aa r X i v : . [ m a t h . R A ] F e b A SANDWICH IN THIN LIE ALGEBRAS
SANDRO MATTAREI
Abstract.
A thin Lie algebra is a Lie algebra L , graded over the positiveintegers, with its first homogeneous component L of dimension two andgenerating L , and such that each nonzero ideal of L lies between consecu-tive terms of its lower central series. All its homogeneous components havedimension one or two, and the two-dimensional components are called dia-monds. We prove that if the next diamond past L of an infinite-dimensionalthin Lie algebra L is L k , with k >
5, then [
Lyy ] = 0 for some nonzero element y of L . Introduction A thin Lie algebra is a graded Lie algebra L = L ∞ i =1 L i with dim( L ) = 2and satisfying the following covering property: for each i , each nonzero z ∈ L i satisfies [ zL ] = L i +1 . This implies at once that homogeneous components of athin Lie algebra are at most two-dimensional. Those components of dimensiontwo are called diamonds, hence L is a diamond, and if there are no otherdiamonds then L is a graded Lie algebra of maximal class, see [CMN97, CN00].However, we adopt the convention of explicitly excluding graded Lie algebrasof maximal class from the definition of thin Lie algebras. We also require thinLie algebras to be infinite-dimensional in this paper.Thus, a thin Lie algebra must have at least one further diamond past L ,and we let L k be the earliest (the second diamond). It turns out that k canonly be one of 3, 5, q , or 2 q −
1, where q is a power of the characteristic p whenthis is positive (with only 3 and 2 q − k is 3 or 5 the classical cases, following a usage in the theoryof (finite-dimensional) simple modular Lie algebras, where the classical onesare those which are analogues of the simple Lie algebras of characteristic zero(including the exceptional ones, as is customary in the modular theory). Themain conclusion of this paper, and its interpretation which we give in Section 3,lends further weight to adopting such terminology.Thin Lie algebras with k equal to 3 or 5, subject to a further restrictiondim( L ) = 1 in the former case, and excluding some small characteristics, were Mathematics Subject Classification. primary 17B50; secondary 17B70, 17B65.
Key words and phrases.
Modular Lie algebra, graded Lie algebra, thin Lie algebra,sandwich. shown in [CMNS96] to belong to (up to) three isomorphism types, associatedto p -adic Lie groups of the classical types A and A (see [Mat99] for matrixrealizations of those groups). In contrast, the values q and 2 q − k oc-cur for two broad classes of thin Lie algebras, of which many were built fromcertain non-classical finite-dimensional simple modular Lie algebras, and alsoto other thin Lie algebras obtained from graded Lie algebras of maximal classthrough various constructions. General discussions of those two classes of thinLie algebras can be found in [AM07] and [CM05], respectively.Our main result is a general fact that was shown so far in ad hoc manners forvarious special instances of the class of thin Lie algebras under consideration. Theorem 1.
Let L be a thin Lie algebra with second diamond L k , where k > .Then there is a nonzero element y of L such that [ Lyy ] = 0 . We use the left-normed convention for iterated Lie products, hence [ abc ]stands for [[ ab ] c ], and so [ Lyy ] = 0 is another way of saying (ad y ) = 0.According to a definition of Kostrikin, for p = 2 the conclusion of Theorem 1says that y is a sandwich element (or simply a sandwich ). Kostrikin introducedsandwich elements in the context of the Burnside problem, see [Kos90]. Fourdecades later they played an important role in the classification theory of finite-dimensional simple modular Lie algebras, especially for small characteristics(five and seven) in [PS97]. One remarkable property of sandwich elements,noted by Kostrikin and proved by Premet in [Pre86], is that their presence in afinite-dimensional modular Lie algebra characterizes those simple Lie algebraswhich are not classical.Although our thin Lie algebras are infinite-dimensional, a connection withfinite-dimensional Lie algebras comes via a loop algebra construction, whichhas been used many times to build thin Lie algebras from finite-dimensionalLie algebras, simple or close to simple, with respect to certain cyclic gradings.In Section 3 we discuss the relevance of the presence of a sandwich element inthis context. 2. A sandwich element in thin Lie algebras A sandwich element of a Lie algebra L is a nonzero element c ∈ L suchthat (ad c ) = 0 and (ad c )(ad z )(ad c ) = 0 for all z ∈ L . If the characteristic isdifferent from two then the latter requirement (from which the name originates)is superfluous, as it follows from the former due to 0 = [ u [ zcc ]] = [ uzcc ] − uczc ] + [ uccz ] for u, z ∈ L .In the context of thin Lie algebras it turns out that a nonzero element y of C L ( L ) satisfies [ Lyy ] = 0 in most cases, as our Theorem 1 states, and so y isa sandwich element if the characteristic is not two. This fact was crucial in thetheory of graded Lie algebras of maximal class, making a theory of constituents possible, although the sandwich point of view was not explicitly mentionedin [CMN97] or [CN00]. In particular, [CMN97, Lemma 3.3] can be restatedas follows: if L is a graded Lie algebra of maximal class, with the centralizer SANDWICH IN THIN LIE ALGEBRAS 3 C L ( L ) spanned by an element y as customary, [ L i − y ] = 0 and [ L i y ] = 0 forsome i , then [ L i +1 y ] = 0, whence [ L i yy ] = 0. A remarkable consequence of thisfact is [ Lyy ] = 0, which means (ad y ) = 0, and so y is a sandwich element ifthe characteristic is not two.Our main goal in this section is showing that such a nonzero element y of C L ( L ) is frequently a sandwich element also in a thin Lie algebra L , undercertain assumptions which we introduce and justify along the way. If L hassecond diamond L , then C L ( L ) = 0, hence we must take dim( L ) = 1 asminimal assumption for our discussion, and then dim (cid:0) C L ( L ) (cid:1) = 1. Takingthen for y a nonzero element of C L ( L ) and extending to a basis x, y of L , aswe do without further mention in this section, we have [ xyy ] = 0.In the case of characteristic two we restrict ourselves to the following simpleobservation. Theorem 2.
Let L be a thin Lie algebra of characteristic two, with dim( L ) =1 , and let y span C L ( L ) . Then [ Lyy ] = 0 .Proof.
Because (ad y ) is a derivation of L , its kernel is a subalgebra. However,because [ xyy ] = 0 both generators x and y of L belong to the kernel, and hencethat is the whole of L . (cid:3) We will show that Theorem 2 extends to arbitrary characteristic if we in-clude a further hypothesis dim( L ) = 1, as in Theorem 1. A justification forthat hypothesis will emerge as we gradually work towards a proof. We startwith suitably extending the conclusion [ L i yy ] = 0 of [CMN97, Lemma 3.3],which was about graded Lie algebras of maximal class, to certain homogeneouscomponents of any thin Lie algebra L with dim( L ) = 1. Lemma 3.
Let L be a thin Lie algebra with dim( L ) = 1 , and let y span C L ( L ) . Let L i be a homogeneous component with [ L i y ] = 0 , and suppose L i − has a nonzero element u with [ uy ] = 0 . Then [ L i yy ] = 0 . Moreover, L i and L i +2 have dimension one.Proof. Because of the covering property, L i is spanned by [ ux ], and hence hasdimension one. From(1) 0 = [ u [ xyy ]] = [ uxyy ] − uyxy ] + [ uyyx ] = [ uxyy ]it follows that [ L i yy ] = 0. Because [ uxy ] = 0 by hypothesis, the coveringproperty implies that L i +2 is spanned by [ uxyx ], and hence has dimensionone. (cid:3) The following immediate consequence of Lemma 3 resembles the formulationof [CMN97, Lemma 3.3] more closely.
Corollary 4.
Let L be a thin Lie algebra with dim( L ) = 1 , and let y span C L ( L ) . If L i − has a nonzero element centralized by y , and L i has none, then L i +1 has such an element as well. SANDRO MATTAREI
Equivalently, in a thin Lie algebra with dim( L ) = 1, at least one of any twoconsecutive homogeneous components has a nonzero element centralized by y .Corollary 4 is often useful as simple but weaker replacement for Lemma 3.However, we need the full strength of Lemma 3 in order to deduce the followingconsequence. Corollary 5.
Let L be a thin Lie algebra with dim( L ) = 1 , and let y span C L ( L ) . Suppose dim( L j ) = 2 for some j > . Then dim( L j − ) = 1 and [ L j − yy ] = 0 . In particular, L j contains a nonzero element centralized by y .Proof. Because of the covering property L j − cannot contain any nonzero el-ement centralized by y . Hence L j − does contain a nonzero element u with[ uy ] = 0. Consequently, v = [ ux ] spans L j − , and [ vyy ] = 0 according toLemma 3. Because dim( L j ) = 2 the element [ vy ] of L j is nonzero, and iscentralized by y as desired. (cid:3) In particular, Corollary 5 implies that two consecutive components in a thinLie algebra L with dim( L ) = 1 cannot both be diamonds. Thus, the distance between any two consecutive diamonds, by which we mean the difference oftheir degrees, is at least two.An easily proved consequence of this fact is that L remains thin under basefield extensions. By contrast, this property fails for a thin Lie algebra L withtwo diamonds occurring as consecutive components, as we show in Remark 6. Infact, the thin Lie algebras studied in [GMY01, ACG + ] where all homogeneouscomponents except L are diamonds, require their base field to have a quadraticfield extension. Remark . Although it is inconsequential for this paper, we explain why in athin Lie algebra L over an algebraically closed field no two diamonds can occuras consecutive components. For a graded Lie algebra L = L ∞ i =1 L i we con-sider the linear maps ψ j : L j → Hom F ( L , L j +1 ) obtained by restriction fromthe adjoint representation. When L is thin with L j and L j +1 both diamonds,the covering property for the homogeneous component L j means that ψ ( L j )is a two-dimensional subspace of Hom F ( L , L j +1 ) where every nonzero elementis a surjective linear map. Upon composing with a linear bijection L j +1 → L (whose choice is immaterial) we may view that as a two-dimensional subspace ofHom F ( L , L ) where every nonzero element has nonzero determinant. In termsof the associated projective spaces, P (cid:0) ψ ( L j ) (cid:1) is then a one-dimensional pro-jective subspace of the three-dimensional projective space P (cid:0) Hom F ( L , L ) (cid:1) ,disjoint from the non-degenerate quadric given by the zeroes of the determi-nant map. That cannot happen if the base field F is algebraically closed.Corollary 5 is relevant in assigning types to certain diamonds of a thin Liealgebra. This was done in different ways according to whether the seconddiamond occurs in degree q or 2 q −
1, in [CM04] and [CM99], respectively.However, in both cases necessary conditions for a diamond L j to be assigned SANDWICH IN THIN LIE ALGEBRAS 5 a type were dim( L j − ) = 1 and [ L j − yy ] = 0. Corollary 5 shows that thoseconditions hold in any thin Lie algebra with dim( L ) = 1.Now we use Lemma 3 to prove that if L is thin with dim( L ) = 1, then[ L i yy ] = 0 for all one-dimensional components L i . Lemma 7.
Let L be a thin Lie algebra with dim( L ) = 1 , and let y span C L ( L ) . Suppose [ L j yy ] = 0 for some j . Then dim( L j ) = 2 .Proof. Note that j >
2. If L j − had a nonzero element centralized by y , thenbecause [ L j y ] = 0 by hypothesis, Lemma 3 would apply with i = j and yield[ L j yy ] = 0, a contradiction. Therefore, L j − does not have any nonzero elementcentralized by y . Then because of Corollary 4 each of L j − and L j has such anelement. In particular, because [ L j y ] = 0 we must have dim( L j ) = 2. (cid:3) Thus, the task of proving [
Lyy ] = 0 in a thin Lie algebra L has now beenreduced to showing [ L j yy ] for the diamonds L j . We pause to note that L havingany two diamonds at distance two implies [ Lyy ] = 0. In fact, if L j and L j +2 are diamonds, in a thin Lie algebra L with dim( L ) = 1, then the coveringproperty easily implies [ L j y ] = 0, whence [ L j y ] = L j +1 because L j +1 is one-dimensional. Consequently, [ L j yy ] = [ L j +1 y ] = 0, because [ L j +1 y ] and [ L j +1 y ]span the diamond L j +2 .The following remark describes one special thin Lie algebra with dim( L ) = 1having certain diamonds at distance two. Remark . The proof of [CMNS96, Theorem 2(a)] implies that in characteristiceither zero or larger than five there is a unique (infinite-dimensional) thin Liealgebra L having second diamond L . That result, which predates the study ofthin Lie algebras per se, only claims uniqueness for the graded Lie algebra L associated to the lower central series of a infinite thin pro- p group with seconddiamond in weight 5, for p >
5. However, its proof applies to any thin Liealgebra L with second diamond L , for p > p = 0. A construction forthat thin Lie algebra L was also given in [CMNS96], and that is valid in everycharacteristic except for characteristic two. The diamonds of L occur in eachdegree congruent to ± L i − yy ] = 0for each i multiple of six. Uniqueness of L fails in characteristic three because5 = 2 q − q = 3, and in characteristic five because we may have 5 = q then, and so lots of other thin Lie algebras enter those cases (see Remark 12).Our next remark describes how in characteristic three one can produce un-countably many thin Lie algebras with dim( L ) = 1 having diamonds at dis-tance two. Remark . According to [CMN97, Section 9], for each power q of the (positive)characteristic there are uncountably many infinite-dimensional graded Lie al-gebras of maximal class M with precisely two distinct two-step centralizers andconstituent sequence beginning with 2 q, q, q . As shown in [CM99, Section 5],each such M has a maximal subalgebra L which becomes thin under a new grad-ing, and its diamonds start with L , L q − , L q − , L q − . In characteristic three SANDRO MATTAREI and taking q = 3 those diamonds are L , L , L , L . Consequently, [ L yy ] = 0and [ L yy ] = 0.The thin Lie algebras of the above remarks suggest that we might be able toobtain more information on the earliest diamond L j with [ L j yy ] = 0 than on anarbitrary one, hence we will do that in preparation for a proof of Theorem 11. Lemma 10.
Let L be a thin Lie algebra with dim( L ) = 1 , and let y span C L ( L ) . Suppose [ Lyy ] = 0 , and let j be minimal such [ L j yy ] = 0 . Then L j − and L j − are one-dimensional and centralized by y .Proof. According to Lemma 7 we have dim( L j ) = 2, and hence dim( L j − ) = 1because of Corollary 5. Note that j >
3. Let t be any nonzero element of L j − .Then [ tyy ] = 0 and [ txyy ] = 0 by minimality of j , and hence0 = [ t [ xyy ]] = [ txyy ] − tyxy ] + [ tyyx ] = − tyxy ] . Because of Theorem 2 we are not in characteristic two, and so we conclude[ tyxy ] = 0. But then [ tyx ] = 0, because L j − contains no nonzero elementcentralized by y . The covering property now implies [ ty ] = 0, otherwise [ tyx ]and [ tyy ], which both vanish, would have to span L j − . Because t was anarbitrary nonzero element of L j − , Corollary 5 implies dim( L j − ) = 1. Finally, L j − has dimension one because it is spanned by [ tx ]. But we know that L j − contains a nonzero element centralized by y , and hence [ L j − y ] = 0. (cid:3) We are ready to prove the harder analogue of Theorem 2 for odd charac-teristics, thus completing a proof of Theorem 1. As our discussion leading toRemark 8 shows, we need an additional assumption to ensure that no diamondsoccur at distance two. As it turns out, assuming dim( L ) = 1 will do, which isanother way of asking that the second diamond of L occurs past L . Theorem 11.
Let L be a thin Lie algebra of odd characteristic, with dim( L ) =dim( L ) = 1 , and let y span C L ( L ) . Then [ Lyy ] = 0 .Proof.
Besides the relation [ yxy ] = 0, which serves to define y up to a scalar,in L we also have [ yxxy ] = 0 and [ yxxxy ] = 0. In fact, the former assertionfollows from 0 = [ yx [ yx ]] = [ yxyx ] − [ yxxy ] = [ yxxy ], and hence [ yxxx ] spans L . To prove the latter, assuming [ yxxxy ] = 0 for a contradiction, it will span L because of our hypothesis dim( L ) = 1. Because 0 = [ yxx [ xyy ]] = [ yxxxyy ]we then have [ L y ] = 0, and hence [ yxxxyx ] spans L . Now the generalizedJacobi identity yields the contradiction 0 = [ y [ yxxxx ]] = − yxxxyx ].Now suppose for a contradiction that [ Lyy ] = 0, and let j be minimal suchthat [ L j yy ] = 0. We will use the relations [ yxxy ] = 0 and [ yxxxy ] = 0 to derivea contradiction. Let t be a nonzero element of L j − . According to Lemma 7 wehave [ ty ] = 0 and [ txy ] = 0, and L j − is spanned by v := [ txx ]. Furthermore,dim( L j ) = 2, and hence [ vyy ] = 0 according to Corollary 5. The calculations0 = [ v [ xyy ]] = [ vxyy ] − vyxy ] , SANDWICH IN THIN LIE ALGEBRAS 7 and0 = [ t [ yxxxy ]] = [ t [ yxxx ] y ] = − txxyxy ] + [ txxxyy ] = [ vxyy ] − vyxy ] , taken together imply [ vxyy ] = [ vyxy ] = 0. Because [ vx ] and [ vy ] span L j weobtain [ L j yy ] = 0, which gives the desired contradiction. (cid:3) We discuss the extent to which hypothesis dim( L ) = 1 of Theorem 11 isnecessary. As we recalled in Remark 8, over a field of characteristic p > L , identified in [CMNS96, Theorem 2(a)]and described in Remark 8. However, if we omit the hypothesis dim( L ) = 1of Theorem 11 in characteristic p = 3, its conclusion [ Lyy ] = 0 is violatedby uncountably many thin Lie algebras with second diamond L and thirddiamond L , which we described in Remark 9. The following remark focuseson the borderline case p = 5. Remark . We discuss how far Theorem 11 may be extended to include thinLie algebras of characteristic five, with second diamond L and an additionalassumption. Countably many thin Lie algebras with second diamond L q wereconstructed in [AM07], for any power q of the odd characteristic. In particular,when q = 5 this gives us a countable family of thin Lie algebras with seconddiamond L . However, all those thin Lie algebras, which have third diamond L q − = L (possibly fake, see [AM07] for what that means), do satisfy theconclusion [ Lyy ] = 0 of Theorem 11.In fact, one can prove that [
Lyy ] = 0 holds for any thin Lie algebra L ofcharacteristic five satisfying dim( L ) = 1, dim( L ) = 2, and [ L y ] = 0. (Herethe hypothesis [ L y ] = 0 implies that neither L nor L is a diamond, andhence the third diamond of L does not occur earlier than L .) A proof ofthis fact follows the general inductive strategy employed in [AM], but is not aformal consequence of [AM], where some generality was sacrificed in favour ofsimpler exposition. In fact, part of that simplification in [AM] relies on usingour Theorem 11.3. Thin loop algebras and the sandwich element
We conclude this paper with a discussion of the significance for thin Lie al-gebras of an important property of sandwich elements. Classical simple Liealgebras do not have sandwich elements, see [Sel67, p. 124]. By contrast, con-firming a conjecture of Kostrikin, Premet proved in [Pre86] that every finite-dimensional simple Lie algebra, over an algebraically closed field of character-istic p >
5, which is not classical, must have sandwich elements (that is, have strong degeneration ).The connection of this characterization of classical Lie algebras with (infinite-dimensional) thin Lie algebras comes from the fact that several of the lat-ter have been constructed as loop algebras of certain finite-dimensional simpleLie algebras, or close to simple. In the simplest setting, one starts from a
SANDRO MATTAREI finite-dimensional simple Lie algebra S over a field F , with a cyclic grading S = L k ∈ Z /N Z S k , and considers the Lie algebra S ⊗ F [ t ] over F , where t isan indeterminate. Its subalgebra L = L k> S ¯ k ⊗ t k , where ¯ k = k + N Z , isnaturally graded over the positive integers, and is called a loop algebra in thiscontext. In certain cases one needs a slightly more general construction involv-ing also a derivation of S (such as that of [AM07, Definition 2.1]), but that isinconsequential for our present observation.It was proved in [CMNS96] that a thin Lie algebra L (of characteristic not2 or 3) having second diamond L and dim( L ) = 1 belongs to one of (up to)two isomorphism types, and a thin Lie algebra L (of characteristic not 2, 3 or5) with second diamond L is unique up to isomorphism (see also Remark 8).Each of those Lie algebras can be realized as a loop algebra of a classical simpleLie algebra of type A or A .Now, the fact that those thin Lie algebras of [CMNS96] are loop algebras ofclassical simple Lie algebras implies that there cannot be any nonzero element y ∈ L ¯1 with (ad y ) = 0, because such y would have the form y = c ⊗ t for somesandwich element c of S , which cannot exist. For the same reason, if a thin Liealgebra L (say of characteristic p >
5) with second diamond past L is a loopalgebra of a simple Lie algebra S , then S cannot be classical, because L has asandwich element y according to Theorem 11, and hence so does S .In fact, all constructions of thin Lie algebras with second diamond q or 2 q − L given bycorresponding uniqueness results shows that the dimension of S ought to bea power of p or one or two less in the various cases. However, the sandwichelement y does provide guidance in identifying the appropriate cyclic gradingof S employed in those constructions. References [ACG + ] M. Avitabile, A. Caranti, N. Gavioli, V. Monti, M. F. Newman, and E. A. O’Brien.Thin subalgebras of Lie algebras of maximal class. preprint, arXiv:2101.11982 .[AJ01] M. Avitabile and G. Jurman. Diamonds in thin Lie algebras. Boll. Unione Mat.Ital. Sez. B Artic. Ric. Mat. (8) , 4(3):597–608, 2001.[AM] M. Avitabile and S. Mattarei. On the structure of Nottingham algebras withdiamonds of infinite type. preprint, arXiv:2011.05491 .[AM05] M. Avitabile and S. Mattarei. Thin Lie algebras with diamonds of finite andinfinite type.
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