orthogonal designs Free nilpotent and H -type Lie algebras. Combinatorial and Journal of Pure and Applied Algebra

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Journal of Pure and Applied Algebra

www.elsevier.com/locate/jpaa

Free nilpotent and H -type Lie algebras. Combinatorial and orthogonal designs

Kenro Furutani^a,1, Irina Markina^b,2, Alexander Vasil’ev^b,2

a DepartmentofMathematics,FacultyofScienceandTechnology,ScienceUniversityof Tokyo, 2641 Yamazaki,Noda,Chiba(278-8510),Japan

bDepartmentofMathematics,UniversityofBergen,POBox7803,BergenN-5020,Norway

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received26January2015 Receivedinrevisedform8April 2015

Availableonline2June2015 CommunicatedbyA.V.Geramita

MSC:

Primary:15A66;17B30;22E25;

secondary:05B30;05C20;05C70;

94B60;30C62

The aimof our paper isto construct pseudo H-type algebras fromthe covering freenilpotenttwo-stepLiealgebraasthequotientalgebrabyanideal.Wepropose an explicit algorithm of construction of such an ideal by makinguse of a non- degeneratescalarproduct.Moreover,asabyproductresult,werecovertheexistence ofa rationalstructure onpseudo H-typealgebras,whichimpliestheexistenceof latticesonthecorrespondingpseudoH-typeLiegroups.Ourapproachsubstantially uses combinatorics and reveals the interplay of pseudo H-type algebras with combinatorialandorthogonaldesigns.OneofthekeytoolsisthefamilyofHurwitz–

Radonorthogonalmatrices.

© 2015ElsevierB.V.All rights reserved.

1. Introduction

Anynilpotent Liealgebraisknown[31]to be obtainedas acosetspaceofafreenilpotentalgebrabya quotientideal.However,itisnotclearhowthisidealcanberecoveredaccordingtothepropertiesofthegiven nilpotentalgebra.Weproposeanexplicitconstructionoftheidealandofthequotientmapbymakinguseofa non-degeneratescalarproducttoobtainpseudoH-typealgebras,i.e.,gradedtwo-stepnilpotentLiealgebras introduced in [4,14,22] and intimately related to representations of Clifford algebras and compositions of quadraticforms.Moreover,asabyproductresult,werecovertheexistenceofarationalstructureonpseudo H-typealgebras, which implies the existenceof latticeson the corresponding pseudo H-type Lie groups, giving a new and essentially different proof of the corresponding result in [9], see also [7,16]. Mal’cev

E-mailaddresses:[email protected](K. Furutani),[email protected](I. Markina), [email protected](A. Vasil’ev).

1 The author wassupported by the Grand-in-aid for Scientific Research (C) No. 23540251 of JSPS (Japan Societyfor the PromotionofScience).

2 TheauthorshavebeensupportedbythegrantsoftheNorwegianResearchCouncil#213440/BG,#239033/F20;andEUFP7 IRSESprogramSTREVCOMS,grantno.PIRSES-GA-2013-612669.

http://dx.doi.org/10.1016/j.jpaa.2015.05.027 0022-4049/© 2015ElsevierB.V.All rights reserved.

introduced in1951[27]anilmanifold asacompactmanifoldwithaconnected,simplyconnectednilpotent Liegroupactingonit.AnilmanifoldisdiffeomorphictothequotientspaceL\GofanilpotentLiegroupG, where L is a uniform lattice. Nilmanifoldsrepresent an example of homogeneous spaces, which plays an importantroleingeometryandharmonicanalysis,aswellasinarithmeticcombinatoricsand,morerecently, ergodictheory.Mal’cev’scriterion[27]guaranteestheexistenceofauniformlatticeL⊂Gprovidedrational structural constants forthe nilpotentLie algebragof the Liegroup G. So thepseudo H-type Lie groups with latticesprovide abroadclassofnilmanifoldswhichcanbe studiedandclassified.

TheproposedapproachisexpectedtocontributetoaclassificationofH-typeandpseudoH-typealgebras, and inparticular,to adescriptionoftheclassesoflatticesthatadmiteverypseudo H-typegroup.To our knowledge, thecomplete classificationoflatticesis knownonlyinthecase ofHeisenberg groups([15], see also[33]).Thiswouldalsoleadtoapossibleclassificationofnon-diffeomorphicnilmanifold.However,these topics exceed the scope of this paper and are the subject of future research. The study of infinitesimal symmetriesofpseudoH-typegroupsandthecomparisonwiththeTanakaprolongationsofsimplealgebras factorized byparabolic subgroupsarecloselylinked.Wehopethatourapproachwillallowustocontribute to thebeautifultheoryofsimplegroups.

At the end, we show relations with combinatorics and orthogonal designs using our construction for pseudo H-typegroupsinsteadof theclassicalapplicationof Cliffordalgebras.Inparticular, we apply our results tothesquare semiregular1-factorizationof acompletegraphK2n,and wefix aGeramita–Seberry Wallis problem on a maximal number of variables in an amicable orthogonal design [13, Problem 5.17], where orthogonalityis understoodwithrespect to anon-degenerate metricwiththeneutralsignature.

The structure ofthe paper isas follows. We giveprecise definitionsand preliminaries inSection2. We provethemain resultsonconstructionofquotient mapsand theexistenceofrationalstructural constants inSection3. Section3.3 isdedicatedtotheclassicalcaseofH-typealgebras withapositivedefinite inner product,andSection3.4generalizesitto pseudoH-typealgebras.Section4givesconnectionsof themain resultsto combinatorialandorthogonaldesigns.

2. Preliminaries

2.1. Cliffordalgebrasandtheir representations

LetV bearealvectorspaceendowedwithanon-degeneratequadraticformQ(v),v∈V,whichdefinesa symmetricbilinearformq(u,v)= ¹₂(Q(u+v)−Q(u)−Q(v)) bypolarization.Hereandfurtheronbysaying scalar product we mean a non-degenerate symmetric real bilinear form and by inner product a positive definite one.TheClifford algebra Cl(V,q),namedafter theEnglishgeometerWilliamKingdonClifford[5], is anassociativeunitalalgebrafreelygeneratedbyV modulotherelations

v²=−Q(v)1=−q(v, v)1 for allv∈V, or uv+vu=−2q(u, v)1 for allu, v∈V.

For anintroductorytext, onemaylook at[11]. Everynon-degeneratequadratic formonann-dimensional vector spaceV isequivalenttothestandarddiagonalform

Qp,q(v) =v₁²+v₂²+· · ·+v_p²−v²_p+1− · · · −v_p+q² , n=p+q.

Using the isomorphism (V,Q) R^p,q = (R^p+q,Qp,q) we will write Cl(V,q) = Clp,q. Starting with an orthonormalbasis₁,. . . ,_n inR^p,q onedefinesabasisofCl_p,q bythesequence

1, . . . ,(_k1·. . .·_kj), 1≤k₁< k₂<· · ·< k_j≤n, j= 1,2, . . . , n.

ItfollowsthatthedimensionofCl_p,q is2ⁿ,n=p+q.

AClifford moduleforCl_p,q isarepresentationoftheCliffordalgebraCl_p,q givenbyafinite-dimensional real space U and a linearmap ρ:Clp,q → End(U), satisfyingthe Cliffordrelation ρ²(v) =−q(v,v)idU or ρ(u)ρ(v)+ρ(v)ρ(u) =−2q(u,v)idU for u,v ∈ R^p,q. An abstract theory of Clifford modules wasfounded in[3].

2.2. Free nilpotentLiealgebras

LetN stand forafree nilpotentLie algebra definedfollowing,e.g.,[20,31].Given areal vectorspace U ofdimensionmwithabasis{e1,. . . ,em},called generators,we constructthespace

U =U⊕(U⊗U)⊕ · · · ⊕(U⊗ · · · ⊗U).

Weintroduceadistributive, non-associativeand non-commutativeoperation ×: U × U → U iteratively as follows.Ife_k ∈U,then

e_i×(e_j1⊗ · · · ⊗e_jk) =

e_i⊗e_j1⊗ · · · ⊗e_jk, ifk≤n−1,

0, ifk=n.

Nextiteratively,ife∈U anda×b isalreadydefinedforafixeda∈ U andforeveryb∈ U,then (e⊗a)×b=e⊗(a×b)−a×(e⊗b).

Letusdenote byW theleftidealof theelementsrepresentedas v×v, forv∈ U. LetN =U/W.Then N carriesthestructureofanilpotentLiealgebrawith theLie product

[ a, b] := a×b,

where a denotes the equivalence class with a representative a ∈ U. The Jacobi identity holds and the nilpotentnesstriviallyfollows from Nⁿ⁺¹ = 0.Abusingnotationswe shallwrite simplyainstead of ain whatfollows.ThealgebraN isafreenilpotent Liealgebraoflengthnwithm generators.

Theorem 1. (See [31].) Let n be a nilpotent Lie algebra of length n generated by m linearly independent elements.Then thereexistsanideal Aof N suchthat nN/A.

Wedenote byπ theprojectionπ:N →n.Inparticular, we areinterested infreenilpotent Liealgebras oflength 2.

2.3. PseudoH-type algebra

Itisconvenient forus tosplitthedefinitionofapseudoH-typealgebraintwo parts.

Definition 1. Wesay thata Lie algebran ≡(n,[·,·],(·,·)) is atwo-step nilpotent metricLie algebra ifit satisfiesthefollowingproperties:

• [[n,n],n]={0};

• thescalarproduct(·,·) isnon-degenerate;

• n=h⊕⊥zwithrespectto(·,·),wherezisthecenterofn;

• therestriction(·,·)_z of(·,·) tozisnon-degenerate.

Definition 2.A two-step nilpotent metric Lie algebran is called apseudo H-type algebraif the operator J : z×h→hdefinedby

(J_zu, v)_h= (z,[u, v])_z, (1)

satisfies thefollowingorthogonalitycondition

(Jzu, Jzv)h= (z, z)z(u, v)h. (2) Given a metric two-step nilpotent Lie algebra n, we call the operator J defined by (1) satisfying (2), a pseudoH-type structureorsimplyanH-typestructureinthecaseofapositivedefinite metric.

This definition for an inner product (positive definite) is equivalent to that found in [22], and for a non-degenerate scalarproduct, in[4,14]. Relation (1) immediately impliesthe following propertiesof the operatorJ.

• TheoperatorJ : z×h→hisbilinear.

• TheoperatorJz∈End(h) foranyfixed z∈zisskewsymmetric

(Jzu, v)h=−(u, Jzv)h, (3)

and

J_z²=−(z, z)_zid_h. (4)

• TheoperatorJ_z:h→hisanisometryforallz∈zwith(z,z)z= 1.

• TheoperatorJ_z:h→hisananti-isometryforallz∈zwith (z,z)_z=−1.

Remark1.Anytwoofproperties(2),(3),and (4)implythethirdproperty.

Proposition 1. (See [4, Proposition 2.2].) Let n be a pseudo H-type algebra with a non-degenerate scalar product(·,·).Ifthescalarproduct(·,·)zisnon-positivedefinite,thennecessarily,thesignatureofthescalar product (·,·)hisneutral.

Proposition2.Letnbeatwo-stepnilpotentmetricLiealgebra,andletz1,. . . ,zpformanorthogonalnon-null basis of z. Let the operatorJ be defined by (1) on the basis vectors of z. The operator Jz, z ∈z, satisfies condition(2),orequivalently,nisapseudoH-typealgebra,ifandonlyif,J_z²_k =−(zk,zk)zidh,andJziJzj =

−J_zjJ_zi.

Proof. Thenecessaryconditionistrivial.Letusfocusonthesufficientpart.Indeed,thedefinition(1)ofthe operatorJ canbeextendedfromthebasisofztothewholezbylinearity,anditimpliestheskew-symmetry (J_zu,v)_h=−(u,J_zv)_h,andlinearitywithrespect toz andu. Therefore,

(J_αz² _1+βz2u, v)h=−(J_αz1+βz2u, J_αz1+βz2v)h=−α²(J_z1u, J_z1v)h−β²(J_z2u, J_z2v)h

=α²(J_z²₁u, v)_h+β²(J_z²₂u, v)_h=−α²(z₁, z₁)_z(u, v)_h−β²(z₂, z₂)_z(u, v)_h

=−(αz1+βz2, αz1+βz2)z(u, v)h,

for any uand v from h and forany basisvectors z₁ and z₂ from z. Hence, the equality J_z² =−(z,z)_zid_h holds foranyvectorz∈z,andtheorthogonalityconditionfollowsfromRemark 1. 2

Let us formulate how the pseudo H-type algebras are related to representations of Clifford algebras.

GivenapseudoH-typealgebran= (z⊕⊥h,[.,.]),theoperatorJ_z isdefinedby(1)satisfying(4)forevery z ∈z, andtherefore, itdefines arepresentationJ:Cl(z,q)→End(h) over thespace h.Here thequadratic form q is defined by the scalar product (.,.)z. Moreover in this case, the scalar product (.,.)h on the representation space is such thatthe operator Jz is skew symmetric with respect to this scalar product, see(3).Following[4]wecall suchapair(h,(.,.)h) anadmissiblerepresentation.Nowletusassumethata representationρ:Cl(Z,q)→End(V) oftheCliffordalgebraCl(Z,q) isgiven, andletussuppose alsothat the representationspace V admits ascalarproduct (.,.)_V,such thatthe restrictionJ =ρ|Z of ρto Z is skewsymmetricwithrespectto(.,.)V,inotherwords(V,(.,.)V) isanadmissiblerepresentation.Then,we definetheLiebrackets[.,.] byformula(1),wherethescalarproduct(.,.)Zisapolarization ofthequadratic formq.TheLiealgebraN = (Z⊕V,[.,.]) willbe apseudoH-typeLiealgebrawiththecenterZ andwith theorthogonaldecompositionZ⊕V with respecttothescalarproduct(.,.)= (.,.)Z+ (.,.)V.

3. H-typealgebrasfromfreealgebras

ThemainideaistoconstructapseudoH-typealgebrabymeansofthefreealgebrastepbysteprestricting thelatterbysatisfyingnecessary conditionsandenrichingitwithadditionalstructures.

3.1. Construction

Firstwe assign the vector space U to be the complementary space to thecenter forthe futurepseudo H-typealgebra,andso wechooseaneven numberm= 2kofgenerators

e₁, . . . , e_k, e_k+1, . . . , e_2k.

A pseudoH-typealgebrais ofsteptwo,sowe areinterestedinalengthtwofreenilpotent LiealgebraN. Letusintroduceascalarproduct(·,·)_N inN ofsomesignature,suchthatthebasisvectorse₁,. . . ,e_2k,e₁× e₂,. . . ,e_2k−1×e_2kareorthogonalandnon-nullwithrespecttothisproduct.Thescalarproduct(·,·)_Nsplits as(·,·)N= (·,·)U+ (·,·)Z,whereZ = span{e1×e2,e1×e3,. . . ,e_2k−1×e2k}.ThiswaytheLiealgebraN splitsinto

N =U⊕⊥Z = span{e₁, . . . , e_2k, e₁×e₂, . . . , e_2k−1×e_2k

k(2k−1)

}.

Define thesignature indexofascalarproduct(·,·)_U by ε_j(s, r) =

1, forj= 1, . . . , s;

−1, forj=s+ 1, . . . , s+r.

Letusnormalize thebasisvectorsso that

• (e_j,e_j)_U =ε_j(s,r)/k forj= 1,. . . ,2k,s+r= 2k,where s,rarenotspecifiedso far;

• |(ei×ej,ei×ej)Z|= 1/k.

IfA⊂ZisanidealofN,suchthatN =U⊕⊥Ω⊕⊥A,andn=N/A,thenh:=U(modA),z:= Ω(modA), thescalarproduct(·,·)N = (·,·)U+ (·,·)Ω+ (·,·)A gives(·,·)n:= (·,·)U + (·,·)Ω.Letus denote ˜u=u+A, andv˜=v+Aforu,v∈U.Thecommutatoronnisdefinedby[˜u,v]˜ =u×v+A.

LetusdefineanoperatorJ_ω:U →U, ω∈Z,by(J_ωu,v)_U = (ω,u×v)_Z. Sofaritisonlyadefinitionof J withoutrequiringtheorthogonalitycondition(2).

Inorderto makenapseudoH-typealgebra,theoperatorJ˜_z,z∈z,z=ω+A, ω∈Ω,mustsatisfy ( ˜J_zu,˜ v)˜_h= (z,[˜u,˜v])_z= (ω+A, u×v+A))_Z = (ω, u×v)_Ω= (J_ωu, v)_U

and theorthogonalitycondition(2).

Thenextstepisto constructtheidealA ofN asanorthogonalcomplementinZ to Ω.Todefine Ω we consider afamilyof partitionsp={pl},l = 1,. . . ,^(2k)!₂kk!,of theset of integers1,2,. . . ,2k inordered pairs (i,j), i < j. For each p_l ∈ pwe construct ordered pairs (e_i,e_j), (i,j) ∈ p_l, the product e_i×e_j, and the vector

ω_l=

(i,j)∈pl

α^l_ij(e_i×e_j)∈Z, α^l_ij ∈R.

TheoperatorJ_ωl:U →U isdefinedbytheequality

(J_ωl(u), v)_U = (ω_l, u×v)_Z, u, v∈U. (5) Letusassumethatthecoefficientsα^l_ij arechosensothatZ = span{ωl:l= 1,. . . ,^(2k)!₂kk!}.Fromthedefinition oftheoperatorJωlandtheproduct(×) itimmediatelyfollowsthatJωlextendstotheoperatorJω:U →U, ω ∈Z bylinearity,and

(Jω(u), v)U =−(u, Jω(v))U.

We restrictthenumberof partitionsfrom passigningspecial valuesto thecoefficientsα^l_ij byrequiring thattheoperatorsJ_ωl actinaspecialwayonthebasis{e₁,. . . ,e_2k}ofU.

Definition 3.By asigned permutation of vectors e₁,. . . ,e_2k from U by theoperator J_ω we understanda permutation where someofresulting vectorsmaychangeorientation.Thatis, forany i= 1,. . . ,2k,there exists j= 1,. . . ,2k,suchthatJ_ωe_i=±e_j.

Proposition 3.Forafixedl,theoperatorJ_ωl:U →U isasignedpermutationof thebasise₁,. . . ,e_2k,ifand only if, forevery fixed a∈ {1,. . . ,2k}, thereexists aunique b∈ {1,. . . ,2k},a=b,suchthat α^l_ab=±1 if a< bor α^l_ba=±1if a> b.

Proof. The sufficient partis trivial. For the necessary part, fix some a∈ {1,. . . ,2k}and l. Then, inthe partitionp_l,there isauniqueindexb∈ {1,. . . ,2k}suchthat(a,b)∈p_l.Forsimplicity assumethata< b.

Thedefinition(5)oftheoperatorJ_ωl immediatelyimplies

(J_ωl(e_a), e_b)_U = (ω_l, e_a×e_b)_Z =±α^l_ab k = 0, and

(Jωl(ea), ej)U = (ωl, ea×ej)Z = 0 for all j∈ {1, . . . ,2k}, j=b,

since ωl doesnot containtheterm ea×ej. Themap Jωl actsby signpermutationof thebasis, therefore, J_ωl(e_a)=±e_b,andthis impliesalsothatα^l_ab=±1.Herethesign±means, + or−. 2

Remark2.ObservethatthemapJωl:U →U isinjectiveinthecaseof signedpermutation.

Thefollowingsimplestatementcanbefoundinstandardtextsincombinatoricsandgraphtheory,e.g.[2].

Proposition4.(See[2].)The setpof all ^(2k)!₂kk! possible partitionscontains asubset p={pl1, . . . ,pl_2k−1},

suchthat each pair(i,j),1≤i< j≤2k,appearsonly oncein thesepartitions.

Thispropositioncanbereformulated intermsofgraphtheory.Letusconsider thecompletegraphK_2k of 2k vertices at the numbers1,. . . ,2k,and recall thata 1-factorizationof agraphis adecomposition of allthe edgesof thegraph into 1-factors,thesets ofk independent edges(withoutcommon vertices). The graph K_2k has exactly(2k−1) 1-factors, see e.g., [17], which in ourcase coincide with a possible set of partitionsp={pl1,. . . ,pl_2k−1}.

Weobserve thatsplitting ofK2k into (2k−1) 1-factorsis notuniqueand maybeeven notisomorphic, e.g.,Kirkman[24]andSteiner[32]tournaments arenon-isomorphic[2],seeexample:

12 38 47 56 13 24 58 67 14 26 35 78 15 28 37 46 16 23 57 48 17 25 34 68 18 27 36 45

and

12 37 45 68 13 27 48 56 14 25 38 67 15 24 36 78 16 28 35 47 17 23 46 58 18 26 34 57 .

Indeed,inthefirstarray,anytwolinessettogetherinonegraphgiveacycleoflength8,whileinthesecond arraythis willgivetwodisjointcyclesof length 4.

Letωlm beconstructedbyplm ∈p,andfromnowon,letthecorrespondingoperatorJω_lm act asasigned basispermutationonU.FixingsomeejweobtainanoperatorJ(ej):Z→U.Weshallomittheupperindex writingsimplyαij insteadofα^l_ij becausethepair(i,j) ismetonlyonceineveryp.

Proposition5.Letp ={pl1,. . . ,pl_2k−1}beasetofpartitions,whereeachpair(i,j),1≤i< j≤2k,appears only once.LettheoperatorJω_lm act overU asasigned permutation.The map

J(ej): span{ωlm=

(i,j)∈plm

αij(ei×ej), m= 1, . . . ,(2k−1)} → {ej}^⊥ isbijective.

Proof. Without loss of generality, fix j = 1. Then Jω_lm(e1) = ±ejm by Proposition 3, where m = 1,. . . ,(2k−1), the index jm ∈ {2,3,. . . ,2k}, and jm = jn if m = n. If ω = 2k−1

m=1cmωlm, then Jω(e1) = 2k−1

m=1 ±cmejm, which vanishes if and only if, all coefficients cm = 0. Therefore, J(e1) acts as anisomorphismoflinearspacesspan_{1≤m≤2k−1}{ωlm}and {e1}^⊥. 2

The scalarproduct onU is defined, and now we fix it on Z requiring (ω_lm,ω_lm)_Z =±1.This canbe alwaysachieved bythefollowing procedure.Fixmand write

(ω_lm, ω_lm)_Z =

(i,j)∈plm

(e_i×e_j, e_i×e_j)_Z.

Wechooseasignature(p,q) of(·,·)_Z,p+q= 2k−1.Thenwedefine(e_i×e_j,e_i×e_j)_Z = 1/kfor(i,j)∈plm, where m = 1,. . . ,p, and (e_i×e_j,e_i×e_j)_Z =−1/k form =p+ 1,. . . ,p+q = 2k−1. We arriveat two options:themetriconZ ispositivedefinite,q= 0,and(ω_lm,ω_lm)_Z = 1,andthemetriconZ is indefinite non-degenerate,q= 0,so that(ω_lm,ω_lm)_Z =−1 for someofl_m.

3.2. Innerproducton Z

Since wehave chosenaninner (positive definite)product onZ, thescalarproduct onU maybyeither an inner product or ascalarnegative definite product,becauseof the propertiesof the operatorJ. Both pictures are isomorphic, hence we choosean inner product onthe whole N. In this section we study the caseofboth metrics(·,·)_U and(·,·)_Z tobepositivedefinite.Inthiscase

J_ω²_l=−idU and Jωl1Jωl2 =−Jωl2Jωl1. (6) To every pl ∈ p, to the corresponding vector ωl, and finally, to the operator Jωl acting by signed permutation, we associate a 2k×2k matrix of coefficients El = {αij} where αij = −αji and αij = 0 if (i,j)∈/pl.Matrices El, l= 1,. . . ,2k−1 areorthogonal withentries 0,±1.So weconstructedaninjective homomorphismof theoperators Jωl tothesetoforthogonal matrices.

We continue with the Hurwitz–Radon–Eckmann theorem, see [8,22]. The Hurwitz–Radon function is definedas ρ(n):= 8α+ 2^β,where nisuniquelyrepresentedbyn=u2^4α+β with uodd,β= 0,1,2 or 3.

Theorem 2. (See [8,18,30].) A family of 2k−1real orthogonal 2k×2k matrices E₁,. . . ,E_2k−1 admits at mostρ(2k)−1matricessatisfyingE_j²=−IandEiEj=−EjEi,whereρ(·)istheHurwitz–Radon function.

Thefunctionρ(n) seemstobequiteirregularfromthefirstglancebutitisnotso.Itisperiodic.Wedid notfindagoodreferenceso letus provethissimplebutusefulfact.

Proposition 6.If n=u2^r,r∈N,thenρ(n+ 2^R)=ρ(n)forany R=r.

Proof. Ifr= 0,thennisoddandρ(n)= 1.Sotheconclusionistrivial.Letr= 4α1+β1>0,n=u12^4α1+β1, and letn+ 2^R be representedasn+ 2^R=u2^4α2+β2,whereαk,βk,k= 1,2 areas above,andbothu1 and u2 areodd.Withoutloss ofgeneralityassumethat4α1+β1≤4α2+β2.Then,

2^4α1+β1 u22^{4(α2−α1)+(β2−β1)}−u1

= 2^r u22^{4(α2−α1)+(β2−β1)}−u1

= 2^R.

SinceR=r,theexpressionu22^{4(α2−α1)+(β2−β1)}−u1isevenwhichispossibleonlyif4(α2−α1)+(β2−β1)= 0.

Since |β₂−β₁| <4,we concludethat α₁ =α₂, and then, β₁ =β₂. The definitionof the Hurwitz–Radon functionimpliesthatρ(n+ 2^R)=ρ(n). 2

Remark3.IfR=r,thentheperiodicitydoesnotholdingeneral,observe e.g.,ρ(12)=ρ(3·2²)= 4= 9= ρ(16)=ρ(3·2²+ 2²).

Thesequenceof values{ρ(n)}n≥1 satisfiesthefollowing telescopicproperty,see Fig. 1.

Corollary 1.Thesequence{ρ(n)}²n=1^r−1 isthesame as{ρ(n)}²n=2^r+1r−+1¹.

Proof. Indeed, thenumber of elements inboth sequences is 2^r−2, forall valuesn = 1,. . . ,2^r−1< 2^r, therefore,

{ρ(n)}²n=1^r−1={ρ(n+ 2^r)}²n=1^r−1={ρ(n)}²n=2^r+1r−1+1

bythepreviousproposition. 2

Theorem 2 wasproved forinteger matricesE_k having entries among {0,±1}, i.e., for aspecial kindof weighing matrices in[12,13].Moreover, theupper boundρ(2k)−1 is achievedfor integerHurwitz–Radon

Fig. 1.Telescopic property.

matrices[13, Theorem 1.6].Relations(6) andTheorem 2reveal thefactthattheHR-familyof orthogonal matricesgivesarepresentation ofaCliffordalgebraCl_p,0.

Theorem3.Givenafreetwo-stepnilpotentLiealgebraN =U⊕⊥Z withaninnerproductanddim(U)= 2k, thereexistsanidealAsuchthatN/AisanH-typealgebran=h⊕⊥z,whereh=U andthecenterisz=Z/A.

Moreover, theidealAcan bechosen suchthatdim(z)takesevery value from{1,. . . ,ρ(2k)−1}.

Proof. We reformulate this statement in the language of partitions as follows. There exists a family of partitions

p={p₁, . . . ,p_ρ(2k)−1} ⊂p,

equivalent to a subset of 1-factorization of the complete graph K2k, such that the corresponding set of vectors {ω1,. . . ,ω_ρ(2k)−1} with the coefficients αij = ±1 defines A as the orthogonal complement to span{ω₁,. . . ,ω_ρ(2k)−1}inZ,and z=Z/Ais thecenterof anH-type algebra.Inwhatfollowswe describe thechoiceof p.

Westartbyanalyzing thecorrespondencebetweentheoperatorsJωl,l∈1,. . . ,2k−1,andtheorthogonal matricesEl.Indeed,anorthogonalmatrixElwithentries0,±1 isapermutationofthestandardbasisofR^2k, if anonly if, eachrow and each columncontains onlyone entry ±1 and others are 0.Respectively, E_l is a signed permutation of the canonical basis, if and only if, each row and each columncontains only one entry1or(−1)andothersare0.TheformofthepartitionspguaranteesthatthematricesElconstructed abovewith the entries αij, (i,j)∈ pl ∈ p for i < j and αij = −αji, are a signpermutation of the basis {e₁,. . . ,e_2k}, i.e., we constructed aone-to-one correspondence between the orthogonal sign permutation matricesEl,satisfyingtheconditionsof Theorem 2, andtheoperatorsJωl.

Given the set of integer matrices {El1,. . . ,Elρ(2k)−1}, representing the maximal family of matrices in Theorem 2,wesetptobethefamilyofpartitionsplk consistingofelements(i,j) suchthatαij isanentry intheuppertriangularpartofthematrixE_lk.Accordingtop,weconstructthevectorsω_l1,. . . ,ω_lρ(2k)−1. Correspondingoperators Jωlsatisfyrelations(6)becausethematricesEl do.Now, weset h=U,andAto be theorthogonalcomplementinZ tospan{ωl1,. . . ,ωlρ(2k)−1}.Then wedefine z=Z/A,and so,h⊕⊥zis anH-typealgebranbyProposition 2. 2

Thefollowingtheorem wasprovedin[7,9,16]bydifferentmethods.

Theorem4.The H-type algebrasadmit rational structureconstants.

Proof. Leth=U,andletz=Z/Abe chosenandspannedbythevectorsω_l=

(i,j)∈pl α_ije_i×e_j modulo theidealA.Then[en,em]=en×em(mod A)∈z.Let(n,m)∈p_l forsomel. Write

ω_l=α_nme_n×e_m+

(i,j)∈(pl\(n,m))

α_ije_i×e_j,

and set

ω_(l;ij)=αnmen×em−αijei×ej, (i, j)∈(p_l \(n, m)) Then, ωl⊥ω_(l;ij)andω_(l;ij)∈A.Sowewrite

en×em(mod A) = 1 k

⎛

⎝α_nmωl+

(i,j)∈(pl\(n,m))

ω_(l;ij)

⎞

⎠(mod A) = αnm

k ωl(mod A). 2 Remark4.Noticethatω_l= ¹₂2k

j=1e_j×E_le_j =¹₂2k

j=1e_j×J_ωle_j.

Asitwasobserved,theexistenceofintegermatrices{E_l1,. . . ,E_lρ(2k)−1}fromtheHurwitz–Radonfamily was proved in [13, Theorem 1.6]. The orthogonality condition assures that they are skew-symmetric and eachofrows andcolumns containsexactlyonenon-zeroelement.

We are not focused on algorithms of calculation of matrices El, vectors ωl and operators Jl in full generalitywhichisthesubjectofcomputationalalgebra,however,welook atanillustrativeexampleinthe nextsubsection.

3.2.1. The casek= 2

Letus haveacloselookatthesimplest exampleof U generated by{e1,e2,e3,e4}and ω₁=α₁₂e₁×e₂+α₃₄e₃×e₄,

ω₂=α₁₃e₁×e₃+α₂₄e₂×e₄, ω₃=α₁₄e₁×e₄+α₂₃e₂×e₃,

where ω_i are orthogonalwith respectto theinner product.Thatis wehavechosen p=p ⊂psuchthat p={p1,p2,p3},where

p1= 12 34, p2= 13 24, p3= 14 23.

DefinitionoftheoperatorJ andthechoiceoftheinnerproductimplythatJ_ω²_m =−idU.Inordertodefine thecoefficientsαweneedtocheck anti-commutativityofJω1,Jω2,and Jω3 onthebasis.Wecalculate

e1 Jω1

−−−→ α12e2 Jω2

−−−→ α12α24e4, e1

Jω2

−−−→ α13e3 Jω1

−−−→ α13α34e4, e₁ −−−→^Jω2 α₁₃e₃ −−−→ −^Jω3 α₁₃α₂₃e₂, e₁ −−−→^Jω3 α₁₄e₄ −−−→ −^Jω2 α₁₄α₂₄e₂, e1

Jω3

−−−→ α14e4 Jω1

−−−→ −α14α34e3, e₁ −−−→^Jω1 α₁₂e₂ −−−→^Jω3 α₁₂α₂₃e₃,

and inorderto satisfyanticommutativitywewritethehomogeneousequations

α₁₂α₂₄+α₁₃α₃₄= 0, α₁₃α₂₃+α₁₄α₂₄= 0, α₁₄α₃₄−α₁₂α₂₃= 0. (7)

Analogouslywestartwiththevectorse₂,e₃,and e₄,and wearriveat threeadditionalequations

α₁₂α₁₃+α₂₄α₃₄= 0, α₁₃α₁₄+α₂₃α₂₄= 0, α₁₂α₁₄−α₂₃α₃₄= 0, (8) which we solve over the integers (1,−1) ∈ Z. However, the systems (7) and (8) are equivalent. Indeed, multiplyingthe firstequation in(8)by α₁₂α₃₄ weobtainthe firstequation in(7),multiplying thesecond equationin(8)byα₁₄α₂₃weobtainthesecondequationin(7),andfinally,multiplyingthethirdequationin (8)byα13α24,weobtainthethirdequationin(7).Thisreflectsthefactthatwecancheckanticommutativity ofJω1,Jω2,and Jω3 startingonlywithonevector,e.g.,e1.

The operators J_ω1, J_ω2, and J_ω3 define theorthogonal matrices E₁, E₂, and E₃ as itwas described in Theorem 3

E1=

⎛

⎜⎜

⎜⎝

0 −α12 0 0

α₁₂ 0 0 0

0 0 0 −α₃₄

0 0 α₃₄ 0

⎞

⎟⎟

⎟⎠, E2=

⎛

⎜⎜

⎜⎝

0 0 −α13 0

0 0 0 −α₂₄

α₁₃ 0 0 0

0 α₂₄ 0 0

⎞

⎟⎟

⎟⎠,

E3=

⎛

⎜⎜

⎜⎝

0 0 0 −α14

0 0 −α23 0

0 α₂₃ 0 0

α₁₄ 0 0 0

⎞

⎟⎟

⎟⎠,

whichsatisfytheconditions

E_j²=−I and EiEj =−EjEi. (9)

Inthiscase,inviewofHurwitz–Radon–Eckmann’sTheorem 2,u= 1,α= 0, β= 2,and ρ(4)−1= 3.

The system (7) has a non-unique solution in integers ±1. This fact can be viewed in the context of algebraic geometry and computational algebra. At the same time, this fact refers to the existence of a specialtypeofweighingmatrices,see[13],inwhicheachlineofE_j containsonlyonenon-zeroentry.Letus defineasymmetricsquareS²(h)=h∨hofthevectorspaceh= span{(x₁,. . . ,x_m)^t}ofdimensionm= 2k, andletP(S²(h)) beitsprojectivizationof(projective)dimension ¹₂(m−1)(m+ 2),

P(S²(h)) ={(y11, y12, . . . , yij, . . . , ymm), yij = 0, i≤j}. TheVeroneseembeddingϕ:P(h)→P(S²(h)) isdefinedby

(x1, . . . , xm)^t→(x²₁, x1x2, . . . , x1xm, x²₂, x2x3. . . , x²_m)^t.

The Veronesevariety V(h), see e.g.,[6], is the image ϕ(P(h))=V(h) ⊂P(S²(h)).The intersection of the VeronesevarietyV(h) witha generic(¹₂(m−1)(m+ 2)−m+ 1)-dimensionalhyperplanehas2^m−1 points, soV(h) isof order2^m−1.

The vector space S²(h) and its projectivization P(S²(h)) can also be realized in terms of symmetric m×m-matrices

y=

⎛

⎜⎜

⎜⎝

y11 y12 . . . y1m

y12 y22 . . . y2m

. . . . . . . . . . . . y_1m y_2m . . . y_mm

⎞

⎟⎟

⎟⎠.

Then theVeroneseembeddingϕ:P(h)→P(S²(h)) isdefinedas

⎛

⎜⎜

⎜⎜

⎝ x₁ x2

... x_m

⎞

⎟⎟

⎟⎟

⎠→

⎛

⎜⎜

⎜⎜

⎝ x₁ x2

... x_m

⎞

⎟⎟

⎟⎟

⎠(x1, x2, . . . , xm) =

⎛

⎜⎜

⎜⎝

x²₁ x1x2 . . . x1xm

x1x2 x²₂ . . . x2xm

. . . . . . . . . . . . x1xm x2xm . . . x²_m

⎞

⎟⎟

⎟⎠.

ApointyisinV(h),ifandonlyif,thematrixcorrespondingtoyisofrank1.Anm×msymmetricmatrix has ₁₂¹(m−1)m²(m+ 1) independent 2-minors, which all vanish for the rank 1 matrixy. Each 2-minor defines a quadric in P(S²(h)) and y ∈ V(h), if and only if, y lies in the intersection of these quadrics.

The linearization algorithm can be viewed as follows, see e.g., [29]. Consider a system of p independent homogeneous quadraticequations f1= 0,. . .,fp= 0 with theprojectivepoint xas anon-trivialsolution.

Thissystemisprojectedtothelinearsystemofpequationsfˆ₁= 0,. . . ,fˆ_p = 0 intermsofvariablesy.Define the projective subspace H ⊂P(S²(h)) asthe intersection ofp hyperplanes definedby fˆ₁ = 0,. . . ,fˆ_p = 0.

So wehave thatthesolutionx isembedded into theintersectionH ∩ V(h). Thusthesolution isobtained by x=ϕ⁻¹(H ∩ V(h)).Since V(h) contains nonon-triviallinear subspaces,thealgorithm is efficient ifH consistsofafinite numberofisolated points.

Let us denote x1 = α12, x2 = α24, x3 = α13, x4 = α34, x5 = α23, x6 = α14, then y12 = α12α24, y13 = α12α13, y24 = α24α34, y34 = α13α34, y35 = α13α23, y26 = α14α24, y46 = α14α34, y15 = α12α23, y₃₆ = α₁₃α₁₄, y₂₅ =α₂₃α₂₄, y₁₆ =α₁₂α₁₄, y₄₅ = α₂₃α₃₄. The equations (9) are equivalent to the linear homogeneoussystemfˆ₁= 0,. . . ,fˆ₈= 0

y11−y22= 0, y11−y33= 0, y11−y44= 0, y11−y55= 0, y11−y66= 0, y12+y34= 0, y35+y26= 0, y46−y15= 0.

Intheprojectivecoordinatesthissystemhas3solutionswhichwewriteinthematrixform as

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎝

1 1 y₁₃ y₁₄ 1 y₁₆ 1 1 y23 y24 y25 −1 y13 y23 1 −1 1 y36

y14 y24 −1 1 y45 1 1 y25 1 y45 1 y56

y₁₆ −1 y₃₆ 1 y₅₆ 1

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎠ ,

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎝

1 1 y₁₃ y₁₄ 1 y₁₆ 1 1 y23 y24 y25 1 y13 y23 1 −1 −1 y36

y14 y24 −1 1 y45 1 1 y25 −1 y45 1 y56

y₁₆ 1 y₃₆ 1 y₅₆ 1

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎠ ,

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎝

1 1 y₁₃ y₁₄ −1 y₁₆ 1 1 y23 y24 y25 −1 y13 y23 1 −1 1 y36

y14 y24 −1 1 y45 −1

−1 y25 1 y45 1 y56

y₁₆ −1 y₃₆ −1 y₅₆ 1

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎠ .

Intheoriginalvariables wehave6solutionsincoordinates

(α12, α24, α13, α34, α23, α14) as

(1,1,1,−1,1,−1), (1,1,−1,1,1,1), (1,1,−1,1,−1,−1), (−1,−1,−1,1,−1,1), (−1,−1,1,−1,−1,−1), (−1,−1,1,−1,1,1).