Cohomology of Restricted Filiform Lie Algebras mλ₂(p)
For the p-dimensional filiform Lie algebra m₂(p) over a field F of prime characteristic p≥5 with nonzero Lie brackets [e₁,eᵢ]=eᵢ₊₁ for 1 < i < p and [e₂,eᵢ]=eᵢ₊₂ for 2 < i < p − 1, we show that there is a family mλ₂(p) of restricted Lie algebra structures parameterized by elements λ ∈ ᵖ...
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2019 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2019
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210293 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Cohomology of Restricted Filiform Lie Algebras mλ₂(p) / T.J. Evans, A. Fialowski // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 15 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859820256190529536 |
|---|---|
| author | Evans, T.J. Fialowski, A. |
| author_facet | Evans, T.J. Fialowski, A. |
| citation_txt | Cohomology of Restricted Filiform Lie Algebras mλ₂(p) / T.J. Evans, A. Fialowski // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 15 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | For the p-dimensional filiform Lie algebra m₂(p) over a field F of prime characteristic p≥5 with nonzero Lie brackets [e₁,eᵢ]=eᵢ₊₁ for 1 < i < p and [e₂,eᵢ]=eᵢ₊₂ for 2 < i < p − 1, we show that there is a family mλ₂(p) of restricted Lie algebra structures parameterized by elements λ ∈ ᵖ. We explicitly describe bases for the ordinary and restricted 1- and 2-cohomology spaces with trivial coefficients, and give formulas for the bracket and [p]-operations in the corresponding restricted one-dimensional central extensions.
|
| first_indexed | 2025-12-07T21:25:03Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 095, 11 pages
Cohomology of Restricted Filiform Lie Algebras mλ
2(p)
Tyler J. EVANS † and Alice FIALOWSKI ‡§
† Department of Mathematics, Humboldt State University, Arcata, CA 95521, USA
E-mail: evans@humboldt.edu
URL: https://sites.google.com/humboldt.edu/tylerjevans
‡ Institute of Mathematics, University of Pécs, Pécs, Hungary
E-mail: fialowsk@ttk.pte.hu
§ Institute of Mathematics Eötvös Loránd University, Budapest, Hungary
E-mail: fialowsk@cs.elte.hu
Received August 19, 2019, in final form November 24, 2019; Published online December 01, 2019
https://doi.org/10.3842/SIGMA.2019.095
Abstract. For the p-dimensional filiform Lie algebra m2(p) over a field F of prime charac-
teristic p ≥ 5 with nonzero Lie brackets [e1, ei] = ei+1 for 1 < i < p and [e2, ei] = ei+2 for
2 < i < p− 1, we show that there is a family mλ2 (p) of restricted Lie algebra structures pa-
rameterized by elements λ ∈ Fp. We explicitly describe bases for the ordinary and restricted
1- and 2-cohomology spaces with trivial coefficients, and give formulas for the bracket and
[p]-operations in the corresponding restricted one-dimensional central extensions.
Key words: restricted Lie algebra; central extension; cohomology; filiform Lie algebra
2010 Mathematics Subject Classification: 17B50; 17B56
We dedicate this paper to Dmitry B. Fuchs
on the occasion of his 80th birthday
1 Introduction
N-graded Lie algebras of maximal class have been intensively studied in the last decade. A Lie
algebra of maximal class is a graded Lie algebra
g = ⊕∞i=1gi
over a field F, where dim(g1) = dim(g2) = 1, dim(gi) ≤ 1 for i ≥ 3 and [g1, gi] = gi+1 for i ≥ 1.
A Lie algebra of dimension n is called filiform if
dim
(
gk
)
= n− k, 2 ≤ k ≤ n, where gk =
[
g, gk−1
]
.
Lie algebras of maximal class with two generators over fields of characteristic zero have been
classified, and exactly three of these algebras are of filiform type [9]. We list them with the
nontrivial bracket structures:
m0 : [e1, ei] = ei+1, i ≥ 2,
m2 : [e1, ei] = ei+1, i ≥ 2,
[e2, ej ] = ej+2, j ≥ 3,
V : [ei, ej ] = (j − i)ei+j , i, j ≥ 1.
This paper is a contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor
of Dmitry Fuchs. The full collection is available at https://www.emis.de/journals/SIGMA/Fuchs.html
mailto:evans@humboldt.edu
https://sites.google.com/humboldt.edu/tylerjevans
mailto:fialowsk@ttk.pte.hu
mailto:fialowsk@cs.elte.hu
https://doi.org/10.3842/SIGMA.2019.095
https://www.emis.de/journals/SIGMA/Fuchs.html
2 T.J. Evans and A. Fialowski
Filiform N-graded Lie algebras g of dimension n over a field of characteristic zero that satisfy
[g1, gi] = gi+1 and dim(gi) = 1 for i < n (which is equivalent to having 2 generators) are
classified in [14]. They include the natural “truncations” of m0(n) and m2(n) obtained by taking
the quotient by the ideal generated by en+1. The algebra V (the Witt algebra) is isomorphic to
the algebra of derivations of the polynomial algebra F[x]. If F has characteristic p > 0, then the
truncation V(p) of V is the derivation algebra of the quotient of F[x] by the ideal generated by
xp − 1. The algebra V(p) is called the (modular) Witt algebra.
The above picture is more complicated in the modular case (that is, over fields of positive
characteristic), see [1, 2, 13], but m0, m2, V and their truncations always show up. We refer
the reader to the book [15] for a general treatment of modular Lie algebras. In this paper, we
show that if the field F has characteristic p ≥ 5, then the Lie algebra m2(p) admits a family
of restricted Lie algebra structures mλ
2(p) parameterized by elements λ ∈ Fp. We describe the
isomorphism classes of these algebras, calculate the ordinary and restricted cohomology spaces
with trivial coefficients Hq
(
mλ
2(p)
)
and Hq
∗
(
mλ
2(p)
)
for q = 1, 2 and give explicit bases for those
spaces. We also give the bracket structures and [p]-operations for the corresponding restricted
one-dimensional central extensions of these restricted Lie algebras.
With this, we complete the description of all three types of truncated filiform restricted
Lie algebras (mλ
0(p), mλ
2(p), and V(p)), their low dimensional cohomology spaces with trivial
coefficients and their restricted one-dimensional central extensions. The algebras mλ
0(p) were
studied in [5], and the algebra V(p) was studied in [6] (where it is denoted by W ).
Remark 1.1. For p = 2 and p = 3, m0(p) = m2(p) so these algebras were treated in [5].
In this paper, all cochain and cohomology spaces are with coefficients in the trivial F-module.
The organization is as follows. In Section 2 we construct the restricted Lie algebra family mλ
2(p),
determine the isomorphism classes of these restricted Lie algebras, and describe both the ordi-
nary and restricted 1- and 2-cochains, including formulas for all differentials. In Section 3 we
calculate both the ordinary and restricted 1-cohomology by giving explicit cocycles. Section 4
contains the calculation of the ordinary and restricted 2-cohomology spaces, again by giving
explicit cocycles. In Section 5 we describe all restricted one-dimensional central extensions and
give their brackets and [p]-operations.
2 Preliminaries
2.1 The Lie algebra m2(p)
Let p ≥ 5 be a prime, and let F be a field of characteristic p. Define the F-vector space
m2(p) = spanF({e1, . . . , ep}),
and define a bracket on m2(p) by
[e1, ei] = ei+1, 1 < i < p,
[e2, ei] = ei+2, 2 < i < p− 1,
with all other brackets [ei, ej ] (for i < j) being 0. Note that m2(p) is a graded Lie algebra
with k-th graded component (m2(p))k = Fek for 1 ≤ k ≤ p. If αi, βi ∈ F and g =
∑p
i=1 αiei,
h =
∑p
i=1 βiei, then
[g, h] = (α1β2 − α2β1)e3 + (α1β3 − α3β1)e4
+
p∑
j=5
((α1βj−1 − αj−1β1) + (α2βj−2 − αj−2β2))ej . (2.1)
Cohomology of Restricted Filiform Lie Algebras mλ
2(p) 3
2.2 The restricted Lie algebras mλ
2(p)
We refer the reader to [12, Chapter V, Section 7] and [15, Section 2.2] for the definition of
a restricted Lie algebra, and for the construction of the [p]-mapping on a given Lie algebra
(m2(p) in the current paper) used below. For any j ≥ 2 and g1, . . . , gj ∈ m2(p), we denote the
j-fold bracket
[g1, g2, g3, . . . , gj ] = [[. . . [[g1, g2], g3], . . . ], gj ].
Since p ≥ 5, (2.1) implies that the center of the algebra is Z(m2(p)) = Fep, and p-fold brackets
are zero. Therefore for each λ = (λ1, . . . , λp) ∈ Fp, setting e
[p]
k = λkep for each k defines
a restricted Lie algebra that we denote by mλ
2(p). Because p-fold brackets in mλ
2(p) are zero, for
all g, h ∈ mλ
2(p), α ∈ F,
(g + h)[p] = g[p] + h[p] and (αg)[p] = αpg[p],
and therefore the [p]-mapping on mλ
2(p) is p-semilinear (see also [15, Chapter 2, Lemma 1.2]).
From this we get that if g =
∑
αkek ∈ mλ
2(p), then
g[p] =
(
p∑
k=1
αpkλk
)
ep. (2.2)
Everywhere below, we write mλ
2(p) to denote both the graded Lie algebra m2(p) and the graded
restricted Lie algebra mλ
2(p) for a given λ ∈ Fp. The Lie brackets and restricted [p]-operators
for these algebras are explicitly given by (2.1) and (2.2), respectively.
Remark 2.1. For p = 2 there are several other possible [2]-mappings, namely any 2-semilinear
transformation on m2(2).
2.3 Isomorphism classes
Proposition 2.2. Let p ≥ 5. If λ, λ′ ∈ Fp, the graded restricted Lie algebras mλ
2(p) and mλ′
2 (p)
are isomorphic if and only if there exists a non-zero µ ∈ F such that λk = µ(k−1)pλ′k for k =
1, . . . , p.
Proof. We only consider isomorphisms that preserve the grading as we are interested in these
algebras as graded restricted Lie algebras. Assume that there exists a graded restricted Lie
algebra isomorphism ϕ : mλ
2(p) → mλ′
2 (p), and let ϕ(e1) = µe1, ϕ(e2) = νe2 for some non-zero
µ, ν ∈ F. Since ϕ preserves the Lie bracket, we must have ϕ(e3) = µνe3, ϕ(e4) = µ2νe4,
ϕ(e5) = µ3νe5. On the other hand, as [e1, e4] = [e2, e3], we also must have ϕ(e5) = µν2e5. From
this it follows that ν = µ2 and ϕ(ek) = µkek for k = 1, . . . , p.
Moreover, ϕ preserves the restricted [p]-structure so that
ϕ(e
[p]
k ) = ϕ(ek)
[p]′
for k = 1, . . . , p (here [p]′ denotes the restricted [p]-structure on mλ′
2 (p)). Now,
ϕ(e
[p]
k ) = ϕ(λkep) = λkµ
pep and ϕ(ek)
[p]′ = (µkek)
[p]′ = µkpλ′kep
so λkµ
p = µkpλ′k, and hence
λk = µ(k−1)pλ′k.
It remains to show that the above condition on λk gives rise to a graded restricted Lie algebra
isomorphism between mλ
2(p) and mλ′
2 (p). If, for 0 6= µ ∈ F, we define ϕ(e1) = µe1, ϕ(ek) = µkek
(2 ≤ k ≤ p), then it is easy to check that the argument above is reversible, and we obtain
a graded isomorphism between the restricted Lie algebras. �
4 T.J. Evans and A. Fialowski
2.4 Cochain complexes with trivial coefficients
For the convenience of the reader and to establish our notations, we briefly recall the definitions
of the cochain spaces used below to compute both the ordinary and restricted Lie algebra 1-
and 2-cohomology. The reader can find more details on these complexes in [3, 4, 5, 10, 11].
2.4.1 Ordinary cochain complex
For ordinary Lie algebra cohomology with trivial coefficients, the relevant cochain spaces from
the Chevalley–Eilenberg complex (with bases) for our purposes are
C0
(
mλ
2(p)
)
= F, {1},
C1
(
mλ
2(p)
)
= mλ
2(p)′,
{
ek | 1 ≤ k ≤ p
}
,
C2
(
mλ
2(p)
)
=
(
∧2mλ
2(p)
)′
,
{
ei,j | 1 ≤ i < j ≤ p
}
,
C3
(
mλ
2(p)
)
=
(
∧3mλ
2(p)
)′
,
{
es,t,u | 1 ≤ s < t < u ≤ p
}
,
(V ′ denotes the dual vector space) and the differentials are defined by
d0 : C0
(
mλ
2(p)
)
→ C1
(
mλ
2(p)
)
, d0 = 0,
d1 : C1
(
mλ
2(p)
)
→ C2
(
mλ
2(p)
)
, d1(ψ)(g, h) = ψ([g, h]),
d2 : C2
(
mλ
2(p)
)
→ C3
(
mλ
2(p)
)
, d2(ϕ)(g, h, f) = ϕ([g, h] ∧ f)− ϕ([g, f ] ∧ h)
+ ϕ([h, f ] ∧ g).
The cochain spaces Cn
(
mλ
2(p)
)
are graded
C1
k
(
mλ
2(p)
)
= span
({
ek
})
, 1 ≤ k ≤ p,
C2
k
(
mλ
2(p)
)
= span
({
ei,j
})
, 1 ≤ i < j ≤ p, i+ j = k, 3 ≤ k ≤ 2p− 1,
C3
k
(
mλ
2(p)
)
= span
({
es,t,u
})
, 1 ≤ s < t < u ≤ p, s+ t+ u = k, 6 ≤ k ≤ 3p− 3,
and the differentials are graded maps. If we adopt the convention that ei,j = 0 whenever j ≤ i,
we can write for 1 ≤ k ≤ p
d1
(
ek
)
= e1,k−1 + e2,k−2. (2.3)
Using the convention that ei,j,k = 0 unless i < j < k, we can write
d2
(
e1,j
)
= −e1,2,j−2, 2 ≤ j,
d2
(
ei,j
)
= e1,i−1,j + e1,i,j−1 + e2,i−2,j + e2,i,j−2, 2 ≤ i < j ≤ p. (2.4)
2.4.2 Restricted cochain complex
The relevant restricted cochain spaces are
C0
∗
(
mλ
2(p)
)
= C0
(
mλ
2(p)
)
,
C1
∗
(
mλ
2(p)
)
= C1
(
mλ
2(p)
)
,
C2
∗
(
mλ
2(p)
)
=
{
(ϕ, ω) |ϕ ∈ C2(mλ
2(p)), ω : mλ
2(p)→ F
has the ∗-property with respect to ϕ
}
C3
∗
(
mλ
2(p)
)
=
{
(ζ, η) | ζ ∈ C3(mλ
2(p)), η : mλ
2(p)×mλ
2(p)→ F
}
.
Cohomology of Restricted Filiform Lie Algebras mλ
2(p) 5
We recall that if ϕ ∈ C2
(
mλ
2(p)
)
, then a map ω : mλ
2(p) → F has the ∗-property with respect
to ϕ if for all α ∈ F and all g, h ∈ mλ
2(p) we have ω(αg) = αpω(g) and
ω(g + h) = ω(g) + ω(h) +
∑
gi=g or h
g1=g, g2=h
1
#(g)
ϕ([g1, g2, . . . , gp−1] ∧ gp). (2.5)
Here #(g) is the number of factors gi equal to g. Moreover, given ϕ, we can assign the values
of ω arbitrarily on a basis for mλ
2(p) and use (2.5) to define ω : mλ
2(p) → F that has the ∗-
property with respect to ϕ (see [5, pp. 249–250]). Recall the space of Frobenius homomorphisms
HomFr(V,W ) from the F-vector space V to the F-vector space W is defined by
HomFr(V,W ) =
{
f : V →W | f(αx+ βy) = αpf(x) + βpf(y)
}
for all α, β ∈ F and x, y ∈ V . A map ω : mλ
2(p) → F has the ∗-property with respect to ϕ = 0
if and only if ω ∈ HomFr(m
λ
2(p),F). In particular, if 1 ≤ k ≤ p, then the map ek : mλ
2(p) → F
defined by
ek
(
p∑
i=1
αiei
)
= αpk,
has the ∗-property with respect to 0.
We will use the following bases for the restricted cochains
C0
∗
(
mλ
2(p)
)
{1},
C1
∗
(
mλ
2(p)
) {
ek | 1 ≤ k ≤ p
}
,
C2
∗
(
mλ
2(p)
) {(
ei,j , ẽi,j
)
| 1 ≤ i < j ≤ p
}
∪
{(
0, ek
)
| 1 ≤ k ≤ p
}
,
where ẽi,j is the map ẽi,j : mλ
2(p) → F that vanishes on the basis and has the ∗-property with
respect to ei,j . More generally, given ϕ ∈ C2(mλ
2(p)), let ϕ̃ : mλ
2(p)→ F be the map that vanishes
on the basis for mλ
2(p) and has the ∗-property with respect to ϕ. The restricted differentials are
defined by
d0
∗ : C0
∗
(
mλ
2(p)
)
→ C1
∗
(
mλ
2(p)
)
d0
∗ = 0,
d1
∗ : C1
∗
(
mλ
2(p)
)
→ C2
∗
(
mλ
2(p)
)
d1
∗(ψ) =
(
d1(ψ), ind1(ψ)
)
,
d2
∗ : C2
∗
(
mλ
2(p)
)
→ C3
∗
(
mλ
2(p)
)
d2
∗(ϕ, ω) =
(
d2(ϕ), ind2(ϕ, ω)
)
,
where ind1(ψ)(g) := ψ
(
g[p]
)
and ind2(ϕ, ω)(g, h) := ϕ
(
g ∧ h[p]
)
. If ψ ∈ C1
∗
(
mλ
2(p)
)
and (ϕ, ω) ∈
C2
∗
(
mλ
2(p)
)
, then ind1(ψ) has the ∗-property with respect to d1(ψ) [7, Lemma 4]. If g =
∑
αiei,
h =
∑
βiei, ψ =
∑
µie
i and ϕ =
∑
σije
i,j , then
ind1(ψ)(g) = µp
p∑
j=1
αpjλj
and
ind2(ϕ, ω)(g, h) =
(
p∑
i=1
βpi λi
)p−1∑
j=1
αjσjp
. (2.6)
Remark 2.3. For a given ϕ ∈ C2
(
mλ
2(p)
)
, if (ϕ, ω), (ϕ, ω′) ∈ C2
∗
(
mλ
2(p)
)
, then d2
∗(ϕ, ω) =
d2
∗(ϕ, ω
′). In particular, with trivial coefficients, ind2(ϕ, ω) depends only on ϕ.
6 T.J. Evans and A. Fialowski
3 The cohomology H1
(
mλ
2(p)
)
and H1
∗
(
mλ
2(p)
)
In this short section we compute, for p ≥ 5, both the ordinary and restricted 1-cohomology
spaces H1(mλ
2(p)) and H1
∗ (m
λ
2(p)), and in particular we show that these spaces are equal.
Theorem 3.1. If p ≥ 5 and λ ∈ Fp, then
H1
(
mλ
2(p)
)
= H1
∗
(
mλ
2(p)
)
and the classes of
{
e1, e2
}
form a basis.
Proof. Easily, the differential (2.3) has a kernel spanned by
{
e1, e2
}
, and d0 = 0, so that
H1
(
mλ
2(p)
) ∼= Fe1 ⊕ Fe2.
As for any restricted Lie algebra, H1
∗
(
mλ
2(p)
)
consists of those ordinary cohomology classes
[ψ] ∈ H1
(
mλ
2(p)
)
for which ind1(ψ) = 0 (see [11, Theorem 2.1] or [7, Theorem 2]). If ψ =∑p
k=1 µke
k is any ordinary cocycle, then µp = 0 (p ≥ 5) so that for any g ∈ mλ
2(p), we have
ind1(ψ)(g) = ψ
(
g[p]
)
= µp
(
p∑
k=1
αpkλk
)
= 0,
and hence H1
∗
(
mλ
2(p)
)
= H1
(
mλ
2(p)
)
. �
Remark 3.2. An alternate proof is the following
H1
(
mλ
2(p)
) ∼= (mλ
2(p)/
[
mλ
2(p),mλ
2(p)
])′
,
where g′ = HomF(g,F) denotes the vector space dual of g and
H1
∗
(
mλ
2(p)
) ∼= (mλ
2(p)/
([
mλ
2(p),mλ
2(p)
]
+ span
(
mλ
2(p)[p]
)))′
(see [8, Proposition 2.7]).
In particular, since span
(
mλ
2(p)[p]
)
⊆ Fep ⊆
[
mλ
2(p),mλ
2(p)
]
, the ordinary and the restricted
1-cohomology spaces coincide.
Remark 3.3. For p ≥ 5, formula (2.3) shows that for 1 ≤ k ≤ p, d1(ek) = e1,k−1 + e2,k−2. For
k ≥ 3, let
ϕk = d1
(
ek
)
= e1,k−1 + e2,k−2.
The set {ϕ3, ϕ4, . . . , ϕp} is a basis for the image d1
(
C1
(
mλ
2(p)
))
.
4 The cohomology H2
(
mλ
2(p)
)
and H2
∗
(
mλ
2(p)
)
4.1 Ordinary cohomology
Lemma 4.1. Let p > 5. For 3 ≤ k ≤ 2p−1, let d2
k : C2
k
(
mλ
2(p)
)
→ C3
k
(
mλ
2(p)
)
denote the restric-
tion of the differential d2 : C2
(
mλ
2(p)
)
→ C3
(
mλ
2(p)
)
to the kth graded component C2
k
(
mλ
2(p)
)
of
C2
(
mλ
2(p)
)
. Then ker
(
d2
k
)
= 0 for k ≥ p+ 2, and for 3 ≤ k ≤ p+ 1, basis elements of ker
(
d2
k
)
are listed in the following table:
Cohomology of Restricted Filiform Lie Algebras mλ
2(p) 7
k = 3 ϕ3 = e1,2
k = 4 ϕ4 = e1,3
k = 5 e1,4, e2,3
k = 6 ϕ6 = e1,5 + e2,4
k = 7 ϕ7 = e1,6 + e2,5, e1,6 + e3,4
8 ≤ k ≤ p+ 1 ϕk = e1,k−1 + e2,k−2
Proof. A direct calculation using (2.4) shows that d2k = 0 for k = 3, 4, 5, d2
6
(
σ1,5e
1,5+σ2,4e
2,4
)
=
(−σ1,5 + σ2,4)e
1,2,3 and d2
7
(
σ1,6e
1,6 + σ2,5e
2,5 + σ3,4e
3,4
)
= (−σ1,6 + σ2,5 + σ3,4)e
1,2,4.
If k ≥ p + 2 and ϕ =
∑
σi,je
i,j ∈ C2
k
(
mλ
2(p)
)
, then (2.4) implies that d2
k(ϕ) will contain the
terms ∑
k−p≤i<s(k)
i+j=k
(σi,j + σi+1,j−i)e
1,i,j−1,
where s(k) = k/2− 1 if k is even and s(k) = (k− 1)/2 if k is odd. If, in addition, ϕ is a cocycle,
then we have the system of equations
σi,j + σi+1,j−i = 0, (4.1)
where k − p ≤ i < s(k) and i + j = k. Note that every coefficient σi,j of ϕ occurs in the
system (4.1). Now, if k < 2p− 2 is even, then d2
k(ϕ) contains exactly one term with e2,k/2−2,k/2,
and hence σk/2−2,k/2+2 = 0. The system (4.1) then implies that σi,j = 0 for all i, j and
hence ϕ = 0. Likewise, if k < 2p − 1 is odd, then e2,(k−1)/2−1,(k−1)/2 occurs once in d2
k(ϕ)
forcing σ(k−1)/2−1,(k−1)/2+2 = 0, and hence ϕ = 0. We can use (2.4) to directly check that
ker
(
d2
2p−2
)
= ker
(
d2
2p−1
)
= 0 so that ker
(
d2
k
)
= 0 for k ≥ p+ 2.
Finally, if 8 ≤ k ≤ p+ 1, then d2
k(ϕ) will also contain the terms∑
2≤i<s(k)
i+j=k
(σi,j + σi+1,j−i)e
1,i,j−1
in addition to the term −σ1,k−1e1,2,k−3. If ϕ is a cocycle, then we have the system of equations
−σ1,k−1 + σ2,k−2 + σ3,k−3 = 0,
σi,j + σi+1,j−i = 0, 3 ≤ i < s(k). (4.2)
The same argument used for k ≥ p + 2 above shows that the system (4.2) implies σi,j = 0 for
i ≥ 3. Therefore σ1,k−1 = σ2,k−2 and ϕk = e1,k−1 + e2,k−2 spans ker
(
d2
k
)
. �
Theorem 4.2. If p = 5, then
dim
(
H2
(
mλ
2(5)
))
= 3
and the cohomology classes of the cocycles
{
e1,4, e1,5 + e2,4, e2,5 − e3,4
}
form a basis.
If p > 5, then
dim
(
H2
(
mλ
2(p)
))
= 3
and the cohomology classes of the cocycles
{
e1,4, e1,6 + e3,4, e1,p + e2,p−1
}
form a basis.
8 T.J. Evans and A. Fialowski
Proof. If p = 5, then the results of Lemma 4.1 still hold except for k = 7, 8 where we have
d2
7
(
σ2,5e
2,5 + σ3,4e
3,4
)
=
(
σ2,5 + σ3,4
)
e1,2,4 and ker
(
d2
8
)
= 0. It follows that{
e1,2, e1,3, e1,4, e2,3, e1,5 + e2,4, e2,5 − e3,4
}
is a basis for ker
(
d2
)
. We can replace e2,3 with ϕ5 = e1,4 + e2,3 in this basis so that, by
Remark 3.3, the classes of
{
e1,4, e1,5 + e2,4, e2,5 − e3,4
}
form a basis for H2
(
mλ
2(5)
)
.
If p > 5, then Lemma 4.1 gives a basis for ker
(
d2
)
. We can again replace e2,3 with ϕ5 =
e1,4 + e2,3 in this basis so that, by Remark 3.3, the classes of
{
e1,4, e1,6 + e3,4, e1,p + e2,p−1
}
form
a basis for H2
(
mλ
2(p)
)
. �
4.2 Restricted cohomology for λ = 0
If λ = 0, then (2.6) shows that ind2 = 0 so that every ordinary 2-cocycle ϕ ∈ C2
(
m0
2(p)
)
gives
rise to a restricted 2-cocycle (ϕ, ϕ̃) ∈ C2
∗
(
m0
2(p)
)
. Moreover, in the case that ϕ = d1(ψ) is
a 1-coboundary, we can replace ϕ̃ with ind1(ψ) and
(
ϕ, ind1(ψ)
)
=
(
d1(ψ), ind1(ψ)
)
= d1
∗(ψ)
is a restricted 1-coboundary as well, and d2
∗(ϕ, ϕ̃) = d2
∗
(
ϕ, ind1(ψ)
)
by Remark 2.3. Finally,
d2
∗
(
0, ek
)
= (0, 0) for all 1 ≤ k ≤ p, and the
(
0, ek
)
are clearly linearly independent. Together
these remarks prove the following
Theorem 4.3. Let λ = 0. If p = 5, then
dim
(
H2
∗
(
m0
2(5)
))
= 8
and the cohomology classes of the cocycles{(
0, e1
)
,
(
0, e2
)
,
(
0, e3
)
,
(
0, e4
)
,
(
0, e5
)
,
(
e1,4, ẽ1,4
)
,
(
ϕ6, ϕ̃6
)
,
(
ξ, ξ̃
)}
form a basis where ξ = e2,5 − e3,4.
If p > 5, then
dim
(
H2
∗
(
m0
2(p)
))
= p+ 3
and the cohomology classes of{(
0, e1
)
, . . . ,
(
0, ep
)
,
(
e1,4, ẽ1,4
)
,
(
η, η̃
)
,
(
ϕp+1, ϕ̃p+1
)}
form a basis where η = e1,6 + e3,4.
Remark 4.4. If p ≥ 5, the maps ϕ̃k are identically zero for k < p+ 1 because the (p− 1)-fold
bracket in (2.5) always gives a multiple of ep so that ϕk vanishes on ep ∧mλ
2(p) when k < p+ 1.
This, in turn, implies that ϕ̃k ∈ HomFr
(
mλ
2(p),F
)
, and since ϕ̃k(ei) = 0 for all i, we have ϕ̃k = 0.
Likewise, ẽ1,4 = 0, η̃ = 0, and ξ̃ = 0 unless p = 5. The restriction of ϕp+1 to ep ∧mλ
2(p) is equal
to e1,p so that ϕ̃p+1 = ẽ1,p. If p = 5, then the restriction of ξ to e5 ∧ mλ
2(5) is equal to e2,5 so
ξ̃ = ẽ2,5. We can then use (2.5) to give explicit descriptions for ϕ̃p+1 and ξ̃ (when p = 5):
ϕ̃p+1
(
p∑
i=1
αiei
)
= ẽ1,p
(
p∑
i=1
αiei
)
= αp−11 α2,
ξ̃
(
5∑
i=1
αiei
)
= ẽ2,5
(
5∑
i=1
αiei
)
=
1
2
α3
1α
2
2.
Remark 4.5. The dimensions in Theorem 4.3 can also be deduced from Theorems 3.1 and 4.2
and the six-term exact sequence in [11] precisely as in [5, Remark 4].
Cohomology of Restricted Filiform Lie Algebras mλ
2(p) 9
4.3 Restricted cohomology for λ 6= 0
If ϕ =
∑
σije
i,j and (ϕ, ω) ∈ C2
∗
(
mλ
2
)
, then (2.6) shows that
ind2(ϕ, ω)(ej , ei) = λiσjp.
Therefore, if λ 6= 0, then d2
∗(ϕ, ω) =
(
d2ϕ, ind2(ϕ, ω)
)
= (0, 0) if and only if d2ϕ = 0 and
σ1p = σ2p = · · · = σp−1p = 0.
Theorem 4.6. If p = 5, then
dim
(
H2
∗
(
mλ
2(5)
))
= 6
and the cohomology classes of{(
0, e1
)
,
(
0, e2
)
,
(
0, e3
)
,
(
0, e4
)
,
(
0, e5
)
,
(
e1,4, ẽ1,4
)}
form a basis.
If p > 5, then
dim
(
H2
∗
(
mλ
2(p)
))
= p+ 2
and the cohomology classes of{(
0, e1
)
, . . . ,
(
0, ep
)
,
(
e1,4, ẽ1,4
)
,
(
η, η̃
)}
form a basis where η = e1,6 + e3,4.
5 One-dimensional central extensions
One-dimensional central extensions E = g⊕ Fc of an ordinary Lie algebra g are parameterized
by the cohomology group H2(g) [10, Chapter 1, Section 4.6], and restricted one-dimensional
central extensions of a restricted Lie algebra g with c[p] = 0 are parameterized by the restricted
cohomology group H2
∗ (g) [11, Theorem 3.3]. If (ϕ, ω) ∈ C2
∗ (g) is a restricted 2-cocycle, then
the corresponding restricted one-dimensional central extension E = g⊕ Fc has Lie bracket and
[p]-operation defined by
[g, h] = [g, h]g + ϕ(g ∧ h)c,
[g, c] = 0,
g[p] = p[p]g + ω(g)c,
c[p] = 0, (5.1)
where [·, ·]g and ·[p]g denote the Lie bracket and [p]-operation in g, respectively [7, equations (26)
and (27)]. We can use (5.1) to explicitly describe the restricted one-dimensional central ex-
tensions corresponding to the restricted cocycles in Theorems 4.3 and 4.6. For the rest of this
section, let g =
∑
αiei and h =
∑
βiei denote two arbitrary elements of mλ
2(p).
Let Ek = mλ
2(p) ⊕ Fc denote the one-dimensional restricted central extension of mλ
2(p) de-
termined by the cohomology class of the restricted cocycle
(
0, ek
)
. Then just as with mλ
0(p)
and V(p) (see [5, Theorem 5.1] and [6, Theorem 3.1]), the
(
0, ek
)
span a p-dimensional subspace
of H2
∗ , and (5.1) gives the bracket and [p]-operation in Ek:
[g, h] = [g, h]mλ2 (p)
,
10 T.J. Evans and A. Fialowski
[g, c] = 0,
g[p] = g
[p]
mλ2 (p) + αpkc,
c[p] = 0.
For restricted cocycles (ϕ, ϕ̃) with ϕ 6= 0, we summarize the corresponding restricted one-
dimensional central extensions E(ϕ,ϕ̃) in the following tables. Everywhere in the tables, we omit
the brackets [g, c] = 0 and [p]-operation c[p] = 0 for brevity.
If λ = 0, then there are three restricted cocycles (ϕ, ϕ̃) with ϕ 6= 0 for a given prime
(Theorem 4.3). We note that if λ = 0, then (2.2) implies g
[p]
mλ2 (p) = 0 for all g ∈ mλ
2(p).
Table 1. Restricted one-dimensional central extensions with ϕ 6= 0 and λ = 0.
p = 5(
e1,4, 0
)
[g, h] = [g, h]mλ2 (p)
+ (α1β4 − α4β1)c
g[p] = 0(
ξ, ξ̃
)
[g, h] = [g, h]mλ2 (p)
+ (α2β5 − α5β2 − α3β4 + α4β3)c
ξ = e2,5 − e3,4 g[p] = 1
2α
3
1α
2
2c
(ϕ6, ϕ̃6) [g, h] = [g, h]mλ2 (p)
+ (α1β5 − α5β1 + α2β4 − α4β2)c
g[p] = 1
2α
4
1α2c
p > 5(
e1,4, 0
)
[g, h] = [g, h]mλ2 (p)
+ (α1β4 − α4β1)c
g[p] = 0
(η, 0) [g, h] = [g, h]mλ2 (p)
+ (α1β6 − α6β1 + α3β4 − α4β3)c
η = e1,6 + e3,4 g[p] = 0
(ϕp+1, ϕ̃p+1) [g, h] = [g, h]mλ2 (p)
+ (α1βp − αpβ1 + α2βp−1 − αp−1β2)c
g[p] = αp−11 α2c
If λ 6= 0 and p = 5, then the only restricted cocycle (ϕ, ϕ̃) with ϕ 6= 0 is (e1,4, 0). If p > 5,
then the restricted cocycles (ϕ, ϕ̃) with ϕ 6= 0 are (e1,4, 0) and (η, 0) (Theorem 4.6).
Table 2. Restricted one-dimensional central extensions with ϕ 6= 0 and λ 6= 0.
p = 5(
e1,4, 0
)
[g, h] = [g, h]mλ2 (p)
+ (α1β4 − α4β1)c
g[p] = g
[p]
mλ2 (p)
p > 5(
e1,4, 0
)
[g, h] = [g, h]mλ2 (p)
+ (α1β4 − α4β1)c
g[p] = g
[p]
mλ2 (p)
(η, 0) [g, h] = [g, h]mλ2 (p)
+ (α1β6 − α6β1 + α3β4 − α4β3)c
η = e1,6 + e3,4 g[p] = g
[p]
mλ2 (p)
Acknowledgements
The authors are grateful to Dmitry Fuchs for fruitful conversations, and the referees whose
comments greatly improved the exposition of this paper.
Cohomology of Restricted Filiform Lie Algebras mλ
2(p) 11
References
[1] Caranti A., Mattarei S., Newman M.F., Graded Lie algebras of maximal class, Trans. Amer. Math. Soc.
349 (1997), 4021–4051.
[2] Caranti A., Newman M.F., Graded Lie algebras of maximal class. II, J. Algebra 229 (2000), 750–784,
arXiv:math.RA/9906160.
[3] Chevalley C., Eilenberg S., Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63
(1948), 85–124.
[4] Evans T.J., Fialowski A., Restricted one-dimensional central extensions of restricted simple Lie algebras,
Linear Algebra Appl. 513 (2017), 96–102, arXiv:1506.09025.
[5] Evans T.J., Fialowski A., Restricted one-dimensional central extensions of the restricted filiform Lie algebras
mλ0 (p), Linear Algebra Appl. 565 (2019), 244–257, arXiv:1801.08178.
[6] Evans T.J., Fialowski A., Penkava M., Restricted cohomology of modular Witt algebras, Proc. Amer. Math.
Soc. 144 (2016), 1877–1886, arXiv:1502.04531.
[7] Evans T.J., Fuchs D., A complex for the cohomology of restricted Lie algebras, J. Fixed Point Theory Appl.
3 (2008), 159–179.
[8] Feldvoss J., On the cohomology of restricted Lie algebras, Comm. Algebra 19 (1991), 2865–2906.
[9] Fialowski A., On the classification of graded Lie algebras with two generators, Moscow Univ. Math. Bull.
38 (1983), 76–79.
[10] Fuks D.B., Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants
Bureau, New York, 1986.
[11] Hochschild G., Cohomology of restricted Lie algebras, Amer. J. Math. 76 (1954), 555–580.
[12] Jacobson N., Lie algebras, Interscience Tracts in Pure and Applied Mathematics, Vol. 10, Interscience
Publishers, New York – London, 1962.
[13] Jurman G., Graded Lie algebras of maximal class. III, J. Algebra 284 (2005), 435–461.
[14] Millionschikov D.V., Graded filiform Lie algebras and symplectic nilmanifolds, in Geometry, Topology, and
Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 212, Amer. Math. Soc., Providence, RI, 2004,
259–279, arXiv:math.RA/0205042.
[15] Strade H., Farnsteiner R., Modular Lie algebras and their representations, Monographs and Textbooks in
Pure and Applied Mathematics, Vol. 116, Marcel Dekker, Inc., New York, 1988.
https://doi.org/10.1090/S0002-9947-97-02005-9
https://doi.org/10.1006/jabr.2000.8316
https://arxiv.org/abs/math.RA/9906160
https://doi.org/10.2307/1990637
https://doi.org/10.1016/j.laa.2016.09.037
https://arxiv.org/abs/1506.09025
https://doi.org/10.1016/j.laa.2018.12.005
https://arxiv.org/abs/1801.08178
https://doi.org/10.1090/proc/12863
https://doi.org/10.1090/proc/12863
https://arxiv.org/abs/1502.04531
https://doi.org/10.1007/s11784-008-0060-y
https://doi.org/10.1080/00927879108824299
https://doi.org/10.2307/2372701
https://doi.org/10.1016/j.jalgebra.2004.11.006
https://doi.org/10.1090/trans2/212/13
https://arxiv.org/abs/math.RA/0205042
1 Introduction
2 Preliminaries
2.1 The Lie algebra m2(p)
2.2 The restricted Lie algebras m2(p)
2.3 Isomorphism classes
2.4 Cochain complexes with trivial coefficients
2.4.1 Ordinary cochain complex
2.4.2 Restricted cochain complex
3 The cohomology H1(to.m2(p))to. and H1*(to.m2(p))to.
4 The cohomology H2(to.m2(p))to. and H2*(to.m2(p))to.
4.1 Ordinary cohomology
4.2 Restricted cohomology for =0
4.3 Restricted cohomology for =0
5 One-dimensional central extensions
References
|
| id | nasplib_isofts_kiev_ua-123456789-210293 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:25:03Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Evans, T.J. Fialowski, A. 2025-12-05T09:23:05Z 2019 Cohomology of Restricted Filiform Lie Algebras mλ₂(p) / T.J. Evans, A. Fialowski // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 15 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B50; 17B56 arXiv: 1901.07532 https://nasplib.isofts.kiev.ua/handle/123456789/210293 https://doi.org/10.3842/SIGMA.2019.095 For the p-dimensional filiform Lie algebra m₂(p) over a field F of prime characteristic p≥5 with nonzero Lie brackets [e₁,eᵢ]=eᵢ₊₁ for 1 < i < p and [e₂,eᵢ]=eᵢ₊₂ for 2 < i < p − 1, we show that there is a family mλ₂(p) of restricted Lie algebra structures parameterized by elements λ ∈ ᵖ. We explicitly describe bases for the ordinary and restricted 1- and 2-cohomology spaces with trivial coefficients, and give formulas for the bracket and [p]-operations in the corresponding restricted one-dimensional central extensions. The authors are grateful to Dmitry Fuchs for fruitful conversations and the referees whose comments greatly improved the exposition of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Cohomology of Restricted Filiform Lie Algebras mλ₂(p) Article published earlier |
| spellingShingle | Cohomology of Restricted Filiform Lie Algebras mλ₂(p) Evans, T.J. Fialowski, A. |
| title | Cohomology of Restricted Filiform Lie Algebras mλ₂(p) |
| title_full | Cohomology of Restricted Filiform Lie Algebras mλ₂(p) |
| title_fullStr | Cohomology of Restricted Filiform Lie Algebras mλ₂(p) |
| title_full_unstemmed | Cohomology of Restricted Filiform Lie Algebras mλ₂(p) |
| title_short | Cohomology of Restricted Filiform Lie Algebras mλ₂(p) |
| title_sort | cohomology of restricted filiform lie algebras mλ₂(p) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210293 |
| work_keys_str_mv | AT evanstj cohomologyofrestrictedfiliformliealgebrasmλ2p AT fialowskia cohomologyofrestrictedfiliformliealgebrasmλ2p |