Clover nil restricted Lie algebras of quasi-linear growth
aa r X i v : . [ m a t h . R A ] A p r CLOVER NIL RESTRICTED LIE ALGEBRAS OF QUASI-LINEAR GROWTH
VICTOR PETROGRADSKY
Abstract.
The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. Theyare natural examples of self-similar finitely generated periodic groups. The author constructed their analoguein case of restricted Lie algebras of characteristic 2 [35], Shestakov and Zelmanov extended this constructionto an arbitrary positive characteristic [50]. Also, the author constructed a family of 2-generated restrictedLie algebras of slow polynomial growth with a nil p -mapping [37].Now, we construct a family of so called clover T (Ξ), where a fieldof positive characteristic is arbitrary and Ξ an infinite tuple of positive integers. All these algebras have anil p -mapping. We prove that 1 ≤ GKdim T (Ξ) ≤
3, moreover, the set of Gelfand-Kirillov dimensions ofclover Lie algebras with constant tuples is dense on [1 , T (Ξ q,κ ), where q ∈ N , κ ∈ R + , with extremely slow quasi-linear growth of type: γ T (Ξ q,κ ) ( m ) = m (cid:0) ln ( q ) m (cid:1) κ + o (1) , as m → ∞ .The present research is motivated by a construction by Kassabov and Pak of groups of oscillatinggrowth [22]. As an analogue, we construct nil restricted Lie algebras of intermediate oscillating growthin [38]. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is ”almostdying” by having ”quasi-linear” growth as above, for infinitely many n the growth function behaves likeexp( n/ (ln n ) λ ), for such periods the algebra is ”resuscitating”. The present construction of 3-generated nilrestricted Lie algebras of quasi-linear growth is an important part of that result, responsible for the lowerquasi-linear bound in that construction. Introduction
Golod-Shafarevich algebras and groups.
The General Burnside Problem asks whether a finitelygenerated periodic group is finite. The first negative answer was given by Golod and Shafarevich, they provedthat there exist finitely generated infinite p -groups for each prime p [14]. As an important instrument, theyfirst construct finitely generated infinite dimensional associative nil-algebras [14]. Using this construction,there are also examples of infinite dimensional 3-generated Lie algebras L such that (ad x ) n ( x,y ) ( y ) = 0, forall x, y ∈ L , the field being arbitrary [15]. Similarly, one easily obtains infinite dimensional finitely generatedrestricted Lie algebras L with a nil p -mapping. This gives a negative answer to the question of Jacobsonwhether a finitely generated restricted Lie algebra L is finite dimensional provided that each element x ∈ L is algebraic, i.e. satisfies some p -polynomial f p,x ( x ) = 0 ([21, Ch. 5, ex. 17]).It is known that the construction of Golod yields associative nil-algebras of exponential growth. Usingspecially chosen relations, Lenagan and Smoktunowicz constructed associative nil-algebras of polynomialgrowth [24], there are more constructions including associative nil-algebras of intermediate growth [7, 25, 52].On further developments concerning Golod-Shafarevich algebras and groups see [55, 11].A close by spirit but different construction was motivated by respective group-theoretic results. A re-stricted Lie algebra G is called large if there is a subalgebra H ⊂ G of finite codimension such that H admitsa surjective homomorphism on a nonabelian free restricted Lie algebra. Let K be a perfect at most count-able field of positive characteristic. Then there exist infinite-dimensional finitely generated nil restricted Liealgebras over K that are residually finite dimensional and direct limits of large restricted Lie algebras [3].1.2. Grigorchuk and Gupta-Sidki groups.
The construction of Golod is rather undirect, Grigorchukgave a direct and elegant construction of an infinite 2-group generated by three elements of order 2 [16].Originally, this group was defined as a group of transformations of the interval [0 ,
1] from which rationalpoints of the form { k/ n | ≤ k ≤ n , n ≥ } are removed. For each prime p ≥
3, Gupta and Sidki gave adirect construction of an infinite p -group on two generators, each of order p [19]. This group was constructedas a subgroup of an automorphism group of an infinite regular tree of degree p . Mathematics Subject Classification.
Key words and phrases. restricted Lie algebras, p -groups, growth, self-similar algebras, nil-algebras, graded algebras, Liesuperalgebra, Lie algebras of differential operators.The author was partially supported by grants CNPq 309542/2016-2, FAPDF 2019/01. The Grigorchuk and Gupta-Sidki groups are counterexamples to the General Burnside Problem. More-over, they gave answers to important problems in group theory. So, the Grigorchuk group and its furthergeneralizations are first examples of groups of intermediate growth [17], thus answering in negative to aconjecture of Milnor that groups of intermediate growth do not exist. The construction of Gupta-Sidki alsoyields groups of subexponential growth [12]. The Grigorchuk and Gupta-Sidki groups are self-similar . Nowself-similar, and so called branch groups , form a well-established area in group theory [18, 30]. There arealso constructions of self-similar associative algebras [4, 51, 41].1.3.
Self-similar nil restricted Lie algebras, Fibonacci Lie algebra.
Unlike associative algebras, forrestricted Lie algebras, natural analogues of the Grigorchuk and Gupta-Sidki groups are known. Namely, overa field of characteristic 2, the author constructed an infinite dimensional restricted Lie algebra L generated bytwo elements, called a Fibonacci restricted Lie algebra [35]. Let char K = p = 2 and R = K [ t i | i ≥ / ( t pi | i ≥ ∂ i = ∂∂t i , i ≥
0. Define the following two derivations of R : v = ∂ + t ( ∂ + t ( ∂ + t ( ∂ + t ( ∂ + t ( ∂ + · · · ))))); v = ∂ + t ( ∂ + t ( ∂ + t ( ∂ + t ( ∂ + · · · )))) . These two derivations generate a restricted Lie algebra L = Lie p ( v , v ) ⊂ Der R and an associative algebra A = Alg( v , v ) ⊂ End R . The Fibonacci restricted Lie algebra has a slow polynomial growth with Gelfand-Kirillov dimension GKdim L = log ( √ / ≈ .
44 [35]. Further properties of the Fibonacci restricted Liealgebra are studied in [40, 42]. On background and some results on Lie algebras of differential operators ininfinitely many variables see [47, 46, 39, 13].Probably, the most interesting property of L is that it has a nil p -mapping [35], which is an analog of theperiodicity of the Grigorchuk and Gupta-Sidki groups. We do not know whether the associative hull A is anil-algebra. We have a weaker statement. The algebras L , A , and the augmentation ideal of the restrictedenveloping algebra u = ωu ( L ) are direct sums of two locally nilpotent subalgebras [40]. In case of arbitraryprime characteristic, Shestakov and Zelmanov suggested an example of a finitely generated restricted Liealgebra with a nil p -mapping [50]. An example of a p -generated nil restricted Lie algebra L , characteristic p being arbitrary, was studied in [43]. These infinite dimensional restricted Lie algebras can have differentdecompositions into a direct sum of two locally nilpotent subalgebras [43].Observe that only the original example has a clear monomial basis [35, 40]. In other examples, elementsof a Lie algebra are linear combinations of monomials, to work with such linear combinations is sometimesan essential technical difficulty, see e.g. [50, 43]. A family of nil restricted Lie algebras of slow growthhaving good monomial bases is constructed in [37], these algebras are close relatives of a two-generated Liesuperalgebra of [36].1.4. Narrow groups and Lie algebras.
Let G be a group and G = G ⊇ G ⊇ · · · its lower centralseries. One constructs a related N -graded Lie algebra L K ( G ) = ⊕ i ≥ L i , where L i = G i /G i +1 ⊗ Z K , i ≥ aG i +1 , bG j +1 ] = ( a, b ) G i + j +1 , where a ∈ G i , b ∈ G j , and ( a, b ) = a − b − ab the groupcommutator.A residually p -group G is said of finite width if all factors G i /G i +1 are finite groups with uniformlybounded orders. The Grigorchuk group G is of finite width, namely, dim F G i /G i +1 ∈ { , } for i ≥ L = L K ( G ) = ⊕ i ≥ L i has a linear growth. Bartholdi presented L K ( G ) as a self-similar restricted Lie algebra and proved that the restricted Lie algebra L F ( G ) is nil while L F ( G ) is not nil [5]. Also, L K ( G ) is nil graded , namely, for any homogeneous element x ∈ L i , i ≥
1, themapping ad x is nilpotent, because the group G is periodic.A Lie algebra L is called of maximal class (or filiform ), if the associated graded algebra with respect tothe lower central series gr L = ∞ ⊕ n =1 gr L n , where gr L n = L n /L n +1 , n ≥
1, satisfiesdim gr L = 2 , dim gr L n ≤ , n ≥ , gr L n +1 = [gr L , gr L n ] , n ≥ , (1)in particular, gr L is generated by gr L . An infinite dimensional filiform Lie algebra L has the smallestnontrivial growth function: γ L ( n ) = n + 1, n ≥
1. In case of positive characteristic, there are uncountablymany such algebras [8]. Nevertheless, in case p >
2, they were classified in [9]. There are generalizationsof filiform Lie algebras. Naturally N -graded Lie algebras over R and C satisfying the condition dim L n +dim L n +1 ≤ n ≥
1, are classified recently by Millionschikov [29]. More generally, an N -graded Lie algebra L = ∞ ⊕ n =1 L n is said of finite width d in the case that dim L n ≤ d , n ≥
1, the integer d being minimal. LOVER NIL RESTRICTED LIE ALGEBRAS OF QUASI-LINEAR GROWTH 3
Pro- p -groups and N -graded Lie algebras cannot be simple. Instead, appears an important notion of being just infinite , namely, not having non-trivial normal subgroups (ideals) of infinite index (codimension). Agroup (algebra) is said hereditary just infinite if and only if any normal subgroup (ideal) of finite index(codimension) is just infinite. The Gupta-Sidki groups were the first in the class of periodic groups to beshown to be just infinite [20]. The Grigorchuk group is also just infinite but not hereditary just infinite [18].Concerning narrow Lie algebras and groups see survey [49].1.5. Lie algebras in characteristic zero.
Since the Grigorchuk group is of finite width, a right analogueof it should be a Lie algebra of finite width having ad-nil elements, in the next result the components are ofbounded dimension and consist of ad-nil elements. Informally speaking, there are no ”natural analogues” ofthe Grigorchuk and Gupta-Sidki groups in the world of Lie algebras of characteristic zero, strictly in termsof the following result.
Theorem 1.1 (Martinez and Zelmanov [28]) . Let L = ⊕ α ∈ Γ L α be a Lie algebra over a field K of charac-teristic zero graded by an abelian group Γ . Suppose that i) there exists d > such that dim K L α ≤ d for all α ∈ Γ , ii) every homogeneous element a ∈ L α , α ∈ Γ , is ad-nilpotent.Then the Lie algebra L is locally nilpotent. Fractal nil graded Lie superalgebras.
In the world of
Lie superalgebras of an arbitrary character-istic , the author constructed analogues of the Grigorchuk and Gupta-Sidki groups [36]. Namely, two Liesuperalgebras R , Q were constructed. Both examples have clear monomial bases. They have slow poly-nomial growth, namely, GKdim R = log ≈ .
26 and GKdim Q = log ≈ .
89. In both examples, ad a is nilpotent, a being an even or odd element with respect to the Z -gradings as Lie superalgebras. Thisproperty is an analogue of the periodicity of the Grigorchuk and Gupta-Sidki groups. The Lie superalge-bra R is Z -graded, while Q has a natural fine Z -grading with at most one-dimensional components. Inparticular, Q is a nil finely graded Lie superalgebra, which shows that an extension of Theorem 1.1 for theLie super algebras of characteristic zero is not valid. Also, Q has a Z -grading which yields a continuum ofdecompositions into sums of two locally nilpotent subalgebras Q = Q + ⊕ Q − . Both Lie superalgebras are self-similar , they also contain infinitely many copies of itself, we call them fractal due to the last property.We construct a more ”handy” 2-generated fractal Lie superalgebra R (the same notation as above but thisis a different algebra) over an arbitrary field [10]. This Lie superalgebra R is Z -graded by multidegree in thegenerators and the Z -components are at most one-dimensional. As an analogue of periodicity, we establishthat homogeneous elements of the Z -grading R = R ¯0 ⊕ R ¯1 are ad-nilpotent. In case of N -graded algebras,a close analogue to being simple is being just infinite. Unlike previous examples of Lie superalgebras [36],we are able to prove that R is just infinite. This example is close to the smallest possible one, because R has a linear growth with a growth function γ R ( m ) ≈ m , as m → ∞ . Moreover, its degree N -grading is offinite width 4 (char K = 2). In case char K = 2, we obtain a Lie algebra of width 2 that is not thin.We also construct a just infinite fractal 3-generated Lie superalgebra Q over arbitrary field, which gives riseto an associative hull A , a Poisson superalgebra P , and two Jordan superalgebras J and K , the latter can bealso considered as analogues of the Grigorchuk and Gupta-Sidki groups in respective classes of algebras [45].2. Basic notions: restricted Lie algebras, Growth
As a rule, K is an arbitrary field of positive characteristic p , h S i K denotes a linear span of a subset S in a K -vector space. Let L be a Lie algebra, then U ( L ) denotes the universal enveloping algebra. Longcommutators are right-normed : [ x, y, z ] := [ x, [ y, z ]]. We use a standard notation ad x ( y ) = [ x, y ], where x, y ∈ L . Also, we use notation [ x k , y ] := (ad x ) k ( y ), where k ≥ x, y ∈ L ; in case k = p l , we have also[ x p l , y ] = [ x [ p l ] , y ], in terms of the p -mapping (see below).2.1. Restricted Lie algebras.
Let L be a Lie algebra over a field K of characteristic p >
0. Then L is called a restricted Lie algebra (or Lie p -algebra ), if it is additionally supplied with a unary operation x x [ p ] , x ∈ L , that satisfies the following axioms [21, 1, 53, 54, 2]: • ( λx ) [ p ] = λ p x [ p ] , for λ ∈ K , x ∈ L ; • ad( x [ p ] ) = (ad x ) p , x ∈ L ; • ( x + y ) [ p ] = x [ p ] + y [ p ] + P p − i =1 s i ( x, y ), for all x, y ∈ L , where is i ( x, y ) is the coefficient of t i − in thepolynomial ad( tx + y ) p − ( x ) ∈ L [ t ]. VICTOR PETROGRADSKY
This notion is motivated by the following construction. Let A be an associative algebra over a field K . Thevector space A is supplied with a new product [ x, y ] = xy − yx , x, y ∈ A , one obtains a Lie algebra denotedby A ( − ) . In case char K = p >
0, the mapping x x p , x ∈ A ( − ) , satisfies three axioms above.Suppose that L is a restricted Lie algebra. Let J be an ideal of the universal enveloping algebra U ( L )generated by { x [ p ] − x p | x ∈ L } . Then u ( L ) = U ( L ) /J is called a restricted enveloping algebra . In thisalgebra, the formal operation x [ p ] coincides with the p th power x p for any x ∈ L . One has an analogue ofPoincare-Birkhoff-Witt’s theorem yielding a basis of the restricted enveloping algebra [21, p. 213]. We shalluse the following version of the formula above:( x + y ) [ p ] = x [ p ] + y [ p ] + (ad x ) p − ( y ) + p − X i =1 s i ( x, y ) , x, y ∈ L, (2)where s i ( x, y ) consists of commutators containing i letters x and p − i letters y .2.2. Growth.
Let A be an associative (or Lie) algebra generated by a finite set X . Denote by A ( X,n ) thesubspace of A spanned by all monomials in X of length not exceeding n , n ≥
0. If A is a restricted Liealgebra, we define A ( X,n ) = h [ x i , . . . , x i s ] p k | x i j ∈ X, sp k ≤ n i K [31]. One obtains a growth function : γ A ( n ) = γ A ( X, n ) := dim K A ( X,n ) , n ≥ . Clearly, the growth function depends on the choice of the generating set X . Let f, g : N → R + be increasingfunctions. Write f ( n ) g ( n ) if and only if there exist positive constants N, C such that f ( n ) ≤ g ( Cn ) for all n ≥ N . Introduce equivalence f ( n ) ∼ g ( n ) if and only if f ( n ) g ( n ) and g ( n ) f ( n ). Different generatingsets of an algebra yield equivalent growth functions [23].It is well known that the exponential growth is the highest possible growth for finitely generated Lieand associative algebras. A growth function γ A ( n ) is compared with polynomial functions n k , k ∈ R + , bycomputing the upper and lower Gelfand-Kirillov dimensions [23]:GKdim A := lim n →∞ ln γ A ( n )ln n = inf { α > | γ A ( n ) n α } ;GKdim A := lim n →∞ ln γ A ( n )ln n = sup { α > | γ A ( n ) < n α } . Solvable finitely generated Lie algebras typically have intermediate growth, see [26, 32, 34].Denote ln ( q ) ( x ) := ln( · · · ln | {z } q times ( x ) · · · ) and exp ( q ) ( x ) := exp( · · · exp | {z } q times ( x ) · · · ) for all q ∈ N . In this paper,we study algebras of quasi-linear growth, growth functions of these algebras A behave as m exp (cid:0) (ln m ) β (cid:1) , β ∈ (0 , m (ln ( q ) m ) β , where q ∈ N , β ∈ R + . Clearly, GKdim A = GKdim A = 1. Inorder to specify parameters q, β define numbers:Ldim A = inf { β ∈ (0 , | γ A ( n ) m exp (cid:0) (ln m ) β (cid:1) } ;Ldim A = sup { β ∈ (0 , | γ A ( n ) < m exp (cid:0) (ln m ) β (cid:1) } ;Ldim q A = inf { β ∈ R + | γ A ( n ) m (ln ( q ) m ) β } , q ∈ N ;Ldim q A = sup { β ∈ R + | γ A ( n ) < m (ln ( q ) m ) β } , q ∈ N . One checks that these numbers are invariants not depending on a generating set. Remark that notations aredifferent from [37].Assume that generators X = { x , . . . , x k } are assigned positive weights wt( x i ) = λ i , i = 1 , . . . , k . Definea weight growth function :˜ γ A ( n ) = dim K h x i · · · x i m | wt( x i ) + · · · + wt( x i m ) ≤ n, x i j ∈ X i K , n ≥ . Set C = min { λ i | i = 1 , . . . , k } , C = max { λ i | i = 1 , . . . , k } , then ˜ γ A ( C n ) ≤ γ A ( n ) ≤ ˜ γ A ( C n ) for n ≥ γ A ( n ) ∼ γ A ( n ). Therefore, we can use the weight growthfunction ˜ γ A ( n ) in order to compute the Gelfand-Kirillov dimensions and Ldim λ A , Ldim λ A as well.Suppose that L is a Lie algebra and X ⊂ L . By Lie( X ) denote the subalgebra of L generated by X . Incase L is a restricted Lie algebra Lie p ( X ) denotes the restricted subalgebra of L generated by X . Similarly,assume that X is a subset in an associative algebra A . Write Alg( X ) ⊂ A to denote an associative subalgebra(without unit) generated by X . LOVER NIL RESTRICTED LIE ALGEBRAS OF QUASI-LINEAR GROWTH 5 Main results: Clover restricted Lie algebras of Quasi-linear Growth
Clover restricted Lie algebras.
Recently, the author introduced a large class of drosophila Liealgebras [38], that generalized some examples of (restricted) Lie (super)algebras considered before [37, 36].In particular, it includes a family of 2-generated restricted Lie algebras studied in [37], now we call suchalgebras as duplex Lie algebras . Example 1 (family of restricted Lie algebras L (Ξ) in [37]) . Let char K = p > . Consider integers Ξ = ( S n , R n | n ≥ , which determine a divided power series ring Ω(Ξ) = h x ( ξ )0 y ( η )0 · · · x ( ξ i ) i y ( η i ) i | ≤ ξ i
Example 2.
Fix the same tuple of integers
Ξ = ( S n , R n | n ≥ and consider another formal divided powerseries ring: R = R (Ξ) := D x ( α )0 y ( β )0 z ( γ )0 · · · x ( α i ) i y ( β i ) i z ( γ i ) i (cid:12)(cid:12)(cid:12) ≤ α i < p S i , ≤ β i , γ i < p R i , i ≥ E . Define pivot elements recursively: v i = ∂ x i + x ( p Si − i y ( p Ri − i v i +1 ; w i = ∂ y i + y ( p Ri − i x ( p Si − i w i +1 ; u i = ∂ z i + z ( p Ri − i x ( p Si − i u i +1 ; i ≥ . (4) We define the clover restricted Lie algebra T (Ξ) := Lie p ( v , w , u ) ⊂ Der R (Ξ) .Remark . Let us draw attention that there is a symmetry between v i and w i , while the remaining u i stayseparate because there is no Z -cyclic symmetry unlike the second example of Lie superalgebras in [36]. Remark . In terminology of [38], species of flies having two flies in some generation either have two fliesin all subsequent generations or go extinct. The goal in introducing the clover specie is to have three fliesin each generation (yielding respective three pivot elements (4)), so that at some moment three flies canproduce a wild specie and the constructed Lie algebra can return to a respective fast intermediate growth.To this end we extend the duplex specie in a specific ”skew” way and obtain the clover specie. This ideaenables us to construct restricted Lie algebras with an oscillating growth in [38] using two theorems below.3.2.
Main results.
As a specific case, we construct restricted Lie algebras of quasi-linear growth. The proofis rather technical and close to that for the 2-generated duplex Lie restricted Lie algebras [37] (Example 1above). The main goal of the paper is to prove the following two theorems, which are an important part ofthe construction of nil restricted Lie algebras of oscillating intermediate growth in [38], namely, the algebrasconstructed below are responsible for periods of quasi-linear growth of that algebras. In comparison with [37],the asymptotic of the next theorem is a little bit more precise.
Theorem 3.1.
Let K be a field, char K = p > , fix κ ∈ (0 , . There exists a tuple of integers Ξ κ such thatthe 3-generated clover restricted Lie algebra T = T (Ξ κ ) = Lie p ( v , w , u ) has the following properties. i) γ T ( m ) = m exp (cid:0) ( C + o (1))(ln m ) κ (cid:1) as m → ∞ , where C := 2(ln p ) − κ /κ κ ; ii) GKdim T = GKdim T = 1 ; iii) Ldim T = Ldim T = κ ; iv) the growth function γ T ( m ) is not linear; v) algebras T (Ξ κ ) for different κ ∈ (0 , are not isomorphic. In comparison with [37], algebras with even slower quasi-linear growth are constructed in the next theorem.
Theorem 3.2.
Let char K = p > , fix q ∈ N , κ ∈ R + . There exists a tuple of integers Ξ q,κ such that the3-generated clover restricted Lie algebra T = T (Ξ q,κ ) = Lie p ( v , w , u ) has the following properties. i) γ T ( m ) = m (cid:0) ln ( q ) m (cid:1) κ + o (1) while m → ∞ ; VICTOR PETROGRADSKY ii) GKdim T = GKdim T = 1 ; iii) Ldim q T = Ldim q T = κ ; iv) the growth function γ T ( m ) is not linear; v) algebras T (Ξ q,κ ) for different pairs ( q, κ ) are not isomorphic.Remark . Similar to [37], we can consider also the associative algebra A = Alg( v , w , u ) ⊂ End R (Ξ) andprove that it has a quasi-quadratic growth, namely γ A ( m ) = m (cid:0) ln ( q ) m (cid:1) κ + o (1) , m → ∞ . Theorem 3.3 ([38]) . Fix char K = p > and a tuple of integers Ξ = ( S n , R n | n ≥ . Consider therespective clover restricted Lie algebra T = T (Ξ) . Then T has a nil p -mapping.Proof. The nillity is proved in a more general setting of so called drosophila Lie algebras with uniformparameters [38]. The proof is a modification of that for the 2-generated duplex Lie algebra L (Ξ) [37]. (cid:3) Let us describe some more interesting results and ideas of the paper. • We describe the structure and construct a clear monomial basis for all clover restricted Lie algebras(Theorem 4.7). • An important instrument is a notion of a weight function, using which we prove that T (Ξ) = T ( v , w , u ) is N -graded by the multidegree in the generators (Theorem 5.3). • We prove that 1 ≤ GKdim T (Ξ) ≤ • In case S i = S and R i = R for all i ≥ T ( S, R ) := T (Ξ).In this case, T ( S, R ) is a self-similar restricted Lie algebra and { GKdim T ( S, R ) | S, R ∈ N } is denseon [1 ,
3] (Corollary 6.5).
Remark . We suggest that the clover restricted Lie algebras T (Ξ) are analogues of the family of theGrigorchuk groups G ω constructed and studied in [17].3.3. Nil Lie algebras of slow polynomial growth.
The Gelfand-Kirillov dimension of an associativealgebra cannot belong to the interval (1 ,
2) [23, Bergman]. One has the same gap for finitely generatedJordan algebras [27, Martinez and Zelmanov]. The author showed that a similar gap does not exist forLie algebras, the Gelfand-Kirillov dimension of a finitely generated Lie algebra can be an arbitrary number { } ∪ [1 , + ∞ ) [33]. The same fact is also established for Jordan superalgebras [44]. Also, an interestingdirection of research is constructing associative nil algebras of different kinds of growth, in particular, ofslow polynomial growth, see [24, 7, 25, 52].Now we get a stronger version of [33], the gap (1 ,
2) can be filled with nil
Lie p -algebras. Namely, usingconstant tuples, we get clover nil restricted Lie algebras which Gelfand-Kirillov dimensions are dense on [1 , Structure of Clover Lie algebras T (Ξ)4.1. Basic relations.
We start with establishing basic relations in clover restricted Lie algebras. In whatfollows, we assume that a field K of characteristic char K = p > S n , R n | n ≥ T = Lie p ( v , w , u ). Lemma 4.1.
Let i ≥ . Then v p m i = ∂ p m x i + x ( p Si − p m ) i y ( p Ri − i v i +1 , ≤ m ≤ S i ; w p m i = ∂ p m y i + y ( p Ri − p m ) i x ( p Si − i w i +1 , ≤ m ≤ R i ; u p m i = ∂ p m z i + z ( p Ri − p m ) i x ( p Si − i u i +1 , ≤ m ≤ R i , where ∂ p Si x i = ∂ p Ri y i = ∂ p Ri z i = 0 above.Proof. Let us prove the first equality by induction on m . The base of induction m = 0 is trivial by (4).Assume that the claim is valid for 0 ≤ m < S i . The summation in (2) is trivial because the second termcannot be used more than once: v p m +1 i = ( v p m i ) p = (cid:16) ∂ p m x i + x ( p Si − p m ) i y ( p Ri − i v i +1 (cid:17) p = ( ∂ p m x i ) p + (cid:0) ad ∂ p m x i (cid:1) p − (cid:16) x ( p Si − p m ) i y ( p Ri − i v i +1 (cid:17) = ∂ p m +1 x i + x ( p Si − p m +1 ) i y ( p Ri − i v i +1 , ≤ m < S i . (cid:3) LOVER NIL RESTRICTED LIE ALGEBRAS OF QUASI-LINEAR GROWTH 7
Lemma 4.2.
Let char K = p > and a tuple Ξ be fixed. Consider the clover restricted Lie algebra T (Ξ) = Lie p ( v , w , u ) . Then i) v p Si i = y ( p Ri − i v i +1 , w p Ri i = x ( p Si − i w i +1 , u p Ri i = x ( p Si − i u i +1 , for all i ≥ . ii) [ w p Ri − i , v p Si i ] = v i +1 , [ v p Si − i , w p Ri i ] = w i +1 , [ v p Si − i , u p Ri i ] = u i +1 , for all i ≥ . iii) v i , w i , u i ∈ T (Ξ) , i ≥ .Proof. Follows from computations of [38]. But let us check the formulas directly. The first claim is a partialcase of Lemma 4.1. The third claim follows from the second. Finally, let us check the second claim.[ w p Ri − i , v p Si i ] = (ad w i ) p Ri − [ w i , v p Si i ]= (ad w i ) p Ri − h ∂ y i + y ( p Ri − i x ( p Si − i w i +1 , y ( p Ri − i v i +1 i = v i +1 . (cid:3) Lemma 4.3.
The subalgebra of T (Ξ) generated by v , w is isomorphic to L (Ξ) defined by (3) .Proof. We observe that v , w have the same presentation as a , b ∈ L (Ξ). (cid:3) Head elements of two types.
We construct a clear monomial basis for T (Ξ) similar to that for itssubalgebra L (Ξ) ∼ = Lie p ( v , w ) ⊂ T (Ξ) found in [37]. In case of a clover algebra, a generation of a pivotelement is also referred to as its length . Consider products of two pivot elements of the same length (4): h i +1 := [ w i , v i ] = x ( p Si − i y ( p Ri − i v i +1 − x ( p Si − i y ( p Ri − i w i +1 ,g i +1 := [ v i , u i ] = x ( p Si − i z ( p Ri − i u i +1 , [ w i , u i ] = 0 , i ≥ . (5) Lemma 4.4 ([37], Lemma 4.2) . For all i ≥ we have the following elements: i) for all ≤ ξ < p S i , ≤ η < p R i (except the case ξ = p S i − and η = p R i − ) we get: h ξ,ηi +1 := [ v ξi , w ηi , h i +1 ] = x ( p Si − − ξ ) i y ( p Ri − − η ) i v i +1 − x ( p Si − − ξ ) i y ( p Ri − − η ) i w i +1 ; (6)ii) The order of the multiplication above is not essential. As partial cases, we get: h , i +1 = h i +1 = [ w i , v i ]; h p Si − ,ηi +1 = y ( p Ri − − η ) i v i +1 , for ≤ η ≤ p R i − as a particular case: h p Si − ,p Ri − i +1 = v i +1 ; h ξ,p Ri − i +1 = − x ( p Si − − ξ ) i w i +1 , for ≤ ξ ≤ p S i − as a particular case: h p Si − ,p Ri − i +1 = − w i +1 ;Thus, for all i ≥
0, we obtain elements (6), called heads of first type of length i + 1: n h ξ,ηi +1 (cid:12)(cid:12)(cid:12) ≤ ξ < p S i , ≤ η < p R i , except ( ξ = p S i − η = p R i − o . (7)Consider (7) as a table of size p S i × p R i , rows and columns being indexed by ξ , η , the lower right corner isempty. The table contains v i +1 and − w i +1 in respective cells.We multiply (5) by v i , u i (the order is not essential) we get heads of second type of length i + 1: n g ξ,ζi +1 := [ v ξi , u ζi , [ v i , u i ]] = x ( p Si − − ξ ) i z ( p Ri − − ζ ) i u i +1 (cid:12)(cid:12)(cid:12) ≤ ξ ≤ p S i − , ≤ ζ ≤ p R i − o . (8)We put (8) in table of size ( p S i − × p R i , rows and columns being indexed by ξ and ζ , the low right cornercontaining u i +1 .4.3. Monomial basis of clover restricted Lie algebras T (Ξ) . Define tails of first and second types: r n ( x, y ) = x ( ξ )0 y ( η )0 · · · x ( ξ n ) n y ( η n ) n ∈ R, ≤ ξ i < p S i , ≤ η i < p R i ; n ≥ r n ( x, y, z ) = x ( ξ )0 y ( η )0 z ( ζ )0 · · · x ( ξ n ) n y ( η n ) n z ( ζ n ) n ∈ R, ≤ ξ i < p S i , ≤ η i , ζ i < p R i ; n ≥ . (9)For n < r n = 1. Another elements of type (9) will be denoted as r ′ n , ˜ r n , r ( k ) n , etc., whilethe lower index denotes the biggest index of variables it depends on. VICTOR PETROGRADSKY
Define standard monomials of first type of length n , n ≥ r n − ( x, y ) h ξ n − ,η n − n = r n − ( x, y ) (cid:16) x ( p Sn − − − ξ n − ) n − y ( p Rn − − − η n − ) n − v n − x ( p Sn − − − ξ n − ) n − y ( p Rn − − − η n − ) n − w n (cid:17) , where 0 ≤ ξ n − < p S n − , ≤ η n − < p R n − , except ( ξ n − = p S n − − η n − = p R n − − . (10)Recall that the heads h ξ n − ,η n − n are described by (6), while the tails r n − ( x, y ) are (9). We call x n − , y n − neck letters . By Lemma 4.4, we get the pivot elements v n , w n , for n ≥
1, as particular cases of suchmonomials. So, we consider that v , w are also the standard monomials of first type of length 0.Define standard monomials of second type of length n , n ≥ r n − ( x, y, z ) g ξ n − ,ζ n − n = r n − ( x, y, z ) x ( p Sn − − − ξ n − ) n − z ( p Rn − − − ζ n − ) n − u n , where 0 ≤ ξ n − ≤ p S n − − , ≤ ζ n − ≤ p R n − − . (11)Recall that the heads g ξ n − ,ζ n − n are described by (8), while the tails r n − ( x, y, z ) are (9). We call x n − , z n − neck letters . By definition, consider that u is a standard monomial of second type of length 0. Theorem 4.5.
A basis of the Lie algebra L = Lie( v , w , u ) (i.e. we use only the Lie bracket) is given by i) the standard monomials of first type of length n ≥ ; ii) the standard monomials of second type of length n ≥ .Proof. The standard monomials of first type form a basis of Lie( v , w ) [37, Theorem 5.1]. Let us provethat the standard monomials of second type belong to L . We proceed by induction on length n . We have u ∈ L . By (8), we get g ξ,ζ = x ( p S − − ξ )0 z ( p R − − ζ )0 u = [ v ξ , u ζ , [ v , u ]] ∈ R , for all 0 ≤ ξ ≤ p S − ≤ ζ ≤ p R −
1. Thus, we have the base of induction for n = 0 ,
1. Now let n ≥
1. By induction hypothesis, r n − ( x, y, z ) z ( p Rn − − n − u n ∈ L . We use recurrence formula for v n − and (5):[ v n − , r n − ( x, y, z ) z ( p Rn − − n − u n ] = h ∂ x n − + x ( p Sn − − n − y ( p Rn − − n − v n , r n − ( x, y, z ) z ( p Rn − − n − u n i = r n − ( x, y, z ) x ( p Sn − − n − y ( p Rn − − n − z ( p Rn − − n − [ v n , u n ]= r n − ( x, y, z ) x ( p Sn − − n − y ( p Rn − − n − z ( p Rn − − n − · x ( p Sn − n z ( p Rn − n u n +1 . By assumption, r n − ( x, y, z ) can have arbitrary powers of its variables. Multiplying by v n = ∂ x n + x ( p Sn − n y ( p Rn − n v n +1 , we can reduce the power of x n above to any desired value. The same argumentapplies to the remaining variables. Thus, all standard monomials of second type belong to L .It remains to show that products of standard monomials are expressed via standard monomials. This istrue for monomials of first type because they form a basis of Lie( v , w ) [37]. Take two standard monomialsof second type, shortly written as d = r n − u n , and d = r m − u m , divided variables will be shortly writtenas x ∗ i . Assume that n ≤ m . Using recurrence presentation, we get[ d , d ] = (cid:2) r n − ( ∂ z n + x ∗ n z ∗ n ( ∂ z n +1 + · · · + x ∗ m − z ∗ m − ( ∂ z m − + x ∗ m − z ∗ m − u m ))) , r m − u m (cid:3) = r n − ∂ z n ( r m − ) u m + r ′ n ∂ z n +1 ( r m − ) u m + · · · + r ′′ m − ∂ z m − ( r m − ) u m . Observe that we get standard monomials of second type above.Consider products of monomials of different types. Write shortly d = r n − ( x, y, z ) u n , d = r m − ( x, y ) h ξ,ηm = r ′ m − ( x, y ) v m − r ′′ m − ( x, y ) w m . Consider the case n ≤ m . Using recurrence presentation and (5),[ d , d ] = h r n − ( x, y, z ) (cid:0) ∂ z n + x ∗ n z ∗ n ( ∂ z n +1 + · · · + x ∗ m − z ∗ m − ( ∂ z m − + x ∗ m − z ∗ m − u m )) (cid:1) , r ′ m − ( x, y ) v m − r ′′ m − ( x, y ) w m i = ¯ r ′ m − ( x, y, z )[ u m , v m ] − ¯ r ′′ m − ( x, y, z )[ u m , w m ] = − ¯ r ′ m − ( x, y, z ) x ( p Sm − m z ( p Rm − m u m +1 , LOVER NIL RESTRICTED LIE ALGEBRAS OF QUASI-LINEAR GROWTH 9 yielding a standard monomial of second type. Consider the case m < n . Then d = n − X j = m (cid:16) r ( j ) j − ( x, y ) ∂ x j − ¯ r ( j ) j − ( x, y ) ∂ y j (cid:17) + r ′ n − ( x, y ) v n − r ′′ n − ( x, y ) w n ;[ d , d ] = n − X j = m (cid:16) r ( j ) j − ( x, y ) ∂ x j − ¯ r ( j ) j − ( x, y ) ∂ y j (cid:17)(cid:16) r n − ( x, y, z ) (cid:17) u n + ˜ r ′ n − ( x, y, z )[ v n , u n ] , the last term yielding a standard monomial of second type by (5). In the preceding sum, the action on vari-ables with indices n − j = n −
1. Recall that d = r n − ( x, y, z ) u n = r n − ( x, y, z ) x αn − z γn − u n ,where 0 ≤ α < p S n − −
1, 0 ≤ γ < p R n − . After the action in the sum above, we again get standard monomialsof second type. (cid:3) By Lemma 4.1, v p i n = ( ∂ p i x n + x ( p Sn − p i ) n y ( p Rn − n v n +1 , ≤ i < S n ; y ( p Rn − n v n +1 , i = S n ,w p i n = ( ∂ p i y n + y ( p Rn − p i ) n x ( p Sn − n w n +1 , ≤ i < R n ; x ( p Sn − n w n +1 , i = R n .u p i n = ( ∂ p i z n + z ( p Rn − p i ) n x ( p Sn − n u n +1 , ≤ i < R n ; x ( p Sn − n u n +1 , i = R n . (12)We refer to nonzero powers of v n , w n as power standard monomials of first type of length n + 1, powers of u n are power standard monomials of second type . One checks that they are linearly independent with thestandard monomials. Lemma 4.6.
Let n ≥ . There are S n + R n power standard monomials of first type and R n power standardmonomials of second type of length n + 1 . Theorem 4.7.
Let T (Ξ) = Lie p ( v , w , u ) be the clover restricted Lie algebra. Then i) A basis of T (Ξ) is given by the standard and power standard monomials of first and second types. ii) We have a semidirect product T (Ξ) = Lie p ( v , w ) ⋌ J, Lie p ( v , w ) ∼ = L (Ξ) , the subalgebra Lie p ( v , w ) is spanned by the standard and power standard monomials of first type,the ideal J is spanned by the standard and power standard monomials of second type.Proof. To get a basis of the p -hull of a Lie algebra we need to add p m -powers, m ≥
1, of its basis [54].Observe that the standard monomials contain non-trivial tails except for the pivot elements. Thus, to geta basis of Lie p ( v , w , u ) we add nontrivial powers of the pivot elements, i.e. power standard monomials.The second claim follows from computations of Theorem 4.5. (cid:3) Weight function, Z -grading, bounds on weights Weights. By pure monomials we call products of divided powers and pure derivations. In particular,if a monomial contains one pure derivation, we get a pure Lie monomial . Set α n = wt ( ∂ x n ) = − wt ( x n ) ∈ C , β n = wt ( ∂ y n ) = − wt ( y n ) ∈ C , γ n = wt ( ∂ z n ) = − wt ( z n ) ∈ C , for all n ≥
0. This values are easily extendedto a weight function on pure monomials, additive on their (Lie or associative) products. Next, considerweight functions such that all terms in recurrence relation (4) have the same weight, thus, attaching thesame value as a weight for the pivot element as well. We get a recurrence relation: α n +1 β n +1 γ n +1 = p S n p R n − p S n − p R n p S n − p R n α n β n γ n , n ≥ . (13)Recurrence relation (13) expresses weights of the pivot elements of generation n + 1 via weights of the pivotelements of generation n . Hence, any weight function satisfying (13) is determined by its values on thezero generation, namely, by wt ( v ) , wt ( w ) , wt ( u ). Also, let wt i ( ∗ ) be a weight function which is equalto zero for all but i -th element in the list { v , w , u } , for i = 1 , ,
3. Compose the multidegree weight function
Gr( v ) := (wt ( v ) , wt ( v ) , wt ( v )), where v is a pure monomial. By definition, Gr( v ) = (1 , , w ) = (0 , , u ) = (0 , , ( ∗ ) , wt ( ∗ ) , wt ( ∗ ). Using 13, we see that Gr( v n ) ∈ N for all n ≥
0. Finally, define a total degree weight function wt( v ) := P j =1 wt j ( v ). Lemma 5.1. wt( v n ) = wt( w n ) = wt( u n ) = n − Q i =0 (cid:0) p S i + p R i − (cid:1) for all n ≥ .Proof. Follows by induction from (4). (cid:3)
Recall that wt( ∗ ) is the total degree function determined by wt( v ) = wt( w ) = wt( u ) = 1. Let wt ( ∗ )be determined by the initial values wt ( v ) = wt ( w ) = 1 and wt ( u ) = 0. Observe that wt ( v ) =wt ( v ) + wt ( v ) and wt( v ) = wt ( v ) + wt ( v ) for any monomial v . Thus, wt ( v ) counts multiplicity of v with respect to v , w and wt ( v ) counts multiplicity with respect to u only. Lemma 5.2.
For all n ≥ we have wt ( u n ) = p R + ··· + R n − , wt ( v n ) = wt ( w n ) = 0;wt ( v n ) = wt ( w n ) = n − Y i =0 (cid:0) p S i + p R i − (cid:1) ;wt ( u n ) = n − Y i =0 (cid:0) p S i + p R i − (cid:1) − p R + ··· + R n − . Proof.
Three formulas are checked by induction. Using wt( ∗ ) = wt ( ∗ ) + wt ( ∗ ) and Lemma 5.1, we getthe last formula. (cid:3) N -gradings. By a generalized monomial a ∈ End R we call any (Lie or associative) product of puremonomials and pivot elements. By construction, actual pivot elements and their products are generalizedmonomials. Observe that generalized monomials are written as infinite linear combinations of pure monomi-als. Our construction implies that these pure monomials have the same weight, we call this value the weightof a generalized monomial. Thus, the weight functions are well-defined on generalized monomials as well.Also, Gr( v ) ∈ N for any generalized monomial v .In many examples studied before [40, 42, 36, 37, 10, 45] we were able, as a rule, to compute explicitlybasis functions for the space of weight functions and study multigradings in more details. Using that baseweight functions and multigradings we were able to get more information about our algebras. In a generalsetting of the present paper it is not possible. Theorem 5.3. i) the multidegree weight function Gr( v ) is additive on products of generalized monomials v, w ∈ End R : Gr([ v, w ]) = Gr( v ) + Gr( w ) , Gr( v · w ) = Gr( v ) + Gr( w ) . ii) T = Lie p ( v , w , u ) , A = Alg( v , w , u ) are N -graded by multidegree in the generators { v , w , u } : T = ⊕ ( n ,n ,n ) ∈ N T n ,n ,n , A = ⊕ ( n ,n ,n ) ∈ N A n ,n ,n . iii) wt( ∗ ) counts the degree of v ∈ T , A in { n , n , n } yielding gradins: T = ∞ ⊕ n =1 T n , A = ∞ ⊕ n =1 A n . Proof.
Claim i) follows from the additivity of the weight function on products of pure monomials. Considerii). Recall that Gr( v ) ∈ N for any generalized monomial v and Gr( ∗ ) is additive on their products. Thus,we get N -gradings on T , A .Let v be a monomial in the generators { n , n , n } each number n i counting entrees of v , w , u , respec-tively. By additivity, Gr( v ) = n Gr( v ) + n Gr( w ) + n Gr( u ) = ( n , n , n ). Hence, T , A are N -gradedby multidegree in the generators. Now, the last claim is evident. (cid:3) LOVER NIL RESTRICTED LIE ALGEBRAS OF QUASI-LINEAR GROWTH 11
Bounds on weights.Lemma 5.4 ([37], Lemma 6.5) . Let w be a (power) standard monomial of first type of length n ≥ . Then wt( v n − ) + 1 ≤ wt( w ) ≤ wt( v n ) . We shall also write a standard monomial w of second type (see (11), (9)) as: w = r n − ( x, y, z ) x ( p Sn − − − α ) n − y ( p Rn − − − β ) n − z ( p Rn − − − γ ) n − · x ( p Sn − − − ξ ) n − z ( p Rn − − − ζ ) n − u n , where 0 ≤ α < p S n − , ≤ β, γ < p R n − , ≤ ζ < p R n − ; and 0 ≤ ξ ≤ p S n − − . (14) Lemma 5.5.
Let w be a (power) standard monomial of second type of length n ≥ . i) Let w be a power standard monomial. Then wt( v n − ) + 1 ≤ wt( w ) ≤ wt( v n ) . ii) Let w be a standard monomial of second type of length n ≥ , then ( p S n − −
1) wt( v n − ) < wt( w ) ≤ wt( v n ) . iii) Let w of second type be presented as (14) , we get more precise bounds: wt( w ) > ( p S n − − α + β + γ ) wt( v n − ) + ( ξ + ζ ) wt( v n − ); (15)wt( w ) ≤ wt( v n − )( ξ + ζ + 2) . (16)iv) Let w be of second type (14) and assume that ξ > or ζ > , then wt( v n − ) + 1 ≤ wt( w ) ≤ wt( v n ) . Proof. i) Weights of power standard monomials of second type (i.e. powers of u n − ) are equal to weights ofpowers of w n − , and we apply Lemma 5.4.ii) Let w be a standard monomial of second type (11) Clearly, weight is bounded by wt( u n ) = wt( v n ).We get a lower bound by taking the maximal allowed powers of variables. Below we get homogeneouscomponents of partial recurrence expansions for u and v , and use that wt v = wt u = 1.wt( w ) ≥ wt (cid:16)(cid:16) n − Y i =0 x ( p Si − i y ( p Ri − i z ( p Ri − i (cid:17) x ( p Sn − − n − z ( p Rn − − n − u n (cid:17) = wt (cid:16)(cid:0) n − Y i =0 x ( p Si − i z ( p Ri − i (cid:1) u n (cid:17) − wt( x (1) n − ) + wt (cid:16) n − Y i =0 y ( p Ri − i (cid:17) = wt u + wt( v n − ) + wt (cid:16) n − Y i =0 y ( p Ri − i (cid:17) = 1 + wt (cid:16)(cid:0) n − Y i =0 x ( p Si − i y ( p Ri − i (cid:1) v n − (cid:17) − n − X i =0 wt( x ( p Si − i )= 1 + wt( v ) + n − X i =0 ( p S i −
1) wt( v i ) ≥ p S n − −
1) wt( v n − ) . iii). The preceding lower bound is given by the maximal allowed powers of the divided variables. Incomparison with that bound we get additional terms ( α + β + γ ) wt( v n − ) and ( ξ + ζ ) wt( v n − ). Recall thatby (8) the head of w is g ξ,ζn = [ v ξn − , u ζn − , [ v n − , u n − ]], the latter multiplicands having the same weight, weget wt( g ξ,ζn ) = wt( v n − )( ξ + ζ + 2). Since tail variables only decrease the weight, we obtain (16).iv). We use iii) and ( ξ + ζ ) wt( v n − ) ≥ wt( v n − ). (cid:3) Growth of general clover restricted Lie algebras T (Ξ)6.1. Arbitrary tuple Ξ .Lemma 6.1. Fix numbers p > and r, s > . Then p s + p r − > p ( s +2 r ) / . Proof.
Assume that s ≥ r , then s ≥ ( s + 2 r ) / ≥ r and p s + p r − ≥ p ( s +2 r ) / + ( p r − > p ( s +2 r ) / .Consider the case s < r , then s < ( s + 2 r ) / < r and p s + p r − > p ( s +2 r ) / + ( p s − > p ( s +2 r ) / . (cid:3) Theorem 6.2.
Let Ξ be an arbitrary tuple of parameters and T (Ξ) the respective clover restricted Liealgebra. Then ≤ GKdim T (Ξ) ≤ . Proof.
By Theorem 4.7, T (Ξ) is a semidirect product of L (Ξ) with the ideal J , where bases of L (Ξ) and J consist of monomials of first and second types, respectively. By [37, Theorem 7.2], GKdim L (Ξ) ≤ u n , n ≥
0) behave like powers of w n , the same estimateon growth of L (Ξ) applies to them.Fix a number m >
1. It remains to derive an upper bound on the number of standard monomials ofsecond type of weight at most m . Let n = n ( m ) be such thatwt( v n − ) < m ≤ wt( v n ) . (17)Put m := wt( v n − ) and m := [ m/m ]. By (5.1) and Lemma 6.1, m = wt( v n − ) = n − Y i =0 ( p S i + p R i − > p ( S + ··· + S n − +2( R + ··· + R n − )) / , n ≥ . (18)Let w be a standard monomial of second type of length n ′ and wt( w ) ≤ m . Assume that n ′ ≥ n + 2. ByClaim ii) of Lemma 5.5, m ≥ wt( w ) > wt( v n ′ − ) ≥ wt( v n ), a contradiction with (17). Hence, w is of lengthat most n + 1.1) We evaluate a number f ( m ) of standard monomials w of second type of length n + 1 satisfyingwt( w ) ≤ m . By claim iv) of Lemma 5.5, ξ = ζ = 0 (i.e. the neck variables reach the maximal values). Thus,we get monomials w = r n − ( x, y, z ) x ( p Sn − − − α ) n − y ( p Rn − − − β ) n − z ( p Rn − − − γ ) n − · x ( p Sn − n z ( p Rn − n u n +1 . (19)Using (9) and (18), we estimate a number of tails r n − ( x, y, z ) in (19) as: p S + ··· + S n − +2( R + ··· + R n − ) < m . (20)Using estimate (15) (the indices are shifted by one!), m ≥ wt( w ) > (1+ α + β + γ ) wt( v n − ). We get estimates0 ≤ α + β + γ ≤ h m wt( v n − ) i − m − , α, β, γ ≥ . (21)A number of possibilities for variables with indices n − α, β, γ satisfying (21), which is equal to (cid:18) m + 23 (cid:19) = m ( m + 1)( m + 2)6 ≤ m , (22)where the last estimate is checked directly for all m ≥
1. Using (20) and (22), we get f ( m ) < m m = ( m m ) ≤ m .
2) Let f ( m ) be a number of standard monomials of second type (14) of length n satisfying wt( w ) ≤ m .Using (18), a number of possibilities for divided powers with indices 0 , . . . , n − p S + ··· + S n − +2( R + ··· + R n − ) < m . (23)Using estimate (15), m ≥ wt( w ) > ( ξ + ζ ) wt( v n − ) . We get estimates0 ≤ ξ + ζ ≤ h m wt( v n − ) i = m , ξ, ζ ≥ . (24)A number of possibilities for the neck letters x n − , z n − in (14) is bounded by a number of pairs of integers ξ, ζ satisfying (24). We get a bound (cid:18) m + 22 (cid:19) ≤ m , m ≥ , (25)where one checks the last estimate directly. Using (23) and (25) we obtain an estimate f ( m ) ≤ m m ≤ m .
3) Let f ( m ) be a number of all standard monomials of second type (14) of length n −
1. Using (18), anumber of possibilities for all divided powers (having indices 0 , . . . , n −
2) is evaluated by f ( m ) ≤ p S + ··· + S n − +2( R + ··· + R n − ) < m ≤ m . LOVER NIL RESTRICTED LIE ALGEBRAS OF QUASI-LINEAR GROWTH 13
A similar estimate on the number of standard monomials of second type of length n − p − than the estimate above. The same applies to lengths n − , . . . ,
0. Let ˜ f ( m ) be the numberof standard monomials of second type (14) of length at most n −
1. We get a bound˜ f ( m ) ≤ n − X i =0 p − i · f ( m ) ≤ m − p − ≤ m < m . Finally, the obtained bounds yield a desired estimate on the number of standard monomial of second typeand weight at most m : f ( m ) + f ( m ) + ˜ f ( m ) ≤ m . (cid:3) Constant tuple Ξ .Theorem 6.3. Let a tuple
Ξ = ( S i , R i | i ≥ be constant: S i = S , R i = R for i ≥ , where S, R ≥ .Denote λ = ( S + 2 R ) ln p ln( p S + p R − . Consider the clover restricted Lie algebra T = T ( S, R ) := T (Ξ) . Then i) GKdim T = GKdim T = λ . ii) C m λ < γ T ( m ) < C m λ for m ≥ , and C , C being positive constants. iii) λ ∈ [1 , .Proof. Fix a number m >
1. Using (5.1), we choose n = n ( m ) satisfyingwt( v n − ) = ( p S + p R − n − < m ≤ wt( v n ) = ( p S + p R − n . (26)Then n < log p S + p R − ( m ) + 1 . (27)Consider a standard monomial w of second type such that wt( w ) ≤ m . Assume that w has length n ′ ≥ n + 2. By claim ii) of Lemma 5.5 wt( w ) > wt( v n ′ − ) ≥ wt( v n ) ≥ m , a contradiction. Hence, w is oflength at most n + 1. Let f ( n ) be a number of standard monomials of second type of length at most n + 1.By (11), (9), and (27), we get f ( n ) < p ( S +2 R )( n +1) ≤ p ( S +2 R )(log pS + pR − ( m )+2) ≤ p S +2 R ) m ( S +2 R ) ln p ln( pS + pR − . Let w be a standard monomial of first type with wt( w ) ≤ m . By the lower bound in Lemma 5.4 and theupper bound in (26), w is of length at most n . Let f ( n ) be the number of standard monomials of first typeof length at most n . By (10), (9), and (27), we get f ( n ) < p ( S + R ) n yielding a smaller bound than above.Let f ( n ) be the number of all power standard monomials of weight at most m . By Lemmas 5.4 and 5.5they are of length at most n . We apply Lemma 4.6 and (27); f ( n ) ≤ n ( S + 2 R ) ≤ (log p S + p R − ( m ) + 1)( S + 2 R ) . Now, the upper bound follows using that γ T ( m ) ≤ f ( n ) + f ( n ) + f ( n ).By (26), n ≥ log p S + p R − ( m ). Consider standard monomials w of second type (11) of length n −
1. Bythe lower bound in (26), wt( w ) ≤ wt( v n − ) < m . We evaluate the number of such monomials by countingtheir different tails r n − ( x, y, z ) (see (9)), yielding a lower bound: γ T ( m ) ≥ p ( S +2 R )( n − ≥ p ( S +2 R )(log pS + pR − ( m ) − = p − S +2 R ) m ( S +2 R ) ln p ln( pS + pR − . The second claim follows by our estimates. The last claim follows from Theorem 6.2. (cid:3)
Lemma 6.4.
Let a tuple Ξ be constant or, more generally, periodic. Then the clover restricted Lie algebra T (Ξ) is self-similar.Proof. The notion of self-similarity for Lie algebras was introduced by Bartholdi [5]. Both Lie superalgebrasof [36] are self-similar. The self-similarity of the two-generated subalgebra L (Ξ) ∼ = Lie p ( v , w ) ⊂ T (Ξ) incase of a periodic tuple Ξ was observed in [37]. We refer a reader to that paper for details. (cid:3) Corollary 6.5.
Let char K = p ≥ . Consider self-similar clover restricted Lie algebras T ( S, R ) given byconstant tuples Ξ determined by two integers S, R . Then { GKdim T ( S, R ) | S, R ∈ N } is dense on [1 , .Proof. By choosing the numbers R = S sufficiently large, we can obtain GKdim T ( S, R ) arbitrarily close to3. By fixing R = 1 and choosing S sufficiently large we can obtain GKdim T ( S, R ) arbitrarily close to 1.Consider a large positive integer S and R ∈ { , . . . , S } . One checks that for R , R ′ = R + 1 the respectiveGelfand-Kirillov dimensions differ by O (1 /S ), S → + ∞ . (cid:3) Proof of main results: Clover restricted Lie algebras of Quasi-linear growth
Proof of Theorem 3.1.
All claims follow from the first one. Recall that we consider the tuple of integersΞ κ := ( S i := [( i + 1) /κ − ] , R i := 1 | i ≥ T = T (Ξ κ ). By Theorem 4.7, T (Ξ)is a semidirect product of L (Ξ) with the ideal J , where bases of L (Ξ) and J consist of monomials of firstand second types, respectively.We start with general estimates used in the proof of the next theorem as well. By Lemma 5.1,wt( v n ) = n − Y i =0 ( p S i + p − > p S + ··· + S n − , n ≥
1; (28)wt( v n ) = n − Y i =0 ( p S i + p − < θp S + ··· + S n − , θ := ∞ Y i =0 (1+ p − S i ) , n ≥ . (29)Indeed, it is well known that convergence of the infinite product is equivalent to convergence of the sum P ∞ i =0 p − S i . We have S i > p i for i ≥ N . Thus, P ∞ i = N p − S i ≤ P ∞ i = N /i < ∞ . We shall use thefollowing well-known estimates:( κ + o (1)) n /κ = n − X i =0 S i < n X i =0 ( i + 1) /κ − = ( κ + o (1)) n /κ , n → ∞ . (30)Let us prove the desired upper bound on the standard monomials of second type. Fix a number m > n = n ( m ) such that wt( v n − ) < m ≤ wt( v n ) . (31)Put m := wt( v n − ) and m := [ m/m ]. By (28) and lower estimate in (30), we get m = wt( v n − ) > p S + ··· + S n − ≥ p ( κ + o (1)) n /κ , n → ∞ ; (32) n ≤ (cid:16) (1 /κ + o (1)) log p m (cid:17) κ ≤ (cid:16) o (1) κ ln p ln m (cid:17) κ , m → ∞ . (33)Let w be a standard monomial of second type with wt( w ) ≤ m . Suppose that it has length n ′ ≥ n + 2. Byclaim ii) of Lemma 5.5, wt( w ) > wt( v n ′ − ) ≥ wt( v n ) ≥ m ≥ wt( w ). The contradiction proves that w is oflength at most n + 1.1) We evaluate a number f ( m ) of standard monomials w of second type of length n + 1 satisfyingwt( w ) ≤ m . By Claim iv) of Lemma 5.5, the head variables in w have the maximal degrees and we getmonomials of the form: w = r n − ( x, y, z ) x ( p Sn − − − α ) n − y ( p Rn − − − β ) n − z ( p Rn − − − γ ) n − · x ( p Sn − n z ( p Rn − n u n +1 . (34)Using (32), we evaluate the number of tails r n − ( x, y, z ) in (34) as: p S + ··· + S n − p R + ··· + R n − ) < m p n . (35)By estimate (15), m ≥ wt( w ) > (1 + α + β + γ ) wt( v n − ). We get estimates0 ≤ α + β + γ ≤ h m wt( v n − ) i − m − , where 0 ≤ β, γ < p R n − = p. (36)So, both β, γ have at most p choices. Now the number of possibilities for variables with indices n − α, β, γ satisfying (36), which is bounded by p m . Combining with thebound on the number of tails (35), we get f ( m ) ≤ p m · m p n ≤ p mp n . (37)2) We evaluate a number f ( m ) of standard monomials of second type and length n such that wt( w ) ≤ m .Using (32), a number of possibilities for variables with indices 0 , . . . , n − p S + ··· + S n − +2( R + ··· + R n − ) < m p n . (38)Using estimate (15), we get m ≥ wt( w ) > ( ξ + ζ ) wt( v n − ) and0 ≤ ξ + ζ ≤ h m wt( v n − ) i = m , where 0 ≤ ζ < p R n − = p. (39) LOVER NIL RESTRICTED LIE ALGEBRAS OF QUASI-LINEAR GROWTH 15
Thus, the number of possibilities for the neck letters x n − , z n − in (14) is bounded by the number of integers ξ, ζ satisfying (39), which is bounded by p ( m + 1). Using the bound on the number of tails (38), we get f ( m ) ≤ p ( m + 1) · m p n ≤ p · mp n . (40)3) We evaluate a number f ( m ) of standard monomials of second type and length n −
1. Using (32), anumber of possibilities for all divided powers, now having indices 0 , . . . , n − f ( m ) ≤ p S + ··· + S n − +2( R + ··· + R n − ) < m p n ≤ mp n . The number of standard monomials of second type of length n − p − thanestimate above. The same applies to lengths n − , . . . ,
0. Let ˜ f ( m ) be the number of standard monomialsof second type (14) of length at most n −
1. Using 33, we get˜ f ( m ) ≤ n − X i =0 p − i f ( m ) ≤ mp n − p − ≤ mp n . (41)Let f ( m ) be the number of power standard monomials w of second type of weight at most m . By (31) andclaim i) of Lemma 5.5, w is of length at most n . By (12), f ( m ) ≤ R + · · · + R n − = n . Combining (37),(40), (41), and using (33), the number of all standard monomials of second type and weight at most m isevaluated by f ( m ) + f ( m ) + ˜ f ( m ) + f ( m ) ≤ ( p + 2 p + 2) mp n + n (42) ≤ ( p + 2 p + 2) m exp (cid:18) p (cid:16) /κ + o (1)ln p ln m (cid:17) κ (cid:19) = m exp (cid:18) p ) − κ + o (1) κ κ (ln m ) κ (cid:19) , m → ∞ . We have an upper bound on the growth of the subalgebra L (Ξ) given by [37, Theorem 9.2], which actuallyyields upper bounds on the number of (power) standard monomials of first type. But now we are proving alittle bit stronger bounds, a reader can either trace that computations or modify more lengthy computationsobtained for the monomials of second type.Finally, let us establish the lower bound. We keep notations (31). Similar to (29) m ≤ wt( v n ) ≤ n − Y i =0 ( p ( i +1) /κ − + p − < θ n − Y i =0 p ( i +1) /κ − , θ := ∞ Y i =0 (1+ p − ( i +1) /κ − ) . (43)Using (43) and the upper bound (30), we get m ≤ θp ( κ + o (1)) n /κ . Hence n ≥ (cid:18) log p ( m/θ ) κ + o (1) (cid:19) κ ≥ (cid:16) o (1) κ ln p ln m (cid:17) κ , m → ∞ . (44)By (29), m = wt( v n − ) = n − Y i =0 ( p S i + p − < θp S + ··· + S n − . (45)Consider standard monomials of second type w = r n − ( x, y, z ) g ξ n − ,ζ n − n (11) of length n . We evaluatethe number of their tails r n − ( x, y, z ) using (45) p S + ··· + S n − p R + ··· + R n − ) > m θ p n − . (46)By our construction and Lemma 5.1 m = h mm i ≤ wt( v n )wt( v n − ) = p S n − + p − . (47)Consider standard monomials of second type w which heads satisfy ξ n − ∈ { , . . . , m − p } , ζ n − = 0.Using (47), we have 0 ≤ ξ n − < p S n − , so, we get standard monomials indeed. Also, using (16), thesemonomials are of weight not exceeding m :wt( w ) ≤ wt( v n − )( ξ n − + ζ n − + 2) ≤ wt( v n − ) m = m m ≤ m. There are m − p + 1 such heads. We multiply this number by the number of different tails (46), usingestimate (44), we obtain the desired lower bound:( m − p +1) m θ p n − ≥ m p θ p n (48) ≥ m p θ · p (cid:0) o (1) κ ln p ln m (cid:1) κ = m exp (cid:18) p ) − κ + o (1) κ κ (ln m ) κ (cid:19) , m → ∞ . (cid:3) Proof of Theorem 3.2.
We use estimates and notations of the previous proof. It is sufficient to prove thefirst claim. Fix the constant λ := (ln p ) /κ ∈ R + . Now we consider the tuple Ξ q,κ = ( S i , R i | i ≥ R i := 1 for all i ≥
0, and define integers S i by induction: S = 1 and S n := [exp ( q ) ( λ ( n + 2))] + 1 − S − · · · − S n − , n ≥ . (49)Let us prove the desired upper bound on the standard monomials of second type. Fix a number m > n = n ( m ) such that wt( v n − ) < m ≤ wt( v n ) . (50)Put m := wt( v n − ) and m := [ m/m ]. By (28) and (49) we get m > m = wt( v n − ) > p S + ··· + S n − ≥ p exp ( q ) ( λn ) ; n < λ ln ( q ) log p ( m ); p n < exp (cid:18) ln p λ ln ( q ) log p ( m ) (cid:19) = (cid:16) ln ( q − log p ( m ) (cid:17) (ln p ) /λ = (ln ( q ) m ) κ + o (1) , m → ∞ . Using estimate (42) on the number of all standard monomials of second type of weight at most m , we getthe desired upper asymptotic on the number of these monomials. Similar bounds are valid for monomials offirst type.Let us check the lower bound. We use notations (50). By estimate (29) and (49) m ≤ wt( v n ) < θp S + ··· + S n − < θp exp ( q ) ( λ ( n +1))+1 ; n > λ ln ( q ) (cid:16) log p ( m/θ ) − (cid:17) − o (1) λ ln ( q +1) ( m ) , m → ∞ ; p n > exp (cid:18) ln p + o (1) λ ln ( q +1) ( m ) (cid:19) = (ln ( q ) m ) κ + o (1) , m → ∞ . Finally, using the lower bound on the number of standard monomials of second type (48) and the boundabove, we obtain the desired lower bound on the growth of T . (cid:3) References [1] Bahturin Yu. A., Identical relations in Lie algebras. VNU Science Press, Utrecht, 1987.[2] Bahturin Yu.A., Mikhalev A.A., Petrogradsky V.M., and Zaicev M. V., Infinite dimensional Lie superalgebras, de GruyterExp. Math. vol. 7, de Gruyter, Berlin, 1992.[3] Bahturin, Yu.A.; Olshanskii, A., Large restricted Lie algebras,
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