Asssociative algebras under multi-commutator

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by  Askar Dzhumadil'daev

ASSSOCIATIVE ALGEBRAS UNDER MULTI-COMMUTATORS
A.S.DZHUMADIL’DAEV
4
1
0
2

n
a Abstract. Foranassociativealgebra A askew-symmetric(symmetric)sum
J of n! products of n elements of A in all possible order is called Lie (Jor-

4 dan) n-commutator. Weconsider A as n-aryalgebraunder n-commutator.
2 Weconstruct n-aryskew-symmetricandsymmetricgeneralizationsofJordan
identity. WeprovethatanyassociativealgebraunderJordan n-commutator

] satisfies a symmetric generalization of Jordan identity. We prove that in
A case of odd n any associative algebra under Lie n-commutator satisfies a
R skew-symmetric generalization of Jordan identity. In case of even n Lie n-
commutator satisfiesthehomotopical n-Lieidentity.
.
h
Wellknownthatanassociativealgebra A underLiecommutatorisLie. Inother
t
a words, a vector space A under commutator [a,b] = ab−ba has skew-symmetric
m multiplication [, ]:∧2A→A, that satisfies the identity, called Jacobi identity
[
[a1,[a2,a3]]−[a2,[a1,a3]]+[a3,[a1,a2]]=0.

2
Well known also, that an associative algebra A under Jordan commutator
v
{a,b} = ab + ba is Jordan. In other words, Jordan commutator is symmetric
2
7 multiplication { , } : S2A → A, that satisfies the identity of degree 4, called
2 Jordan identity
1
. {a1,{a0,{a2,a3}}}+{a2,{a0,{a1,a3}}}+{a3,{a0,{a1,a2}}}
1
0 −{{a ,a },{a ,a }}−{{a ,a },{a ,a }}−{{a ,a },{a ,a }}=0.
0 1 2 3 0 2 1 3 0 3 1 2
4
In our paper we consider multi-versions of these connections. We answer to a
1
question of A.G. Kurosh who asked about identites of multi-associative algebras
:
v under multi-commutator [5]. We show that an associative algebra under skew-
i
X symmetric n-commutator satisfies a homotopy identity (generalisation of Jacobi
identity) if n is even and one skew-symmetric generalization of Jordan identity if
r
a n isodd. Weestablishthatanassociativealgebraundersymmetric n-commutator
satisfies symmetric generalizationof Jordan identity.
To formulate our results we need to introduce some definitions. Let A be a
vector space over a field K. For a multilinear map α : A×···×A → A we say
that A = (A,α) is n-algebra with n-multiplication α. A n-algebra A is said
skew-commutative if α is skew-symmetric,
α(a ,...,a )=signσα(a ,...,a ),
σ(1) σ(n) 1 n
for any permutation σ ∈Sym . Similarly, (A,α) is commutative n-algebra,if
n
α(a ,...,a )=α(a ,...,a ),
σ(1) σ(n) 1 n
1991 Mathematics Subject Classification. 16S32,17B01.
Key words and phrases. Lie algebra, Jordan algebra, multi-operator algebra, multi-
commutator,polynomialidentity.
1
2 A.S.DZHUMADIL’DAEV
for any σ ∈Sym .
n
An absolute free n-algebra (free n-magma) can be defined as algebra of (n-
non-commutative, n-non-associative) n-polynomials Kht ,t ,...i. Denote by ω
1 2
a n-multiplication in free n-magma. To construct n-polynomials we have to
introduce n-monoms.
By definition, any variable t is a n-monom of ω-degree 0. If f is a n-
i i
monom of ω-degree k , and i = 1,...,N, then ω(f ,...,f ) is a n-monom
i 1 N
of ω-degree k +···+k +1. A linear combination of n-monoms is called (n-
1 N
non-commutative, n-non-associative) n-polynomial. A space of n-polynomials
Kht ,t ,...i is defined as a linear space with base generated by n-monoms. A
1 2
multiplication ω on Kht ,t ,...i is defined in a natural way. If g ,...,g ∈
1 2 1 N
Kht ,t ,...i, then by multilinearity ω(g ,...,g ) is a linear combination of n-
1 2 1 N
monoms. Wecanimagine n-monomsasarootedtree,whereeachvertexhas n-in
edges and 1-out edge. Leaves are labeled by elements of algebra and to innner
vertices correspond n-ary products of elements that come by in-edges.
Let f =f(t ,...,t ) be any n-polynomialof Kht ,t ,...i. Let (A,α) be any
1 k 1 2
n-algebra with n-multiplication α. For any k elements a ,...,a ∈ A one can
1 k
make substitutions t := a and ω := α in polynomial f and consider another
i i
element f(a ,...,a ) of A wheremultiplicationsaremadeintermsofmultiplica-
1 k
tion α insteadof ω. Wesaythat f =0 isa n-identityon A if f(a ,...,a )=0
1 k
for any a ,...,a ∈A.
1 k
In case of n = 2 we obtain usual algebras. In 2-algebras multiplications are
usually denoted as a ◦ b a × b a + b, etc, instead of α(a,b). The notions of
2-polynomialsand 2-polynomialidentitiesarecoincidewithusualnotionsofpoly-
nomials and polynomial identities
Let
s =s (t ,...,t )= signσt ···t
n n 1 n σ(1) σ(n)
σ∈XSymn
be a standard (associative, non-commutative) skew-symmetric polynomial. Then
anyassociativealgebra A with 2-multiplication ab canbeendowedbyastructure
of n-algebra given by n-multiplication s (a ,...,a ). Call
n 1 n
[a ,...,a ]=s (a ,...,a )
1 n n 1 n
asLie n-commutator. Notethat Lie 2-commutatorcoincides withusualLie com-
mutator,
s (a ,a )=a a −a a .
2 1 2 1 2 2 1
Let
s+ =s+(t ,...,t )= t ···t
n n 1 n σ(1) σ(n)
σ∈XSymn
be a standard (associative, non-commutative) symmetric polynomial. We can en-
dow any associative algebra A with 2-multiplication ab by a structure of n-
algebra given by Jordan n-commutator
{a ,...,a }=s+(a ,...,a ).
1 n n 1 n
Note that Jordan 2-commutator coincides with usual Jordan commutator,
s+(a ,a )=a a +a a .
2 1 2 1 2 2 1
In our paper we study n-polynomial identities of the algebra (A,s ), and
n
(A,s+) if 2-algebra A is associative. In fact we construct generalizations of Lie
n
ASSSOCIATIVE ALGEBRAS UNDER MULTI-COMMUTATORS 3
andJordanidentities that hold for total associativealgebrasunder Lie and Jordan
n-commutators.
1. Formulations of main results
Let Sym be set of all permutations on [n]={1,2,...,n}. Let
n
S ={σ ∈Sym |σ(1)<...<σ(k),σ(k+1)<···<σ(k+l)}
k,l k+l
S ={σ ∈Sym |σ(1)<...<σ(n−1),
n−1,n−1,n 3n−2
σ(n)<···<σ(2n−2), σ(2n−1)<···<σ(3n−2)},
S ={σ ∈Sym |σ(1)<...<σ(n−2),
n−2,n,n 3n−2
σ(n−1)<···<σ(2n−2), σ(2n−1)<···<σ(3n−2), σ(n−1)<σ(2n−1)}.
are subsets of shuffle-permutations
If n-multiplication ω(t ,...,t ) is skew-symmetric, then there exists only one
1 n
n-monomial of ω-degree 2
H(t ,...,t )=ω(t ,...,t ,ω(t ,...,t )).
1 2n−1 1 n−1 n 2n−1
Let
1
h(t ,...,t )= ω(t ,...,t ,ω(t ,...,t ))
1 2n−1 (n−1)!n! [1 n−1 n (2n−1)]
be its skew-symmetrisation by all parameters. Note that,
h(t ,...,t )= signσω(t ,...,t ,(t ,...,t )).
1 2n−1 σ(1) σ(n−1) σ(n) σ(2n−1)
σ∈SyXmn−1,n
For skew-commutative n-algebras there are two n-monomials of ω-degree 3
F (t ,...,t )=(t ,t ,...,t ,(t ,...,t ,(t ,...,t )))
1 1 3n−2 1 2 n−1 n 2n−2 2n−1 (3n−2)
and
F (t ,...,t )=(t ,t ,...,t ,(t ,...,t ),(t ,...,t )))
2 1 3n−2 1 2 n−2 n−1 2n−2 2n−1 (3n−2)
Let us introduce their skew-symmetric sums by all parameters except t ,
1
1
F[2](t ,...,t )= ω(t ,t ,...,t ,ω(t ,t ,...,t ,ω(t ,...,t ))),
1 1 3n−2 (n−1)!(n−2)!n! [2 3 n [1] n+1 2n−2 2n−1 (3n−2)]
1
F[2](t ,...,t )= ω(t ,t ,...,t ,ω(t ,t ,...,t ),ω(t ,...,t )).
2 1 3n−2 (n−2)!(n−1)!n! [2 3 n−1 [1] n 2n−2 2n−1 (3n−2)]
Upper index s in F[s] corresponds to the ω-place where t is and lower index l
l 1
corresponds to n-bracketing types of ω-degree 3. We have,
F[2](t ,...,t )=
1 1 3n−2
signσ(t ,...,t ,(t ,t ,...,t ,(t ,...,t ))),
σ(1) σ(n−1) 1 σ(n+1) σ(2n−2) σ(2n−1) σ(3n−2)
σ∈Symn−1X,n−1,n,σ(n)=1
F[2](t ,...,t )=
2 1 3n−2
signσ(t ,...,t ,(t ,t ,...,t ),(t ,...,t ))).
σ(1) σ(n−2) 1 σ(n) σ(2n−2) σ(2n−1) σ(3n−2)
σ∈Symn−2X,n,n,σ(n−1)=1
Let
[2] [2] [2]
f =F +λF .
λ 1 2
4 A.S.DZHUMADIL’DAEV
Thesenotionshavesymmetricanaloges. We savethe samenotationsasinskew-
symmetric case. Just change brackets of the form [ , ] to { , }.
For commutative n-algebras there are two n-monomials of ω-degree 3
F+(t ,...,t )=(t ,t ,...,t ,(t ,...,t ,(t ,...,t )))
1 1 3n−2 1 2 n−1 n 2n−2 2n−1 (3n−2)
and
F+(t ,...,t )=(t ,t ,...,t ,(t ,...,t ),(t ,...,t )))
2 1 3n−2 1 2 n−2 n−1 2n−2 2n−1 (3n−2)
Their symmetric sums by all parameters except t are defined by
1
1
F{2}(t ,...,t )= ω(t ,t ,...,t ,ω(t ,t ,...,t ,ω(t ,...,t ))),
1 1 3n−2 (n−1)!(n−2)!n! {2 3 n {1} n+1 2n−2 2n−1 (3n−2)}
1
F{2}(t ,...,t )= ω(t ,t ,...,t ,ω(t ,t ,...,t ),ω(t ,...,t )).
2 1 3n−2 (n−2)!(n−1)!n! {2 3 n−1 {1} n 2n−2 2n−1 (3n−2)}
Upperindex s in F{s} correspondstotheplaceof ω, where t isandlowerindex
l 1
l corresponds to n-bracketing types of ω-degree 3. We have,
F{2}(t ,...,t )=
1 1 3n−2
(t ,...,t ,(t ,t ,...,t ,(t ,...,t ))),
σ(1) σ(n−1) 1 σ(n+1) σ(2n−2) σ(2n−1) σ(3n−2)
σ∈Symn−1X,n−1,n,σ(n)=1
F{2}(t ,...,t )=
2 1 3n−2
(t ,...,t ,(t ,t ,...,t ),(t ,...,t ))).
σ(1) σ(n−2) 1 σ(n) σ(2n−2) σ(2n−1) σ(3n−2)
σ∈Symn−2X,n,n,σ(n−1)=1
Let
f{2} =F{2}+λF{2}.
λ 1 2
Let A be n-algebra with n-multiplication (a ,...,a ). Denote by [A] an
1 n
algebra with vector space A and n-multiplication
[a ,...,a ]=(a ,...,a )= signσ(a ,...,a )
1 n [1 n] σ(1) σ(n)
σ∈XSymn
(Lie n-commutator). Similarly, denote by {A} an algebra with vector space A
and n-multiplication
{a ,...,a }=(a ,...,a )= (a ,...,a )
1 n {1 n} σ(1) σ(n)
σ∈XSymn
(Jordan n-commutator).
Recall that A is called total associative [3] if
(a ,...,a ,(a ,a ,...,a ),a ,a ,...,a )=
1 i i+1 i+2 i+n i+n+1 i+n+2 2n−1
(a ,...,a ,a ,(a ,...,a ,a ),a ,...,a ),
1 i i+1 i+2 i+n i+n+1 i+n+2 2n−1
forany 1≤i≤n−2. Anyassociativealgebra A under n-multiplication (a ,...,a )7→
1 n
a1·····a became total associative.
n
The following skew-symmetric ω-degree 2 polynomial is called homotopical n-
Lie
homot(t ,...,t )= signσω(t ,...,t ,ω(t ,...,t )).
1 2n−1 σ(1) σ(n−1) σ(n) ω(2n−1)
σ∈SXn−1,n
An n-ary algebra (A,ω) is called homotopical n-Lie, if it satisfies the identity
homot=0 [4].
ASSSOCIATIVE ALGEBRAS UNDER MULTI-COMMUTATORS 5
Theorem 1.1. Let A be total associative n-algebra. If n is even or if charK =
p>0 and n isdivisable by p, thenits n−commutatorsalgebra [A] is homotopical
n-Lie.
Theorem 1.2. Let A be total associative algebra. Then its Lie n-commutators
algebra [A] satisfies the identity f[2] =0.
−1
Theorem1.3. Let A betotalassociativealgebra. ThenitsJordan n-commutators
algebra {A} satisfies the identity f{2} =0.
−1
Remarks. The fact that 3-commutatiors algebra [A] has no identity of ω-
degree 2 was noticed by A.G. Kurosh in [5]. The identity f[2] = holds for any
−1
n-commutators algebra, but this identity in general is not minimal. If n is even
or if the characteristic of main field is p > 0 and n ≡ 0(modp), then one can
find for [A] the identity of ω-degree 2, for example, homot = 0. We think that
homot=0 foreven n and f[2] =0 forodd n areminimalidentitiesthatholdfor
−1
any Lie n-commutatoralgebras [A], if charK =0. We think also that f{2}=0 is
−1
minimal identity that hold for any Jordan n-commutators algebra {A}.
The case n = 3 was considered by M. R. Bremner [1], [2]. He proved that
f[2] = 0 and f{2} = 0 are identities for Lie and Jordan 3-commutators of total
−1 −1
associative algebras and he established the minimality of these identities.
In case of n=2 the polynomial f{2} coincides with usual Jordan polynomial.
−1
2. Proof of Theorem 1.1
Let A be a free total associative n algebra with n-multiplication ω and [ω]
be its n-commutator,
[ω](t ,...,t )= signσω(t ,··· ,t )
1 n σ(1) σ(n)
σ∈XSymn
We have to prove that X =0, where
X =X(t ,...,t )= signσ[ω](t ,...,t ,[ω](t ,...,t )).
1 2n−1 σ(1) σ(n−1) σ(n) σ(2n−1)
σ∈SXn−1,n
Expand n-commutators [ω] in terms of associative n-multiplication ω. We see
that X is a sum of elements of a form
±ω(t ,...,t ,ω(t ,...,t ),t ,...,t ).
i1 is is+1 is+n is+n+1 2n−1
Since A is total associative, this sum is reduced to a sum of elements of a form
±ω(t ,...,t ,ω(t ,...,t )).
j1 jn−1 jn j2n−1
Let µ ∈ K be the coefficient of X at ω(t ,...,t ,ω(t ,...,t )) Since
1 n−1 n 2n−1
X(t ,...,t ) isskewsymmetricbyallarguments t ,...,t toprove X =0
1 2n−1 1 2n−1
it is enough to establish that µ=0.
Note that the element Q := ω(t ,...,t ,ω(t ,...,t )) may enter with
1 n−1 n 2n−1
non-zero coefficient only in summands of X of a form
R :=[ω](t ,...,t ,[ω](t ,...,t )),
n−1 1 n−1 n 2n−1
R :=[ω](t ,...,t ,t ,[ω](t ,...,t )), ...,
n−2 1 n−2 2n−1 n−1 2n−2
R :=[ω](t ,...,t ,[ω](t ,...,t )).
0 n+1 2n−1 1 n
6 A.S.DZHUMADIL’DAEV
The element
R =[ω](t ,...,t ,t ,...t ,[ω](t ,··· ,t )), 0≤i≤n−1,
i 1 i n+i+1 2n−1 i+1 i+n
enter to X with coefficient that is equal to signature of the permutation
1 ··· i i+1 ··· n−1 n ··· 2n−1
γ = ∈S
i (cid:18) 1 ··· i n+i+1 ··· 2n−1 i+1 ··· i+n (cid:19) n−1,n
We have
signγ =(−1)(n−i−1)n.
i
Toobtaina component Q from R wehaveto permutethe part ω(t ,...,t )
i i+1 i+n
of R (n−i−1) times,
i
[ω](t ,...,t ,t ,...t ,[ω](t ,··· ,t )) ···
1 i n+i+1 2n−1 i+1 i+n
(−1)n−1−iω(t ,...,t ,ω(t ,...,t ),t ,...,t ).
1 i i+1 i+n i+n+1 2n−1
Therefore
n−1 n−1
µ= signγ (−1)n−i−1 = (−1)(n−1−i)(n+1).
i
Xi=0 Xi=0
Note that
0 if n is even
µ=
(cid:26) n if n is odd
Hence, X =0, if n even or charK =p>0, n is odd and n is divisable by p.
3. Proof of Theorem 1.2
Note that
F[2](t ,...,t )=
1 1 3n−2
σ∈Symn−1X,n−1,n,σ(n)=1
signσ[ω](t ,...,t ,[ω](t ,t ,...,t ,[ω](t ,...,t ))),
σ(1) σ(n−1) 1 σ(n+1) σ(2n−2) σ(2n−1) σ(3n−2)
[2]
F (t ,...,t )=
2 1 3n−2
σ∈Symn−2X,n,n,σ(n−1)=1
signσ[ω](t ,...,t ,[ω](t ,t ,...,t ),[ω](t ,...,t ))).
σ(1) σ(n−2) 1 σ(n) σ(2n−2) σ(2n−1) σ(3n−2)
For any permutation i i ...i ∈Sym set
1 2 3n−2 3n−2
e(i ...i ):=ω(t ,...,t ,ω(t ,...,t ,ω(t ,...,t )))
1 3n−2 i1 in−1 in i2n−2 i2n−1 i3n−2
and
[e](i ...i ):=[ω](t ,...,t ,[ω](t ,...,t ,[ω](t ,...,t ))).
1 3n−2 i1 in−1 in i2n−2 i2n−1 i3n−2
For any 1≤i≤3n−2 let
e =e(2,...,i,1,i+1,...,3n−2)
i
Theindex i correspondstheplacewhereis 1. Forexample, e =e(1,2,...,3n−2),e =
1 2
e(2,1,3,...,3n−2),e =e(2,...,3n−2,1).
3n−2
Since A is total associative, for any s = 1,2, the element F[2](t ,...,t )
s 1 3n−2
can be presented as a sum of elements e(i ,...,i ), where i ...i is a
1 3n−2 1 3n−2
permutation of the set [3n−2]={1,2,...,3n−2}.
Let µ beacoefficientof F[2](t ,...,t ) at e =e(2,...,i,1,i+1,...,3n−
s,i s 1 3n−2 i
2), where 1 ≤ i ≤ 3n − 2. Since F[2](t ,t ,...,t ) is skew-symmetric by
s 1 2 3n−2
[2]
all variables except t , the element F (t ,t ,...,t ) is uniquily defined by
1 s 1 2 3n−2
ASSSOCIATIVE ALGEBRAS UNDER MULTI-COMMUTATORS 7
coefficient µ , where 1 ≤ i ≤ 3n − 2. Then the condition that f[2] = 0 is
s,i −1
identity on [A] is equivalent to the following relations
(1) µ =µ , 1≤i≤3n−2.
1,i 2,i
We will establish the following common values for µ and µ .
1,i 2,i
Let for even n
(−1)(i+1)⌊i+1⌋ if i≤n
2

µi = (−1)i+1(n2 +2⌊n−2i−1⌋) if n+1≤i≤2n−2
(−1)i⌊3n−i⌋ if 2n−1≤i≤3n−2
and for odd n  2
(−1)(i+1) (2n−i−1)i if i≤n
2

µi = (−1)(i+1)(n(n2−1) +(i−n)(−2n+i+1)) if n+1≤i≤2n−2
Note that (−1)i (3n−i−21)(n−i) if 2n−1≤i≤3n−2
µ =µ , 1≤i≤3n−2.
i 3n−1−i
Let [i,j] = {s ∈ Z|i ≤ s ≤ j} be segment with endpoints i,j and [i,j) =
{s ∈ Z|i ≤ s < j}, (i,j] = {s ∈ Z|i < s ≤ j}, (i,j) = {s ∈ Z|i < s < j}
be semi-segments. Note that semi-segment [i,j) has endpoints i and j−1 and
similarly,endpointsof (i,j] is i+1 and j. Numberofelementsof(semi)-segment
is called length. For example, |[i,j]|=j−i and |[i,j)|=j−1−i, if j >i. Say
that [i ,j ]⊆[i,j] is subsegment if i≤i <j ≤j.
1 1 1 1
Lemma 3.1. Let µ be the coefficient at e of the element F[2](t ,...,t ).
1,i i 1 1 3n−2
Then
µ =µ ,
1,i i
for any 1≤i≤3n−2.
Proof. Consider in the segment P = [2,3n−2]= {2,...,3n−2} chain with
1
two subsegements
P ⊂P ⊂P , |P |=3n−3,|P |=2n−2,|P |=n
3 2 1 1 2 3
Denote endpoints of P ,P ,P as A ,B A ,B and A ,B . Then
1 2 3 1 1 2 2 3 3
P =[2,3n−2],P =[p+1,p+2n−2],P =[q,q+n−1]
1 2 3
for some 1 ≤ p < q ≤ 2n−1 and the points A ,A ,A ,B ,B ,B on R has
1 2 3 3 2 1
coordinates 2,p+1,q,q+n−1,p+2n−2,3n−2. Note that
(2) 1≤p≤n, p<q ≤2n−1, q ≤p+n−1
Then
P =[A ,A )∪P ∪(B ,B ],
1 1 2 2 2 1
P =[A ,A )∪P ∪(B ,B ].
2 2 3 3 3 2
8 A.S.DZHUMADIL’DAEV
Let us introduce the following subsets of increasing integers
X =[A ,A )∪(B ,B ],
1 1 2 2 1
X ={1}∪[A ,A )∪(B ,B ],
2 2 3 3 2
X =P =[A ,B ].
3 3 3 3
In the following picture parts of X ,X ,X are marked equally.
1 2 3
A A A B B B
1 2 3 3 2 1
r r r r r r
∼∼∼∼∼∼∼∼∼ ∼∼∼∼∼∼∼∼∼∼
2 p+1 q q+n−1 p+2n−2 3n−2
So, for any such chain P ⊃ P ⊃ P one corresponds a sequence of elements
1 2 3
X X X where in each part X elements are written in increasing order and X
1 2 3 i 2
begins by 1. In other words, any chain P ⊃P ⊃P defines in a unique way an
1 2 3
element [e](X X X ). More exactly,
1 2 3
[e](X X X )=
1 2 3
[ω](t ,...,t ,t ,...,t ,[ω](t ,t ,...,t ,t ...,t ,[ω](t ,...,t ))).
2 p p+2n−1 3n−2 1 p+1 q−1 q+n p+2n−2 q q+n−1
Signature of the permutation
X X X =2...pp+2n−1 ...3n−21p+1...q−1q+n...p+2n−2q...q+n−1
1 2 3
is equal to
(−1)(p−1)+(n−p)(2n−1)+(n−q+p−1)n.
So,
(3) sgnX X X =(−1)(p+1−q)n+1
1 2 3
For 1 ≤ i ≤ 3n−2 and 1 ≤ p ≤ n, 0 < q−p ≤ n−1, denote by µ(p,q), the
1,i
coefficient at e of the element
i
[ω](t ,...,t ,t ,...,t ,[ω](t ,t ,...,t ,t ...,t ,[ω](t ,...,t ))).
2 p p+2n−1 3n−2 1 p+1 q−1 q+n p+2n−2 q q+n−1
In case of p=1, by µ(p,q) we understand the coefficient at e of the element
1,i i
[ω](t ,...,t ,[ω](t ,t ,...,t ,t ...,t ,[ω](t ,...,t ))).
2n 3n−2 1 2 q−1 q+n p+2n−2 q q+n−1
In case of p=n, by µ(p,q) we mean the coefficient at e of the element
1,i i
[ω](t ,...,t ,[ω](t ,t ,...,t ,t ...,t ,[ω](t ,...,t ))).
2 n 1 n+1 q−1 q+n p+2n−2 q q+n−1
For any 1≤i≤3n−2 denote by µ(0,q) the coefficient at e of the element
1,i i
ω(t ,...,t ,ω(t ,t ,...,t ,ω(t ,...,t ))).
2n 3n−2 1 n+2 2n−1 2 n+1
(0,q)
Then µ =0, if i≤n or i≥2n−1.
1,i
Below we use the following notation Y Z that means that Z is obtained
from Y by using skew-symmetry property of [ω]
We have,
[ω](t ,...,t ,t ,...,t ,[ω](t ,t ,...,t ,t ...,t ,ω(t ,...,t )))
2 p p+2n−1 3n−2 1 p+1 q−1 q+n p+2n−2 q q+n−1
(−1)(p−q+n−1)[ω](t ,...,t ,t ,...,t ,[ω](t ,t ,...,t ,ω(t ,...,t ),t ...,t ))
2 p p+2n−1 3n−2 1 p+1 q−1 q q+n−1 q+n p+2n−2
ASSSOCIATIVE ALGEBRAS UNDER MULTI-COMMUTATORS 9
(−1)(q−1)[ω](t ,...,t ,[ω](t ,t ,...,t ,ω(t ,...,t ),t ...,t ),t ,...,t )
2 p 1 p+1 q−1 q q+n−1 q+n p+2n−2 p+2n−1 3n−2
(−1)(q−1)ω(t ,...,t ,[ω](t ,t ,...,t ,ω(t ,...,t ),t ...,t ),t ,...,t )
2 p 1 p+1 q−1 q q+n−1 q+n p+2n−2 p+2n−1 3n−2
Nowexpand [ω] in [ω](t ,t ,...,t ,ω(t ,...,t ),t ...,t ). Then
1 p+1 q−1 q q+n−1 q+n p+2n−2
t might be in i-th place only in the following cases
1
(4) ω(t ,t ,...,t ,ω(t ,...,t ),t ...,t ), i=p,
1 p+1 q−1 q q+n−1 q+n p+2n−2
(5)
(−1)i−pω(t ,...,t ,t ,...,t ,ω(t ,...,t ),t ...,t ), p+1≤i≤q−1,
p+1 i 1 q−1 q q+n−1 q+n p+2n−2
(6)
(−1)n−1−p+iω(t ,...,t ,ω(t ,...,t ),t ...,t ,t ,...,t ), q+n−1≤i≤p+2n−2
p+1 q−1 q q+n−1 q+n i 1 p+2n−2
Note that µ(p,q) =0, if i∈/ [A ,A )∪(B ,B ]. Therefore, by (4), (5) and (6),
1,i 2 3 3 2
(7)
0 if i<p or q ≤i≤q+n−1 or p+2n−2≤i≤3n−2

µ(1p,i,q) = (−1)i+1+p−q if p≤i≤q−1
(−1)n+p+i−q if q+n−1≤i≤p+2n−2
Note also1≤p≤n,p<q ≤p+n−1. Hence q ≤2n−1.
The element e , where 1≤i≤3n−2, may appear in expanding of
i
[ω](t ,...,t ,t ,...,t ,[ω](t ,t ,...,t ,t ...,t ,[ω](t ,...,t )))
2 p p+2n−1 3n−2 1 p+1 q−1 q+n p+2n−2 q q+n−1
with the coefficient
(8) µ = µ(p,q).
1,i 1,i
Xp,q
Let i≤n. Thenthe case q+n−1≤i, 1≤p<q is impossible. Therefore,by
(1), (3), (7),
i p+n−1 i p+n−1 i p+n−1
µ = µ(p,q) = (−1)(p+1−q)n+1(−1)p−q+i+1 = (−1)p−q+i+pn+qn+n.
1,i 1,i
Xp=1q=Xi+1 Xp=1q=Xi+1 Xp=1q=Xi+1
So, for even n,
i p+n−1 i p+n−1
i+1
µ = (−1)p−q+i =(−1)i (−1)p−q =(−1)i+1⌊ ⌋.
1,i
2
Xp=1q=Xi+1 Xp=1q=Xi+1
For odd n
i p+n−1 i p+n−1
(2n−i−1)i
µ = (−1)i+1 =(−1)i 1=(−1)i+1 .
1,i
2
Xp=1q=Xi+1 Xp=1q=Xi+1
10 A.S.DZHUMADIL’DAEV
Consider the case n+1≤i≤2n−2. By (1), (3), (7),
n p+n−1 n i−n
µ = (−1)(p+1−q)n+1(−1)i+1+p−q+ (−1)(p+1−q)n+1(−1)n+p+i−q
1,i
Xp=1q=Xi+1 Xp=0q=Xp+1
So, if n is even, then
n p+n−1 n i−n
µ = (−1)i+p−q− (−1)i−q+p =
1,i
Xp=1q=Xi+1 Xp=0q=Xp+1
n p+n−1 n i−n
(−1)i( (−1)p−q− (−1)q−p)=
Xp=1q=Xi+1 Xp=0q=Xp+1
n n−i−1
(−1)i+1( +2⌊ ⌋).
2 2
If n is odd, then
n p+n−1 n i−n
µ = (−1)i+1+ (−1)i−1 =
1,i
Xp=1q=Xi+1 Xp=0q=Xp+1
n(n−1)
(−1)i+1( +(i−n)(−2n+i+1)).
2
Consider the case 2n−1 ≤ i ≤ 3n−2. Then all cases except q+n−1 ≤ i ≤
p+2n−2 are not possible. Therefore, i−2n+2≤p<q ≤i−n+1, and
n i−n+1 n i−n+1
µ = µ(p,q) = (−1)(p+1−q)n+1(−1)n+p+i−q
1,i 1,i
p=iX−2n+2q=Xp+1 p=iX−2n+2q=Xp+1
Hence, for even n
n i−n+1 n i−n+1
3n−i
µ = (−1)i+p−q+1 =(−1)i+1 (−1)p−q =(−1)i⌊ ⌋.
1,i
2
p=iX−2n+2q=Xp+1 p=iX−2n+2q=Xp+1
and for odd n
n i−n+1 n i−n+1
(3n−i−1)(n−i)
µ = (−1)i+1 =(−1)i+1 1=(−1)i .
1,i
2
p=iX−2n+2q=Xp+1 p=iX−2n+2q=Xp+1
Lemma 3.1 is proved.
Lemma 3.2. Let µ be the coefficient at e of the element F[2](t ,...,t ).
2,i i 2 1 3n−2
Then
µ =µ ,
2,i i
for any 1≤i≤3n−2.
Proof. Consider in the segment P = [2,3n−2] = {2,...,3n−2} two non-
1
intersecting subsegements of length n−1 and n
P ⊃P , P ⊃P , |P |=3n−3,|P |=n−1,|P |=n.
1 2 1 3 1 2 3
Denote endpoints of P ,P ,P as A ,B A ,B and A ,B . Then
1 2 3 1 1 2 2 3 3
P =[2,3n−2], P =[p+1,p+n−1], P =[q,q+n−1]
1 2 3
for some 1≤p≤2n−1 and 2≤q ≤2n−1.