No Lie $p$-algebras of cohomological dimension one

0.1 MB·

by  Pasha Zusmanovich

NO LIE p-ALGEBRAS OF COHOMOLOGICAL DIMENSION ONE
PASHAZUSMANOVICH
ABSTRACT. We prove that a Lie p-algebra of cohomological dimension one is one-dimensional, and
discussrelatedquestions.
0. INTRODUCTION
6
1 AcohomologicaldimensionofaLiealgebraLoverafieldK,denotedbycd(L),isdefinedastheright
0 projective dimension of the trivial L-module K, i.e., the minimal possible length of a finite projective
2
resolution

g
u (1) ···→P →P →P →K
2 1 0
A
consisting of right projective modules P over the universal enveloping algebra U(L), or infinity if no
i
0
such finite resolution exists. Since for every projectiveresolution (1) and every L-moduleM, the coho-
2
mologyoftheinducedcomplex

]
A 0→M =Hom (K,M)→Hom (P ,M)→Hom (P ,M)→...
U(L) U(L) 1 U(L) 2
R
of L-modules coincides with the Chevalley–Eilenberg cohomology H•(L,M), L has cohomological di-
.
h mension n if and only if there is an L-module M such that Hn(L,M) 6= 0, and one of the following
t
a equivalentconditionsholds:
m
(i) Hi(L,M)=0 foranyL-moduleM and anyi>n;
[
(ii) Hn+1(L,M)=0 forany L-moduleM.

2
v Asimilarnotionmaybedefinedforotherclassesofalgebraicsystemswithgoodcohomologytheory,
2 e.g., forgroupsand associativealgebras.
5
The Shapiro lemma about cohomology of a coinduced module implies that if S is a subalgebra of a
3
0 Lie algebra L, then cd(S)≤cd(L). As cohomological dimensionof the one-dimensional Lie algebra is
0 equaltoone, thecohomologicaldimensionofanynonzeroLiealgebrais≥1. In particular,theclassof
.
1 Liealgebras ofcohomologicaldimensiononeisclosedwithrespect to subalgebras.
0
Due to the standard interpretation of the second cohomology, the condition for a Lie algebra L to be
6
1 ofcohomologicaldimensiononeisequivalentto theconditionthateach short exactsequence
:
v
0→·→·→L→0
i
X
of L-modules splits. The latter condition holds for a free Lie algebra, due to its universal property, and
r
a hence a free Lie algebra (of any rank) has cohomological dimension one. The same is true for free
groupsand free associativealgebras.
ThecelebratedStallings–Swantheoremsaysthatforgroupstheconverseistrue: agroupofcohomo-
logical dimension one is free (cf., e.g., [Co]). A question by Bourbaki ([B, Chapitre II, §2, footnote to
Exercice9])askswhetherthesameistrueforLiealgebras,i.e.,whetheraLiealgebraofcohomological
dimensiononeisfree.
Feldman [Fe] answered this question affirmatively in the case of 2-generated Lie algebras. For a
while, it was widely believed that the answer is affirmativein general (the author has witnessed several
attempts of the proof), until Mikhalev, Umirbaev and Zolotykh constructed an example of a non-free
Liealgebraofcohomologicaldimensiononeoverafield ofcharacteristic>2(cf. [MUZ]; notethatthe
Date:FirstwrittenJanuary3,2016;lastrevisedAugust10,2016.
2010MathematicsSubjectClassification. Primary17B50,17B55.
arXiv:1601.00352.
1
NOLIE p-ALGEBRASOFCOHOMOLOGICALDIMENSIONONE 2
cases of characteristic zero and characteristic 2 remain widely open). This example is not a p-algebra,
andatthesamepapertheymadethefollowingconjecture: aLie p-algebraofcohomologicaldimension
one is a free Lie p-algebra ([MUZ, Conjecture 2]). As stated, the conjecture is somewhat misleading,
for a free Lie p-algebra is not of cohomological dimension one: its cohomological dimension is equal
to infinity. Indeed, for any nonzero element x of such an algebra, the elements x,x[p],x[p]2,... span
an infinite-dimensional abelian subalgebra, whose cohomological dimension is equal to infinity (cf.
Lemma1 below).
This conjecture may be repaired in two ways. First, one may merely ask about description of Lie
p-algebras of cohomological dimension one. A (trivial) answer to this question is given in §1: such
algebrasareone-dimensional. Anotherpossibilityistoreplacecohomologicaldimensionwithrestricted
cohomological dimension; this is discussed in §2. Also, §1 contains auxiliary results and conjectures
related to the old Jacobson conjecture about periodic Lie p-algebras, and to the problem of description
ofLiealgebras allwhosepropersubalgebrasareone-dimensional.
1. LIE p-ALGEBRAS OF COHOMOLOGICAL DIMENSION ONE, ALMOST-PERIODIC ALGEBRAS, AND
ALGEBRAS WITH ONE-DIMENSIONAL SUBALGEBRAS
Thefollowinglemmaiselementarybut useful.
Lemma 1.
(i) Cohomologicaldimensionofan abelianLiealgebrais equaltoitsdimension.
(ii) Cohomologicaldimensionofthetwo-dimensionalnonabelianLie algebraisequal to2.
Proof. Itis clearthat cohomologicaldimensionofaLiealgebra doesnotexceed itsdimension.
(i)ForanabelianLiealgebra,wehaveHn(L,K)=( nL)⋆ foranyn(⋆ denotesthedualvectorspace).
V
(ii) Let L be the two-dimensional nonabelian Lie algebra with a basis {x,y}, [x,y] =x. For an one-
dimensionalmoduleKvwithan L-action x•v=0,y•v=−v, wehavedimH2(L,Kv)=1. (cid:3)
Corollary. A Lie algebra of cohomological dimension one does not contain a two-dimensional subal-
gebra.
Theorem. A Lie p-algebraofcohomologicaldimensiononeisone-dimensional.
Proof. Let L be a Lie p-algebra of cohomological dimension one. For any x∈L, we have [x,x[p]]=0,
and byCorollary to Lemma1,
(2) x[p] =l (x)x
forsomel (x)∈K.
Suppose L is of dimension > 1, and pick two linearly independent elements x,y ∈ L. By [Fe], the
subalgebra of L generated by x,y is free. But according to (2), (adx)p(y)=l (x)[y,x], a contradiction.
(cid:3)
Let us now reflect on the condition (2). This condition reminds of various conditions on the p-map
studied by Jacobson and others. The majoropen problem in this area is the conjectureof Jacobson that
aperiodicLie p-algebraisabelian(cf. [J,ChapterV,Exercise16]). RecallthataLiealgebraLiscalled
periodic if for any x∈L there is integer n(x)>0 such that x[p]n(x) =x. The strongest result toward this
conjecturebelongstoPremet: aperiodicfinite-dimensionalLiealgebraisabelian ([P,Corollary 1]).
Generalizing the condition of periodicity, let us call a Lie p-algebra L almost periodic, if for any
x∈L, thereisan integern(x)>0and an elementl (x)∈K suchthat
(3) x[p]n(x) =l (x)x.
Theelementsforwhich l (x)=0, i.e.,x[p]n(x) =0, willbecalled p-nilpotent.
Proposition1. LetLbeanalmostperiodicLie p-algebraofdimension>1overanalgebraicallyclosed
field, withalln(x)’s bounded. Then Lcontainsa nonzero p-nilpotentelement.
NOLIE p-ALGEBRASOFCOHOMOLOGICALDIMENSIONONE 3
Notesomeotherrelated resultsconnectingproperties ofLie(p-)algebras and itselements:
(i) Chweprovedin[Ch2]thataLie p-algebraoveranalgebraicallyclosedfieldwithanondegenerate
p-map is abelian.
(ii) Farnsteiner investigated in [Fa1] Lie p-algebras in which some power [p]n of the p-map is pn-
semilinear. Thecondition(3)issomewhatreminiscentofsemilinearity(insomesensestronger,in
somesenseweaker).
(iii) Itiswell-knownthatanyfinite-dimensionalLiealgebraoveranalgebraicallyclosedfieldcontains
a nilpotent element. (For Lie p-algebras, this follows from the Seligman–Jordan–Chevalley de-
composition,cf.,e.g.,[P,ProofofTheorem3],andforashortelementaryproofvalidforarbitrary
Lie algebras, cf. [BI]). Lemma 1 establishes a similar result for not necessary finite-dimensional
Liealgebras, butsubjectto astrongconditionofbounded p-periodicity.
Note also that the condition of the ground field being algebraically closed cannot be dropped from
the Proposition, for any non-split 3-dimensional simple Lie algebra over a field of characteristic p>0
provides a counterexample: it satisfies the condition x[p] =l (x)x for any nonzero element x, but does
nothavenonzero p-nilpotentelements(i.e., l (x)6=0forany x6=0).
ProofofProposition1. Sincen(x) are bounded,we mayassumethat
(4) x[p]n =l (x)x
forsomefixedn(forexample,bylettingntobetheproductofalldistinctn(x)’s,andredenotingl (x)’s
appropriately).
Pick any two linearly independent elements x,y∈L, and set j (t)=l (x+ty), fort ∈K. Using the
xy
well-known Jacobson binomial formula for the p-map (strictly speaking, its generalization for the nth
powerofthe p-map – cf., e.g., [Fa1,§1]), wehave
pn−1 pn−1
(5) j (t)(x+ty)=(x+ty)[p]n =x[p]n+tpny[p]n+ (cid:229) tis (x,y)=l (x)x+tpnl (y)y+ (cid:229) tis (x,y),
xy i i
i=1 i=1
where s (x,y) are certain Lie monomials in x,y. Completing x,y to a basis of L, writing s (x,y)’s as
i i
linear combinations of basis elements, and collecting all coefficients of x in (5), we get that j (t) is a
xy
polynomialint withthefree terml (x).
Suppose that there is a pair x,y such that j (t) is not constant. Since the ground field K is alge-
xy
braically closed,j (t)has arootx . Thismeansthat thenonzero elementx+x y isnilpotent.
xy
Suppose now that for any pair x,y∈L, j (t) is constant, i.e., j (t)=l (x). This means that l (x+
xy xy
ty)=l (x)foranylinearlyindependentx,y∈L,andanyt ∈K,and,consequently,l (x)=l isconstant.
If l 6=0, then substituting in (4) a x instead of x, we get that a pn =a for any a ∈K, i.e., K is a finite
field, acontradiction. Hencel =0,and everyelement ofLis nilpotent. (cid:3)
Proposition1canbeusedtogiveanalternativeproofoftheTheorem,notutilizingtheFeldmanresult
about 2-generated Lie algebras, albeit in the case of algebraically closed ground field only (note that,
in general, the cohomological dimension may increase when extending the ground field). For that, we
need anotherelementarylemmas.
Lemma 2. Let x,y be two elements of a Lie algebra without two-dimensional subalgebras, such that
(adx)ny=0 forsomen. Then x,yarelinearlydependent.
Proof. Applyingrepeatedlytheconditionofabsenceoftwo-dimensionalsubalgebras,wecanlowerthe
degreen. Indeed,(adx)ny=[(adx)n−1(y),x]=0implies[(adx)n−2(y),x]=(adx)n−1(y)=l xforsome
l ∈K, what, in turn, impliesl =0. Repeating this process, we get eventually[y,x]=0, and hence x,y
are linearlydependent. (cid:3)
Lemma 3. A Lie p-algebra of dimension >1 over an algebraicallyclosed field contains a two-dimen-
sionalsubalgebra.
NOLIE p-ALGEBRASOFCOHOMOLOGICALDIMENSIONONE 4
Thislemmamaybeconsideredasageneralizationofanelementaryfactthatafinite-dimensionalLie
algebraofdimension>1overanalgebraicallyclosedfield containsatwo-dimensionalsubalgebra. We
donotassumefinite-dimensionality,butthepresenceof p-structureisaconditionstrongenoughtoinfer
thesameconclusion.
Proof. Let L be a Lie p-algebra without two-dimensional subalgebras. By the same reason as in the
proofoftheTheorem,Lsatisfiesthecondition(2). AccordingtoProposition1(withn(x)=1forallx),
L is either one-dimensional, or contains a nonzero nilpotent element. In the latter case by Lemma 2, L
is one-dimensionaltoo,acontradiction. (cid:3)
NowLemma3,togetherwithCorollarytoLemma1, providesanalternativeproofoftheTheoremin
thecaseofalgebraically closedgroundfield.
Could in this proof the condition of algebraic closedness of the ground field be removed? Note that
it cannot be removed from Lemma 3, with the same counterexample as in the case of Proposition 1: a
non-split3-dimensionalsimpleLiealgebra. Thisis,however,theonlycounterexampleknownto us.
Conjecture 1. A Lie p-algebraof dimension>3containsa two-dimensionalsubalgebra.
NotethatthereareseveralintriguingopenquestionsaboutLiealgebrasallwhosepropersubalgebras
areone-dimensional. Itisnotknownwhethersuchinfinite-dimensionalalgebras(Lie-algebraicanalogs
of Tarski’s monsters in group theory constructed by Olshanskii)exist. The finite-dimensional situation
is, naturally, understood much better: the combination of classification of simple Lie algebras over an
algebraically closed field of positive characteristic, and the standard Galois-cohomological machinery
for determining forms of algebras, immediately imply that over a perfect field of characteristic 6= 2,3
any finite-dimensional Lie algebra all whose proper subalgebras are one-dimensional, is (non-split) 3-
dimensionalsimple. However,ifthegroundfieldisnotperfect(inwhichcasetheGalois-cohomological
machinery is not available), or is of characteristic equal to 2 or 3 (in which case the classification
of simple Lie algebras is presently absent), the question about existence of such finite-dimensional
algebras in dimension >3 is open. Conjecture 1 implies that there are no such algebras in the class of
Lie p-algebras (bothfinite-and infinite-dimensional).
2. LIE p-ALGEBRAS OF RESTRICTED COHOMOLOGICAL DIMENSION ONE
Whenspeakingaboutcohomologicaldimension,weconsiderthecategoryofallLiealgebramodules,
including infinite-dimensional ones. If we restrict ourselves with, say, finite-dimensional Lie algebras
and the category of finite-dimensional modules, the whole subject, both in results and methods em-
ployed, becomes quite different. In fact, we cannot longer speak about cohomological dimension, as
vanishingofallcohomologyinagivendegreedoesnotimplyvanishinginhigherdegrees. Asampleof
results in this domain: in characteristic zero, an “almost” converse of the classical Whitehead Lemmas
holds ([Z1], [Z2]), and in positivecharacteristic, for any degree less than the dimension of the algebra,
amodulewithnon-vanishingcohomologyexists([D]and [FS]).
Still,insteadofthecategoryofall moduleswecan considerasmallersubcategoryofmoduleswitha
good-behavingcohomologytheory: forexample,restrictedmoduleswithrestrictedcohomology. Recall
that for a Lie p-algebra L, and a bimodule M over its restricted universal enveloping algebra u(L), we
have
(6) Hn(L,Mad)≃HHn(u(L),M),
∗
where H and HH stand for therestricted cohomologyof a Lie p-algebra, and Hochschildcohomology
∗
ad
of an associative algebra, respectively, and M is a restricted L-module structure on M defined via
x•m=xm−mx forx∈L, m∈M.
The definition of a restricted cohomological dimension of L (notation: cd (L)) repeats the definition
∗
oftheordinarycohomologicaldimension,withprojectiveresolutions(1)areconsideredinthecategory
ofrestricted modulesoveru(L).
NOLIE p-ALGEBRASOFCOHOMOLOGICALDIMENSIONONE 5
As in the unrestricted case, Shapiro’s lemma for restricted cohomology implies that the restricted
cohomological dimension does not increase when passing to subalgebras. In particular, a subalgebra
of a Lie p-algebra ofrestricted cohomologicaldimensiononeis of restricted cohomologicaldimension
oneorzero. A free Lie p-algebra has restrictedcohomologicaldimensionone.
IfLisafinite-dimensionaltorus,i.e., anabelianLie p-algebrasuchthatanyelementx∈Lisalinear
combination of its p-powers x[p]k, k = 1,2,... (over a perfect field this is equivalent to the condition
that L consists of p-semisimple elements), then u(L) is a commutative semisimple algebra, and hence
cd (L) = 0. Conversely, the main theorem of [Hoch] (cf. also [S, Satz 10] and [Fa2, Theorem 3.1])
∗
amounts to saying (in a different terminology) that any finite-dimensional Lie p-algebra of restricted
cohomological dimension zero is a torus. Moreover, according to [W, Theorem 9.2.11], the restricted
cohomological dimension zero implies finite-dimensionality, so Lie p-algebras of restricted cohomo-
logicaldimensionzero areexactlyfinite-dimensionaltori.
Existence of nontrivial Lie p-algebras of restricted cohomological dimension zero allows, by the
extension procedure, to get new Lie p-algebras of restricted cohomological dimension one out of old
ones.
Lemma 4. Let I bea p-idealof a Lie p-algebraL. Then:
(i) cd (L)≤cd (I)+cd (L/I)+1;
∗ ∗ ∗
(ii) if cd (I)=0, thencd (L)=cd (L/I);
∗ ∗ ∗
(iii) if cd (L/I)=0,then cd (L)=cd (I).
∗ ∗ ∗
Part (i)isarestricted analogueof[BK, Theorem3.11.9]†.
Proof. This follows immediately from the Lyndon–Hochschild–Serre spectral sequence converging to
HHs+t(u(L),M)and havingtheE term
2
Est =HHs(u(L/I),HHt(u(I),M))
2
(here M is an arbitrary u(L)-module). If HHt(u(I),M)=0 for any t >n, and HHs(u(L/I),M)=0 for
any s>m, then Est =0 for any s+t >n+m+1, and hence HHi(u(L),M)=0 for any i>n+m+1,
2
what proves(i).
To prove (ii), note that cd (I)=0 implies that the only non-vanishing E terms are Es0, the spectral
∗ 2 2
sequences stabilizes at E , and HHn(u(L),M) ≃ En0 = HHn(u(L/I),MI). Since any restricted L/I-
2 2
modulecan beliftedto arestrictedL-moduleby lettingI act trivially,thedesired equalityfollows.
Part (iii) is established similarly: the condition cd (L/I)=0 implies that the only non-vanishing E
∗ 2
terms are E0t, and hence HHn(u(L),M)≃HHn(u(I),M)L/I. The latter isomorphism implies cd (L) ≤
2 ∗
cd (I). On theotherhand, cd (I)≤cd (L), whatimpliesthedesired equality. (cid:3)
∗ ∗ ∗
Parts (ii) and (iii) of Lemma 4 show in particular, that extending a Lie p-algebra of restricted coho-
mologicaldimensionzerobyaLie p-algebraofrestrictedcohomologicaldimensionone,or,viceversa,
aLie p-algebra ofrestrictedcohomologicaldimensiononeby aLie p-algebraofrestricted cohomolog-
icaldimensionzero,wegetaLie p-algebraofcohomologicaldimensionone. AsanyextensionofaLie
p-algebra ofrestricted cohomologicaldimension≤1 splits,any algebrawhich can beobtained starting
from afree Lie p-algebra bysuccessivelyapplyingsuch extensions,has thefollowingform:
(7) (...((L ⊲⊳T )⊲⊳T )...)⊲⊳T ,
1 2 n
whereL isafreeLie p-algebra, T ,...,T arefinite-dimensionaltori,andeach symbol⊲⊳ standseither
1 n
for ⋊(action of theleft-hand sideon theright-handside), orfor ⋉ (action oftheright-handsideon the
left-hand side).
Conjecture 2. AnyLie p-algebraof restrictedcohomologicaldimensiononeisof theform (7).
† The formulationof [BK, Theorem3.11.9]containsan obvioustypo: the summand“+1” at the right-handside of the
inequality,similarlywiththeinequalityinLemma4(i),ismissing.
NOLIE p-ALGEBRASOFCOHOMOLOGICALDIMENSIONONE 6
In particular, this conjecture implies that a Lie p-algebra of restricted cohomological one has a free
Lie p-subalgebraoffinitecodimension.
LetusestablishsomefactsaboutLie p-algebrasofrestrictedcohomologicaldimensionone,providing
a(limited)evidencein supportoftheconjecture.
The following fact was established in [Ch1, Theorem 5.1] using a not entirely trivial result from
homological algebra due to Kaplansky. We give an alternative, more elementary proof – a mere refor-
mulationofknown(and easy)resultsaboutcohomologyofcommutativeassociativealgebras.
Lemma5. Arestrictedcohomologicaldimensionofafinite-dimensionalLie p-algebraiseitherzeroor
infinity.
Proof. Let L be a finite-dimensional Lie algebra of restricted cohomological dimension >0, i.e. not a
torus. Since Lisnot atorus,thereisx∈L satisfyingtherelationoftheform
(8) l x[p]+l x[p]2+···+l x[p]n =0
1 2 n
for some n ≥ 1 and l ,l ,...,l ∈ K, l 6= 0. For the p-subalgebra (x) generated by x, we have
1 2 n n p
u((x) ) ≃ K[x]/(f), where the polynomial f is obtained from the left-hand side of (8) by replacing
p
p-powers in a Lie p-algebra by the ordinary p-powers in a polynomial algebra: f(t)=l tp+l tp2+
1 2
···+l tpn.
n
The Hochschild cohomology of quotients of polynomial algebras is well understood, cf., e.g., [Hol]
and references therein. In particular, in [Hol, Proposition 2.2] a periodic free resolution of such alge-
bras is constructed, from which it follows that the complex computing the Hochschild cohomology of
K[x]/(f)isoftheform
0 f′ 0 f′
K[x]/(f)−→K[x]/(f)−→K[x]/(f)−→K[x]/(f)−→...
Since f′ (the formal derivative of f) vanishes, HHn(K[x]/(f),K[x]/(f)) does not vanish for any n.
As K[x]/(f) is commutative, K[x]/(f)ad, as an (x) -module, is the direct sum of pn copies of the
p
trivial (x) -module K, and due to isomorphism (6), Hn((x) ,K) is nonzero for any n. Consequently,
p ∗ p
therestrictedcohomologicaldimensionof(x) , and thusofL, is equalto infinity. (cid:3)
p
Proposition2. A p-subalgebraofaLie p-algebraoffiniterestrictedcohomologicaldimensioniseither
a torus,orisinfinite-dimensional.
Proof. FollowsfromLemma5. (cid:3)
In particular, in a Lie p-algebra L of finite restricted cohomological dimension, every nonzero p-
algebraic element is p-semisimple. This can be considered as a Lie-p-algebraic analog of the well-
knownfactthatgroupsoffinitecohomologicaldimensionaretorsion-free(cf., e.g.,[Co,p.6,Corollary
2]).
Proposition3. An abelian p-subalgebraof a Lie p-algebra of restrictedcohomologicaldimensionone
is eithera torus,or isisomorphicto thedirect sumof a torusand thefreeLie p-algebraof rankone.
Proof. Let L be an abelian subalgebra of a Lie p-algebra of restricted cohomological dimension one.
The restricted cohomological dimension of L is either equal to zero, in which case L is a torus, or is
equal toone. In thelattercase, assumefirst thatLdoes nothavenonzero p-algebraicelements.
To prove that L is a free Lie p-algebra of rank one, it is enough to prove that any two commuting
elements of L, say, x and y, can be represented as p-polynomials of a third element. Suppose the con-
trary. ByProposition2,eachofx,ygeneratethefreeLie p-algebraofrankone,andhencetherestricted
universal enveloping algebra of the p-subalgebra S of L generated by x,y, is isomorphicto the polyno-
mialalgebraintwovariablesK[x,y]. Thelatteralgebrahasnon-vanishing2ndHochschildcohomology
(forexample,HH2(K[x,y],K[x,y])≃ 2(Der(K[x,y]))⊗ K by theHochschild–Kostant–Rosenberg
K[x,y]
V
theorem), and reasoning as at the end of the proof of Lemma 5, we get that H2(S,K) does not vanish,
∗
whencecd (L)≥cd (S)≥2,a contradiction.
∗ ∗
NOLIE p-ALGEBRASOFCOHOMOLOGICALDIMENSIONONE 7
In the general case, consider the set T of all p-semisimple elements of L. Obviously, T forms a
proper subalgebra, and hence a proper ideal, of L. By Lemma 4(ii), the quotient L/T is an abelian Lie
p-algebra of restricted cohomological dimension one. Since L/T does not have nonzero p-algebraic
elements, L/T is isomorphic to the free Lie p-algebra of rank one by above. The extension obviously
splits,and thedesiredconclusionfollows. (cid:3)
Thenextlemmashowsthatthe(ordinary)cohomologyofLie p-algebras ofrestrictedcohomological
dimensiononebehavesin aratherpeculiarway.
Lemma 6. Let L be a Lie p-algebra L of restricted cohomological dimension one, and M a restricted
L-moduleM. Then
(9) Hn(L,M)≃ nL ⋆⊗ML ⊕ n−1L ⋆⊗H1(L,M)
(cid:16)(cid:16)^ (cid:17) (cid:17) (cid:16)(cid:16)^ (cid:17) ∗ (cid:17)
foranyn≥1.
Proof. ThisfollowsfromaparticularformoftheGrothendieckspectralsequencerelatingrestrictedand
ordinary cohomology. Namely, for a Lie p-algebra and a restricted L-module M, there is a spectral
sequencewiththeE term
2
t ⋆
Est =Ct(L,Hs(L,M))≃ L ⊗Hs(L,M)
2 ∗ (cid:16)^ (cid:17) ∗
convergingto Hs+t(L,M) (cf. [FP, Proposition5.3]; notethat thestandingassumptionin [FP]offinite-
dimensionality of algebras and modules is not relevant here; cf. also [Fa2, Theorem 4.1] and [M,
Corollary 1.3]). HereCn(V,W)≃( nV)⋆⊗W denotes,as usual,thespaceofskew-symmetricn-linear
V
mapsfrom onevectorspaceto another.
IfHs(L,M)=0fors≥2,theonlynonvanishingE termsareE0t andE1t. Hencethespectralsequence
∗ 2 2 2
stabilizesat E , Hn(L,M)≃E0n⊕E1,n−1 foranyn≥1,and (9)follows. (cid:3)
2 2 2
Lemma 6 provides a yet another proof of the fact that a Lie p-algebra L of restricted cohomological
dimension one is infinite-dimensional (what follows also from Lemma 5), without appealing to any
computation of Hochschild cohomology. Indeed, suppose the contrary, and take in (9) n = dimL+1.
Then the left-hand side and the first direct summand at the right-hand side of the isomorphism vanish,
and the second direct summand is isomorphic to H1(L,M). Therefore, H1(L,M)=0 for any restricted
∗ ∗
L-moduleM, i.e.,L isofrestricted cohomologicaldimensionzero, acontradiction.
Moreover,a strongerstatementholds:
Proposition 4. A Lie p-algebra of restricted cohomological dimension one has infinite (ordinary) co-
homologicaldimension.
Proof. LetL beaLiealgebraofrestrictedcohomologicaldimensionone. Takingin (9)M =K, weget
Hn(L,K)≃ nL ⋆⊕ n−1L ⋆⊗H1(L,K) .
(cid:16)^ (cid:17) (cid:16)(cid:16)^ (cid:17) ∗ (cid:17)
Either by Lemma 5, or by the reasoning above, L is infinite-dimensional, and thus nL, and hence
Hn(L,K),does notvanishforany n≥1. V (cid:3)
ACKNOWLEDGEMENTS
Thanks are due to Jo¨rg Feldvoss and Oleg Gatelyuk for useful discussions. The hospitality of Uni-
versity of Sa˜o Paulo during the early stage of this work is gratefully acknowledged. This work was
supported by the Statutory City of Ostrava (grant 0924/2016/SaSˇ), and the Ministry of Education and
Science oftheRepublicofKazakhstan (grant 0828/GF4).
NOLIE p-ALGEBRASOFCOHOMOLOGICALDIMENSIONONE 8
REFERENCES
[BI] G.BenkartandI.M.Isaacs,Ontheexistenceofad-nilpotentelements,Proc.Amer.Math.Soc.63(1977),39–40.
[BK] L.A.Bokut’andG.P.Kukin,AlgorithmicandCombinatorialAlgebra,Kluwer,1994.
[B] N.Bourbaki,GroupesetAlge`bresdeLie,Chapitres2et3,Hermann,Paris,1972;reprintedbySpringer,2006.
[Ch1] B.-S. Chwe, Relative homologicalalgebra and homologicaldimension of Lie algebras, Trans. Amer. Math. Soc.
117(1965),477–493.
[Ch2] ,OnthecommutativityofrestrictedLiealgebras,Proc.Amer.Math.Soc.16(1965),547.
[Co] D.E.Cohen,Groupsofcohomologicaldimensionone,Lect.NotesMath.245(1972).
[D] A.S.Dzhumadil’daev,CohomologyoftruncatedcoinducedrepresentationsofLiealgebrasofpositivecharacteristic,
Mat.Sbornik180(1989),456–468(inRussian);Math.USSRSbornik66(1990),461–473(Englishtranslation).
[Fa1] R.Farnsteiner,RestrictedLiealgebraswithsemilinearp-mappings,Proc.Amer.Math.Soc.91(1984),41–45.
[Fa2] ,Cohomologygroupsofreducedenvelopingalgebras,Math.Z.206(1991),103–117.
[FS] andH.Strade,Shapiro’slemmaanditsconsequencesinthecohomologytheoryofmodularLiealgebras,
Math.Z.206(1991),153–168.
[Fe] G.L.Feldman,EndsofLiealgebras,UspekhiMat.Nauk38(1983),N1,199–200(inRussian);Russ.Math.Surv.
38(1983),N1,182–184(Englishtranslation).
[FP] E.M. Friedlander and B.J. Parshall, Modular representation theory of Lie algebras, Amer. J. Math. 110 (1988),
1055–1093.
[Hoch] G. Hochschild, Representations of restricted Lie algebras of characteristic p, Proc. Amer. Math. Soc. 5 (1954),
603–605.
[Hol] T.Holm,Hochschildcohomologyringsofalgebrask[X]/(f),Beitra¨geAlgebraGeom.41(2000),291–301.
[J] N.Jacobson,LieAlgebras,IntersciencePubl.,1962;reprintedbyDover,1979.
[MUZ] A.A.Mikhalev,U.U.Umirbaev,andA.A.Zolotykh,ALiealgebrawithcohomologicaldimensiononeoverafield
ofprimecharacteristicisnotnecessarilyfree, FirstInternationalTainan-MoscowAlgebraWorkshop(ed.Y. Fong
etal.),DeGruyter,1996,257–264.
[M] M.Muzere,RelativeLiealgebracohomologyrevisited,Proc.Amer.Math.Soc.108(1990),665–671.
[P] A.A. Premet, On CartansubalgebrasofLie p-algebras,Izv.Akad.NaukSSSR Ser. Mat. 50(1986),788–800(in
Russian);Math.USSRIzvestija29(1987),145–157(Englishtranslation).
[S] H.Strade,EinigeVereinfachungeninderTheoriedermodularenLie-Algebren,Abh.Math.Sem.Univ.Hamburg54
(1984),257–265.
[W] C.Weibel,AnIntroductiontoHomologicalAlgebra,CambridgeUniv.Press,1994.
[Z1] P. Zusmanovich, A converse to the Second Whitehead Lemma, J. Lie Theory 18 (2008), 295–299; Erratum: 24
(2014),1207–1208;arXiv:0704.3864.
[Z2] ,AconversetotheWhiteheadTheorem,J.LieTheory18(2008),811–815;arXiv:0808.0212.
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