K ⊆ G
the commutator [H, K] is the subgroup generated by all elements [h, k] with h ∈ H and k ∈ K.
Recall that a central series G = (Gi )i∈N in a (topological) group G is a decreasing sequence
of (closed) normal subgroups Gi E G, with G = G1 and such that [Gi , G] ⊆ Gi+1 for all i ∈ N.
The lower central series of a topological group G is the (decreasing) sequence Gmin = (G[i] )i∈N
of subgroups of G defined recursively by G[1] := G and G[i+1] := [G, G[i] ]. Observe that each
G[i] is a characteristic subgroup of G; i.e. globally preserved by any automorphism of G. Further, for any central series G = (Gi )i∈N in G we have, by construction, G[i] 6 Gi and moreover
one may prove that [G[i] , G[j] ] 6 G[i+j] for all i, j ∈ N (see e.g. [3, Corollary 0.31]). A group
G is called nilpotent if G[d] = {1} for some d ∈ N; in this case the (nilpotency) class of G is
defined as the number cl(G) := min{d | G[d] = {1}}− 1. Note that in the special case where G
is a connected, simply connected, nilpotent Lie group, the algebraically defined lower central
series automatically consists of closed subgroups [6, Section 1.2]. In the remainder of this
section we recall some fundamental results about nilpotent (torsion-free) groups and Lie algebras, and introduce the class of groups that will be our primary object of study in the sequel.
Recall that for any Lie algebra g, the lower central series is defined (analogously to the
definition for groups) by g[i+1] = spanR [g, g[i] ]. Let G be a connected, simply connected

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DAVID KYED AND HENRIK DENSING PETERSEN

(henceforth abbreviated ‘csc’), nilpotent Lie group, and denote its Lie algebra g. Then for
each i, one has that G[i] is a Lie sub-group of G with Lie algebra g[i] . Moreover, for such
G, the exponential map exp : g → G is a global diffeomorphism onto G, and it therefore also
induces a diffeomorphism g[i] /g[i+1] → G[i] /G[i+1] for each each i. A (strong) Mal’cev basis
for g (with respect to the lower central series) is a linear basis (Xi,j )(i,j)∈Brk(G) of g, such that
for each i, Xi,j ∈ g[i] for all j, and the set {Xi,j }j projects to a linear basis of g[i] /g[i+1] (see
Section 2 for a definition of the multiindex Brk(G) ). Such a basis always exists [6, Section 1.2]
and once a Mal’cev basis is chosen, the map
Y
X
exp(ti,j Xi,j ) ∈ G,
(4.1)
ti,j Xi,j 7−→
g∋
(i,j)∈Brk(G)

(i,j)∈Brk(G)

is a diffeomorphism as well [6, Section 1.2], and the induced global coordinate system on G
is called (the system of) Mal’cev coordinates relative to the chosen Mal’cev basis. Abusing
terminology slightly, we will therefore also refer to the family {gi,j := exp(Xi,j ) | (i, j) ∈
t so that each element i g ∈ G can
Brk(G) } as a Mal’cev basis of G and denote exp(tXij ) by gij
be uniquely written as
Y
t
giji,j ,
tij ∈ R.
g=
(i,j)∈Brk(G)

Here, and above, we use the ordered product notation introduced in the Section 2. Given any
Mal’cev basis (Xi,j )(i,j)∈Brk(G) of g, for all (i, j), (s, t) ∈ Brk(G) there are constants ci,j,s,t
∈R
k,l
such that
X
ci,j,s,t
[Xi,j , Xs,t ] =
k,l Xk,l .
(k,l)∈Brk(G)

The (indexed) collection of all the constants ci,j,s,t
is called the set of structure constants
k,l
of g (with respect to the chosen basis), and in his groundbreaking paper [14], Mal’cev proved
the following result:

Theorem 4.1 (Mal’cev). A csc nilpotent Lie group G has a lattice if and only if it has a
Mal’cev basis with rational structure constants. Furthermore, every lattice Γ in G is cocompact
and there exists a Mal’cev basis (Xij )(i,j)∈Brk(G) which is based in Γ, in the sense that




Y
m
Γ=
gij ij mij ∈ Z .


(i,j)∈Brk(G)

Any lattice in a csc nilpotent Lie group is necessarily torsion-free and finitely generated,
and Mal’cev also proved that the converse is true:

Theorem 4.2 (Mal’cev). Let Γ be a finitely generated, torsion-free (discrete) nilpotent group.
Then there exists a csc nilpotent Lie group G such that Γ embeds as a lattice in G. Furthermore,
the embedding is unique up to natural isomorphism; that is, given any two such embeddings
i : Γ → G and j : Γ → H, there is an isomorphism ψ : G → H intertwining i and j.
The csc nilpotent Lie group G in the theorem is called the Mal’cev completion of Γ, and is
occasionally denoted Γ ⊗ R. For a proof of the theorem see [14], or for an alternative approach
[3] (which, in turn, is based on [9]); we will indicate an approach below as well. Finally,
we note that any locally compact, compactly generated, totally disconnected nilpotent group
contains a neighbourhood basis of the identity consisting of compact open normal subgroups

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

7

[18]. Thus, up to quotienting out a totally disconnected compact group, such groups can
be studied via discrete nilpotent groups. The techniques developed in this paper, based on
cohomology with coefficients in vector spaces, essentially do not “see” compact subgroups; this
motivates the following definition:
Definition 4.3 (Mal’cev group). Let G be a locally compact, compactly generated topological
group. We will say that G is a Mal’cev group if it satisfies either of the following two equivalent
criteria:
(i) G embeds as a closed, cocompact subgroup in a csc nilpotent Lie group,
(ii) G is a torsion free, nilpotent Lie group.
If G is a Mal’cev group then the csc nilpotent Lie group into which it embeds cocompactly
is uniquely determined up to isomorphism and is, in analogy with the discrete case, called
the (real) Mal’cev completion of G and denoted G ⊗ R. If G is a Mal’cev group then Gi :=
G ∩ (G ⊗ R)[i] defines a central series in G which we will refer to as the Mal’cev central series.
Remark 4.4. Since csc nilpotent Lie groups are torsion free, the implication from (i) to (ii)
is clear and the fact that (ii) implies (i) is due to Mal’cev in the case when G is discrete and
Wang in general (see comments right before Proposition 4.6 in [17]).
We will need the following additional facts about the class of Mal’cev groups.
(1) Reasoning exactly as in the discrete case (see e.g. [6, Chapter 5]) one may prove that
any Mal’cev group G admits a Mal’cev basis for G ⊗ R (strongly) based in G; that is,
there exists a Mal’cev basis (Xij )(i,j)∈Brk(G⊗R) such that




Y
G=
exp(mij Xij ) mij ∈ Zij ,


(i,j)∈Brk(G⊗R)

where the sets Zij ⊂ R are equal to either Z or R. Abusing notation slightly, we will
refer to the elements gij := exp(Xij ) as a Mal’cev basis for G.
(2) By fixing a Mal’cev basis for G we also obtain isomorphisms of abelian groups
′

i
Gi /Gi+1 ≃ ⊕nj=1
Zij ≃ Zmi ⊕ Rmi ,

where mi , m′i ∈ N0 sum up to ni := dim(g[i] /g[i+1] ); here g denotes the Lie algebra of
G ⊗ R and (Gi )i is the Mal’cev central series defined above.
(3) Since a Mal’cev group G is nilpotent, it always has non-trivial center, and upon choosing a Mal’cev basis for G, one can always find a central subgroup Z such that Z is of
m
the form Z := {gcl(G),j
| m ∈ Zcl(G),j }, where Zcl(G),j is either Z or R and such that
G/Z is again a Mal’cev group with Mal’cev basis (ḡij )(i,j)6=(cl(G),j) . In particular, we
get a natural, continuous cross section σ : G/Z → G of the quotient homomorphism
by setting
Y
Y
t
t
σ:
ḡijij 7−→
gijij
(i,j)6=(cl(G),j)

(i,j)6=(cl(G),j)

This will be of importance in the sequel, as it is a necessary requirement for using the
Hochschild-Serre spectral sequence in group cohomology [8, III, no 5.1].

Definition 4.5 (length and rank). Let G be a Mal’cev group. We denote the length of
the (Mal’cev-, equivalently lower-) central series by cl(G).We denote by rk(G) the rank of

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DAVID KYED AND HENRIK DENSING PETERSEN

G, defined by rk(G) := dimR g/g[2] , where g is the Lie algebra of G ⊗ R. That is, we have
′
G1 /G2 ∼
= Rm1 × Zm1 for some uniquely determined m1 , m′1 ∈ N0 and rk(G) = m1 + m′1 .
For n, d ∈ N, we denote by Fn (d) the free nilpotent group of class d on n generators. That
is, denoting by Fn the free group on n generators f1 , . . . , fn , we set Fn (d) := Fn /(Fn )[d+1] .
This group satisfies the following universal condition: given any nilpotent group Γ of class at
most d and any g1 , . . . , gn ∈ Γ, there is a unique group homomorphism ϕ : Fn (d) → Γ such
that ϕ(fi ) = gi , for i = 1, . . . , n. Similarly, we denote by fn (d) the free nilpotent Lie algebra of
class d on n generators Xi , i = 1, . . . , n, and by Nn (d) the associated csc (nilpotent) Lie group.
Since the structure constants of fn (d) with respect to a canonical Mal’cev basis consisting of (a
subset of all of the) commutators of the Xi are rational, such a basis determines a cocompact
lattice in Nn (d) by Mal’cev’s theorem, and this is precisely the free nilpotent group Fn (d).
In particular, using this we can now indicate a construction of the Mal’cev completion Γ ⊗ R
of any finitely generated, torsion-free nilpotent group Γ. Let F be a free nilpotent group
surjecting onto Γ, say ϕ : F → Γ, and let N be the associated free nilpotent Lie group. (Note
in particular that, by the above, we have a priori N = F ⊗R.) Let K := ker ϕ. By [6, Theorem
5.4.3] there is a unique smallest Lie group L 6 N containing K, and K is cocompact in L.
(Note, again, that thus we have in fact L = K ⊗ R.) The cocompactness ensures that, since
Γ is torsion-free, L ∩ F = K, and it follows from this that Γ is a cocompact lattice in the csc
nilpotent Lie group N/L.
4.1. Cohomology of Mal’cev groups. In this section we gather the results needed about
the cohomological properties of Mal’cev groups, which turn out, not surprisingly, to be very
much alike those for csc nilpotent Lie groups.
Definition 4.6. We say that a lcsc group G is cohomologically finite dimensional if Hn (G, E)
is finite dimensional for every n ∈ N whenever E is a finite dimensional continuous G-module.
Note that any finitely generated discrete group whose classifying space is a finite dimensional
CW complex is an example of a cohomologically finite dimensional group. So are connected Lie
groups (e.g. by the van Est theorem [8, III, Corollaire 7.2]), and, as the following proposition
shows, so are Mal’cev groups.
Proposition 4.7. Mal’cev groups are cohomologically finite dimensional.
Proof. Let G be a Mal’cev group and let G̃ be the csc, nilpotent Lie group in which G is
cocompact. First note that the cohomology of G stops after degree d := dim(G̃); indeed, for
a Fréchet G-module E, the Shapiro lemma [8, III, Proposition 4.1] gives
Hn (G, E) ≃ Hn (G̃, Ind(E)),
and for n > d the right hand side vanishes (for instance by the van Est theorem [8, III,
Corollaire 7.2]). Let V be a continuous, finite dimensional G-module; we prove the statement
by induction on d = dim(G̃). In the case d = 1, we have G ≃ R or G ≃ Z and both of these
are cohomologically finite dimensional. For the inductive step, let G be a Mal’cev group with
d-dimensional ambient Lie group and take a central subgroup Z 6 G, isomorphic to either Z
or R, such that G/Z is again a Mal’cev group whose Mal’cev completion has dimension d − 1.
Then, as we just saw, Z is cohomologically finite dimensional and thus Hq (Z, V ) is finite
dimensional and, in particular, Hausdorff [8, III, Proposition 3.1], so the Hochschild-Serre
spectral sequence exists [8, III, no 5.1] and has E2 -term
E2pq = Hp (G/Z, Hq (Z, V )).

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9

So E2pq = 0 whenever q > 2 or p > d − 1 and all non-vanishing terms are finite dimensional
by the induction hypothesis; thus also Hn (G, V ) is finite dimensional.

We now recall some well-known results concerning the continuous cohomology of nilpotent
groups [2, 7, 16]. In order to have the standard operator theoretic tools at our disposal and
to comply with the standing assumption in [8] that vector spaces are complex, in the rest of
this section the terminology ‘H is a unitary Hilbert G-module’ will mean that H is a complex
Hilbert space with a continuous, unitary G-action. This, however, is not a serious restriction
since in all our applications we will be able to pass from the setting of real topological vector
spaces to the complex ditto via the standard complexification procedure since Hn(d) (G, E ⊗R
C) ∼
= Hn(d) (G, E) ⊗R C.
Definition 4.8 (Property HT [16]). Let G be a lcsc group and let M be a subclass
of the class of continuous G-modules. Given I ⊆ N we say that G has property HT (I) with
respect to M , if for every G-module H in M with HG = 0 and for every n ∈ I we have
Hn (G, H) = 0.
When M is the class of continuous unitary Hilbert G-modules, we will often suppress the
explicit reference to M and simply say that G has property HT (I), or even just property HT
if I = N and no confusion can arise.
Next we discuss a well known theorem, essentially due to Delorme, concerning the vanishing
of cohomology for nilpotent (Lie) groups. The classical form of the statement is the following:
Theorem 4.9 ([4, Theorem 10.1]). Let G be a csc nilpotent Lie group. For every irreducible,
continuous unitary Hilbert G-module H such that HG = 0, we have Hn (G, H) = 0 for all
n ∈ N0 . In particular G has property HT (N).
Note that the latter statement in the theorem above does indeed follow from the former,
since any unitary representation is a direct integral of irreducible representations, and property
HT therefore follows from [4, Theorem 7.2].
It will be convenient for us to have the following alternate form of Theorem 4.9, which at
the same time generalizes the statement to the class of Mal’cev groups.
Theorem 4.10. If G is a Mal’cev group and H is a unitary G-module, then there exists
an increasing sequence (Hi )i∈N of closed, G-invariant subspaces of H with dense union, such
that HG ⊂ Hi for each i ∈ N and such that the inclusion map induces an isomorphism
Hn (G, HG ) ≃ Hn (G, Hi ) for each n ∈ N.
Note that it is part of the conclusion that Hn (G, Hi ) is Hausdorff if HG is finite dimensional,
since, in this case, Hn (G, HG ) is finite dimensional (and thus Hausdorff [8, III, Proposition
3.1]) by Proposition 4.7. For the proof of Theorem 4.10 the following lemma is convenient.
Lemma 4.11. Let G be lcsc group, H be a unitary G-module and assume that there exists a
central element z ∈ G such that the corresponding unitary u ∈ B(H) satisfies that T := u − 1
is injective. Then there exists an increasing sequence (Hi )i∈N of closed, G-invariant subspaces
with dense union and such that Hn (G, Hi ) = 0 for each i ∈ N and n ∈ N0 .
Proof. Denote the representation by π. Since u := π(z) is unitary, the operator T := u − 1
is normal and hence the Borel functional calculus may be applied to T . As z is central,
T commutes with π(G) and hence so do its spectral projections. Because T is assumed to

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DAVID KYED AND HENRIK DENSING PETERSEN

be injective, its spectral projections En := χσ(T )\{z||z|61/n} (T ) are increasing an converging
strongly to 1, and since each Pn commutes with π(G), its range Hn := Pn (H) is a closed
G-invariant subspace; we denote the restricted representation of G on Hn by πn . Since u is
unitary, the injectivity of T implies that its range is dense and from this it follows that the
operator πn (z) − 1, which is simply T |Hn , has dense range as well. The operator πn (z) − 1 is
furthermore bounded away from zero, and thus invertible on Hn , and by [8, III, Proposition
3.1] this implies that Hk (G, Hn ) vanishes for each k ∈ N0 and n ∈ N.

We are now ready to give the proof of Theorem 4.10. In the proof we will several times use
the fact that for a unitary Z-module H, having HZ = {0} is equivalent with u1 − 1 acting
injectively (here u1 is the unitary corresponding to 1 ∈ Z), a fact that not true for unitary Rmodules, which accounts for the distinction between discrete and continuous one dimensional
subgroups present in the proof.
Proof. By splitting H as H = HG ⊕ HG⊥ , it suffices to treat the case where HG = {0}.
Denote by G̃ the csc, nilpotent Lie group in which G embeds cocompactly; we now prove the
statement by induction on d := dim(G̃). For d = 1, the group G is isomorphic to either R or
Z, and in the latter case the statement follows directly from Lemma 4.11. If G ≃ R, consider
the subgroup Z corresponding to Z 6 R and split H as HZ ⊕ HZ⊥ . Since G is abelian,
this is a splitting of H as a unitary G-module and by Lemma 4.11 we now get increasing,
closed, G-invariant subspaces Ki 6 HZ⊥ with dense union an vanishing cohomology. Put
Hi := HZ ⊕ Ki . Then we have Hn (G, Hi ) = Hn (G, HZ ), so our task is to prove that the later
vanishes in all degrees. To this end, note that
(
HZ if q = 0, 1
Hq (Z, HZ ) =
{0} otherwise
and, in particular, Hq (Z, HZ ) is Hausdorff for all q ∈ N0 . The Hochschild-Serre spectral
sequence therefor exists [8, III, no 5.1] and has E2 -term
(
Hp (G/Z, HZ ) if q = 0, 1
pq
E2 = Hp (G/Z, Hq (Z, HZ )) =
{0}
otherwise
However, since G/Z ≃ S 1 is compact, we have Hp (G/Z, HZ ) = {0} when p > 0 [8, III,
Corollaire 2.1] and in degree zero we have

G/Z
= HG = {0}.
H0 G/Z, HZ = HZ
Thus, E2pq = {0} for all p, q > 0 and hence Hn (G, HZ ) = {0}, as claimed.
For the inductive step, let G be a Mal’cev group with dim(G̃) = d and choose central
subgroups Z 6 Z ′ 6 G such that Z ≃ Z, K := Z ′ /Z is compact and G/Z ′ is again a Mal’cev
group whose ambient csc nilpotent Lie group has dimension d − 1; this is always possible since
Gcl(G) ≃ Zk ⊕ Rl for some k, l ∈ N0 so we have that K is either trivial or S 1 . Now decompose
H = HZ ⊕ HZ⊥ and note that the decomposition respects the G-action since Z is central.
For the restricted action G y HZ⊥ we have, by construction, a central element such that
the corresponding unitary acts without fixed points, so by Lemma 4.11 we get a sequence of
closed G-equivariant subspaces Ki ⊂ HZ⊥ with dense union and such that Hn (G, Ki ) = {0}
for all n ∈ N0 and i ∈ N. Next split HZ = (HZ )K ⊕ (HZ )K⊥ , and since K is central in G/Z
this decomposition respects the natural G/Z-action. On (HZ )K we get an induced action of

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

11

(G/Z)/K = G/Z ′ without non-trivial fixed points, so the induction takes over and provides
us with an increasing family of closed G/Z ′ -invariant subspaces subspaces Li 6 (HZ )K for
which Hn (G/Z ′ , Li ) = {0} for all i ∈ N and n ∈ N0 . We now define
K⊥
K
L⊥
Hi := Li ⊕ HZ
⊕ Ki 6 H Z
⊕ HZ
⊕ HZ = H,

and note that, as Hn (G, Ki ) = {0}, we have to show that Hn (G, Li ⊕ (HZ )K⊥ ) = {0} for all
i ∈ N and n ∈ N0 . As in the case d = 1, this can be deduced by a spectral sequence argument:
Since Li ⊕ (HZ )K⊥ 6 HZ we have
(


Li ⊕ (HZ )K⊥ if q = 0, 1
Hq Z, Li ⊕ (HZ )K⊥ =
{0}
otherwise

so the Hochschild-Serre spectral sequence exists [8, III, no 5.1] and has E2 -term
(




Hp G/Z, Li ⊕ (HZ )K⊥ if q = 0, 1
pq
q
Z K⊥
p
=
E2 = H G/Z, H Z, Li ⊕ (H )
{0}
otherwise

Since K 6 G/Z is central and compact another application of the Hochschild-Serre spectral
sequence (similar to the one carried out above in the case d = 1) yields that




K 

Hp G/Z, Li ⊕ (HZ )K⊥ ≃ Hp (G/Z)/K, Li ⊕ (HZ )K⊥
= Hp G/Z ′ , Li = {0}.

Thus E2pq = {0} for all p, q > 0 and we conclude that Hn (G, Hi ) = {0} for all n ∈ N0 and
i ∈ N, as desired.

Corollary 4.12. Mal’cev groups have property HT (N).

Proof. Let G be a Mal’cev group and let H be a unitary Hilbert G-module without fixed points,
and choose, according to Theorem 4.10, an increasing sequence Hi 6 H of closed, G-invariant
subspaces with vanishing cohomology and dense union. Denote by Pi the orthogonal projection
onto Hi ; then the sequence (Pi )i∈N converges strongly to 1, and hence the convergence also
holds uniformly (in the Hilbert space norm) on compact subsets of H. Fix an n ∈ N and a
continuous n-cocycle c : Gn → H. Since Hi is G-invariant, the projection Pi commutes with
the G-action, so the map ci := Pi c(−) : Gn → Hi is again a cocycle and hence inner by
the defining properties of Hi . Viewing ci as a sequence of cocycles with values in H, we are
therefore done if we can show that (ci )i converges to c in the standard topology on Zn (G, H)
given by uniform convergence on compact subsets. For a compact set K ⊂ Gn , by continuity
of c the subset c(K) ⊂ H is also compact and thus
sup kc(g) − ci (g)k = sup k(1 − Pi )ξk → 0.

g∈K



ξ∈c(K)

Observe that, in the proof just given, we only used the fact that Mal’cev groups satisfy the
conclusion of Theorem 4.10 to conclude that they have property HT (N), and for the sake of
generality it is convenient to promote this property to a definition:
Definition 4.13. Let G be a lcsc groups, let I ⊆ N be any subset and let M be a class of
topological G-modules. Then G is said to have strong property HT (I) with respect to M if
for any E ∈ M there exists an increasing sequence of closed, G-invariant subspaces with dense
union, for which EG ⊂ Ei for each i ∈ N, and such that the inclusion induces an isomorphism
Hn (G, EG ) ≃ Hn (G, Ei ) for all n ∈ I. As with the ordinary property HT , we will often suppress

12

DAVID KYED AND HENRIK DENSING PETERSEN

the reference to the class M when it is the class of unitary Hilbert G-modules, and when I = N
we will also often suppress the reference to it, and simply say that G has strong property HT .
The following corollary provides a very direct and useful extension of Theorem 4.9.
Corollary 4.14. Let G be cohomologically finite dimensional lcsc group with strong property
HT (N) with respect to the class of unitary Hilbert G-modules. If H is a continuous, unitary
Hilbert G-module with dimR HG < ∞, and F is a continuous, finite dimensional G-module
with F = FG(d) for some d ∈ N, then the natural inclusion map HG ⊗ F → H ⊗ F induces an
isomorphism
Hn (G, HG ⊗ F)

∼
=

/ Hn (G, H ⊗ F) .

Proof. Indeed, denoting (HG )⊥ by K we have the following decomposition (respecting the
topology)
Hn (G, H ⊗ F) = Hn (G, K ⊗ F) ⊕ Hn (G, HG ⊗ F).
and since dim(HG ⊗F) < ∞, Hn (G, HG ⊗F) is also finite dimensional and hence automatically
reduced ([8, III, Proposition 3.1]). Thus, we have to show that Hn (G, K ⊗ F) = 0. Applying
Theorem 4.10, we obtain an increasing sequence (Ki ) of closed, G-invariant subspaces of K
with dense union, and we now prove, by induction on d, that Hn (G, Ki ⊗ F) = {0} for all i.
Indeed, if d = 1 then F = FG , and the action therefore trivial, and since Hn (G, Ki ) = 0 and
Ki ⊗ F is, as a G-module, just a finite direct sum of copies of Ki , we also have Hn (G, Ki ⊗ F).
For the inductive step, consider the short exact sequence
0 → FG(d−1) ⊗ Ki → FG(d) ⊗ Ki → FG(d) /FG(d−1) ⊗ Ki → 0.
The induction hypothesis implies that Hn (G, FG(d−1) ⊗ Ki ) = {0}, and the induced action
on the quotient FG(d) /FG(d−1) is easily seen to trivial so, as in the case d = 1, we also get
Hn (G, FG(d) /FG(d−1) ⊗ Ki ) = {0}. Since F is assumed to be equal to FG(d) , the long exact
sequence in cohomology now shows that also Hn (G, F ⊗ Ki ) = {0}. Finally, since K ⊗ F
admits a continuous G-equivariant projection PKi ⊗ 1 onto Ki ⊗ F for all i, and we have that
PKi ⊗ 1 converges strongly to 1 ⊗ 1, we conclude, as in the proof of Corollary 4.12, that
Hn (G, K ⊗ F) = 0.

5. Polynomial cohomology in terms of ordinary cohomology
The main aim of this section is the following proposition which gives a description of polynomial cohomology in terms of ordinary cohomology. In the statement we write Pold−1 (G) for
the space C(G, R)G(d) , where the G(d)-invariants are taken with respect to the right-regular
representation; see Section 6 below, for an explanation of this terminology.
Proposition 5.1. Let G be a lcsc group and let d ∈ N. Then:
∼

(i) There are isomorphisms τ • : H• (G, Pol d−1 (G)) −→ H•(d) (G, R), given on continuous
cochains by
(τ n ξ)(g0 , . . . , gn ) = ξ(g0 , . . . , gn )(1),
and with inverse defined, also at the level of continuous cochains, by
(τ n )−1 (ξ)(g0 , . . . , gn )(t) = ξ(t−1 g0 , . . . , t−1 gn ).

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

13

(ii) More generally, let G1 = G2 = G and let E be a continuous G-module. Considering
C(G, E) as a G1 × G2 -module with the action (g1 , g2 ).f (g) := g1 .f (g1−1 gg2 ) there is
an isomorphism
 ∼

χ• : H• G1 , C(G, E)G2 (d) −→ H•(d) G2 , C(G, E)G1 = H•(d) (G, E).
(5.1)

Proof. We first prove (ii). For the sake of clarity, denote by X a third copy of G and write
the coefficient module as C(X, E). Now define two complexes (C n , dnC )n∈N0 and (D n , dnD )n∈N0
of G1 × G2 -modules as follows:
C n := C(Gn+1
, C(X, E)) and D n := C(Gn+1
, C(X, E)),
1
2
with G1 × G2 actions
((g, h).f )(g0 , . . . , gn )(x) := g(f (g −1 g0 , . . . , g−1 gn )(g −1 xh)), f ∈ C n ;
((g, h).f )(h0 , . . . , hn )(x) := g(f (h−1 h0 , . . . , h−1 hn )(g −1 xh)), f ∈ D n .
In both cases, the coboundary maps are the standard inhomogeneous ones; i.e.
dnC (f )(g0 , . . . , gn+1 )

=

n+1
X

f (g0 , . . . , ĝi , . . . , gn+1 ),

i=0

and similarly for dnD . Augmenting (C • , d•C ) with εC : C(X, E) → C 0 given by εC (ξ)(g) := ξ
and similarly for (D • , d•D ) we obtain two complexes of G1 × G2 -modules:
0 → C(X, E) → (C • , d•C )
0 → C(X, E) → (D

•

, d•D )

(5.2)
(5.3)

Considering C n as a G2 -module, the action is only from the right in the X-variable, so


n+1
n G2 (d)
G2 (d)
.
(C )
= C G1 , C(X, E)

Thus, as a complex of G1 -modules, ((C n )G2 (d) , dnC |) is exactly the standard, relative injective
resolution of the G1 -module C(X, E)G2 (d) , and we have therefore proved:
Claim 1. Upon passing to G1 -invariants and cohomology, the complex ((C n )G2 (d) , dnC |) computes Hn (G1 , C(X, E)G2 (d) ).
Similarly, passing to G1 -invariants in the (5.3) we see that
(D n )G1 = C(Gn+1
, C(X, E)G1 ),
2
and hence, as a complex of G2 -modules, ((D • )G , d•D |) is the standard, relatively injective
resolution of the G2 -module C(X, E)G1 . The latter G2 -module identifies with E (as a Gmodule) via the map α : E → C(X, E)G1 given by α(ξ)(x) := xξ. This proves:
Claim 2. Upon passing to G2 (d)-invariants and cohomology, the complex ((D n )G1 , dnD |) computes Hn(d) (G, E).
Lastly we want to relate the two complexes; this is done by means of
Claim 3. The map κn : C n → D n given by κn (f )(h0 , . . . , hn )(x) := f (xh0 , . . . , xhn ) is an
isomorphism of G1 × G2 -complexes.

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DAVID KYED AND HENRIK DENSING PETERSEN

To see this, we first note that a direct computation shows that κ• is indeed a map of complexes commuting with the G1 × G2 -actions, and that the map (κn )−1 : D n → C n given by
(κn )−1 (f )(g0 , . . . , gn )(x) = f (x−1 g0 , . . . , x−1 gn )(x) is its inverse. Thus, by composition we get
an isomorphism:


G1


• G2 (d)
•
n
G2 (d)
n
n
(C )
, dC |
(Claim 1)
=H
τ : H G1 , C(X, E)


 G1
κ• n
• G2 (d)
•
≃H
(D )
, dD |
(Claim 3)


G2 (d) •
, dD |
= Hn (C • )G1
= Hn(d) (G, E).

(Claim 2)

This proves (ii), and to obtain (i) we simply put E = R and note that for an inhomogeneous
cochain
ξ : Gn+1 → Pold−1 (G) = C(X, R)G2 (d)
the class τ n ([ξ]) is represented by the cocycle
ev1 ◦κ(ξ)(h0 , . . . , hn ) = ξ(h0 , . . . , hn )(1).
Conversely, for an inhomogeneous polynomial cocycle ξ : Gn+1 → R = C(X, R)G2 the image
under (τ n )−1 is represented by the inhomogeneous cocycle (κn )−1 (ξ) ∈ C(Gn+1
, C(X, R)G2 (d) ) =
1
n+1
n
−1
−1
−1
C(G
, Pold−1 (G)) given by (κ ) (ξ)(g0 , . . . , gn )(x) = ξ(x g0 , . . . , x gn )(x).

6. Polynomial maps on groups
In this section we study the space of polynomials on a group (already ad hoc introduced in
the previous section) in greater detail. The main result is that H1(d) (G, R) can be described
directly in terms of the polynomials on G, but before restricting attention to the case of trivial
coefficients, we first prove at few preparatory results of general nature, with the aim of providing a simpler picture of the space of polynomial 1-cocycles, corresponding to the description
of classical cohomology in terms of inhomogeneous cocycles.

6.1. Inhomogeneous polynomial 1-cocycles. Let G be a lcsc group, E a continuous Gmodule, and consider the standard relatively injective resolution introduced in Section 3.
0

/E

d−1

/ C(G, E)

d0

/ C(G2 , E)

d1

/ C(G3 , E)

/ ··· .

For functions ξ : G → E we define the unitized difference operator (see also Definition 3.5 for
notation) by
(∂¯ g ξ)(h) := (∂ g ξ)(h) − (∂ g ξ)(1).
Lemma 6.1. For ξ ∈ C(G, E) and d ∈ N the following are equivalent:
(i) For all g1 , . . . , gd one has ∂¯g1 ◦ · · · ◦ ∂¯gd ξ = 0
(ii) For all g1 , . . . , gd one has that ∂g1 ◦ · · · ◦ ∂gd ξ is a constant function into E.

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

15

Proof. For d = 1, we have ∂¯g1 ξ := ∂g1 ξ − ∂g1 ξ(1), so if ∂¯g1 ξ = 0 then clearly ∂g1 ξ is constant,
and, conversely, if ∂g1 ξ is constant then it equals ∂g1 ξ(1), so ∂¯g1 (ξ) = 0. For the general case,
one first observes that
∂g1 ◦ · · · ◦ ∂gd (ξ) = ∂g1 ◦ · · · ◦ ∂gd−1 (∂gd (ξ) − ∂gd (ξ)(1) + ∂gd (ξ)(1))
= ∂g1 ◦ · · · ◦ ∂gd−1 (∂¯gd (ξ)) + ∂g1 ◦ · · · ◦ ∂gd−1 (∂gd (ξ)(1))
{z
}
|
constant as function into E

and by iterating this argument we see that ∂ g1 ◦ · · · ∂ gn (ξ) and ∂¯ g1 ◦ · · · ◦ ∂¯ gd (ξ) differ by a
constant function. Thus, if ∂¯ g1 ◦· · ·◦∂¯ gd (ξ) = 0 then ∂ g1 ◦ · · · ∂ gn (ξ) is constant and, conversely,
if ∂ g1 ◦ · · · ∂ gn (ξ) then so is ∂¯ g1 ◦ · · · ◦ ∂¯ gd (ξ), and since the latter function is normalized to be
0 at 1, it follows that ∂¯ g1 ◦ · · · ◦ ∂¯ gd (ξ) = 0.

Note that when the G-action on E is trivial, condition (ii) in Lemma 6.1 is equivalent to
ξ ∈ C(G, E)G(d+1) . We will also need a bit of information regarding the kernel of d1 . To this
end, note that d1 (ξ)(s, t, u) := ξ(t, u) − ξ(s, u) + ξ(s, t) so d1 (ξ) = 0 implies
ξ(s, u) = ξ(s, t) + ξ(t, u).

(6.1)

Using this, it easily follows that for ξ ∈ ker(d1 ) ⊂ C(G2 , E) we have
ξ(1, 1) = 0

(6.2)

ξ(g, h) = ξ(g, 1) + ξ(1, h)

(6.3)

ξ(1, g) = −ξ(g, 1)

(6.4)

The following proposition now generalizes the usual description of inhomogeneous 1-cocycles;
¯ = ξ(1, g).
there and in the sequel we will for ξ ∈ C(G2 , E) denote by ξ̄ ∈ C(G, E) the map ξ(g)
Proposition 6.2. The map C(G2 , E) ∋ ξ 7→ ξ¯ ∈ C(G, E) restricts to a continuous bijection
from Z1(d) (G, E) := C(G2 , E)G(d) ∩ ker(d1 ) onto
P := {η ∈ C(G, E) | η(1) = 0 and ∂¯g1 ◦ · · · ◦ ∂¯gd η = 0 for all g1 , . . . , gd ∈ G}.
Proof. Denote the map ξ 7→ ξ¯ by β. Since the topology on C(G2 , E) and C(G, E) is given by
uniform convergence on compact subsets the continuity of β is clear. It is furthermore injective
on ker(d1 ), because if d1 (ξ) = 0 and ξ¯ = 0 then ξ(1, g) = 0 and, by (6.4), also ξ(g, 1) = 0.
Thus, by (6.3), ξ(g, h) = ξ(g, 1) + ξ(1, h) = 0. We now need to prove that β takes values in
the prescribed set. So, let ξ ∈ Z1(d) (G, E) and g1 , . . . , gd ∈ G be given. Since ξ(1, 1) = 0, we
have ξ̄(1) = 0 so we only need to prove that ∂¯g1 ◦ · · · ◦ ∂¯gd ξ = 0. Using the three basic cocycle
properties above, we now get:
∂g ξ(h) := ∂g ξ(1, h) = gξ(g −1 , g−1 h) − ξ(1, h) =


= g ξ(g−1 , 1) + ξ(1, g −1 h) − ξ(1, h)

= gξ(1, g −1 h) − ξ(1, h) − gξ(1, g −1 ) + ξ(1, 1)
¯
= ∂g ξ̄(h) − ∂g ξ(1)
= ∂¯g (ξ̄)(h).

Inductively we therefore get that
∂g1 ◦ · · · ◦ ∂gd (ξ) = ∂¯g1 ◦ · · · ∂¯gd (ξ̄),

(6.5)

16

DAVID KYED AND HENRIK DENSING PETERSEN

for ξ ∈ ker(d1 ). Thus, if ξ ∈ Z1(d) (G, E) then the left hand side of (6.5) vanishes and hence
so does the right hand side; i.e. ξ̄ ∈ P. To prove that β is surjective, let η ∈ P be given
and set η̃(g, h) := η(h) − η(g). Then clearly η̃ ∈ C(G2 , E) and a direct computation shows
that d1 η̃ = 0. Since η(1) = 0, it is furthermore clear that η̃¯ = η so all we have to prove is
that η̃ ∈ C(G, E)G(d) . Since we have already established that η̃ ∈ ker(d1 ) we may use (6.5)
to conclude that ∂g1 ◦ · · · ◦ ∂gd (η̃) = 0. However, as η̃ ∈ ker(d1 ) and ker(d1 ) is a G-invariant
subspace, also ∂g1 ◦ · · · ◦ ∂gd (η̃) ∈ ker(d1 ), and since β is injective on ker(d1 ) we conclude that
∂g1 ◦ · · · ◦ ∂gd (η̃) = 0 as desired.

Example 6.3 (quadratic 1-cocycles). By the Lemma 6.1 and Proposition 6.2, we may describe
the inhomogeneous “quadratic” 1-cocycles ξ : G → E as precisely those unital maps for which,
for all g, h ∈ G, (∂ g ◦ ∂ h )ξ is constant. Computing this we get
(∂ g ◦ ∂ h )(ξ)(k) = gh.ξ((gh)−1 k) − g.ξ(g−1 k) − h.ξ(h−1 k) + ξ(k)
= gh.ξ((gh)−1 ) − g.ξ(g−1 ) − h.ξ(h−1 ),
where the second equality follows by letting k = 1. This can be rewritten as
ξ(ghk) = ξ(gh) + g.ξ(hk) + ghg −1 .ξ(gk) − ghg −1 .ξ(g) − g.ξ(h) − gh.ξ(k),

g, h, k ∈ G. (6.6)

6.2. Polynomial maps and polynomial cohomology with trivial coefficients. In this
section we focus on polynomial 1-cohomology with trivial coefficients; i.e. the case of E = R
with trivial action. In this case, the maps satisfying the conditions in 6.1 are called polynomial
maps on G, see e.g. [13] (the (abstract) notion of polynomial maps on groups seems to go back
at least to [1, 12]). We now formally define the space of polynomial maps:
Definition 6.4 (polynomial maps). Let G be a lcsc group and let ξ ∈ C(G, R) \ {0}. We say
that ξ is a polynomial of degree at most d ∈ N0 if for all g1 , . . . , gd+1 ∈ G we have
(∂ g1 ◦ · · · ◦ ∂ gd+1 )(ξ) = 0,

(6.7)

where C(G, R) is considered a G-module for the left regular action. The degree deg ξ of a
polynomial map ξ is the smallest number d such that ξ satisfies (6.7) for all g1 , . . . , gd+1 ∈ G.
Moreover, the zero-map is formally included in the set of polynomials and assigned the degree
−∞. We denote the set of polynomials of degree at most d by Pold (G) := C(G, R)G(d+1) and
by Pol(G) the set ∪d∈Z∗ Pold (G). Lastly, a polynomial ξ is said to be unital of ξ(1) = 0; we
denote the set of unital polynomials by Pol0 (G) and those of degree at most d by Pol0d (G).
Remark 6.5. We record the following basic facts concerning polynomials:
(1) When G is equal to R (or more generally Rn ) the above definition recovers the classical
notion of polynomials and their degrees; we leave the argument as an exercise.
(2) The set Pol0 (G) consists of the constant functions on G and the set Pol1 (G) consists
of functions of the type ξ = ϕ + r where ϕ : G → R is a (continuous) homomorphism
and r ∈ R is a constant.
(3) When G is compact the only polynomials are the constant functions. For polynomials
of degree 1 this is clear from the description just given, since the image of G under a
continuous homomorphism is a compact, additive subgroup of R and hence equal to
{0}. The general case now follows by induction on the degree.

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

17

In the language just introduced, Lemma 6.1 and Proposition 6.2 simply say, that (with
trivial coefficients) the space of homogeneous 1-cocycles of polynomial degree d in the standard resolution is isomorphic to the space of unital polynomial maps of degree at most d.
The following proposition describes (again for trivial coefficients) the space of polynomial
1-coboundaries:
Proposition 6.6. Let G be a lcsc group and let d ∈ N. Under the map β : ξ 7→ ξ̄ defined above,
the set of polynomial 1-coboundaries B1(d) (G, R) := d0 (C(G, R)G(d) ) maps bijectively onto the
space of (continuous) unital polynomial maps of degree at most d − 1; that is we obtain an
isomorphism
H1(d) (G, R) ∼
= Pold (G)/ Pol d−1 (G).
In particular, H1(d) (G, R) is Hausdorff and the natural G-action on Z1(d) (G, R) induces the
trivial action on H1(d) (G, R).
We remark that all statements in Proposition 6.6 are trivial when d = 1, as H1 (G, R) ≃
Hom(G, R) and the induced is already trivial already at the level of inhomogeneous cocycles
(i.e. G-invariant functions).
Proof. Since B1(d) (G, R) := d0 (C(G, R)G(d) ) is a subset of Z1(d) (G, R), on which we know that
the map β is already injective and takes values in Pol0d (G), we have to prove that β restricted
to the coboundaries takes values in Pold−1 (G, R) and is surjective onto this set. Recall that
d0 (η)(g, h) := η(h) − η(g), so if ξ = d0 (η) for η ∈ C(G, R)G(d) then we have
β(ξ)(g) = β(d0 η)(g) = d0 (η)(1, g) = η(g) − η(1).

(6.8)

Thus,
∂g1 ◦ · · · ∂gd (β(ξ)) = ∂g1 ◦ · · · ∂gd (η − η(1)) = ∂g1 ◦ · · · ∂gd (η) = 0,
and hence deg(β(ξ)) 6 d − 1. On the other hand, given η ∈ Pol0d−1 (G) ⊂ C(G, R)G(d) , the
computation (6.8) shows that β(d0 η) = η, and hence β is surjective from B1(d) (G, R) onto
Pol0d−1 (G). Moreover since β : Z1(d) (G, R) → Pold (G, R) is continuous, it follows from this
that B1(d) (G, R) = β −1 (Pold−1 (G)) is closed in Z1(d) (G, R) since Pold−1 (G) is closed in Pold (G),
and hence that H1(d) (G, R) is Hausdorff. Moreover, β induces an isomorphism of topological
vector spaces
H1(d) (G, R) ≃ Pol0d (G)/Pol 0d−1 (G) ≃ Pold (G)/Pol d−1 (G)
as claimed, where the latter isomorphism is induced by the (split) inclusion ι : Pol0 (G) →
Pol(G). The induced action on the right hand side is trivial, since for ξ ∈ Pold (G) and g ∈ G
we have ∂g ξ ∈ Pold−1 (G). However, ι ◦ β is not quite a G-equivariant map at the level of
cocycles, but we now show that the induced map H1(d) (G, R) → Pold (G)/Pol d−1 (G) is. More
precisely, we show that ι ◦ β(g.f ) − g.(ι ◦ β(f )) differ by a constant map — in particular the
difference is in Pold−1 (G). This follows from (6.4), (6.3) and (6.1), since for f ∈ ker(d1 ) we
have
(ι◦β(g.f )−g.(ι◦β(f ))(x) = f (g −1 , g−1 x)−f (1, g −1 x) = f (g −1 , g−1 x)+f (g −1 x, 1) = f (g −1 , 1).

Corollary 6.7. Let G be a cohomologically finite dimensional lcsc group. Then Pold (G) is
finite dimensional for all d ∈ N0 .

18

DAVID KYED AND HENRIK DENSING PETERSEN

Proof. By induction on d. For d = 0 this is trivial, and for d = 1 we observe that
dimR Pol1 (G) = 1 + dimR H1 (G, R) < ∞.
The inductive step follows from part (i) of Proposition 5.1.



Remark 6.8. We now return to the isomorphism τ 1 : H1 (G, Pol d−1 (G)) → H1(d) (G, R) given
by part (i) of Proposition 5.1, with the aim of providing a more explicit description of this
map in terms of the description of H1(d) (G, R) by means of polynomials on G. For notational
convenience, we denote (τ 1 )−1 by τ ′ . We first describe τ 1 at the level of inhomogeneous
cocycles, i.e. from Z1 (G, Pold−1 (G)) to Pold (G). Given an inhomogeneous 1-cocycle c : G →
Pold−1 (G), then the standard map back to the homogeneous picture sends c to c̃ : G × G →
Pold−1 (G) given by c̃(g0 , g1 ) := g0 c(g0−1 g1 ) (see e.g. [8, I, n◦ 3.2 & III, n◦ 1.3 ]) and thus
τ 1 (c̃)(g0 , g1 ) = (g0 c(g0−1 g1 ))(1) = c(g0−1 g1 )(g0−1 ).
This is then a homogeneous polynomial 1-cocycle, and to get back to the inhomogeneous
picture (i.e. the description using polynomials) we need to apply the ‘bar-map’ β defined
above. That is, we fix the first variable g0 = 1 and obtain the map g 7→ c(g)(1), and this is
then the polynomial in Pold (G) representing τ 1 ([c]) in H1(d) (G, R) = Pold (G)/Pol d−1 (G). To
get an explicit description of the inverse map τ ′ , consider a polynomial ξ ∈ Pold (G). The
corresponding homogeneous polynomial 1-cocycle is given by ξ̂(g0 , g1 ) := ξ(g1 ) − ξ(g0 ) (cf. the
proof of Proposition 6.6) and therefore
τ ′ (ξ̂)(g0 , g1 )(t) = ξ̂(t−1 g0 , t−1 g1 ) = ξ(t−1 g1 ) − ξ(t−1 g0 )
The inhomogeneous 1-cocycle corresponding to τ ′ ([ξ̂] is then obtained by fixing the variable
g0 = 1, and hence τ ′ [ξ] is represented the 1-cocycle c : G → Pold−1 (G) given by
c(g)(t) = ξ(t−1 g) − ξ(t−1 ) = ∂g (ξ̃)(t),
where ξ̃(g) := ξ(g −1 ). That is, c(g) = ∂g ξ̃.
Corollary 6.9. The map π ∗ : H1 (G, Pol d (G)) → H1 (G, H1(d) (G, R)), induced by the quotient
map π : Pold (G) → H1(d) (G, R), is injective
Proof. Considering the short exact sequence
0 → Pold−1 (G) → Pold (G) → H1(d) (G, R) → 0,
and the corresponding long exact sequence in cohomology, the statement is seen to be equivalent to showing that ι : Pold−1 (G) → Pold (G) induces the zero map in 1-cohomology. To this
end, we first prove that the following diagram commutes
H1 (G, Pold−1 (G))

ι∗

≃ τ1

≃ τ1



H1(d) (G, R)

/ H1 (G, Pold (G))

(⊂)∗


/ H1
(G,
R),
(d+1)

where (⊂)∗ is the map induced by the inclusion Pold (G) ⊂ Pold+1 (G) and τ 1 is the isomorphism given by Proposition 5.1. For this, we will use the explicit description of τ 1 and
τ ′ := (τ 1 )−1 at the level of cocycles discussed in Remark 6.8. Let η ∈ Pold (G) be given.

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

19

Since everything depends only on the class [η] ∈ H1(d) (G, R), by subtracting a constant polynomial we may assume that η(1) = 0. Now consider the cocycle g 7→ ∂g η̃ representing
τ ′ ([η]) ∈ H1 (G, Pold−1 (G)), where η̃(g) := η(g−1 ). Composing with ι just gives the same
cocycle now considered as taking values in Pold (G), and applying τ 1 amounts to evaluating
at 1; that is, τ 1 ◦ ι∗ ◦ τ ′ ([η]) ∈ H1(d+1) (G, R) is represented by the polynomial:
g 7→ (∂g η̃)(1) = η̃(g−1 ) − η̃(1) = η(g) − η(1) = η(g).
Hence the map τ 1 ◦ ι∗ ◦ τ ′ agrees with (⊂)∗ : H1(d) (G, R) → H1(d+1) (G, R), and the latter map
is clearly zero.

In what follows we investigate alternative definitions of the space of polynomials and show
that they agree with the one given in Definition 6.4. We first show that there is no difference
between the polynomials defined by left- and right regular action. To this end define, for any
function ξ : G → R, the right-difference operator by
( ∂g ξ)(h) := ξ(hg) − ξ(h).
Then, by induction on d, it is easy to see that ξ ∈ C(G, R) is a polynomial map of degree at
most d if and only if for all g1 , . . . , gd+1 ∈ G we have
( ∂g1 ◦ · · · ◦ ∂gd+1 )(ξ) = 0.
For d = 0 this is clear. For the inductive step, assume that ξ is a (left) polynomium of degree
at most d. Then for g0 , . . . , gd ∈ G we have (∂g0 ◦ · ◦ ∂gd−1 )(∂gd ξ) = 0 so by the inductive
hypothesis we know that ∂gd ξ is a right polynomium of degree at most d − 1 for every gd ∈ G.
Since ∂ and ∂ commute this means that for all g0 , . . . , gd−1 ∈ G
0 = ( ∂g0 ◦ · ◦ ∂gd−1 )(∂gd ξ) = ∂gd (( ∂g0 ◦ · ◦ ∂gd−1 )(ξ)).
Thus ( ∂g0 ◦ · ◦ ∂gd−1 )(ξ) is constant so
∂gd ◦ ∂g1 ◦ · · · ◦ ∂gd−1 (ξ) = 0.
That is, ξ is a right degree polynomium of degree at most d. By symmetry this proves that
the the right polynomials of degree at most d coincide with the left polynomials of degree at
most d. That is, left-polynomials and right-polynomials coincide [13, Corollary 2.13], and the
degree of a polynomial map ξ into R coincides for left- and right-differentiation. For a more
general result, see [13, Proposition 3.16].
Observation 6.10. Let G be a lcsc group and ξ : G → R be a G(d)-invariant Borel map;
i.e. satisfying that ∂g1 ◦ · · · ◦ ∂gd ξ = 0 for all g1 , . . . , gd ∈ G. We claim that ξ is in fact
continuous; i.e. that the class of measurable polynomials is nothing but the class of continuous
polynomials. To prove this we proceed by induction on d. For d = 1 this is trivial since ξ
is then constant. For d = 2, ξ is an affine homomorphism, that is, a homomorphism plus a
constant translation. In this case the claim follows from a well-known argument of Banach. In
fact, recall that if η : G → E is a Borel 1-cocycle into a continuous, separable Fréchet G-module
E, then ξ is continuous (we will apply a very similar argument below, and will therefore not
recall Banach’s argument here; see e.g. [10, 9.10]). For the induction step, consider the map
αξ : g 7→ ∂g ξ taking values, by the induction hypothesis, in C(G, R). A direct computation
shows that, when considering C(G, R) with the right-regular representation ρ, αξ satisfies
the 1-cocycle identity: αξ (gh) = ρ(g).αξ (h) + αξ (g) for all g, h ∈ G. We consider C(G, R)
with the Fréchet topology of uniform convergence on compacts, for which every continuous

20

DAVID KYED AND HENRIK DENSING PETERSEN

functional ϕ ∈ C(G, R)∗ is given by integration against a compactly supported Radon measure
on G. From this one sees that αξ is weakly Borel, that is, for each ϕ ∈ C(G, R)∗ the function
g 7→ ϕ(αξ (g)) is Borel. But since C(G, R) is a separable Fréchet space, the weak and strong
Borel structures coincide (by Petti’s measurability theorem), and so αξ is strongly Borel. By
the argument of Banach alluded to above, it follows that αξ is continuous. Now let K ⊂ G
be a compact set of positive (Haar) measure such that ξ|K is continuous; the existence of K
is guaranteed by Lusin’s theorem. If xn →n x in K then, by continuity of αξ , we get
ξ(1) + ξ(1) −→ ∂x−1 ξ(1) + ξ(1) = ξ(x−1 ),
ξ(x−1
n ) = ∂x−1
n
n→∞

so ξ is continuous on K −1 as well. We now claim that for sequences (xn )n ⊂ K and (yn )n ⊂
K −1 converging to x and y, respectively, we have ξ(xn yn ) →n ξ(xy). By a ‘subsequences of
subsequences’ argument, using the compactness of K, this is easily seen to imply that ξ is
continuous on KK −1 , which, as it turns out, is what is needed to finish the proof. To see the
claim, we write
(6.9)
ξ(xn yn ) = ( ∂yn ξ)(xn ) + ξ(xn ).
By the continuity of αξ (and compactness of K) it follows that ( ∂yn ξ)(xn ) →n ( ∂y ξ)(x), and
so we conclude that ξ(xn yn ) →n ξ(xy) as desired. By a result of Weil (see e.g. [15, Theorem
2.2]), KK −1 contains an open neighbourhood of the identity. Thus, ξ is continuous at 1, and
using (6.9) it follows that ξ continuous on all of G.
Remark 6.11. As follows from Remark 6.8, the map ∂ : C(G, R) → C(G, C(G, R)) given by
f 7→ (g 7→ ∂ g f ), when pre-composed with inv∗ : C(G, R) → C(G, R) : f 7→ (g 7→ f (g −1 )), in∼
=

duces an isomorphism ∂ ◦ inv∗ : Pold (G)/ Pol d−1 (G) −
→ H1 (G, Pold−1 (G)). However, observe
∗
that since inv induces a degree-preserving linear automorphism of Pol(G), it follows that
the map ∂¯ : Pold (G)/ Pol d−1 (G) → H1 (G, Pol d−1 (G)), mapping ξ ∈ Pold (G) to the equivalence class of the cocycle g 7→ ∂ g ξ, is an isomorphism as well. Below we will freely identify
Pold (G)/Pold−1 (G) with H1 (G, Pol d−1 (G)) and, unless explicitly specified otherwise we will
¯ Note that, more generally, given any continuous G-module E
do so using the isomorphism ∂.
G(d+1)
and any ξ ∈ E
, the map g 7→ ∂ g .ξ is a cocycle into EG(d) ; so this is consistent with the
above.
The remainder of this section is devoted to a more detailed analysis of the degree function
and its interplay with the differentiation operators, and for this we will use the extended
addition and subtraction on Z∗ = N0 ∪ {−∞} defined in the Section 2.
Proposition 6.12. Let G be a group and ξ : G → R be a polynomial map of degree d > 1.
Then for every s ∈ G, the map ϕξ,s : g 7→ (∂ g ξ)(s) is a polynomial map of degree d, and so is
g 7→ ( ∂g ξ)(s).
Proof. For any function ξ : G → R, a direct computation verifies that the differential satisfies
∂ gh ξ = (∂ g ◦ ∂ h )(ξ) + ∂ g ξ + ∂ h ξ,

g, h ∈ G.

(6.10)

Thus for any h ∈ G we have
( ∂h ϕξ,s )(g) = ϕξ,s (gh) − ϕξ,s (g)
= (∂ g ◦ ∂ h )(ξ)(s) + (∂ h ξ)(s)
= ϕ∂ h ξ,s (g) + ϕξ,s (h).

(6.11)

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

21

If d = 1 then ϕ∂ h ξ,s = 0 so ( ∂h ϕξ ) is constant equal to ϕξ,s (h), and hence deg(ϕξ,s ) 6 1.
However, since d = 1, ∂ h ξ is constant for every h ∈ G and for some h0 this constant is nonzero. So, ∂h0 ϕξ,s 6= 0 proving that deg(ϕξ,s ) = 1. The general case now follows by induction
on d. For the inductive step, assume that the statement is true for d − 1 and that ξ has
degree d > 2. For h ∈ G, ∂h ξ has degree at most d − 1, so the induction takes over and
gives deg(ϕ∂ h ξ,s ) 6 d − 1 and since ϕξ,s (h) is constant in the variable g, deg(ϕξ,s ) 6 d by the
computation (6.11). But for some h0 , ∂h0 ξ has degree equal to d − 1 > 1 and hence, by the

induction, so does ϕ∂ h0 ξ,s (−) + ϕξ,s (h) = ∂h0 ϕξ,s ; thus, deg(ϕξ,s ) = d.
Let G be a lcsc group and let ξ, η : G → R be (continuous) polynomial maps on G. Then it
is easy to see that the pointwise product ξ ·η : g 7→ ξ(g)η(g) is again a (continuous) polynomial
map with deg(ξ · η) 6 deg ξ +̇ deg η; indeed, we have
∂g (ξ · η)(h) = ξ(hg) · ( ∂g η)(h) + ( ∂g ξ)(h) · η(h) and
∂ g (ξ · η)(h) = ξ(g−1 h) · (∂ g η)(h) + (∂ g ξ)(h) · η(h),

(6.12)

from (either of) which the inequality follows by induction on deg ξ +̇ deg η; we shall show
below that equality holds for Mal’cev groups. In particular the multiplication map induces a
linear map
H1(d) (G, R) ⊗ H1(d′ ) (G, R) → H1(d+d′ ) (G, R)
for each pair (d, d′ ) ∈ N2 . Note also, that equation (6.12) implies that we have the following
version of Leibnitz’ rule for the differentials:
∂g (ξη) = ∂g (ξ) ∂g (η) + ∂g (ξ)η + ξ ∂g (η),

(6.13)

and similarly for ∂. We can now give a sharper estimate on the degree of ∂ g ξ for a polynomial
map ξ. To do so, the following definition will be convenient.
Definition 6.13 (degree wrt. a central series). Let G be a (lcsc) group and G a central series
in G of finite length. For every g ∈ G we define the degree degG g of g with respect to the
central series G by
degG g := max{i | g ∈ Gi , g ∈
/ Gi+1 }.
When G is a Mal’cev group, the degree deg g of an element g in G, will refer to the degree
with respect to the Mal’cev central series, unless explicitly stated otherwise.
Lemma 6.14. Let G be a group and let ξ ∈ Pol(G). Then for g ∈ G[k] we have deg( ∂g ξ) 6
deg ξ −̇k. Hence, deg( ∂g ξ) 6 deg ξ −̇ deg g, when deg(g) is taken with respect to the lower
central series.
We remark that, in the statement of Lemma 6.14, the group G is not a priori assumed
to carry a topology and the lower central series is therefore to be understood in the purely
algebraic sense. Note, however, that if G is a csc Lie group, then the algebraically defined
lower central series automatically consists of closed subgroups [6] and hence, in this case, there
is no difference between the topological and algebraic lower central series. More generally, if G
is a lcsc group (possibly not of Lie type) and ξ ∈ Pold (G), then Lemma 6.14 shows that ∂g ξ ∈
Pold−̇k (G) for all g ∈ G[k] (the algebraically defined lower central series). Moreover, the map
g 7→ ∂g ξ is a measurable 1-cocycle into C(G) (endowed with the right regular representation)
and therefore automatically continuous (see [10, 9.10] and Observation 6.10 for more details)
and since Pold−̇k (G) is a closed subspace in C(G), this shows that ∂g ξ ∈ Pold−̇k (G) also for

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DAVID KYED AND HENRIK DENSING PETERSEN

g in the closure G[k] ; i.e., the statement of Lemma 6.14 holds true in the topological context
as well.
Proof of Lemma 6.14. We prove the statement by induction on k. If k = 1 then the statement
is true by the definition of a polynomial map. Assume now that the statement is true for
k − 1 > 1 and let x ∈ G[k] be given. Assume first that x = g−1 h−1 gh with g ∈ G[k−1]
and h ∈ G. Then, as G[k−1] is normal in G, by computing modulo Pold−̇k (G) (symbolically
represented by ‘≡’) we get
∂x ξ = ∂g−1 (h−1 gh)
= ∂g−1 ◦ ∂h−1 gh ξ + ∂g−1 (ξ) + ∂h−1 gh (ξ)

(by (6.10))

≡ ∂g−1 (ξ) + ∂h−1 gh (ξ)
≡ ∂g−1 (ξ) + ∂h−1 ◦ ∂gh (ξ) + ∂h−1 (ξ) + ∂gh (ξ)


= ∂g−1 (ξ) + ∂h−1 ∂g ◦ ∂h (ξ) + ∂g (ξ) + ∂h (ξ) + ∂h−1 (ξ) + ∂g ◦ ∂h (ξ) + ∂g (ξ) + ∂h (ξ)
≡ ∂g−1 (ξ) + ∂h−1 ◦ ∂h (ξ) + ∂h−1 (ξ) + ∂g (ξ) + ∂h (ξ)
= ∂g−1 (ξ) + ∂hh−1 (ξ) + ∂g (ξ)
= ∂gg−1 (ξ) − ∂g ◦ ∂g−1 (ξ)
= − ∂g ◦ ∂g−1 (ξ) ≡ 0.
A completely analogous computation shows that also ∂x−1 ξ ≡ 0 and from (6.10) if follows
that ∂x ξ ≡ ∂y (ξ) ≡ 0 implies that ∂xy ξ ≡ 0. Hence ∂z (ξ) ≡ 0 for all z ∈ G[k] as desired. 
As a consequence of Lemma 6.14, we also record the following result due to Leibman:
Corollary 6.15 ([13]). If ξ ∈ Pol0d (G) then ξ vanishes on G[d+1] .
Proof. For g ∈ G[d+1] we have deg( ∂g ξ) 6 d−̇(d + 1) = −∞ so ∂g ξ = 0. Thus
0 = ∂g (ξ)(1) = ξ(g) − ξ(1) = ξ(g).



We end this section with a small lemma to be used in the section to follow.
Lemma 6.16. Let G be a lcsc group and let Z 6 G be a normal subgroup isomorphic to R.
If z0 ∈ Z \ {1G } and ξ ∈ Pol(G) satisfies ∂z0 ξ = 0 then ξ descends to a polynomial ξ̄ on G/Z
of the same degree.
Proof. To see that ξ̄ is well defined we need to show that ξ is constant on the cosets of Z.
For g ∈ G, the left translate g −1 .ξ is again a polynomial and hence so is the restriction
η := (g−1 .ξ)|Z . By assumption we have ξ(hz0 ) = ξ(h) for all h ∈ G and hence ξ(gz02 ) =
ξ(gz0 ) = ξ(g) and, recursively, ξ(gz0n ) = ξ(g) for all n ∈ N. The map η is therefore a
polynomial on Z ≃ R which is constant on an infinite set, and since the polynomials on R are
exactly the classical polynomials this can only happen if η = 0. That is, ξ(gz) = ξ(g) for all
z ∈ Z and therefore ξ̄ : G/Z → R is well defined. We furthermore have
∂z0 ◦ ∂ g (ξ) = ∂ g ◦ ∂z0 (ξ) = 0 for all g ∈ G,
and hence ∂g ξ is well defined as well, and a direct computation verifies that
∂ḡ1 ◦ · · · ◦ ∂ḡd ξ¯ = ∂g1 ◦ · · · ◦ ∂gd ξ,
from which it follows that ξ¯ is a polynomial of degree deg(ξ).



POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

23

Remark 6.17. The proof Lemma 6.16 also shows the following general fact: if G is a group
and H 6 G is a normal subgroup, then any ξ ∈ Pol0 (G) with the property that ∂h ξ = 0 for
all h ∈ H descends to a polynomial ξ̄ ∈ Pol0 (G/H) of the same degree.
6.3. Polynomials on Mal’cev groups. Building on the general results in the previous section, we can now give a complete description of the polynomials on a Mal’cev group. Let
therefore G be a Mal’cev group and let (gi,j ) be a Mal’cev basis of G. Then for each pair
(i0 , j0 ) ∈ Brk(G) we consider the map ζgi0 ,j0 : G → R given by
ζgi0 ,j0 :

Y

t

gi,ji,j 7−→ ti0 ,j0 ,

(6.14)

(i,j)∈Brk(G)

where ti,j ranges over the set Zi,j , either equal to Z or R (see Section 4 for this and Section 2
for the definition of the multiindex notation).
More generally, we will need the following notation: for any multi-index d ∈ Dd,dim(G) we
define
Y
Y
Y
d
t
ti,jij .
gi,ji,j 7−→
ζgi,j (−)di,j :
ζd :=
(i,j)∈Brk(G)

(i,j)∈Brk(G)

(i,j)∈Brk(G)

With this notation at our disposal, we can now give the promised description of polynomials
on Mal’cev groups.
Theorem 6.18. Let G be a Mal’cev group and let (gi,j )i,j be a Mal’cev basis. Then for all
(i0 , j0 ) ∈ Brk(G) the map ζgi0 ,j0 defined above is a polynomial map of degree deg ζgi0 ,j0 = i0 .
Furthermore, the set {ζd | d ∈ Dd,dim(G) } is a linear basis of Pold (G).
Proof. Assume first that G is a csc nilpotent Lie group. We shall then show, by induction on
m := dim(G), that deg(ζgij ) = i; the case of m = 1 being trivial. For the inductive step, we
need a bit of notation. We first recall Leibman’s definition of lc-polynomials
from [13]: if G is nilpotent with cl(G) = c and H is any group then ϕ : H → G is an
lc-polynomium of lc-degree at most (1, 2, · · · , c) if for all i = 1, . . . , c and h1 , . . . , hi+1 ∈ H:
∂h1 ◦ · · · · ◦ ∂hi+1 (ϕ)(H) ∈ G[i+1] , where the differentiation operator is defined as ∂h (ϕ)(g) :=
ϕ(g)−1 ϕ(gh). In particular, since G[c+1] = {1} this forces ϕ to be a “ polynomium of degree
at most c”; i.e.to satisfy ∂h1 ◦ · · · ◦ ∂hc+1 ξ = 1G .
The main virtue of the class of lc-polynomials is that they, by [13, Theorem 3.2], form
a group under pointwise multiplication. Now, if (gij ) is a Mal’cev basis and ξ : G → R is
ξ(h)
a polynomium of degree at most i, then the map ϕ : G → G given by ϕ(h) = gij is an
lc-polynomium with lc-degree at most (1, . . . , c). To see this, note that
∂g ◦···◦ ∂gl+1 (ξ)(h)
∂g1 ◦ · · · ◦ ∂gl+1 (ϕ)(h) = gij 1

(6.15)

R 6 G
and since gij
[i] we have ∂g1 ◦ · · · ◦ ∂gl+1 (ϕ)(h) ∈ G[i] 6 G[l+1] when l + 1 6 i and
∂g1 ◦ · · · ◦ ∂gl+1 (ϕ)(h) = 1 ∈ G[l+1] when l + 1 > i = deg(ξ). We are now ready to return to
the inductive step. Fix some z := gc,j0 ∈ G[c] 6 Z(G) and denote G/z R by Ḡ and the quotient
map G → Ḡ by π. Note that the gij ’s with (i, j) 6= (c, j0 ) project onto a Mal’cev basis ḡij
for Ḡ and hence ζḡij has degree i by the induction hypothesis. But ζgij = ζḡij ◦ π so we also

24

DAVID KYED AND HENRIK DENSING PETERSEN

obtain deg(ζgij ) = i. Thus, we only have to prove that deg(ζz ) = c. To this end, write
−1

Y
(h)
ζ
g
z ζz (h) = h · 
(6.16)
gi,ji,j  .
(i,j)6=cl(G),j0

As already mentioned, by [13, Theorem 3.2] the set of lc-polynomials from G to G of degree at
most (1,. . . , c) is a group under pointwise multiplication, so since the identity map is clearly
ζg

(h)

such an lc-polynomial, if we can prove that each of the factors h 7→ gi,ji,j
in the product has
ζ
z (h)
lc-degree at most (1, . . . , c) we obtain that also ϕ : h 7→ z
has lc-degree at most (1, . . . , c).
ζg (h)

has
However, as we just saw, when (i, j) 6= (c, j0 ), ζgij has degree i and hence h 7→ gij ij
lc-degree at most (1, . . . , c) as desired , and from this it follows, using (6.15), that ζz : G → R
has degree at most c. We still need to prove that deg(ζz ) = c, but if deg(ζz ) 6 c − 1 then, by
Corollary 6.15, the (unital) polynomium ζz must vanish on G[c] , which cannot be the case as
z ∈ G[c] and ζz (z) = 1.
For the second part of the statement, denote again by c = cl(G) the class of G and define
for d > 0

Ad := spanR ζd | d ∈ Dd,dim(G)
P
(recall that d = (dij ) ∈ Dd,dim(G) iff
ij idij 6 d) and put A−∞ = {0}. We need to
prove that Pold (G) = Ad . Here the inclusion “⊇” follows from what was already proven and
the general estimate deg(ξη) 6 deg(ξ) + deg(η). To see the opposite inclusion, we run an
induction on m = dim(G), in which the the base case m = 1 is trivial. For the inductive step,
let ξ ∈ Pold (G) be given and fix z := gc,j0 ∈ G[c] 6 Z(G); we now run a finite subinduction
(n)

on the minimal number n ∈ N such that ∂z (ξ) = 0. If n = 1, then ∂z ξ = 0 and hence, by
Lemma 6.16, induces a polynomium ξ̄ on Ḡ := G/z R which, by the primary induction, can be
written as a linear combination of products of (ζḡi,j )(i,j)6=(c,j0 ) . This means that ξ ∈ Ad as it
can be written as a linear combination of the ζd ’s even without using ζz . For n = 2, a direct
computation shows that
∂z ( ∂z (ξ)ζz − ξ) = 0.
Thus, by the n = 1 case just covered, this means that ∂z (ξ)ζz − ξ ∈ Ad . Moreover, by Lemma
6.14, deg( ∂z (ξ)) 6 d−̇c and since ∂z ( ∂z (ξ)) = 0, the n = 1 case gives that ∂z ξ ∈ Ad−̇c . Hence
∂z (ξ)ζz ∈ Ad and thus also ξ ∈ Ad . The general case is a bit more involved, but overall builds
on the same idea used for n = 2, and for that we need som more detailed information about
differentiation and integration with respect to z, contained in the following three claims.
Claim 1. We have ∂z (ζzk ) ∈ spanN {ζzl | 0 6 l 6 k − 1}.
Proof of Claim 1. For k = 1, ∂z (ζz ) = 1 and for k = 2 we have ∂z (ζz2 ) = 1 + 2ζz . The general
case follows inductively: assuming the result for k − 1 we have, using the Leibnitz rule (6.13),
that
∂z (ζzk ) = ∂z (ζzk−1 ζz ) = ∂z (ζzk−1 ) · 1 + ∂z (ζzk−1 ) · ζz + ζzk−1 · 1
By the induction hypothesis, ∂z (ζzk−1 ) ∈ spanN {ζzl | 0 6 l 6 k − 2} and thus ∂z (ζzk−1 ) · ζz ∈
spanN {ζzl | 0 6 l 6 k − 1} and hence also ∂z (ζzk ) ∈ spanN {ζzl | 0 6 l 6 k − 1}.

Claim 2. For each k ∈ N0 there exists Υk ∈ spanR {ζzl | 0 6 l 6 k + 1} such that ∂z Υk = ζzk .

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

25

Proof of Claim 2. For k = 0 this is clear as ∂z ζz = 1 and for k = 1 we have ∂z (ζz2 ) = 1 + 2ζz
so Υ1 := 12 ζz2 − 21 ζz does the job. The general case follows inductively: assume Claim 2 true
for k − 1. By Claim 1, we get a0 , . . . , ak ∈ N such that
∂z (ζzk+1 ) = a0 + a1 ζz + · · · + ak ζzk ,
and since ak ∈ N we have
ζzk

1
=
ak
= ∂z

∂z (ζzk+1 )

−

k−1
X

ai ζzi

i=0
k−1
X

1
1 k+1
ζz −
ak
ak
|
{z

=:Υk

i=0

!

ai Υi

1
=
ak
!

∂z (ζzk+1 )

−

k−1
X
i=0

ai ∂z Υi

!

.

}



Claim 3. For every ξ ∈ Ad−̇c there exists Ξ ∈ Ad such that ∂z Ξ = ξ maps Ad .
Proof of Claim 3. For ξ ∈ Ad−̇c , by grouping summands together according to their power of
ζz we can write it as
m
X
rk ηk ζzk
ξ=
k=0

P
(for some m ∈ N and rk ∈ R) where ηk ∈ Ad−̇c−̇kc and ∂z (ηk ) = 0. Putting Ξ := m
k=0 rk ηk Υk ,
where the Υk ’s are as in Claim 2, we get, using the Leibnitz rule (6.13), that ∂z (Ξ) = ξ.
Moreover, by Claim 2 we have Υk ∈ spanR {ζzl | 0 6 l 6 k + 1} and since ηk ∈ Ad−̇c−̇kc we
conclude that Ξ ∈ Ad as desired.

We can now finish the (sub-)induction argument, which is running over the minimal n such
(n)
that ∂z (ξ) = 0. Given ξ ∈ Pold (G), ∂z ξ falls under the induction hypothesis and has degree
at most d−̇c by Lemma 6.14. Hence ∂z (ξ) ∈ Ad−̇c , so by Claim 3 there exists Ξ ∈ Ad such
that ∂z Ξ = ∂z (ξ). Thus, ∂z (Ξ − ξ) = 0, and by the base case (n = 1) this means that
Ξ − ξ ∈ Ad . By construction, Ξ ∈ Ad and hence also ξ ∈ Ad . This finishes the proof that
that Pold (G) = Ad , and the linear independence of the polynomials {ζd | d ∈ Dd,dim(G) }
is clear, since they pull back to linearly independent polynomials on Rdim g via the Mal’cev
coordinates. This completes the proof in the case where G is a csc nilpotent Lie group.

In the general case, we know that G is cocompactly embedded in its Mal’cev completion
L, and that we may choose a Mal’cev basis for L based in G. Denote the dimension of L
by n. Then the Mal’cev coordinates gives a diffeomorphism L ≃ Rn which identifies G with
′
a (cocompact) subset of the form Zm × Rm where m + m′ = n. Moreover, by the what
was just proven we know that polynomials on L pull back to polynomials on Rn via the
′
Mal’cev coordinates, and polynomials on Rn are uniquely determined on the subset Zm × Rm .
Thus, polynomials on L are uniquely determined by their values on G, so the restriction map
Pold (L) → Pold (G) is injective for all d ∈ N. We now need to prove that it is also surjective.
We first note that this is trivially the case when d = 0, and we now proceed by induction on

26

DAVID KYED AND HENRIK DENSING PETERSEN

d. Assuming this to be true up to d − 1 we have
C ⊗R (Pold (G)/Pol d−1 (G)) ≃ C ⊗R H1 (G, Pold−1 (G))

(Proposition 5.1)

≃ H1 (G, C ⊗R Pold−1 (G))
≃ H1 (G, C ⊗R Pold−1 (L))

≃ H1 L, L2 (L/G) ⊗C C ⊗R Pold−1 (L)
≃ H1 (L, C ⊗R Pold−1 (L))

≃ C ⊗R (Pold (L)/Pold−1 (L)) .

([8, III, n◦ 4])
(Corollary 4.14)
(Proposition 5.1)

Note that we may indeed apply Corollary 4.14 to obtain the penultimate equality, because
L2 (L/G)L = C.1L/G and H1 (G, Pold−1 (G)), and hence also H1 (L, L2 (L/G)⊗C C⊗R Pold−1 (L)),
is finite dimensional (by Proposition 4.7 and Corollary 6.7) which implies that the latter is
automatically Hausdorff [8, III, Proposition 3.1]. From this we conclude that Pold (G) and
Pold (L) have the same (finite) linear dimension and hence the restriction map Pold (L) →
Pold (G) must be surjective as well. The only thing left to prove is that deg(ζgij ) = i when ζgij
is considered as a polynomial on G. However, by considering ζgij as a polynomial on L and
pulling it back via the Mal’cev coordinates, we already know that it is a polynomial of degree
i on Rn , which therefore cannot coincide with a polynomial of lower degree on the cocompact
′

subset Zm × Rm corresponding to G. Hence deg(ζgij ) = i as a polynomial on G.
Corollary 6.19. For a Mal’cev
group G and ξ, η ∈ Pol(G) we have deg(ξ·η) = deg(ξ)+deg(η).
P
In particular, deg(ζd ) = i,j idij .

Proof. As we saw above, the inequality ‘6’ is true for any group G so we only need to prove
the opposite. Put d := P
deg(ξ) and d′ := deg(η) andPpick a Mal’cev basis for G. By Theorem
sc ζc . When multiplying the
6.18, we can write ξ = d∈Dd,dim(G) rd ζd and η = c∈D ′
d ,dim(G)

two sums only the degree d and degree d′ terms matter for the degree of the product (due to
the linear independence of the ζd ’s) and hence the only way that we can have deg(ξη) < d + d′
is if
X
X
rd sc ζd ζc = 0.
c∈D=
d∈D=
d,dim(G)
d′ ,dim(G)

However, by pulling the polynomials back to polynomials on Rdim g via the Mal’cev coordinates,
it follows that this can only happen if one of the pullbacks is the zero polynomial, which
contradicts the choice of d and d′ . The last statement follows by applying the first statement
multiple times using the fact that deg(ζgij ) = i.

Corollary 6.20. Let G be a Mal’cev group and let ξ ∈ Pol(G). Then for every g ∈ G we
have deg ∂ g ξ 6 deg ξ −̇ deg g and analogously for ∂g ξ, where deg(g) is the degree with respect
to the Mal’cev central series.
Proof. The Mal’cev central series is given as Gi = G ∩ (G ⊗ R)[i] and hence deg(g) is the
same whether we compute it with respect to the Mal’cev central series in G or the lower
central series in G ⊗ R. Furthermore, by Theorem 6.18 we know that the restriction map
Pold (G ⊗ R) → Pold (G) is bijective and degree preserving and the result therefore follows
from Corollary 6.14.

We will need also the following uniqueness results for polynomials.

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

27

Lemma 6.21. Let G be a Mal’cev group and let (gi,j )i,j be a Mal’cev basis for G. Denote
n := rk G and let G0 be the (not necessarily closed) subgroup of G generated (algebraically) by
S := {g1,1 , . . . , g1,n }. Finally, denote by S 6d the set of words on S of length at most d ∈ N.
Then for d ∈ N and ξ, η ∈ Pold (G) we have
ξ = η ⇔ ξ|S 6d = η|S 6d .
In particular, any polynomial map on G is uniquely determined by its values on G0 .
Proof. By Theorem 6.18, the polynomials on G and on its Mal’cev completion are the same,
so by passing to the Mal’cev completion we may assume that G is a csc, nilpotent Lie group.
As in [13, Proposition 1.15], we see that if ξ|S 6d = η|S 6d then ξ|G0 = η|G0 , so the lemma will
follow if we show that any polynomial vanishes on G if it vanishes on G0 . Denote the closure
of G0 by H and note that H surjects onto G/G[2] by construction. By [6, Theorem 5.4.13]
(and the generalizing remarks following it in section 5.5), this implies that H is cocompact in
G. In other words, H is Mal’cev group with Mal’cev completion G and we therefore know that
polynomials are uniquely determined by their values on H and, by continuity, on its dense
subgroup G0 .

Lemma 6.22. Let G be a Mal’cev group, (gi,j )i,j a Mal’cev basis and let ξ, η ∈ Pol(G). If
ξ(1) = η(1) and ∂g1,j ξ = ∂g1,j η for all j = 1, . . . , rk(G) then ξ = η.
The statement may be deduced from the previous lemma, by showing that ξ(g) = η(g) for
all g ∈ G0 , by induction on word-length. Here is an alternative argument:
Proof. For any f ∈ C(G, R), the function g 7→ ∂g f satisfies the 1-cocycle identity, when
C(G, R) is considered a G-module for the right regular action. Hence we conclude that ∂g (ξ −
η) = 0 for all g ∈ G0 , the subgroup of G generated by (g1,j )j . By Proposition 6.12, the map
g 7→ ∂g (ξ − η)(1) is itself a polynomial map on G, and since it vanishes on G0 it vanishes on
all of G by the previous lemma. Thus, ξ(g) = η(g) as desired.

7. The algebra of polynomial maps
The space of polynomial maps Pold (G) introduced in the previous section may be seen as
containing certain “d’th order dual structure”. For instance, Pol1 (G), being essentially (that
is, up to addition of some constant) the space of continuous group homomorphisms into R,
contains very precise information about the (torsion-free part of the) abelianization of G. In
this section we elaborate on these considerations and Theorem 7.9 below makes precise in
which way Pol(G) should be considered a dual object.
Remark 7.1. If G is a csc nilpotent Lie group, when thinking of G as the set of real points on
an algebraic group, it follows from Theorem 6.18 that Pol(G) is the set of regular functions on
G, in the sense of algebraic geometry [5]. Many of the results deduced in this section therefore
also follow from well-known results in algebraic geometry (e.g. the fact that Pol(G) is a Hopf
algebra [5, Chapter 1]), but for the sake of completeness, and since we wish to keep track of
the degree of polynomials, which is not covered by algebraic geometry, we include the details
below. Here, and in what follows, the symbol “⊗” is used for the algebraic tensor product of
real vector spaces.

28

DAVID KYED AND HENRIK DENSING PETERSEN

Lemma 7.2. Let G and H be Mal’cev groups. The map α : C(G)⊗C(H) → C(G×H) given by
α(ξ ⊗ η)(g, h) := ξ(g)η(h) restricts to an algebra-isomorphism Pol(G) ⊗ Pol(H) ≃ Pol(G × H)
which respects the grading given by the polynomial degree; that is deg α(ξ ⊗ η) = deg ξ + deg η.
Proof. If (gij ) is a Mal’cev basis for G and (hkl ) is one for H, then the set ((gij , 1), (1, hkl ))i,j,k,l
is a Mal’cev basis for G × H and a direct computation verifies that α(ζgij ⊗ 1) = ζ(gij ,1) and
α(1 ⊗ ζhkl ) = ζ(1,hkl ) . From this it follows that α, which is easily seen to be an algebra
homomorphism, maps Pol(G) ⊗ Pol(H) to Pol(G × H) and as Pol(G × H) is generated, as an
algebra, by (ζ(gij ,1) , ζ(1,hkl ) )ijkl (Theorem 6.18) the restriction of α is surjective. Furthermore,
by Theorem 6.18 the elements ζd ⊗ ζc with d ∈ Dd,dim(G) and c ∈ Dd′ ,dim(H) constitute a
basis for Pold (G) ⊗ Pold′ (H), and since α(ζd ⊗ ζc ) ∈ {ζb | Dd+d′ ,dim(G×H) } it follows that
α is injective on Pold (G) ⊗ Pold′ (G) for any d, d′ ∈ N, and hence globally. That α is degree
preserving can be seen by the same argument used to prove Corollary 6.19.

The previous lemma, in particular, shows that, given Mal’cev groups G and H, any linear
map Ψ : Pol(G) → Pol(H) satisfying that deg Ψ(ξ) 6 deg ξ for all ξ ∈ Pol(G), induces
a map Ψ ⊗ Ψ : Pol(G × G) → Pol(H × H) given by Ψ(ξ ⊗ η) = Ψ(ξ) ⊗ Ψ(η) such that
deg(Ψ ⊗ Ψ)(ξ ′ ) 6 deg ξ ′ .
Definition 7.3 (degree-preserving maps). Let G and H be lcsc groups. We will say that a
linear map Ψ : Pol(G) → Pol(H) is degree-preserving if deg Ψ(ξ) 6 deg ξ for all ξ ∈ Pol(G),
and properly degree-preserving if equality holds.
Definition 7.4 (strongly unital maps). We say that a linear map Ψ : Pol(G) → Pol(H) is
strongly unital if it is unital and if Ψ(ξ)(1) = ξ(1) for all ξ ∈ Pol(G).
Remark 7.5. Let G be a Mal’cev group with multiplication m : G × G → G. We note that,
by [13], m∗ is a well-defined (that is, m∗ ξ is a polynomial for every polynomial ξ map. Indeed,
we claim that multiplication m : G × G → G is a polynomial map of lc-degree (cf. [13, Section
3]) lc-deg m = (1, . . . , cl(G)). To see this, let πi : G × G → G, i = 1, 2, denote the projections
on the first and second factor, respectively. Then m(g) = π1 (g) · π2 (g) is a pointwise product
of homomorphisms, so the claim follows by [13, Theorem 3.2]. In fact, we now show that m∗
is properly degree preserving. this, we first show that if (ξi )∞
i=1 is basis for Pol(G), chosen
such that ξ0 = 1, ξi (1) = 0 for i > 1 and {ξi | deg(ξi ) 6 d} is a basis for Pold (G) for every
d ∈ N0 , and m∗ (ξ) is written as
m
X
ξi ⊗ ηi ,
m∗ (ξ) =
i=0

with ηi ∈ Pol(G) then deg(ξi ) ∔ deg(ηi ) 6 deg(ξ). When deg(ξ) = 0 this is basic linear
algebra, and the general case now follows by induction on n := deg(ξ): a direct computation
shows that
m
X
ξi ⊗ ∂g ηi
(7.1)
m∗ ( ∂g (ξ)) =
i=0

so the induction hypothesis gives that

deg(ξi ) ∔ deg( ∂g ηi ) 6 deg( ∂g ξ) 6 deg(ξ) − 1 = n − 1, for all g ∈ G.

(7.2)

For each non-constant ηi there exists a g ∈ G such that deg( ∂g ηi ) = deg(ηi ) − 1 > 0 and
hence
deg(ξi )+̇ deg( ∂g ηi ) = deg(ξi ) + deg(ηi ) − 1 6 n − 1.

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

29

Thus, for those i we have deg(ξi )+̇ deg(ηi ) 6 n and deg(ξi ) 6 n − 1. We may write
X
X
ξ(g) = m∗ (ξ)(g, 1) =
ξi (g)ηi (1) +
ξi (g)ηi (1),
i:deg(ηi )>0

i:deg(η)60

and since ξ can be uniquely expressed as a linear combination of the elements {ξi | deg(ξi ) 6
n}, if deg(ξi ) > n for some i, then ηi (1) = 0 and ηi is constant by (7.2); thus in this case
deg(ξi ) ∔ deg(ηi ) = −∞ 6 n which proves the claim. In fact, when ξ is a unital polynomial
the pull back takes the form
X
m∗ (ξ) = 1 ⊗ ξ +
ξi ⊗ ξi′ + ξ ⊗ 1,
(7.3)
i

ξi , ξi′

′
where
are unital,
P non-constant polynomials with deg(ξi ) + deg(ξi ) 6 deg(ξ). To see this,
∗
we expand m (ξ) = i ξi ⊗ ηi according to the basis (ξi )i chosen above and, by what was just
proven, this means that deg(ξi )+̇ deg(ηi ) 6 deg(ξ). Then write ηi = ξi′ + ri 1 with ξi′ unital
and ri ∈ R, and note that since ξ and (ξi )i>0 are unital and ξ0 = 1 we have

0 = ξ(1) = m∗ (ξ)(1, 1) = η0 (1) = r0 ;
X
ξ(g) = m∗ (ξ)(1, g) = η0 (g) +
ξi (1)ηi (g) = η0 (g);
i>0

∗

ξ(g) = m (ξ)(g, 1) = 1 ⊗ η0 (1) +

X

ξi (g)ξi′ (1) +

i>0

Thus,
m∗ (ξ) = 1 ⊗ η0 +

X

ξi ⊗ ξi′ +

i>0

X

X

ξi (g)ri =

i>0

ri ξ i ⊗ 1 = 1 ⊗ ξ +

i>0

and restricting the last sum to those i for which

X

ξi (g)ri

i>0

X

ξi ⊗ ξi′ + ξ ⊗ 1,

i>0

ξi′

6= 0 we get the decomposition (7.3).

By the above remark, the multiplication map m : G×G → G dualizes to a degree preserving
map at the level of polynomial algebras, and hence the following definition makes sense.
Definition 7.6 (co-multiplicativity). Let G, H be Mal’cev groups. We say that a linear map
Ψ : Pol(G) → Pol(H) is co-multiplicative if the following diagram commutes:
Pol(G)

m∗

/ Pol(G2 ) .

(7.4)

Ψ⊗Ψ

Ψ



Pol(H)

m∗


/ Pol(H 2 )

Remark 7.7. By Lemma 7.2 and Remark 7.5, the multiplication map m : G × G → G on
a Mal’cev group dualizes to a (degree preserving) map m∗ : Pol(G) → Pol(G) ⊗ Pol(G) and
it is now straight forward to check that Pol(G) is a commutative Hopf-algebra with comultiplication m∗ , antipode inv∗ and counit ev1 . Note also, that in this terminology a strongly
unital, comultiplicative algebra homomorphism between polynomial algebras is nothing but a
morphism in the category of unital Hopf algebras.
Lemma 7.8. Let G and H be Mal’cev groups and let Ψ1 , Ψ2 : Pol(G) → Pol(H) be strongly
unital, co-multiplicative, algebra homomorphisms. Let (gi,j ) be a Mal’cev basis for G and (hk,l )
be a Mal’cev basis for H and suppose that for all ℓ = 1, . . . , rk(H) and all i, j we have
(Ψ1 ζgi,j )(h1,ℓ ) = (Ψ2 ζgi,j )(h1,ℓ ).

(7.5)

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DAVID KYED AND HENRIK DENSING PETERSEN

Then Ψ1 = Ψ2 .
Proof. We show that Ψ1 (ξ) = Ψ2 (ξ) by induction on d := deg ξ. The case d = 0 is clear and
the case d = 1 follows directly from the hypotheses using Theorem 6.18. Let d > 1 be given.
Suppose that (ΨP
1 ξ)(h) = (Ψ2 ξ)(h) and (Ψ1 ξ)(k) = (Ψ2 ξ)(k) for some h, k ∈ H. Writing
∗
m (ξ) = 1 ⊗ ξ + i ξi ⊗ ξi′ + ξ ⊗ 1 as in (7.3), the induction hypothesis gives:
Ψ1 (ξ)(hk) = m∗ (Ψ1 (ξ))(h, k)

= (Ψ1 ⊗ Ψ1 )(m∗ (ξ))(h, k)
X
=
Ψ1 (ξi )(h) · Ψ1 (ξi′ )(k) + Ψ1 (ξ)(h) + Ψ1 (ξ)(k)
i

=

X

Ψ2 (ξi )(h) · Ψ2 (ξi′ )(k) + Ψ2 (ξ)(h) + Ψ2 (ξ)(k)

i

= (Ψ2 ⊗ Ψ2 )(m∗ (ξ))(h, k)
= Ψ2 (ξ)(hk).
Using this computation repeatedly, the assumption (7.5) implies that (Ψ1 ξ)(h) = (Ψ2 ξ)(h)
for all words in h1,ℓ , and by Lemma 6.21 it follows that Ψ1 ξ = Ψ2 ξ.

Observe that if ϕ : H → G is a homomorphism then it induces a strongly unital, comultiplicative (degree-preserving) algebra homomorphism ϕ∗ : Pol(G) → Pol(H). The next
result gives a converse to this, in the spirit that Pol(G) acts as a “total” dual space of G. As
mentioned already, by using the identification of Pol(G) with the algebra of regular functions
the result can also be deduced from classical results in the theory of algebraic groups [5,
Chapter 1].
Theorem 7.9. Let G and H be csc nilpotent Lie groups and suppose that Ψ : Pol(G) →
Pol(H) is a strongly unital, co-multiplicative, algebra homomorphism. Then there is a unique
continuous group homomorphism ϕ : H → G such that Ψ is induced by ϕ. Further, ϕ is an
isomorphism if and only if Ψ is bijective.
Proof. s. Fix Mal’cev bases {gi,j } and {hi,j } for G and H, respectively, and let F be the free
nilpotent Lie group of class cl(F ) = max{cl(G), cl(H)} with rk(H) generators f1,1 , . . . , f1,rk(H) .
Then there are unique Lie group homomorphisms ϕH : F → H and ϕG : F → G with closed
images, defined on the generators by
ϕH (f1,ℓ ) := h1,ℓ
ϕG (f1,ℓ ) :=

Y

(Ψζgi,j )(h1,ℓ )

gi,j

(i,j)∈Brk(G)

Since H is generated, as a topological group, by (h1,ℓ )ℓ and ϕH has closed image, it is actually
surjective and induces an isomorphism F/ ker(ϕH ) ≃ H. A direct computation shows that
(ϕ∗H ◦ Ψ)(ζgij )(f1l ) = ϕ∗G (ζgij )(f1l ) and thus, by Lemma 7.8, we get ϕ∗H ◦ Ψ = ϕ∗G . Let f ∈ F
be in ker(ϕH ). Then for every ζ ∈ Pol0 (G) we have, since Ψ is assumed strongly unital, that
0 = Ψ(ζ)(1) = Ψ(ζ)(ϕH (f )) = (ϕ∗H ◦ Ψ)(ζ)(f ) = ϕ∗G (ζ)(f ) = ζ(ϕG (f )),

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

31

and since Pol0 (G) separates points in G (Theorem 6.18) we conclude that ϕG (f ) = 1. Thus
ϕG induces a map ϕ̄G : F/ ker(ϕH ) → G and we therefore obtain a homomorphism
ϕ̄

G
ϕ : H ≃ F/ ker(ϕH ) −→
G

Note that, since ϕG has closed image the same is true for ϕ. By construction we have ϕ(h1l ) =
ϕG (f1l ) and a direct computations now shows that ϕ∗ (ζgij )(h1l ) = Ψ(ζgij )(h1l )) for all i, j and
l and by Lemma 7.8 we conclude that ϕ∗ = Ψ. This also proves the uniqueness of ϕ, because
if ψ were another homomorphism predual to Ψ then for every h ∈ H and every ξ ∈ Pol(G)
we have
ξ(ψ(h)) = ψ ∗ (ξ)(h) = Ψ(ξ)(h) = ϕ∗ (ξ)(h) = ξ(ϕ(h)),
and since Pol(G) separates points in G we conclude that ϕ(h) = ψ(h). If Ψ is moreover
assumed bijective, then Ψ−1 is also a strongly unital, comultiplicative algebra homomorphism
and is therefore induced by a unique group homomorphism ψ : H → G. Again by the uniqueness of the homomorphism, it follows that ψ ◦ ϕ = idG and ψ ◦ ϕ = idH .

Remark 7.10. If G and H are Mal’cev groups then by Theorem 6.18, the restriction map
Pol(G ⊗ R) → Pol(G) is a strongly unital, comultiplicative (degree preserving) algebra isomorphism. Thus, if Ψ : Pol(G) → Pol(H) is a strongly unital, comultiplicative, algebra homomorphism then, by Theorem 7.9, it is induced by a group homomorphism ψ : H ⊗ R → G ⊗ R
at the level of Mal’cev completions which is an isomorphism exactly when Ψ is bijective.
References
[1] Ararat Babakhanian, Cohomological Methods in Group Theory, 1972. ↑16
[2] Uri Bader and Christian Rosendal and Roman Sauer, On the cohomology of weakly almost periodic group
representations, J. Topology and Analysis 6 (2014), 153–165. ↑9
[3] Gilbert Baumslag, Lecture Notes on Nilpotent Groups, 1971. ↑5, 6
[4] Philippe Blanc, Sur la cohomologie continue des groupes localement compacts, Ann. Sci. École Norm. Sup.
12 (1979), 137–168. ↑9
[5] Armand Borel, Linear algebraic groups, Second, Graduate Texts in Mathematics, vol. 126, Springer-Verlag,
New York, 1991. ↑27, 30
[6] Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups and their applications, Part I: Basic theory and examples, 1990. ↑5, 6, 7, 8, 21, 27
[7] Patrick Delorme, 1-Cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations, Bull. Soc. Math. France 105 (1977), 281–336. ↑9
[8] Alain Guichardet, Cohomologie des groupes topologies et des algèbres de Lie, 1980. ↑2, 3, 4, 5, 7, 8, 9, 10,
11, 12, 18, 26
[9] Stephen A. Jennings, The group ring of a class of infinite nilpotent groups, Canad. J. Math. 7 (1955),
169–187. ↑6
[10] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, Springer-Verlag,
New York, 1995. ↑19, 21
[11] Alexander Kirillov, An Introduction to Lie Groups and Lie Algebras, Cambridge Studies in Advanced
Mathematics, vol. 113, Cambridge University Press, 2008. ↑5
[12] Michel Lazard, Sur les groupes nilpotents et les anneaux de Lie, Annales Scientifiques de l’É.N.S. 71
(1954), 101–190. ↑16
[13] Alexander Leibman, Polynomial mappings of groups, Israel Journal of Mathematics 129 (2002), 29–60.
Erratum available. ↑16, 19, 22, 23, 24, 27, 28
[14] Anatoly Mal’cev, On a class of homogeneous spaces, AMS Translation 39 (1951). ↑6
[15] Christian Rosendal, Automatic continuity of group homomorphisms, Bulletin of Symbolic Logic 15 (2009),
no. 2, 184–214. ↑20
[16] Yehuda Shalom, Harmonic analysis, cohomology, and large-scale geometry of amenable groups, Acta Math.
192 (2004), 119–185. ↑9

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[17] Hsien-Chung Wang, discrete subgroups of solvable Lie groups, I, Annals of Mathematics 64 (1956), 1–19.
↑7
[18] George Willis, Totally disconnected, nilpotent, locally compact groups, Bulletin of the Australian Mathematical Society 55 (1997), 143–146. ↑7
David Kyed, Department of Mathematics and Computer Science, University of Southern
Denmark, Campusvej 55, DK-5230 Odense M, Denmark
E-mail address: dkyed@imada.sdu.dk

