1. Introduction
One of the features of Hamiltonian operators is their connection with certain algebraic structures [
1,
2,
3,
4,
5,
6,
7]. In 1976, I.M. Gel’fand and L.A. Dikii [
1] introduced formal variational calculus and found some interesting Poisson structures when studying Hamiltonian systems related to some nonlinear partial differential equations such as the Korteweg–de Vries equations. A little later, I.M. Gel’fand and I.Ya. Dorfman [
2] found more connections between Hamiltonian operators and some algebraic structures. From 1983 to 1985, B.A. Dubrovin, A.A. Balanskii and S.P. Novikov [
3,
4,
5] studied similar Poisson structures from another point of view. One of the algebraic structures in [
2,
5], introduced in connection with Poisson brackets of hydrodynamic type, was called a Novikov algebra by J.M. Osborn [
8,
9,
10].
We recall some important results on the solvabilty and nilpotency of Novikov algebras. In 1987, E.I. Zelmanov [
11] proved that if
N is a finite dimensional right nilpotent Novikov algebra, then
is nilpotent. In 2001, V.T. Filippov [
12] proved that any left-nil Novikov algebra of bounded index over a field of characteristic zero is nilpotent. Recently, I. Shestakov and Z. Zhang [
13] showed that a Novikov algebra is solvable if and only if it is right nilpotent.
Symmetries of algebraic systems are called automorphisms. The most famous example of symmetry in algebra is related to the action of the symmetric group on a polynomial algebra in the variables . The algebra of invariants of this action is a polynomial algebra generated by elementary symmetric polynomials.
Let R be an algebra over a field F. For any automorphism of R the set of fixed elements is a subalgebra of R, and is called the subalgebra of invariants of . An automorphism is called regular if . For any group G of automorphisms of R the subalgebra of invariants is defined similarly.
In 1957, G. Higman [
14] published a classical result on Lie algebras which says that if a Lie algebra
L has a regular automorphism
of prime order
p, then
L is nilpotent. It was also shown that the index of nilpotency
of
L depends only on
p. An explicit estimation of the function
was found by A.I. Kostrikin and V.A. Kreknin [
15] in 1963. A little later, V.A. Kreknin proved [
16] that a finite dimensional Lie algebra with a regular automorphism of an arbitrary finite order is solvable. In 2005, N. Yu. Makarenko [
17] proved that if a Lie algebra
L admits an automorphism of a prime order
p with a finite-dimensional fixed subalgebra of dimension
t, then
L has a nilpotent ideal of finite codimension with the index of nilpotency bounded in terms of
p and the codimension bounded in terms of
t and
p.
In 1973, G. Bergman and I. Isaacs [
18] published a classical result on the actions of finite groups on associative algebras. Let
G be a finite group of automorphisms of an associative algebra
R and suppose that
R has no
-torsion. If the subalgebra of invariants
is nilpotent, then the Bergman–Isaacs Theorem [
18] states that
R is also nilpotent. Since then, a very large number of papers have been devoted to the study of automorphisms of associative rings. The central problem of these studies was to identify the properties of rings that can be transformed from the ring of invariants to the whole ring. In 1974, V. K. Kharchenko [
19] proved if
is a PI-ring, then
R is a PI-ring under the conditions of the Bergman–Isaacs Theorem.
The Bergman–Isaacs Theorem was partially generalized by W.S. Martindale and S. Montgomery [
20] in 1977 to the case of a finite group of
Jordan automorphisms, that is, a finite group of automorphisms of the adjoint Jordan algebra
.
An analog of Kharchenko’s result for Jordan algebras was proved by A. P. Semenov [
21] in 1991. In particular, A. P. Semenov proved that if
is a solvable algebra over a field of characteristic zero, then so is the Jordan algebra
J. His proof uses a deep result by E.I. Zel’manov [
22], which says that every Jordan nil-algebra of bounded index over a field of characteristic zero is solvable. If a Jordan algebra
J over a field of characteristic not equal to
admits an automorphism
of the second order with solvable
, then
J is solvable [
23].
In the case of alternative algebras, one cannot expect that nilpotency of the invariant subalgebra implies the nilpotency of the whole algebra. There is an example (see in [
24,
25]) of a solvable non-nilpotent alternative algebra with an automorphism of order two such that its subalgebra of invariants is nilpotent. A combination of Semenov’s result [
21] and Zhevlakov’s theorem [
26] gives for an alternative algebra
A over a field of characteristic zero that the solvability of the algebra of invariants
for a finite group
G implies the solvability of
A. It is also known [
27] that if
A is an alternative algebra over a field of characteristic not equal to 2 with an automorphism
of order two, then the solvability of the algebra of invariants
implies the solvability of
A. In [
28], M. Goncharov proved that an alternative
-graded algebra
over a field of characteristic not equal to
is solvable if
is solvable.
Notice that an algebra A over a field containing all nth roots of unity admits an automorphism of order n if and only if A admits a -grading.
In this paper, we study the conditions of solvability of graded Novikov algebras from the point of view of the Bergman–Isaacs Theorem. We prove that a -graded Novikov algebra over a field of characteristic not equal to 3 is solvable if is solvable. We also show that a -graded Novikov algebra over a field of characteristic not equal to 2 is solvable if is solvable. This implies that for every n of the form that every -graded Novikov algebra N over a field of characteristic not equal to is solvable if is solvable.
The paper is organized as follows. In
Section 2, we give some preliminary facts.
Section 3 is devoted to the study of
-graded Novikov algebras with solvable even part. In
Section 4, we construct some ideals of
-graded Novikov algebras. The solvability of
-graded Novikov algebras with solvable 0-component is proven in
Section 5.
2. Preliminary Calculations
An algebra
N over a field
F is called a
Novikov algebra if it satisfies the following identities:
where
is the associator of the elements
.
It follows from (
2) that every Novikov algebra satisfies the identity
Let A be an arbitrary algebra and be subsets of A. Denote by the linear span of all products and by the linear span of all commutators where . Furthermore, denote by the linear span of all associators where .
Lemma 1. Let N be a Novikov algebra, B be its subalgebra, and be B-subbimodules of N. Then, , i.e., is a B-bimodule. In particular, if are ideals of N, then is an ideal of N (see also in [13]). Proof. We have
by (
2). By (
1) we get
as
M and
L are
B-subbimodules of
N. □
Let
be the additive cyclic group of order
n. Let
be a
-graded Novikov algebra. Then, the 0-component
of
N is a subalgebra on
N and for convenience of notation we often denote this subalgebra by
.
Lemma 2. Let N be a -graded Novikov algebra and I be an ideal of A. Assume thatfor some . Then, Proof. Let
and
. We write
if
. By Lemma 1,
is an ideal of
A. Then,
It remains to consider the product
. We have
Notice that in the last calculation we used twice that
, proven above. □
The derived powers of an arbitrary algebra A are defined by induction on s as follows: and for any positive integer . If A is a Novikov algebra, then is an ideal of A for all by Lemma 1. An algebra A is called solvable if for some s. If s is the minimal number such that , then s is called the length of solvability of A.
Corollary 1. Let N be a -graded Novikov algebra. Then,for any non-negative integer s and for any . Proof. The statement of the corollary is obviously true for
. Assume that
and
As
and
is an ideal of
A, it follows from Lemma 2 that
□
3. -Graded Novikov Algebra with Solvable Even Part
If N is a -graded Novikov algebra, then we write by setting and . Notice that the study of -graded algebras is more popular as it is related to the study of superalgebras. In this case, A is called the even part and M is called the odd part of N.
In this section, we prove that every Novikov algebra over a field of characteristic with solvable even part is solvable. First, formulate one important corollary of Lemma 2.
Corollary 2. Let be a -graded Novikov algebra. Then,is a -graded ideal of N. Proof. We write
if
. Let
and
. First, we prove that
J is a right ideal of
N. We have
as
. By Lemma 2,
.
Now we prove that
J is a left ideal. We have
By Lemma 2
.
Thus, J is a -graded ideal of N. □
Proposition 1. Let F be a field of characteristic and let be a -graded Novikov algebra. Suppose that . Then, for some positive integer n.
Several lemmas precede the proof of this proposition. These lemmas are formulated under the conditions of Proposition 1.
Lemma 3. Let . Then, I is a -graded ideal of algebra .
Proof. It is clear that
. Then
. As
, using (
2) we get
Thus,
Therefore,
I is a right ideal of
.
Now, we prove that
I is a left ideal of
. By (
1) we have
Hence
. From here we obtain that
Therefore,
Thus, I is a -graded ideal of the algebra . □
Lemma 4. We can assume that .
Proof. Let
. As
, applying Lemma 3 we get
Therefore, the algebra
N is solvable if the algebra
is solvable. By (
2), we have
Considering
I instead of
N, we can assume that
. □
Lemma 5. We can assume that for all .
Proof. Let
. First we show that
. Let
. By lemma 4, we obtain
since
. Therefore,
for all
. By Lemma 4, we have
, since
.
Therefore, the vector space spanned by all commutators , where , lies in the annihilator of the algebra N. Therefore, the algebra N is solvable, if the quotient algebra of N is a solvable algebra. Changing N to , we can assume that for all . □
The Proof of Proposition 1. By Lemma 4, we can assume that . Let . Then, . Therefore, it is sufficient to prove that the algebra I is solvable.
Let
,
. By (
2), we have
. Using (
1) and
we obtain that
Therefore,
. By Lemma 5,
for all
. Using this we get
over a field of characteristic
.
Consequently,
Thus,
.
Thus, the algebra N is solvable. □
Theorem 1. Every -graded Novikov algebra with solvable even part over a field of characteristic is solvable.
Proof. Let be a -graded Novikov algebra with solvable even part A. Let n be the length of solvability of A. We prove the statement of the theorem by induction on n. If , then N is solvable by Proposition 1.
By Corollary 2, is a -graded ideal of N. Notice that the even part of J is solvable with solvability length . By the induction proposition, J is solvable, that is, for some positive integer s. Moreover, the even part of the quotient algebra has trivial multiplication. Consequently, is solvable by Proposition 1, that is, for some positive integer t. Then, . □
Recall that the powers of an arbitrary algebra A are defined inductively by and for all integers . An algebra A is called nilpotent if for some positive integer n. Obviously, every nilpotent algebra is solvable. The converse is not true in the case of Novikov algebras.
Example 1. [13] Let be a vector space of dimension 2. The product on N is defined asIt is easy to check that N is a solvable Novikov algebra but not nilpotent as . Moreover, N is a -graded Novikov algebra with and . The even part of N is nilpotent. This means that in the formulation of Theorem 1, solvability cannot be replaced by nilpotency.
5. -Graded Novikov Algebras with Solvable 0-Component
In this section, we show that -graded Novikov algebras over a field of characteristic with solvable 0-component are solvable. We start with the case when the length of solvability of the 0-component is 1.
Proposition 2. Let F be a field of characteristic and let be a -graded Novikov algebra. Suppose that . Then N is a solvable algebra.
We give several lemmas prior to the proof of this proposition. These lemmas are formulated under the conditions of Proposition 2. The 0-component of N is usually denoted by .
First formulate a direct corollary of Lemma 6.
Corollary 3. The vector space is a -graded ideal of . Moreover, .
For any elements define .
Lemma 8. Let and . Then, . Moreover, we can assume that for any and
Proof. Let
and
. Then,
since
. Similarly,
Consequently, for all .
By Corollary 3, I is an ideal of and . Consequently, the algebra N is solvable if and only if I is solvable. Replacing N by I, we may assume that for all and . □
Lemma 9. The following equalities hold in N:where . Proof. Let
and
. By Lemma 8,
and
. Then, we obtain
Therefore,
. Thus,
Similarly,
Now we will show that
. By (
2),
As
and
by Lemma 8, it follows that
Notice that
From this, using (
1) and (
2), we get
since
. As
, applying (
2) we obtain
Similarly,
and
. □
Lemma 10. We can assume that .
Proof. First we prove that is a -graded ideal of the algebra N and .
We have
by Lemma 1. Therefore,
. Since
it follows that
. By Lemma 8 and (
2), we get
Therefore,
, since
.
Applying (
1) and (
2), we see that
Similarly, one can prove that
Therefore,
.
Consequently, K is a -graded ideal of the algebra N.
Let , where are the images of in the quotient algebra , respectively. Then, Therefore, Thus, and N is a solvable if and only if K is solvable.
Let . We show that is an ideal of K.
We have
by (
2). By Lemma 1, we get
. Therefore,
, i.e.,
is a subalgebra of
K. Using (
1) and (
2), we also have
Therefore,
is a ideal of
K. Moreover,
. Thus, the algebra
N is solvable if and only if
is solvable. Therefore, replacing
N by
, we can assume that
since
. In this case we have
Thus, we can assume that in N. □
Lemma 11. We can assume that .
Proof. First we prove that the vector space is a -graded ideal of N and .
We have by Lemma 8. Then, Lemma 9 and Lemma 1 give that .
We prove that
By Lemma 10, we have
. As
, using (
2) and Lemma 9 we get
Therefore,
.
Applying (
2), Lemma 9, and Lemma 8, we obtain that
Consequently,
and
.
We prove that
By Lemma 10, Lemma 8, and (
2), we get
Since
and
by Lemma 9, using (
2) we obtain
Therefore,
K is an ideal of the algebra
N.
Applying Lemma 9, (
1), and Lemma 10, we also get
Therefore,
As
it follows that
As
, using Lemma 9 we get
. Therefore, by Lemmas 8 and 9, we also get
Then
Similarly, is a ideal of the algebra N and . Therefore, is a solvable ideal of the algebra N. From the solvability of the quotient algebra follows the solvability of N. We have in the quotient algebra , where is the image of in , .
Therefore, we can assume that in the algebra N. □
The Proof of the Proposition 2. Let
,
. Then,
Thus,
. Therefore,
. Then,
by Lemma 11. Similarly,
and
Consequently, and . Moreover, Similarly, .
Let . By Lemma 1, and . By Corollary 3 and Lemma 8, we get . Therefore, the algebra N is solvable. □
Theorem 2. Let F be a field of characteristic and let be a -graded Novikov algebra. Suppose that is a solvable algebra. Then, N is a solvable algebra.
Proof. Let be a solvable algebra with solvability length . If , then N is solvable by Proposition 2. Suppose that , that is . Let I be the ideal of from Lemma 6. Recall that . Therefore, it is sufficient to prove that I is a solvable ideal of N.
Let K be the ideal of I from Lemma 7. As , the quotient algebra is again solvable by Proposition 2. Therefore, for some positive integers s.
Notice that the 0-component of K is and has the solvability length . Leading an induction on n we may assume that K is solvable. Consequently, I and N are both solvable. □
Corollary 4. Let n be a positive integer of the form for some non-negative integers . Let N be a -graded Novikov algebra over a field of characteristic . If is solvable, then N is solvable.
This is a standard corollary of Theorems 1 and 2 (see, for example, [
28]).
The right powers of an arbitrary algebra
A are defined inductively by
and
for all integers
. An algebra
A is called
right nilpotent if
for some positive integer
n. I. Shestakov and Z. Zhang recently proved [
13] that every solvable Novikov algebra is right nilpotent.
Corollary 5. Let n be a positive integer of the form for some non-negative integers . Let N be a -graded Novikov algebra over a field of characteristic . If is right nilpotent, then N is right nilpotent.