Elementary gradings on the Lie algebra UT Journal of Algebra

Contents lists available atScienceDirect

Journal of Algebra

www.elsevier.com/locate/jalgebra

Elementary gradings on the Lie algebra U T

DepartmentofMathematics,StateUniversityofCampinas,651SergioBuarque de Holanda,13083-859Campinas,SP,Brazil

a r t i c l e i n f o a bs t r a c t

Article history:

Received29July2015

Availableonline4November2016 CommunicatedbyVeraSerganova ForPlamenSiderov,inmemoriam MSC:

17B01 17B60 17B70 16R10 16R99

Keywords:

Gradedpolynomialidentities Uppertriangularmatrices GradedLiealgebras

ThealgebrasU Tn(K) oftheuppertriangularmatricesovera fieldKareofsignificantimportanceinthetheoryofalgebras withpolynomialidentities.Groupgradingsonalgebrasappear invariousareasandprovideanindispensabletoolinthestudy ofthealgebraicandcombinatorialpropertiesofthealgebrasin question.InthispaperweconsidertheLiealgebraU Tn(K)⁽⁻⁾ ofalluppertriangularmatricesofordern.Westudythegroup gradingsonthisalgebra.Itturnsoutthatthegradingsonthe Lie algebra U Tn(K) aremuch moreintricate than those in theassociativecase.Inthispaperwedescribetheelementary gradingsontheLiealgebraU Tn(K)⁽⁻⁾.Finallywestudythe canonical gradingon U Tn(K)⁽⁻⁾ by thecyclic groupZn of ordern.Weproducea(finite)basisofthegradedpolynomial identitiessatisfiedbythisgrading.

©2016ElsevierInc.Allrightsreserved.

Introduction

Graded algebras first appeared in Commutative algebraas anatural generalization of propertiesofpolynomialrings.Gradings onalgebras appearedinacelebratedpaper

* Correspondingauthor.

E-mailaddresses:plamen@ime.unicamp.br(P. Koshlukov),ra091138@ime.unicamp.br(F. Yukihide).

1 PartiallysupportedbyFAPESPgrantNo.2014/09310-5,andbyCNPqgrantNo.304632/2015-5.

2 SupportedbyPhDgrant2013/22802-1fromtheSãoPauloResearchFoundation(FAPESP).

http://dx.doi.org/10.1016/j.jalgebra.2016.10.028 0021-8693/©2016ElsevierInc.Allrightsreserved.

ofWall[29] wherehe describedthefinite dimensionalgradedsimplealgebras whenthe gradinggroupis Z2, thecyclicgroupof order 2.Later ongradingsonalgebras became anobjectofextensivestudy.Around1985,Kemerdevelopedthestructuretheoryofthe T-ideals(ideals of identities) inthe free associativealgebra, see for instance [18]. One ofthe principalingredients ofthattheory isthe study ofZ2-gradedalgebras and their graded identities. Thetheory developedby Kemer has sinceimmensely influenced the research inPI theory,and motivated furtherstudy of gradings and on graded polyno- mialidentitiesinassociativealgebras.Onemayconsultthepapers [5,6],andtherecent monograph[16] andthebibliographythereinforfurtherandmoredetailedinformation concerning gradings onalgebras. Werecall some ofthe cornerstone results inthearea thatwillbeusedintheexpositionbelow.Inordertodoitinaprecisewayweintroduce nowsomeofthemainnotionswewillneed.

LetGbe agroup,K afield,and A analgebraover K. ThealgebraA needsnotbe associativenorcommutative.The algebraAis G-graded ifA=⊕g∈GAg where theAg

are vector subspacesof A suchthat AgAh ⊆Agh for allg, h ∈G. An ideal I of A is graded(alsocalledhomogeneous)ifI=⊕g∈GI∩A_g,analogouslyforgradedsubalgebras and vector subspaces of A. Let X_g, g ∈ G be countable infinite sets and denote by X = X_G = ∪g∈GX_g their union. Form the free algebra K{X_G} over K that is freely generated by X, its elements are called polynomials. This free algebra has a natural G-gradingby assigningdegreeg to allelements from Xg. Apolynomialf ∈K{XG}is agraded identity for theG-graded algebra A when f vanishes underevery evaluation onArespecting thegrading.Inother wordsf liesinthekernels ofallhomomorphisms from K{X_G} to A that respect the grading. Such homomorphisms are called graded homomorphisms.

The classification of the gradings on the (associative) matrix algebras of order n was obtained by Bahturin and Zaicev, see for example [7]. Later on this description was extended, in the case of abelian groups, to simple associative algebras having a minimalonesidedideal(overanalgebraicallyclosedfield),see[16,pp.27,28].Lateron similarresultswereobtainedforsimpleLiealgebras,seeforexamplethemonograph[16]

for an extensive collection of the state-of-art; simple Jordan algebras, see [8] and its bibliography.

We note that relatively little is known about the classification of the gradings on important algebras that are not simple. TheGrassmann (or exterior) algebraappears naturally in various branches of Mathematics and Physics but the gradings on this algebra are known in several limited cases, see for example [15]. The gradings on the associativealgebraoftheuppertriangularmatricesofanyorderwereclassifiedbyValenti andZaicevin[26]. ThegradingsontheJordanalgebraofthe2×2 matricesoforder2 weredescribed in[22]. Notethatthe latteralgebrais aJordan algebraof asymmetric bilinear form though the form is degenerate. Recently Bahturin studied gradings on nilpotentalgebrasthatarefreeinavarietyof algebras[9].

Asdiscussed abovegradings onalgebras are importantwhen studying theirpolyno- mial identities.As arule thegraded identities satisfiedbyan algebraprovide valuable

information about theordinarypolynomial identities,and veryoften, about thestruc- ture of the algebra. Since the homogeneous componentsin the grading on an algebra are notas “large”as the wholealgebraitis “easier”to study graded identitiesinstead of ordinaryones.Herewerecallthatthepolynomialidentitiesoftheassociativematrix algebras M_n(K) overafieldK areknownonly forsmall valuesofn.Namely theiden- titiesofM_n(K) areknown,whenK isafinitefield,forn≤4,see[24,17],andwhen K is infinite and of characteristic different from 2,for n= 2,see [25,19]. It isnot known whetherthepolynomialidentitiesofM2(K) arefinitelygeneratedwhenKisaninfinite field of characteristic 2.Neither are knownthe identitiesof Mn(K),n ≥3 when K is infinite.

OntheotherhandthegradedidentitiesofM_n(K) arewellunderstood.Thisalgebra admitsnaturalgradingsbyZandbyZn,therespectivegradedidentitiesweredescribed in [27,28]in characteristic 0,and in[3,4] when K is infinite. The graded identitiesfor the naturalgradings on theso-called T-prime algebras also weredescribed, seefor ex- ample [14] and its bibliography for the exact references. It was proved in [23] that if thegroupGisabelianandfinitethentwofinite dimensionalgradedsimplealgebrasare isomorphic as graded algebras if and only if they satisfy the samegraded polynomial identities. The same result was extended in [1] to any finite (not necessarily abelian) group. Here we draw thereader’s attentionthatagraded simple algebrais analgebra withoutnontrivialhomogeneousideals,thusitneednotbesimpleasanalgebrawithout the grading. We direct the interested reader to [23] and its references for the known resultsconcerningrecognitionofsimplealgebrasintermsoftheirpolynomialidentities.

Theelementarygradings(sometimescalledgoodgradings)appearintheclassification of the gradings on matrix algebras over a field. A grading on Mn(K) is elementary whenever the matrix units eij are all homogeneous in the grading. There are other, equivalent definitionsofanelementarygrading,seebelow.Thenotionofanelementary grading can be transferred in a natural manner to subalgebras and vector subspaces of M_n(K). One can find additional information about gradings on matrix algebras in thesurvey [7], inthepaper[5],andalsointherecentmonograph [16].

The elementarygradingson theassociativealgebraU Tn(K) of theuppertriangular matrices of order n over a fieldK were described in [12]. Under somerestrictions on the base field it was proved there that one can distinguish the elementary gradings onU T_n(K) knowingtheirgradedpolynomialidentities.

In this paper we study the elementary gradings on the Lie algebra of the upper triangularmatricesofordernoveraninfinitefield.Weclassifythesegradings.Moreover weconsiderthecanonicalZn-gradingonthisLiealgebraanddescribethecorresponding idealofgraded identities.

Here we feel we have to comment on the following phenomenon. The associative algebraU T_n aswellastheLiealgebraU Tn⁽⁻⁾ areconsideredasrelativelyeasyfrom the pointofviewoftheirpolynomialidentities.Theiridentitieshavebeendescribedtightly.

OntheotherhandthegradingsonU T_nandthecorrespondinggradedidentitiesarenot that immediate,see for example[12].We recall thatthe involutions onU Tn havealso

beendescribed,see[13],butthecorrespondingidentitieswithinvolution areknownfor n= 2,andforn= 3 incharacteristic0,[13].ItoftenhappensthattheLieanaloguesfor somerelativelyeasyresultsintheassociativecaseeitherarenottrueoriftruetheyare muchmoreintricatetoobtain.Werefertheinterestedreadertocomparethedescription ofthe gradedidentitiesfor theassociativealgebraM₂ (donein[11]incharacteristic 0, in[2] inpositivecharacteristic different from 2, and finally in[10] when thealgebra is over an infinite integraldomain insteadof afield) on one hand,and theLie analogue given in [21]. While the former takes at most halfa page and is quite elementary the latteroccupiesmorethan10pagesandappliesmethodsfromInvarianttheory.Alsothe identities for the canonical grading on M_n are well understoodover any infinite field, see[28,3]whilethegradedidentitiesforthecanonicalgradingonsln arenotknownyet whenevern>2.

In the case of U Tn and U Tn⁽⁻⁾ the situation is not much different. In the associa- tivecaseallgradingsare elementarywhile intheLie casethereappearnon-elementary gradingsevenwhenthegroupisfiniteandabelian.

In this paper we study the elementary gradings on U Tn⁽⁻⁾ and describe the graded identitiesforthecanonicalcase.Inasubsequentpaper weshallstudy generalgradings onthisLiealgebra.

1. Preliminaries

InwhatfollowsGstandsforamultiplicativegroupwithunitelement 1. Weusethe multiplicativenotationevenintheabeliancase.WefixaninfinitefieldKandweconsider all algebras and vector spaces over K. Let A be a G-graded algebra (not necessarily associative), A = ⊕g∈GAg, the elements a ∈ Ag are homogeneous of (homogeneous) degreeg.Wedenote thisasG-dega=g,orifnotambiguous,simplyasdega=g.

Definition 1.1. Let A = ⊕g∈GA_g and B = ⊕g∈GB_g be two G-graded algebras, the map f: A → B is ahomomorphism (endomorphism, automorphism, isomorphism) of G-gradedalgebrasiffisahomomorphism(endomorphism,automorphism,isomorphism, respectively)of algebrassuchthatf(Ag)⊂Bg forallg∈G.

The algebras A and B are G-graded isomorphic if there exists an isomorphism of G-gradedalgebrasf:A→B.

ThedefinitionsaysthatAandBareisomorphicasG-gradedalgebraswheneverthere isanisomorphismof algebrasfrom AtoB whichrespectsthegradings.

AssumeAisaG-gradedalgebraandB ⊂A isagradedsubspace.Then B admitsa naturalG-gradinginducedbythegrading ofA. IfB is anidealof Athenthe quotient A/B alsoadmitsaninducedG-grading.

Here we recall the definition of the free G-graded algebra. Let g ∈ G and let X_g := {x^(g)₁ ,x^(g)₂ ,. . .} be a set of symbols (variables). Put X = ∪g∈GX_g, and form thefreealgebra(associative,orLie,orwhateverelse)K{XG}inthevariablesofX.One

definesnaturallyaG-gradingonK{XG}byassigningdegreeg toallvariablesfromXg, degxi(g) =g,and thenextendingthistoallmonomialsm∈K{XG}bymeansof

degm=

g, ifm=x^(g)_i , (degm1)(degm2), ifm=m1m2.

In thecase of associativeor Lie algebras, wedenote the corresponding free algebraby KX_Gand LX_G,respectively.

Definition 1.2.Letf(x^(g₁¹⁾,. . . ,x^(gm^m))∈K{X_G}andletAbe aG-gradedalgebrainthe class of K{XG}. Then f is a G-graded polynomial identity for A iff(a1,. . . ,am) = 0 for alla1, . . . ,am∈A suchthatdeg(ai)=gi forall i.We denote byTG(A) theset of G-graded identitiesforA.

ClearlyT_G(A) isanidealwhichisclosedunderendomorphismsofK{X_G}thatrespect the grading. Conversely every such ideal is the ideal of G-graded identities for some G-graded algebra.

Recall thateij stand for the matrix units: eij has an entry 1 at position (i,j) and 0 elsewhere. If a, b ∈ U Tn⁽⁻⁾ denote [a,b] = ab−ba. The Lie brackets (without inner brackets)areassumedleft normed,thatis[a,b,c]= [[a,b],c] andso on.

LetGbeagroup,aG-gradingontheassociativealgebraMn(K) iselementarywhen- evereachmatrixuniteij ishomogeneousinthegrading.Thiscanberestatedasfollows.

Assume V is a K-vector space, dimV = n, and let V = ⊕g∈GVg be a direct sum of vectorsubspacesindexed bytheelementsofG.OnecanconsiderthistobeaG-grading onV.OnedefinesaG-grading onthealgebraE(V) ofthelineartransformationsonV as follows.Takeahomogeneousbasisv1,. . . ,vn ofV,degvi=gi.Definealineartrans- formationbyEij(vk)=δjkvi whereδjk istheKronecker delta.TheG-gradingonE(V) is definedbydegE_ij =g_ig_j⁻¹.IdentifyingE(V) andM_n(F) byfixingthebasisgivesan elementarygradingonM_n(F).Theconversealsoholdsand iseasytodeduce.

2. Elementarygradings

Letus restatetheaboveforthecaseofU Tn⁽⁻⁾.

Definition 2.1.Let Gbeagroup.A G-grading onU Tn⁽⁻⁾is elementaryifthereexists a sequence (g1,g2,. . . ,gn)∈Gⁿ suchthateverymatrixuniteij ∈U Tn⁽⁻⁾is homogeneous and degeij =gig⁻¹_j .

It is well known thatinthe associativecase theelementary G-gradings on U Tn are in1–1correspondencewiththesetGⁿ⁻¹,see [26].Weshallseethatthisisnotthecase forU Tn⁽⁻⁾.Namelyweshallseethatthereversesequence(g_n−1,. . . ,g₂,g₁) producesthe sameelementarygradingasthesequence(g1,g2,. . . ,gn−1)∈Gⁿ⁻¹.

Belowwe provethatthis istheonlyway toobtainisomorphicgradings. Tothis end we define an equivalence relation on theset Gⁿ⁻¹ as follows. If η = (g1,g2,. . . ,gn−1), μ∈Gⁿ⁻¹thenη∼μwheneverμ=η orμ= (g_n−1,. . . ,g₂,g₁).Thusweshallshowthat thereisa1–1correspondencebetweentheG-gradingsonU Tn⁽⁻⁾ andGⁿ⁻¹/∼.

Welistseveralbasicandwellknownfactsaboutelementarygradings.Notethatthey alsoholdfortheassociativecase.Asthesefactsareeasyweomittheirproofs.

Lemma 2.2. Let U Tn⁽⁻⁾ be endowed with an elementary G-grading. Then degeii = 1 for all i. Moreover the sequence dege12, dege23, . . . , degen−1,n determines uniquely thegrading onU Tn⁽⁻⁾. The latterelements generate an abelian subgroup containingthe supportof thegrading.

Thuswecanassume,withoutloss ofgeneralitythatthegroupGisabelian.

Remark.Ifη= (g₁,g₂,. . . ,g_n−1)∈Gⁿ⁻¹weobtainanelementarygradingonU Tn⁽⁻⁾by puttingdegei,i+1=gi foralli.

Wedenoteby(U Tn⁽⁻⁾,η) theLiealgebraU Tn⁽⁻⁾endowedwiththisG-grading.

Conversely given an elementary G-grading on U Tn⁽⁻⁾ we form the sequence η = (dege12,dege23,. . . ,degen−1,n), then the G-grading thus obtained coincides with (U Tn⁽⁻⁾,η).

Definition 2.3. Given asequence η = (g1,g2,. . . ,gm) ∈ G^m, we define the reverse se- quencerevη = (gm,gm−1,. . . ,g2,g1).

Lemma 2.4. If η ∈ Gⁿ⁻¹ then (U Tn⁽⁻⁾,η) is isomorphic, as a G-graded algebra, to (U Tn⁽⁻⁾,revη).

Proof. Let ψ: U Tn → U Tn be the unique linear map on U Tn such that ψ(eij) =

−en−j+1,n−i+1. Then it is immediate that ψ: (U Tn⁽⁻⁾,η) → (U Tn⁽⁻⁾,revη) is an iso- morphismofG-gradedalgebras. 2

Asintheassociativecase(see[12,Def.2.1]),wedefine goodand badsequences.

Definition2.5.

1. The sequenceμ = (g1,g2,. . . ,gm)∈G^m is agoodsequence with respect tothe el- ementary grading defined by the sequence η (η-good sequence for short) if there exist strictly upper triangular matrix units r1, r2,. . . , rm ∈ U Tn⁽⁻⁾ such that degri=gi for i = 1, 2, . . . , m, and [r1,r2,. . . ,rm] = 0. If μ is not η-good we call itη-bad.

2. Giveng∈Gandm∈Nwedefine

f_m^(g)=

x^(g)m, ifg= 1,

[x⁽¹⁾_2m−1, x⁽¹⁾_2m], ifg= 1 . 3. Givenμ= (g1,g2,. . . ,gm) wedefine fμ= [f₁^(g1),f₂^(g2),· · ·,fm^(gm)].

Lemma 2.6. The polynomial fμ corresponding to the sequence μ ∈ G^m is a G-graded identity for(U Tn⁽⁻⁾,η)if andonlyif μis η-badsequence.

Proof. Thepolynomialfμ ismultilinear andonecanrepeatword bywordtheproof of Proposition 2.2in[12]. 2

Now we develop some combinatorial properties of permutations. We denote by S_m thesymmetricgrouppermutingthesymbols 1,2,. . . ,m.

Definition 2.7.Lettbe aninteger,1≤t≤m.Apermutationσ∈S_m satisfiesthet-Lie ordering conditionif:

(i) σ(t)= 1;

(ii) Ifk1,k2≥0 aresuchthatk1+k2< nand

{σ(t−k1), σ(t−k1+ 1), . . . , σ(t+k2)}={1,2,· · ·, k1+k2+ 1},

then t−k1 −1 ≥ 1 and σ(t−k1−1) = k1+k2+ 2 or t+k2 + 1 ≤ n and σ(t+k2+ 1)=k1+k2+ 2.

We denote _Tm^(t) = {σ ∈ S_m | σsatisfies thet-Lie ordering condition} and _Tm =

∪^mt=1Tm^(t).

Wedrawthereader’sattentionthatingeneral_TmisnotevenasubgroupofS_m.But Tmenjoysinterestingpropertiesthatwewillexploitlateron.

Remark.Let θ∈Sm be writteninthetwo rowsnotation(that iswrite inthefirst row the integersfrom 1to m, andwrite θ(i) beloweachi). Thenθ ∈Tm ifand onlyiffor every r, 1 ≤ r ≤ m, the elements 1, 2, . . . , r appear “together”, in a “block” in the second row.

Lemma 2.8. Letr1,r2, . . . ,rm be strictly uppertriangularmatrix units suchthat their associative productr1r2· · ·rm= 0.Then

(i) r_σ−1(1)r_σ−1(2)· · ·r_σ−1(m)= 0 if andonlyif σ= 1;

(ii) [r_σ−1(1),r_σ−1(2),· · ·,r_σ−1(m)]= 0 ifandonly ifσ∈Tm. Proof. (i)Thefirststatementofthelemmaisimmediate.

(ii)Thesecond followsbythedefinitionof Tmandby(i). 2

Wegiveacoupleofexamplesinorder toshedsomeadditionallightonthenotionof t-Lieordering.

Examples.

1. Considerthefollowingtwopermutationsσ, τ∈S6. σ=

1 2 3 4 5 6

4 1 2 3 5 6

, τ =

1 2 3 4 5 6

3 2 1 6 5 4

.

Onecancheck easily(either directlyorbyusing theRemarkabove)thatσ∈T₆⁽²⁾ butτ /∈T6.

2. Let r1, r2, . . . , rmbe strictly upper triangularmatrices such thattheirassociative product r1r2· · ·rm= 0. Thenr_σ(1)r_σ(2)· · ·r_σ(m)= 0 ifandonlyifσ= 1.

Nowconsider theLiecase.Letσ∈S_m besuchthat

[r_σ(1), r_σ(2), . . . , r_σ(m)]= 0 (1) Assume t = σ(1). Since [r_σ(1),r_σ(2)] = 0, it follows that either σ(2) = t+ 1 or σ(2)=t−1.Byiteratingthisandbyanobviousinductionweobtainthatσ∈Tm. Thesameideacanbeusedtoprovetheconverse,thatis(1)holdsforeachσ∈Tm. 3. Considerthepermutations

σ1=

1 2 3 2 1 3

, σ2=

1 2 3 3 1 2

, σ2◦σ1=

1 2 3 1 3 2

.

Thenσ1,σ2∈T3butσ2◦σ1∈/T3.HenceTmisnotingeneralasubgroupofSm. Definition2.9.Letη = (g1,g2,. . . ,gm)∈G^mandσ∈Sm,wedefinetheleftaction ofσ onη byση:= (g_σ−1(1),g_σ−1(2),. . . ,g_σ−1(m)).

Remark.Itmakesnodifference inthedefinitionwhethertheelementsof thesequences belong to a group. The same action can be defined for any sequence of any symbols;

weshallindeedconsider thesequenceas asequenceofsymbols.Note thatthefollowing lemmaalsoholdsinsuchageneralsetting.Thelemma wasprovedin[30].

Lemma 2.10. ([30]) Let η, μ∈G^m be twosequences. Then η =μ,or η =revμ if and only if for every choice of σ, τ ∈Tm there exist σ, τ ∈ Tm such that ση=σμ and τ η=τμ.

ItfollowsfromLemmas 2.10 and2.8thatapolynomialf_μ,μ∈Gⁿ⁻¹isnotaG-graded identityfor(U Tn⁽⁻⁾,η) ifandonlyifμ=σηforsomeσ∈Tm.Thisisthemainresultof thissection

Corollary 2.11. Suppose η, μ ∈ Gⁿ⁻¹. Assume further μ = η and μ = revη. Then (U Tn⁽⁻⁾,μ)(U Tn⁽⁻⁾,η).

Proof. Accordingto Lemma 2.10there exists τ ∈T_n−1 suchthatτ μ=τη forallτ∈ Tn−1 (changingη and μ,ifnecessary). Therefore τ μis agoodsequencefor (U Tn⁽⁻⁾,μ) butitisabadsequencefor(U Tn⁽⁻⁾,η).Thisimpliesthatf_{τ μ}isaG-gradedidentityfor (U Tn⁽⁻⁾,η) but notfor(U Tn⁽⁻⁾,μ),thusprovingtheclaim. 2

Here we summarize our main results in this section. We formulate these results so thattheybecomeindependentoftheremainderof thetext.

Theorem 2.12. Let U Tn⁽⁻⁾ be endowed with an elementary G-grading, then necessarily thesupport ofthegradinggenerates an abeliangroup.

If thegroupGisabelian,we definean equivalence relation∼on thesetof sequences Gⁿ⁻¹: given η and μ = (g1,g2,. . . ,gn−1) ∈ Gⁿ⁻¹ then η ∼μ if and only if η = μ or η = (gn−1,gn−2,. . . ,g1).

Thenthenon-isomorphicelementaryG-gradingsonU Tn⁽⁻⁾arein1–1correspondence with theelements oftheset Gⁿ⁻¹/∼.

3. Zn-gradedidentitiesforU T_n⁽⁻⁾

In this section we consider a particular but important Zn-grading on U Tn⁽⁻⁾. In it everymatrixunite_ij ishomogeneousanddege_ij =j−i∈Zn.Thisisindeedagrading whichiscalled thecanonicalZn-gradingofU Tn⁽⁻⁾.Here wenotethatthesamegrading canbedefinedonU Tn asanassociativealgebra.

Remark.WhenitisconvenientweshallidentifyZn with{0,1,2,. . . ,n−1}⊂Z,andas itiscustomary,withoutmakinganymention.Thusweshallpasstotheadditivenotation insteadofthemultiplicativeonewheneveritismoresuitableforourpurposes.

OnecandefinethecanonicalgradingonU Tn⁽⁻⁾bytheinfinitecyclicgroupZinexactly the samemanner(that isassuming allcomponentsof homogeneousdegreeg suchthat g <0 andg > n−1 vanish).Infactthisrepresentsanotherwaytoformalizingtheabove identification. Since such anidentification is trouble-freewe shall workfreely inZn or inZ.

TheT_Zn-idealofgradedidentitiesofU Tn overaninfinitefieldwasdescribedin[20].

Theorem 3.1. Letthe base field be infinite, and let U Tn be equipped with the canonical Zn-grading.Then theT_Zn-ideal of identitiesof U T_n is generated (asan ideal of graded identities) by

x⁽ⁱ⁾₁ x^(j)₂ = 0, i+j ≥n [x⁽⁰⁾₁ , x⁽⁰⁾₂ ] = 0 .

Remark. The graded identities listed above are the “obvious” ones. They express the factthattwodiagonalmatrices commute,and thatifJ standsfortheJacobsonradical ofU Tn thenJⁿ= 0.

OurgoalinthissectionistoproveasimilarresultfortheLiecaseU Tn⁽⁻⁾.Asin[20]

weshallworkoverafixed infinitefieldK.

We shall denote by _C = [x⁽⁰⁾₁ ,x⁽⁰⁾₂ ]^TZn ⊂KX_Zn the T_Zn-ideal generated by the commutator[x⁽⁰⁾₁ ,x⁽⁰⁾₂ ] inthefreeassociativeZn-gradedalgebra.DenotefurtherbyD⊂ LX_Zn the T_Zn-idealin thefree Zn-graded Lie algebrathat is generated by thesame polynomial.DefineKX_Zn=KX_Zn/C,LX_Zn=LX_Zn/Dandletp: KX_Zn→ KX_Znbethecanonicalprojection.

Weshallconsiderthefollowingset ofassociativemonomials

x⁽ⁱ⁾₁ x^(j)₂ |i+j≥n

, (2)

andalsothefollowingset ofLiemonomials

[x⁽ⁱ⁾₁ , x^(j)₂ ]|i+j≥n

. (3)

SinceKX_Znisarelativelyfreegradedassociativealgebraitmakessenseconsidering T_Zn-idealsinKX_Zn.ItalsomakessenseinthecaseofthegradedLiealgebraLX_Zn. WeshallspeakofT_Zn-idealswhenwerefertosuchidealsinKX_ZnorinLX_Zn.For agiven (associative or Lie) algebra A we denote by T_Zn(A) theT_Zn-ideal of identities of A inKX_Zn or inLX_Zn, depending on A being consideredas an associativeor Liealgebra,respectively.Clearlyinorder todothatAmustsatisfythegradedidentity [x⁽⁰⁾₁ ,x⁽⁰⁾₂ ]= 0.

Thenextlemma isabasicbutveryimportantfact.

Lemma3.2. LX_Zn⊂KX_Zn.Inparticular _D=_C

LX_Zn.

Proof. Let A be anassociative algebra and letϕ be ahomomorphism of Lie algebras ϕ: LX_Zn→A⁽⁻⁾. Wedefine ϕ:¯ KX_Zn→A as theunique algebrahomomorphism satisfying

¯

ϕ(x⁽ⁱ⁾_j ) =ϕ(x⁽ⁱ⁾_j ).

Inorderto provethatthedefinitionofϕ¯iscorrectweshallshowthatonecaninduceϕ¯ onKX_Zn.Tothisend itsufficestoseethatϕ(¯C)= 0.

LetB =KX0⊂KX_ZnbethesubalgebrageneratedbyallvariablesofZn-degree 0, and let C be the associative subalgebra of A generated by ϕ(x⁽⁰⁾_i ), i ≥ 1. Since [ϕ(x⁽⁰⁾_i ),ϕ(x⁽⁰⁾_j )] = ϕ[x⁽⁰⁾_i ,x⁽⁰⁾_j ] = 0 we obtain thatC is a commutative algebra.Note thatϕ(B)¯ ⊂C.

Let

f =

i

f_i[f_i, g_i]g_i∈C

where f_i,g_i areelementsofZn-degree0inKX_Zn.Itiseasytoseethatnecessarilyf_i, g_i∈B.Thereforebythedefinitionofϕ¯we musthave

¯ ϕf =

i

¯

ϕ(f_i[fi, gi]g_i) =

i

¯

ϕ(fi)[ ¯ϕ(fi) ∈C

,ϕ(g¯ i) ∈C

] ¯ϕ(g_i) = 0.

Thus ϕ¯canbe induced onKX_Zn, and itis ahomomorphismof associativealgebras thatextendsϕ.

ThisimpliesthattheuniversalenvelopingalgebraofLX_ZnisKX_Zn.Inparticular we havethe inclusion LX_Zn ⊂KX_Zn. Moreover accordingto theconstruction of thesets wehavetheequality_D=_C

LX_Zn. 2

Lemma 3.3. The inverse image of p in KX_Zn of the T_Zn-ideal in KX_Zn generated by (2)coincides withT_Zn(U T_n),theT_Zn-ideal of graded(associative)identitiesof U T_n. In particular the ideal of Zn-graded identities of U T_n inside the algebra KX_Zn is generated by thegradedidentities{x⁽ⁱ⁾₁ x^(j)₂ |i+j ≥n}.

Proof. Theproof follows immediately bythefactthatthe projectionis canonical,and by the isomorphism theorem combined with the main result of [20] (see Theorem 3.1 above).Wedrawthereaders’attentiontothefactthathereweworkmodulothegraded identity [x⁽⁰⁾₁ ,x⁽⁰⁾₂ ], and that is why it does not appear among the list of generating identities. 2

Before stating the next lemma we observe the following. The Zn-graded algebra KX_Zn has a basis (as a vector space) consisting of homogeneous monomials since the freegraded algebrahassuchabasis. Letf ∈KX_Zn,then f canbe writtenas a sumof(different)monomialsand suchasumiscanonicalandunique.

Lemma 3.4.

1. Letf =

imi∈KX_Znbewrittenasasumofmonomials.Thenf isaZn-identity forU Tn if andonlyif each mi isaZn-identityforU Tn.

2. A monomial m = x^(j_i1¹⁾x^(j_i2²⁾· · ·x^(j_im^m) is a Zn-identity if and only if the inequality j₁+j₂+· · ·+j_m≥nholds.

Proof. 1.Itfollowsfrom Lemma 3.3thatwehavef =

if_i(f_ig_i)g_i whereZn- degf_i+ Zn- deggi≥n.Writingfi,f_i,gi,g_iassumsofmonomials(intheabovediscussedunique and canonicalway),wehavetheclaim.

2.Theproofofthis claimis nowimmediate. 2

UsingtheinclusionLX_Zn⊂KX_Zn,wehaveananaloguetotheaboveintheLie case.

Lemma3.5.

1. Letm∈LX_Znbea Liemonomial andwritem=

ifi.Hereeach fi∈KX_Zn is an associative monomial (as mentioned above this can be done in a unique and canonicalway).Then misa(Lie) Zn-identityforU Tn⁽⁻⁾ ifandonly ifat leastone of thef_i isaZn-identityforU T_n (if andonly ifallf_i areZn-identities).

2. Letf =

imi∈LX_ZnbewrittenasasumofmonomialsinLX_Zn(thereexists such a sum, but there is no canonicalway to do this). Then f isa Zn-identity for U Tn⁽⁻⁾ if andonlyif each mi isaZn-identity forU Tn⁽⁻⁾.

Proof. 1.Thisclaimfollowsimmediately fromLemma 3.4.

2. The claim is a consequence of the first claim of the lemma and of Lemma 3.4, claim 1. 2

Inthiswaywereachatthefollowingcorollary.

Corollary3.6.Letm∈LX_ZnbeaLiemonomialthatisaZn-identityforU Tn⁽⁻⁾.Then misa consequenceofthemonomials(3).

Proof. Clearly m cannot be of (usual) degree 1 therefore we can write m = [u,v].

Combining Claim 1 of Lemma 3.5 and Claim 2 of Lemma 3.4 we see that necessar- ilyZn- degu+Zn- degv≥n.Thisprovesthelemma. 2

Nowwestatethemaintheorem inthissection.

Theorem 3.7. Assume K is an infinite field. Then the Zn-graded identities of the Lie algebra U Tn⁽⁻⁾ of theuppertriangularn×n-matrices overK followfrom

[x⁽ⁱ⁾₁ , x^(j)₂ ], i+j≥n

[x⁽⁰₁, x⁽⁰⁾₂ ] (4)

Proof. We shall use the notation introduced in this section. Consider the canonical projections:

LX_Zn →LX_Zn/D→ LX_Zn/D T_Zn(U Tn⁽⁻⁾).

CombiningLemma 3.5(Claim2) andCorollary 3.6, weobtainthatT_Zn(U Tn⁽⁻⁾) isgen- eratedbymonomialsoftheform(3).Bytheisomorphismtheoremandbythefactthat

all projections are canonical we get that T_Zn(U Tn⁽⁻⁾) ⊂ LX_Zn is generated by the monomials in(4). 2

4. Furtherdiscussion

There areseveralnaturalquestionsrelatedto theresultsinthispaper.

1.LetGbeagroup.AreallG-gradingsonU Tn⁽⁻⁾G-gradedisomorphictoanelemen- tary grading?

Intheassociativecase,ValentiandZaicev(see[26])provedthatthisindeedhappens.

InthecaseofLiealgebrastheanswerisnegative.Thereareexamplesofgroupgradings onU Tn⁽⁻⁾thatarenotelementarynorareisomorphictoelementaryones.Wewilldiscuss suchexamplesandclassificationsofthegradingsonU Tn⁽⁻⁾indetailelsewhere.

2.GivenanelementaryG-gradingonU Tn⁽⁻⁾,dotheidentitiesT_G(U Tn⁽⁻⁾) followfrom theelements fμ whereμrunsoverthebadsequences?

ThisisthecasefortheassociativealgebraU Tn,establishedin[12].FortheLiealgebra U Tn⁽⁻⁾ itisverydifficulttoeitherproveorgiveacounter-exampletosuchastatement.

Acknowledgment

TheReferee’scommentshelpedusimprovethereadabilityofthepaperandhavebeen incorporated intothetext,withgratitude.

References

[1]E.Aljadeff,D.Haile,SimpleG-gradedalgebrasandtheirpolynomialidentities,Trans.Amer.Math.

Soc.366(2014)1749–1771.

[2]S.Azevedo,P.Koshlukov,GradedidentitiesforT-primealgebrasoverfieldsofpositivecharacter- istic,IsraelJ.Math.128(2002)157–176.

[3]S.S. Azevedo,Graded identities for thematrix algebra of order n overan infinitefield, Comm.

Algebra30 (12)(2002)5849–5860.

[4]S.S.Azevedo,Abasisfor Z-gradedidentitiesofmatricesoverinfinite fields,SerdicaMath.J.29 (2003)149–158.

[5]Yu.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2)(2001)677–698.

[6]Yu.A.Bahturin,M.V.Zaicev,Groupgradingsonmatrixalgebras,Canad.Math.Bull.45 (4)(2002) 499–508.

[7]Yu.A.Bahturin,M.V.Zaicev,Gradedalgebrasandgradedidentities,in:PolynomialIdentitiesand CombinatorialMethods,Pantelleria,2001,in:Lect.NotesPureAppl.Math.,vol. 235,Dekker,NY, 2003,pp. 101–139.

[8]Yu.A.Bahturin,I.P.Shestakov,M.V.Zaicev,GradingsonsimpleJordanandLiealgebras,J.Algebra 283 (2)(2005)849–868.

[9]Yu.A.Bahturin,Groupgradingsonfreealgebras ofnilpotentvarietiesof algebras,Serdica Math.

J.38 (1–3)(2012)1–6.

[10]A.P.BrandãoJr.,P.Koshlukov,A.Krasilnikov,Gradedcentralpolynomialsforthematrixalgebra ofordertwo,Monatsh.Math.157 (3)(2009)247–256.

[11]O.M.DiVincenzo,OnthegradedidentitiesofM1,1(E),IsraelJ.Math.80 (3)(1992)323–335.

[12]O.M.DiVincenzo,P.Koshlukov,A.Valenti,Gradingsonthealgebraofuppertriangularmatrices andtheirgradedidentities,J.Algebra275 (2)(2004)550–566.

[13]O.M.Di Vincenzo,P. Koshlukov,R. La Scala,Involutionsforupper triangular matrixalgebras, Adv.inAppl.Math.37 (4)(2006)541–568.

[14]O.M.DiVincenzo,P. Koshlukov,E.SantuloJr.,Gradedidentitiesfortensorproductsofmatrix (super)algebrasovertheGrassmannalgebra,LinearAlgebraAppl.432 (2–3)(2010)780–795.

[15]O.M.Di Vincenzo, V.R.T.Silva,On Z2-gradedpolynomialidentitiesof theGrassmannalgebra, LinearAlgebraAppl.431(2009)56–72.

[16]A. Elduque, M. Kochetov, Gradings on SimpleLie Algebras, Math. SurveysMonogr., vol. 189, Amer.Math.Soc.,AARMC,Providence,RI,Halifax,NovaScotia,2013.

[17]G.Genov,P.Siderov,Abasisforidentitiesofthealgebraoffourth-ordermatricesoverafinitefield I,II,Serdica8(1982)313–323,351–366(inRussian).

[18]A.Kemer,IdealsofIdentitiesofAssociativeAlgebras,Transl.Math.Monogr.,vol. 87,Amer.Math.

Soc.,Providence,RI,1991.

[19]P.Koshlukov,Basisoftheidentitiesofthematrixalgebraofordertwooverafieldofcharacteristic p= 2,J.Algebra241(2001)410–434.

[20]P.Koshlukov,A.Valenti,Gradedidentitiesforthealgebraofn×nuppertriangularmatricesover aninfinitefield,Internat.J.AlgebraComput.13 (5)(2003)517–526.

[21]P.Koshlukov,GradedpolynomialidentitiesfortheLiealgebrasl₂(K),Internat.J.AlgebraComput.

18 (5)(2008)825–836.

[22]P.Koshlukov,F.Martino,PolynomialidentitiesfortheJordanalgebraofuppertriangularmatrices oforder 2,J.PureAppl.Algebra216 (11)(2012)2524–2532.

[23]P.Koshlukov,M.V.Zaicev,Identitiesandisomorphismsofgradedsimplealgebras,LinearAlgebra Appl.432 (12)(2010)3141–3148.

[24]Yu.N.Malcev,E.N.Kuzmin,Abasisfortheidentitiesofthealgebraofsecondordermatricesover afinitefield,AlgebraLogika17 (1)(1978)28–32(inRussian).EnglishTransl.: AlgebraLogic17 (1) (1978)18–21.

[25]Yu.Razmyslov,IdentitiesofAlgebrasandTheirRepresentations,Transl.Math.Monogr.,vol. 138, Amer.Math.Soc.,Providence,RI,1994.

[26]A.Valenti,M.V.Zaicev,Groupgradingsonuppertriangularmatrices,Arch.Math.89 (1)(2007) 33–40.

[27]S.Yu.Vasilovsky,Z-gradedpolynomialidentitiesofthefullmatrixalgebra,Comm.Algebra26 (2) (1998)601–612.

[28]S.Yu.Vasilovsky,Zn-gradedpolynomialidentitiesofthefullmatrixalgebraofordern,Proc.Amer.

Math.Soc.127 (12)(1999)3517–3524.

[29]C.T.C.Wall,GradedBrauergroups,J.ReineAngew.Math.213(1963/1964)187–199.

[30] F.Yasumura,E.A.Hitomi,OnthecombinatoricsofcommutatorsofLiealgebras,https://arxiv.org/

abs/1609.09407,submittedforpublication.