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PAULI GRADINGS ON LIE SUPERALGEBRAS AND GRADED CODIMENSION GROWTH 7 1 DUSˇAND.REPOVSˇANDMIKHAILV.ZAICEV 0 2 n Abstract. WeintroducegradingoncertainfinitedimensionalsimpleLiesu- a peralgebras of type P(t) by elementary abelian 2-group. This grading gives J rise to Pauli matrices and is a far generalization of (Z2×Z2)-grading on Lie algebra of (2×2)-traceless matrices.We use this gradingfor studying numer- 4 ical invariants of polyomial identities of Lie superalgebras. In particular, we 2 compute gradedPI-exponent correspondingtoPauligrading. ] A R . 1. introduction h t a In this paper we study algebras over a field F of characteristic zero. Group m gradedalgebrashavebeenintensivelystudiedinthe lastdecades(see,forexample, [ [3, 5, 6, 10, 11, 18, 19, 26]). All possible gradings on matrix algebras over an algebraically closed field were described in [3, 6]. Recently, all gradings by a finite 1 abelian groups on finite dimensional simple real algebras have also been classified v 4 in [7, 23]. Many authors have also paid attention to grading on Lie algebras [5, 4 8, 11, 19]. Both, in associative and Lie case, an exceptional role is played by 8 gradings which cannot be ”refined” – in particular, gradings whose homogeneous 6 components are one-dimensional [3, 6, 8, 19]. Classification of group gradings on 0 Liesuperalgebrasis onlyinits initialstages(see,e.g.,[4]). Thereforeanimportant . 1 role is played by new examples of gradings on Lie superalgebras. 0 It is well known that abelian gradings are closely connected to automorphism 7 and involution actions on algebra (see, for example, [3]), hence the knowledge of 1 : gradingsgivesusanimportantinformationaboutthe groupofautomorphismsand v antiautomorphisms of an algebra. Another application of gradings is the study of i X graded and non-graded identities and their numerical invariants. r Given an algebra A, one can associate to it an infinite sequence of non-negative a integers c (A) , n=1,2,..., n { } called codimensions of A. The study of asymptotic behavior of c (A) is one of n { } themostimportantandcurrentapproachesinthemodernPI-theory[14]. Inmany cases codimension growth is exponentially bounded. In particular, dimA=d< c (A) dn+1 n ∞ ⇒ ≤ Date:January25,2017. 2010 Mathematics Subject Classification. Primary17B01,16P90; Secondary15A30,16R10. Key words and phrases. Polynomial identities, Lie superalgebras, graded algebras, codimensions,exponential growth,Pauligradings. The first author was supported by the Slovenian Research Agency grants P1-0292-0101, J1- 7025-0101, J1-6721-0101 and J1-5435-0101. The second author was supported by the Russian ScienceFoundation, grant16-11-10013. Wethankthereviewersforcomments. 1 2 D.D.REPOVSˇANDM.V.ZAICEV (see [2] and also [15, Proposition 2]). If, in addition, A is endowed with a grading by a group G then one can also define the graded codimension sequence cG(A). n For a finite dimensional algebra A, graded and ordinary codimensions satisfy the following inequalities: (1) c (A) cG(A) (dimA)n+1 n ≤ n ≤ (see [2]). As a rule,aninvestigationofasymptoticsofgradedcodimensionsis mucheasier than a study of non-graded codimensions. This fact was used in our previous papersforobtainingtheresultsonbothgradedandnon-gradedcodimensiongrowth [16, 20, 21, 22]. If A is a finite dimensional graded simple algebra then there exist the limits (2) exp(A)= lim n c (A), expG(A)= lim n cG(A) n n n→∞ n→∞ q p and according to (1) we have (3) exp(A) expG(A) dimA. ≤ ≤ Itiswellknownthatinmanymostimportantcasesofalgebras(associative,Lie, Jordan, alternative, etc.) (4) exp(A)=dimA, provided that A is simple and F is algebraically closed [12, 13, 25]. In this case expG(A) is also equal to dimA for any grading on A. If A is graded simple but not simple in the usual sense then graded and non-graded exponents can differ. For example, if G is a finite abelian group of order G = m and A is its group | | algebra, A = FG, then exp(A) = 1 whereas expG(A) = m. Clearly, if A is simple in non-graded sense then A is also graded simple for any G-grading. Relations (3) and(4)showthatthe conjecturethatexp(A)=expG(A)holdsforassociative,Lie, Jordan and alternative algebras over an algebraically closed field. Nevertheless, in the Lie superalgebra case there exist simple algebras such that exp(A) and expG(A) exist and are strictly less than dimA (see [16, 22]). Here we are talking about canonical Z -grading on Lie superalgebras. Therefore the study 2 of relations between graded and non-graded PI-exponents is of interest in the gen- eralcase. Inparticular,if the conjecture that exp(A)=expG(A) is confirmedthen it would give us a powerful tool for computing precise asymptotics of codimension growth. Another consequence would be the independence of expG(A) on the particular G-grading. The goal of the present paper is twofold. In the first part we define the so- called Pauli G-grading on the simple Lie superalgebra of the type L = P(t) (in the notation of [17], for general material on Lie superalgebras see alo [24]), where t is the power of 2 and G is an elementary abelian 2-group. This grading posesses many remarkable properties. In fact, it is induced from the grading on simple 3- dimensionalLiealgebrasl (F)byPaulimatricesandiscompatiblewiththecanon- 2 ical Z -grading. All non-zero homogeneous components of L are one-dimensional. 2 Also, any even homogeneous element 0 = a L is a non-degenerate matrix and g 6 ∈ for any homogeneous elements a L ,b L their Lie supercommutator is either g h ∈ ∈ zero or non-degenerate. In the second part of the paper we investigate the graded codimension growth of L. We show that all computations are much easier than in the non-gradedcase due to the remarkable properties of Pauli grading. PAULI GRADINGS ON LIE SUPERALGEBRAS AND GRADED CODIMENSION GROWTH 3 OurmainresultisTheorem1below,statingthatexpG(P(t))=t2 1+t√t2 1. − − NotethatTheorem1istruefort=2althoughP(2)isnotsimpleandexpG(P(2))= 3+2√3 holds for both Pauli grading and the canonical Z -grading (see [20]). 2 Theorem 1. Let L be a Lie superalgebra of the type P(t), t = 2q,q 1, equipped ≥ with G-grading given in Proposition 2. Then G-graded PI-exponent of L exists and expG(L)=t2 1+t t2 1. − − p 2. Pauli gradings Let L be an algebra over a field F and let G be a group. One says that L is G-graded if L has a vector space decomposition L= L g gM∈G suchthatL L L forallg,h G. SubspacesL ,g G,arecalledhomogeneous g h gh g ⊆ ∈ ∈ components of L. Any element a L is called homogeneous of degree dega = g. g ∈ The subset Supp L= g GL =0 g { ∈ | 6 } is said to be the support of the grading. A subspace V L is called homogeneous ⊆ if V = V L . g ∩ gM∈G LetAandBbetwoassociativealgebrasandletGandH betwogroups. Suppose that A and B are endowed by G- and H-gradings, respectively, A= A , B = B . g h gM∈G hM∈H Then one can introduce G H-grading on the tensor product A B by setting × ⊗ (A B) =A B . gh g h ⊗ ⊗ An associative algebra R is said to be a superalgebra if R has some Z -grading, 2 that is R=R(0) R(1), R(0)R(0)+R(1)R(1) R(0), R(0)R(1)+R(1)R(0 R(1). ⊕ ⊆ ⊆ A special case of associativesuperalgebraswhich we will use later is the Z -graded 2 n n matrix algebra R=M (F) with k,l × A B A 0 0 B R= =R(0) R(1), R(0) = , R(1) = (cid:26)(cid:18) C D (cid:19)(cid:27) ⊕ (cid:26)(cid:18) 0 D (cid:19)(cid:27) (cid:26)(cid:18) C 0 (cid:19)(cid:27) wheren=k+l, A,B,C,D arek k,k l,l k andl l matrices,respectively. In particular, when k = l we have Z×-grad×ing o×n M (F×) which will be used for the 2 2k definition of Lie superalgebra P(k). Recall now that Z -graded non-associative algebra L = L(0) L(1) is called a 2 ⊕ Lie superalgebra if it satisfies homogeneous relations ab+( 1)|a||b|ba=0, a(bc)=(ab)c+( 1)|a||b|b(ac)=0 − − 4 D.D.REPOVSˇANDM.V.ZAICEV for all a,b,c L(0) L(1) where x = 0 if x L(0) and x = 1 if x L(1). In ∈ ∪ | | ∈ | | ∈ particular, any associative superalgebra R = R(0) R(1) with the new product ⊕ called supercommutator, defined for homogeneous elements as [a,b]=ab ( 1)|a||b|ba − − becomes a Lie superalgebra. Let L(0) L(1) be a Lie superalgebra and let G be a group. Then a G-grading ⊕ L= L g∈G g ⊕ is called compatible with Z -grading of L if L L(0) or L L(1) for all g G. 2 g g ⊆ ⊆ ∈ FordefiningthePauligradingontheassociativematrixalgebraM2q(F)westart with q =1. Consider 2 2 matrices × (5) 1 0 1 0 0 1 0 1 σ = ,σ = ,σ = ,σ = . 0 (cid:26)(cid:18) 0 1 (cid:19)(cid:27) 1 (cid:26)(cid:18) 0 1 (cid:19)(cid:27) 2 (cid:26)(cid:18) 1 0 (cid:19)(cid:27) 3 (cid:26)(cid:18) 1 0 (cid:19)(cid:27) − − Matrices (5) are closely related to Pauli matrices. 0 1 0 i 1 0 σx =(cid:26)(cid:18) 1 0 (cid:19)(cid:27),σy =(cid:26)(cid:18) i −0 (cid:19)(cid:27),σz =(cid:26)(cid:18) 0 1 (cid:19)(cid:27). − It is well-known that the linear span L =< σ ,σ ,σ > is closed under Lie com- x y z mutator and L su(2) as Lie algebra whereas the span < σ ,σ ,σ ,σ > as an 0 1 2 3 ≃ associative algebra is isomorphic to M (F). Denote by G =< a > < b > 2 2 2 × the product of two cyclic groups of order 2 with generators a and b, respectively. Clearly, G is isomorphic to Z Z and the decomposition 2 2 × (6) R=M (F)=R R R R 2 e a b ab ⊕ ⊕ ⊕ is a G-grading, where R =<σ >,R =<σ >,R =<σ >,R =<σ >. e 0 a 1 b 2 ab 3 We call the grading (6) on M (F) Pauli grading on M (F). 2 2 We generalize this construction to matrices of arbirary size 2q,q 2 in the ≥ following way. Let R = R R where all R ,...,R are isomorphic to the 1 q 1 q ⊗···⊗ 2 2 matrix algebra M (F). Let also 2 × (7) G =G ... G ,G =<a > <b > Z Z ,j =1,...,q. 0 1 q j j 2 j 2 2 2 × × × ≃ ⊕ Then R has a basis consisting of elements (8) c=x x 1 q ⊗···⊗ whereallx ,...,x areofthetype(5). ThenintheKroneckerrealizationoftensor 1 q product of matrices for transpose involution T we have cT =(x x )T =xT xT. 1⊗···⊗ q 1 ⊗···⊗ q In particular, the element c of the type (5) is symmetric if and only if the number of matrices σ among x ,...,x is even and cT = c if and only if the number of 3 1 q − σ is odd. 3 All R ,...,R have Pauli grading as defined earlier and we can extend these 1 q gradings to their tensor product R. Then we obtain G -grading on R 0 R= R g gM∈G0 PAULI GRADINGS ON LIE SUPERALGEBRAS AND GRADED CODIMENSION GROWTH 5 where R =<x x > and all x ,...,x are of the type (5). Moreover, we g 1 q 1 q ⊗···⊗ have (9) deg(x x )=degx degx 1 q 1 q ⊗···⊗ ··· where e , if x =σ i i 0 (10) degx = ai, if xi =σ1, i  bi, if xi =σ2, a b , if x =σ i i i 3 and σ ,σ ,σ ,σ are defined in(5). 0 1 2 3 Combining all previous arguments we get the following. Proposition 1. The following assertions hold: 1) Relations(5),(9),(10)defineG0-gradingonthematrixalgebraR=M2q(F), where G is the elementary abelian 2-group defined in (7); 0 2) dimR =1 for every g G ; g 0 ∈ 3) R has a homogeneous in G -grading basis consisting of products (8) and 0 any basis element is either symmetric or skew-symmetric under transpose involution; 4) Every non-zero homogeneous element is invertible; and 5) Lie subalgebra sl2q of traceless matrices is homogeneous in this grading. (cid:3) Applying Proposition 1, we construct a grading on some simple Lie superalgebras. Recall that P(t) (in the notation [17]) is a Lie superalgebra L M (F) t,t ⊂ with A 0 0 B L(0) = , L(1) = (cid:26)(cid:18) 0 AT (cid:19)(cid:27) (cid:26)(cid:18) C 0 (cid:19)(cid:27) − where A,B and C are t t matrices, trA = 0, BT = B,CT = C and X XT × − → is the transpose involution on M (F). We equip L with an abelian grading in the t following way. Let t=2q, R=R R , R = =R =M (F) 1 q 1 q 2 ⊗···⊗ ··· and let G be as in (7). We extend G to 0 0 G=<a > G (Z )2q+1 0 2 0 2 × ≃ and define G-grading on L compatible with canonical Z -grading. If X R is 2 g ∈ homogeneous,degX =g G , then g 0 ∈ X 0 (11) Y = g , (cid:26)(cid:18) 0 XT (cid:19)(cid:27) − g is homogeneous in L, degY =g for all Xg sl2q(F) R, ∈ ⊂ 0 X (12) if X is symmetric then Y = g g (cid:26)(cid:18) 0 0 (cid:19)(cid:27) is homogeneous, degY =a g 0 0 0 (13) if X is skew then Y = g (cid:26)(cid:18) Xg 0 (cid:19)(cid:27) 6 D.D.REPOVSˇANDM.V.ZAICEV is homogeneous, degY = a g. The following proposition is an immediate conse- 0 quence of Proposition 1 and multiplication rule of L. Proposition 2. Let G =<a > <b > <a > <b > 0 1 2 1 2 q 2 q 2 × ×···× × and G=<a > G 0 2 0 × be elementary abelian 2-groups. Then (11), (12) and (13) define a G-grading on L=P(2q) compatible with the canonical Z -grading. All homogeneous components 2 of L are 1-dimensional. If g =a g ,h=a h ,g ,h G , 0=X L ,X L 0 0 0 0 0 0 0 g g h h ∈ 6 ∈ ∈ and both X ,X are either of the type (12) or of the type (13) then [X ,X ] = 0. g h g h In all other cases [Xg,Xh] is an invertible element of M2q(F). (cid:3) 3. Graded PI-exponent We recall some key notions from the theory of identities and their numerical invariants. We refer the reader to [1, 9, 14] for details. Consider an absolutely free algebra F X with a free generating set { } X = X , X = for any g G. g g | | ∞ ∈ g[∈G Onecandefine aG-gradingonF X bysetting deg x=g,whenx X ,andex- { } G ∈ g tendthisgradingtotheentireF X inthenaturalway. Apolynomialf(x ,...,x ) 1 n { } in homogeneous variables x X ,...,x X is called a graded identity of a 1 ∈ g1 n ∈ gn G-graded algebra A if f(a ,...,a ) = 0 for any a A ,...,a A . The set 1 n 1 ∈ g1 n ∈ gn IdG(A) of all graded identities of A forms an ideal of F X which is stable under { } graded homomorphisms F X F X . { }→ { } First, let G be finite, G= g ,...,g and 1 k { } X =X ... X . g1 gk [ [ DenotebyP thesubspaceofF X ofmultilinearpolynomialsoftotaldegree n1,...,nk { } n=n + +n in variables 1 k ··· x(1),...,x(1) X ,...,x(k),...,x(k) X . 1 n1 ∈ g1 1 nk ∈ gk Then the value P c (A)=dim n1,...,nk n1,...,nk P IdG(A) n1,...,nk ∩ is called a partial codimension of A while n (14) cG(A)= c (A) n (cid:18)n ,...,n (cid:19) n1,...,nk n1+·X··+nk=n 1 k is called a graded codimension of A. Recall that the support of the grading is the set Supp A= g GA =0 . g { ∈ | 6 } Note that if Supp A=G, say, Supp A= g ,...,g , d<k, then the value 1 d 6 { } n P (15) dim n1,...,nd (cid:18)n ,...,n (cid:19) P IdG(A) n1+·X··+nd=n 1 d n1,...,nd ∩ PAULI GRADINGS ON LIE SUPERALGEBRAS AND GRADED CODIMENSION GROWTH 7 coincides with (14). Denote P (16) P (A)= n1,...,nk . n1,...,nk P IdG(A) n1,...,nk ∩ For finding a lower bound for PI-exponent we need the following observation. Lemma 1. Let A be a G-graded algebra with the support SuppA= g ,...,g 1 d { }⊆ G. Let also dimA =1 for any g SuppA. Then g ∈ (1) if P (A)=0 then dimP (A)=1, n1,...,nd 6 n1,...,nd (2) dimP (A)=1 if and only if there exist u A ,...,u A and a n1,...,nd 1 ∈ g1 d ∈ gd monomial m(u ,...,u )=m=0 on u ,...,u such that every u appears 1 d 1 d j 6 in m exactly n times, j =1,...,d. j Proof. First, let P (A) =0. Then there exists a multilinear homogeneous n1,...,nd 6 polynomial f =f(x(1),...,x(1),...,x(d),...,x(d)) P 1 n1 1 nd ∈ n1,...,nd which is not an identity of A. That is, one can find u A ,...,u A such 1 ∈ g1 d ∈ gd that f(u ,...,u )=0. If 1 d 6 g =g(x(1),...,x(1),...,x(d),...,x(d)) P IdG(A) 1 n1 1 nd ∈ n1,...,nd \ then g(u ,...,u ,...,u ,...,u )=λf(u ,...,u ,...,u ,...,u ) 1 1 d d 1 1 d d for some scalar λ since dimA = 1 for g = gn1 gnd. Hence g λf 0 is an g 1 ··· d − ≡ identity of A. This proves (1). Now let dimP (A)=1, that is P (A)=0. Then there exist n1,...,nd n1,...,nd 6 f =f(x(1),...,x(1),...,x(d),...,x(d)) P IdG(A) 1 n1 1 nd ∈ n1,...,nd \ and u A ,...,u A such that 1 ∈ g1 d ∈ gd f(u ,...,u ,...,u ,...,u )=0 1 1 d d 6 n1 nd | {z } | {z } in A. Hence, at least one monomial of f has a non-zero value under evaluation ϕ:F X A, where { }7→ ϕ(x(i))=u , 1 i d, 1 j n . j i ≤ ≤ ≤ ≤ i This implies (2), and have we completed the proof. (cid:3) Corollary 1. n (17) cG = n (cid:18)n ,...,n (cid:19) X 1 d where the sum in (17) is taken over all tuples (n ,...,n ) such that 1 d (18) P (A)=0. n1,...,nd 6 Moreover, for the inequality (18) it suffices to check the condition (2) of Lemma 1. 8 D.D.REPOVSˇANDM.V.ZAICEV (cid:3) Now we go back to the Lie superalgebra L=L(0) L(1) =P(t), t=2q, ⊕ with the G-grading presented in Proposition 2. First, we give an upper bound for expG(L). Note that Stirling formula for factorials implies the inequalities 1 n (19) Φ(n;n ,...,n )n nΦ(n;n ,...,n )n nd 1 d ≤(cid:18)n ,...,n (cid:19)≤ 1 d 1 d where Φ(n;n1,...,nd)=(nn1)−nn1 ···(nnd)−nnd and n=n + +n . 1 d ··· Denote t(t+1) t(t 1) a= ,b= − ,c=t2 1,d=a+b+c=dimL. 2 2 − The algebra L has a natural Z-grading L= −1 0 1 L ⊕L ⊕L where 0 0 A 0 0 B = , =L(0) = , = . L−1 (cid:26)(cid:18) C 0 (cid:19)(cid:27) L0 (cid:26)(cid:18) 0 AT (cid:19)(cid:27) L1 (cid:26)(cid:18) 0 0 (cid:19)(cid:27) − Allremaining components , k =0, 1,arezero. Clearly, P (L)=0 only if Lk 6 ± n1,...,nd 6 (20) n + +n n n 1 1 a a+1 a+b | ··· − ···− |≤ where g ,...,g G is the support SuppL. It followsfrom Corollary1 and (19) 1 d { }⊆ that 1 (21) max Φ(n;n ,...,n )n cG(L) ndmax Φ(n;n ,...,n )n nd { 1 d }≤ n ≤ { 1 d } where the maximum is taken among all n ,...,n satisfying (20). 1 d First, consider the case where the left side of (20) is equal to zero. Then we rewrite Φ(n;n ,...,n )=Φ(x ,...,x ) 1 d 1 d where x + +x =1, x ,...,x 0, 1 d 1 d ··· ≥ (22) Φ(x ,...,x )=x−x1 x−xd 1 d 1 ··· d and x + +x =x + +x . 1 a a+1 a+b ··· ··· It is easy to see that the maximal value of the function (22) is achieved when x = =x , x = =x , x = =x . 1 a a+1 a+b a+b+1 a+b+c ··· ··· ··· Denote x=x ,y =x ,z =x . Then (22) does not exceed 1 a+b a+b+c Φ=Φ(x,y,z)=x−axy−byz−cz and x,y,z satisfy the relateionseax=by, ax+by+cz =1. These relations imply Φ−1 =z(t2−1)z(1 (t2 1)z)(1−(t2−1)z)(t2(t2 1))(t2−12)z−1 − − − as a functioneof z. Then 1 g(z)=lnΦ−1 =czlnz+(1 cz)ln(1 cz) (1 cz)ln(ct2). − − − 2 − e PAULI GRADINGS ON LIE SUPERALGEBRAS AND GRADED CODIMENSION GROWTH 9 Direct calculations show that g′(z)=0 only if z =z =(t2 1+t t2 1)−1 0 − − p and g′′(z )>0. Hence, in z the function g(z) has a local mnimum. Moreover, 0 0 g(z )= ln(t2 1+t t2 1). 0 − − − p It follows that Φ t2 1+t t2 1 ≤ − − p and e (23) qn cGn(L)≤nnd(t2−1+tpt2−1) as follows from (21) in the case n + +n =n + +n . 1 a a+1 a+b ··· ··· If n + +n n n = 1 then 1 a a+1 a+b ··· − −···− − n n+1 (cid:18)n ,...,n (cid:19)≤(cid:18)n +1,n ,...,n (cid:19) 1 d 1 2 d and (24) qn cGn(L)≤(n+1)n+d1(t2−1+tpt2−1). Similarly, if n + +n n n =1 then 1 a a+1 a+b ··· − −···− (25) qn cGn(L)≤(n−1)n−d1(t2−1+tpt2−1) since n n 1 n − . (cid:18)n ,...,n (cid:19)≤ (cid:18)n ,...,n ,n 1,n ,...,n (cid:19) 1 d 1 a+b−1 a+b a+b+1 d − Inequalities (23), (24) and (25) give us the following. Lemma 2. expG(L) t2 1+t t2 1. ≤ − − p Now we will get the same lower bound. Lemma 3. (26) expG(L) t2 1+t t2 1. ≥ − − p Proof. Recall that L is Z-graded algebra, L = , and a = dim , −1 0 1 1 L ⊕L ⊕L L b=dim , c=dim . Consider a collection −1 0 L L X = x ,...,x ,...,x ,...,x 1 1 a a { } b b where x ,...,x are homogeneo|usi{nzG-g}radin|gele{mzent}s with pairwise distinct 1 a 1 L degree in G-grading. Similarly, we take Y = y ,...,y ,...,y ,...,y , 1 1 b b { } a a with homogeneous y ,...,y | {z, de}g y |are{dzisti}nct. Renaming elements of 1 b −1 G i ∈ L X,Y we write X = x(1),...,x(ab) , Y = y(1),...,y(ab) . { } { } 10 D.D.REPOVSˇANDM.V.ZAICEV We remark that any x ,1 i a appears among x(1),...,x(ab) exactly b times. i ≤ ≤ Similarly,anyy ,1 j b,appearsamongy(1),...,y(ab) exactlyatimes. Consider j ≤ ≤ supercommutators z =[x(1),y(1)],...,z =[x(ab),y(ab)]. 1 ab ByProposition2allz areinvertibleinM (F)matriceshomogeneousinG-grading i 2t of L. Also, z ,...,z L(0) sl (F). 1 ab 2t ∈ ≃ Note that xy = yx for any homogeneous x,y L(0). It follows that for any ± ∈ i=1,...,ab there exists z′ L(0) homogeneous in G-grading such that i ∈ [z′,z ]=2z′z =0 i i i i 6 where the product z′z is taken in the associative algebra M (F). Hence, the i i 2t left-normed Lie commutators z(i) =[z′,z ,...,z ]=2kz′zk, k =1,2,..., k i i i i i k | {z } are non-zero homogeneous elements of L(0). Asbefore,onecanfindhomogeneousu ,...,u L(0) andlinearlyindependent 1 ab ∈ homogeneous v ,...,v L(0) such that 1 c ∈ w =[z(1),u ,z(2),u ,...,z(ab),u ]=0 k k 1 k 2 k ab 6 and w =[w ,w′,v ,...,v ,w′,v ,...,v ,...,w′,v ,...,v ]=0 k,s k 1 1 1 2 2 2 c c c 6 s s s for some homogeneous w′|,...{,zw′ } L(0)|. {z } | {z } 1 c ∈ Ifuisamonomialonx ,...,x ,y ,...,y ,v ,...,v inLthenwewilldenoteby 1 a 1 b 1 c Deg u,Deg u,Deg u the total number of factors x ,y and v in u, respectively. xi yi vi i i i Then Deg w kb for all i=1,...,a, xi k,s ≥ Deg w ka for all i=1,...,b, yi k,s ≥ Deg w s for all i=1,...,c. vi k,s ≥ Total degrees Deg on x ,y ,v are as follows: α β γ { } Degz(i) =2k+1,Degw =2ab(k+1),Degw =2ab(k+1)+c(s+1)=n. k k k,s Denote (27) n =Deg w , i=1,...,a, i xi k,s (28) n =Deg w , i=1,...,b, a+i yi k,s (29) n =Deg w , i=1,...,c. a+b+i vi k,s If m = =m =kb,m = =m =ka,m = =m =s, 1 a a+1 a+b a+b+1 a+b+c ··· ··· ··· and m=m + +m =2abk+cs 1 a+b+c ···