Journal of Lie Theory Volume 12 (2002) 369–395 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

Polynomial Identities in Smash Products

Yuri Bahturin^{∗} and Victor Petrogradsky^{†}

Communicated by E. Zelmanov

Abstract. Suppose that a group G acts by automorphisms on a (restricted) Lie algebra L over a field K of positive characteristic. This gives rise to smash products U(L)#K[G] and u(L)#K[G]. We find necessary and sufficient conditions for these smash products to satisfy a nontrivial polynomial identity.

1. Introduction: polynomial identities in group algebras and enveloping algebras

The first observation on the polynomial identities in enveloping algebras was made by V. N. Latyshev [9]. He proved that the universal enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero satisfies a nontrivial identical relation if and only if this Lie algebra is abelian. Later Yu. Bahturin has noticed in [1] that the condition of finite dimensionality is inessential.

D. S. Passman has obtained the complete description of group algebras satisfying polynomial identities.

Theorem 1.1. ([11]) The group algebra K[G] of a group G satisfies a nontrivial polynomial identity if and only if the following conditions are satisfied:

1. there exists a normal subgroup A ⊂ G of finite index;

2. A is abelian if char K = 0, and the commutator subgroup A′ is a finite abelian p-group if char K = p > 0.

Yu. Bahturin has settled the problem of the existence of nontrivial identities for the universal enveloping algebra fields of positive characteristic.

Theorem 1.2. ([2]) Let L be a Lie algebra over a field K of positive characteristic. Then the universal enveloping algebra U(L) satisfies a nontrivial polynomial identity if and only if the following conditions are satisfied:

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag 1. there exists an abelian ideal H ⊂ L of finite codimension;

∗ The authors acknowledge partial support by Dean of Science Start-up Funds, Memorial University of Newfoundland

† The second author is partially supported by RFBR grants # 98-01-01020, # 99-01-00233; Ministry of Education of Russia grant # 98-7

2. all inner derivations ad x, x ∈ L are algebraic of bounded degree.

Petrogradsky [15] and Passman [14] have specified restricted Lie algebras (also called Lie p-algebras) L such that the restricted enveloping algebra u(L) satisfies a nontrivial polynomial identity.

Theorem 1.3. ([14], [15]) Let L be a Lie p-algebra. Then the restricted enveloping algebra u(L) satisfies a nontrivial polynomial identity if and only if there exist restricted ideals Q ⊂ H ⊂ L such that:

1. dim L/H < ∞, dim Q < ∞;

2. H/Q is abelian;

3. Q is abelian with nilpotent p-mapping.

See also further developments for Lie p-superalgebras [16] and color Lie p-superalgebras [3].

The main body of this paper consists of the proof of our main result which completely describes smash products u(L)#K[G] that are PI rings (Theorem 3.1). We start with establishing some identical relations that nicely suit our purposes (Section 4). As an important ingredient we begin developing a delta-theory for smash products (Section 5). Next we describe the structure of delta-sets in our case (Section 6).

But the first results in this paper (Section 2) deal with necessary and sufficient conditions under which the smash product U(L)#K[G] satisfies a nontrivial identity (Theorem 2.3). Actually, this result, as well as Theorem 2.1 could be derived from a result on general smash products by Handelman - Lawrence - Schelter (see [7]) and probably by Passman [13]. But we prefer to keep our proofs here since they are relevant to the techniques of delta-sets used in the next sections for the proof of the main result (Theorem 3.1).

2. Polynomial identities in smash products U(L)#K[G]

We denote the ground field by K. Suppose that a group G acts on an associative algebra A by automorphisms: φ : G → Aut(A), φ(g) : x ↦ φ(g)(x), g ∈ G, x ∈ A. We set g ∗ x = φ(g)(x). Now one can form the smash product R = A#K[G]. This is a vector space R = A ⊗_{K} K[G] endowed with multiplication

(a_{1}, g_{1}) · (a_{2}, g_{2}) = (a_{1}(g_{1} ∗ a_{2}), g_{1}g_{2}), a_{1}, a_{2} ∈ A, g_{1}, g_{2} ∈ G.

By linearity also the group ring K[G] acts on A:

(α_{1}g_{1} + ⋯ + α_{m}g_{m}) ∗ a = α_{1}(g_{1} ∗ a) + ⋯ + α_{m}(g_{m} ∗ a), g_{i} ∈ G, α_{i} ∈ K, a ∈ A.

Now suppose that G acts on a Lie algebra L by automorphisms. Then this action is naturally extended to the action on the universal enveloping algebra U(L) and we can form the smash product U(L)#K[G]. Such algebras are important because each cocommutative Hopf algebra over an algebraically closed field of characteristic zero can be presented as a smash product U(L)#K[G] (Kostant, Cartier, et al., see [10]).

The conditions for the existence of nontrivial identities for the smash products U(L)#K[G] can be derived from [7]. Our next formulation is from [8].

Theorem 2.1. Let G be a group, L a Lie algebra over a field K of characteristic 0, and G acts on L by automorphisms. Then U(L)#K[G] satisfies a nontrivial polynomial identity if and only if the following conditions are satisfied:

1. L is abelian;

2. there exists an abelian normal subgroup A ⊂ G of finite index;

3. A acts trivially on L.

The proof in [8] as well as the original proof of the theorem about the identical relations in U(L) [2] (see also [1], [3]) is based on the following classical result.

Theorem 2.2. (Posner, [5]) Let R be a prime algebra with unit over a field satisfying some nontrivial polynomial identity. Let C be the center of R and Q the field of quotients of C. Then the algebra R^{Q} = Q ⊗_{C} R of central quotients of R is finite-dimensional central simple over Q and coincides with the left and the right classical rings of quotients of R. Moreover, R and R^{Q} satisfy the same identities.

The goal of this section is to prove a result similar to the one just formulated in the case of the fields of positive characteristic. Again we mention a possibility of deriving this result from [7].

Theorem 2.3. Let G be a group, L a Lie algebra over a field K of characteristic p > 0 and G act on L by automorphisms. Then U(L)#K[G] satisfies a nontrivial polynomial identity if and only if the following conditions are satisfied:

1. there exists an abelian G-invariant ideal H ⊂ L of finite codimension and all derivatives ad x, x ∈ L are algebraic of bounded degree;

2. there exists a normal subgroup A ⊂ G of finite index with the commutator subgroup A′ being a finite abelian p-group.

3. A acts trivially on L.

Let us comment on this result. We observe that K[G] and U(L) are the subrings of U(L)#K[G] and thus Theorems 1.1 and 1.2 apply. This gives us the structure for G and L described in the first two claims except for the fact that H is G-invariant. So, the most essential here is the third claim.

We start with recalling the notion of delta-sets. They provide us with the key instrument to the study of identities of enveloping algebras. One defines the sets of elements of “finite width”

δ_{n}(L) = {x ∈ L | dim[x, L] ≤ n}, n ∈ ℕ;
δ(L) = ∪_{n=1}^{∞} δ_{n}(L).

These sets have appeared in [2] as Lie algebra analogues of delta-sets for the groups. Those delta-sets were crucial in the study identical relations for group rings [11], [12]. Namely, if G is a group then one can define the sets of elements having finitely many conjugates as follows

δ_{n}(G) = {g ∈ G | |g^{G}| ≤ n}, n ∈ ℕ;
δ(G) = ∪_{n=1}^{∞} δ_{n}(G).

Lemma 2.4. ([3]) Let L be a Lie algebra. Then the delta-sets have the following properties.

1. if x ∈ δ_{i}(L), y ∈ δ_{j}(L) then αx + βy ∈ δ_{i+j}(L), α, β ∈ K;

2. if x ∈ δ_{i}(L), y ∈ L then [x, y] ∈ δ_{2i}(L);

3. let x ∈ δ_{i}(L) and suppose that L is a restricted Lie algebra. Then x^{[p]} ∈ δ_{i}(L);

4. δ_{i}(L) is invariant under the automorphisms of L, i ∈ ℕ;

5. δ(L) is a (restricted) invariant ideal of L.

Recall that a subalgebra is called invariant if it is stable under all automorphisms and restricted if closed under the p-map.

We use this lemma to prove Theorem 2.3

Proof. First, let us check that our conditions are sufficient. Let f_{1}, …, f_{k} form a basis of L modulo H. By the hypothesis, each ad f_{i} annihilates some nonzero polynomial q_{i}(t), i = 1, …, k. Recall that a polynomial of the form q(t) = ∑_{i=0}^{s} α_{i}t^{p^{i}} is called a p-polynomial [6]. Such polynomials have the following property. Let x be an element of an associative algebra over the field of characteristic p, viewed as a Lie algebra under the bracket operation [a, b] = ab − ba. Then

ad(q(x)) = ad(∑_{i=0}^{s} α_{i}x^{p^{i}}) = ∑_{i=0}^{s} α_{i}(ad x)^{p^{i}} = q(ad x).  (1)

Any polynomial is a divisor of some nonzero p-polynomial [6]. So, we may assume that q_{i}(t) are some p-polynomials. By (1) z_{i} = q_{i}(f_{i}), i = 1, …, k are central elements in U(L). Let d_{i}, i = 1, …, k, be the degrees of polynomials q_{i}. We denote by B the ring generated by U(H) along with z_{1}, …, z_{k}. Then _{B}U(L) is a free B -module with a finite basis {f_{1}^{i_{1}} ⋯ f_{k}^{i_{k}}|0 ≤ i_{j} < d_{j}, 1 ≤ j ≤ k} [6]. Let g_{1}, …, g_{s} be the right coset representatives of A in G. Since A acts trivially on the whole of L we obtain that B̃ = B ⊗ K[A] is a commutative subring of R = U(L)#K[G]. Also, B̃R is a free module with a basis {f_{1}^{α_{1}} ⋯ f_{k}^{α_{k}}g_{j}|0 ≤ α_{i} < d_{i}, 1 ≤ j ≤ s} and rank t = d_{1} ⋯ d_{k}s.

We identify any x ∈ R with an operator of the right multiplication: R → R, a ↦ ax, a ∈ R. This yields an embedding of R into a matrix ring over the commutative ring B̃:

R ⊂ EndB̃ R ≅ M_{t}(B̃) ≅ M_{t}(K) ⊗_{K} B̃.

By Regev’s Theorem about tensor products of PI-rings [17] we conclude that R = U(L)#K[G] is a PI-ring.

Now suppose that U(L)#K[G] satisfies a nontrivial polynomial identity. First, let us prove that there exists a G-invariant abelian ideal H ⊂ L of

finite codimension. We need to recall the steps of the proof of Theorem 1.2 in [2] (see also this construction in [15], [16], and [3]).

1) The existence of a nontrivial identity in U(L) implies that for some number m we have δ(L) = δ_{m}(L) and dim L/δ(L) < ∞. We set D = δ(L).

2) We apply P.M.Neumann’s Theorem on bilinear maps (see Theorem 6.7 below) and conclude that the commutator subalgebra D^{2} = [D, D] is finitedimensional. We set C = C_{D}(D^{2}) = {x ∈ D|[x, D^{2}] = 0}. Then dim D/C < ∞, and C^{3} = 0.

3) We again use the identity in the enveloping algebra for C and prove that dim C/H < ∞, where H = Z(C) is the center of C.

One can trace these steps and see by Lemma 2.4 that all these subalgebras are invariant ideals. Hence, we obtain the G-invariant abelian ideal H ⊂ L of finite codimension.

Next, we apply Theorem 1.1 and obtain a subgroup of finite index B ⊂ G such that the commutator subgroup B′ is a finite abelian p-group. Now our task is reduced to the following. We consider the smash product U(L)#K[B] and find a subgroup of finite index A ⊂ B acting trivially on the whole of L. Let {U_{n}(L)|n = 0, 1, 2 … } be the standard filtration on the universal enveloping algebra. Then it induces a filtration on the smash product, the associated graded algebra is also PI and

gr{U_{n}(L)#K[B]|n = 0, 1, 2, … } ≅ (gr{U_{n}(L)|n = 0, 1, 2, … })#K[B].

But gr{U_{n}(L)|n = 0, 1, 2, … } ≅ U(L̄), where L̄ is an abelian Lie algebra with the same vector space L. So, we may assume that L is abelian so that U(L) is a polynomial ring.

Let g ∈ B be an element of infinite order. We claim that there exists m > 0 such that g^{m} acts trivially on L. Let ⟨g⟩ be the cyclic subgroup generated by g. By way of contradiction suppose that ⟨g⟩ acts faithfully on L. One easily verifies that the ring R = U(L)#⟨g⟩ has no zero divisors, hence is prime. Let C be the center of R and consider some central element c = ∑_{i} c_{i}g^{i} ∈ C, c_{i} ∈ U(L). Suppose that there exists c_{j} ≠ 0, j ≠ 0. Since ⟨g⟩ acts faithfully on L we find a ∈ L with g^{j} ∗ a ≠ a. Remark that (g^{j} ∗ a − a)c_{j} ≠ 0 because U(L) has no zero divisors. Then

[c, a] = ∑_{i} c_{i}g^{i}a − a ∑_{i} c_{i}g^{i} = ∑_{i}(g^{i} ∗ a − a)c_{i}g^{i} ≠ 0.  (2)

This is a contradiction with the fact that c is central. Therefore c_{i} = 0 for i ≠ 0, so C ⊂ U(L). Let Q be the field of fractions for C. Then the elements g^{i}, i ∈ ℕ are linearly independent over Q in the ring of fractions for R. This contradicts Posner’s theorem. Hence for any g ∈ B there exists m such that g^{m} acts trivially on L.

Let us consider an arbitrary finitely generated subgroup W ⊂ B. Suppose that the action of W on L is faithful. Then our argument implies that the generating elements of W are of finite order. In view of the structure of B we conclude that W is finite. Now we consider the ring R_{0} = U(L)#K[W] and apply one result on the smash products of type A#H, where H is a finite dimensional Hopf algebra [10, p. 55]. Namely, A#H is a prime ring if and only if A is a faithful left and right A#H -module and the invariant subring A^{H} is prime. Of course, in our case U(L)^{K[W]} is prime. Let us check the conditions of faithful action. If H = K[W] then the left and right actions are defined by

(a#g) · b = a(g ∗ b),  a, b ∈ A, g ∈ W;
a · (b#g) = g^{−1} ∗ (ab),  a, b ∈ A, g ∈ W.

Suppose that a nonzero element

a_{1}g_{1} + ⋯ + a_{m}g_{m},  0 ≠ a_{i} ∈ U(L), g_{i} ∈ W,  (3)

acts trivially on the left on U(L). Let m be taken minimal among nonzero elements (3) that act trivially on U(L). Then

a_{1}(g_{1} ∗ x) + ⋯ + a_{m}(g_{m} ∗ x) = 0, x ∈ U(L).  (4)

We replace x by xy and multiply (4) by g_{1} ∗ y on the right. Here we also use the commutativity of U(L).

a_{1}(g_{1} ∗ x)(g_{1} ∗ y) + ⋯ + a_{m}(g_{m} ∗ x)(g_{m} ∗ y) = 0, x, y ∈ U(L);
a_{1}(g_{1} ∗ x)(g_{1} ∗ y) + ⋯ + a_{m}(g_{m} ∗ x)(g_{1} ∗ y) = 0, x, y ∈ U(L);

a_{2}((g_{2} − g_{1}) ∗ y)(g_{2} ∗ x) + ⋯ + a_{m}((g_{m} − g_{1}) ∗ y)(g_{m} ∗ x) = 0, x, y ∈ U(L).

Since U(L) has no zero divisors and by the choice of m in (3) we conclude that (g_{2} − g_{1}) ∗ y = 0 for all y ∈ U(L), contradicting to the fact that W acts faithfully on L. We can check that the right action is faithful in the same way.

Now we can apply Posner’s theorem. The same computation (2) shows that the center C of R is contained in U(L).

Let Q be the field of fractions for C. Again all elements g ∈ W are linearly independent over Q in the ring of quotients R_{Q} for R. Let d be the degree of a nontrivial polynomial identity satisfied by the smash product U(L)#K[G]. By Posner’s theorem |W | ≤ dim_{Q} R_{Q} ≤ [d/2].

Let now W be an arbitrary finitely generated subgroup of B. We set St_{W} L = {w ∈ W |w ∗ x = x, x ∈ L}. Then by the above arguments |W : St_{W} L| ≤ s = [d/2]. Consider A_{1} = St_{B} L = {b ∈ B|b ∗ x = x, x ∈ L}. We claim that |B : A_{1}| ≤ s. By way of contradiction suppose that |B : A_{1}| > s, then we can take elements g_{1}, …, g_{s+1} ∈ B lying in different left classes modulo A_{1}. We consider the subgroup W generated by g_{1}, …, g_{s+1}. Then these elements belong to different cosets of St_{W} L ⊂ A_{1}, proving |W : St_{W} L| > s, a contradiction with the above. Thus we should have |B : A_{1}| ≤ s and by construction A_{1} acts trivially on L.

To finish the proof it is enough to choose a normal subgroup of finite index A ⊂ A_{1}.

3. Polynomial identities in the smash products u(L)#K[G]

The main goal of this paper is to prove the following result.

Theorem 3.1. Suppose that a group G acts by automorphisms on a Lie palgebra L. Then u(L)#K[G] is a PI-algebra if and only if

1. there exist G-invariant restricted subalgebras Q ⊂ H ⊂ L with

(a) dim L/H < ∞, dim Q < ∞;

(b) [H, H] ⊂ Q;

(c) Q is abelian with a nilpotent p-mapping.

2. there exists a subgroup A ⊂ G with

(a) |G : A| < ∞;

(b) the commutator subgroup A′ is a finite abelian p-group;

3. A acts trivially on H/Q.

We remark that K[G] and u(L) are the subrings of u(L)#K[G] and we can apply Theorems 1.1 and 1.3. This gives us the structure of G and L described in the first two claims except for the fact that H, Q are G-invariant. But the most difficult here is the third claim about the action of G on L.

While studying polynomial identities for u(L), a crucial example is the infinite-dimensional Heisenberg algebra. By δ_{ij} we mean the Kronecker symbol.

Example 3.2. ([15, 16, 3]) We consider the infinite-dimensional Heisenberg Lie algebra

L = ⟨x_{1}, x_{2}, …, y_{1}, y_{2}, …, z|[x_{i}, y_{j}] = δ_{ij}z, [x_{i}, z] = [y_{j}, z] = 0, i, j ∈ ℕ⟩_{K}.

Then the existence of a nontrivial identity for u(L) depends on the value of the p-map on the central element z

1. if z^{[p]} = 0, then u(L) satisfies a nontrivial identity (XY − Y X)^{p} ≡ 0;

2. if z^{[p]} = z, then u(L) does not satisfy any nontrivial identity.

Let us illustrate our main result by examples. These examples are similar to the Heisenberg algebra, we only need to remember that G acts by automorphisms on L.

Example 3.3. Let L = ⟨y, x_{j}, x_{j}^{[p]}, …, x_{j}^{[p^{k}]}, … |j = 1, 2, …; y^{[p]} = 0⟩_{K} be an abelian restricted Lie algebra and the group G = (ℤ_{p})ℕ acts on L as

g_{i} ∗ x_{j} = x_{j} + δ_{ij}y; g_{i} ∗ x_{j}^{[p^{k}]} = x_{j}^{[p^{k}]}, k ≥ 1; g_{i} ∗ y = y;

where g_{i} = (0, …, 0, 1, 0, …), with 1 on i -th place, i ∈ ℕ. We consider the smash product R = u(L)#K[G]. Then

1. R satisfies the identity (XY − Y X)^{p} ≡ 0;

2. G acts faithfully on L.

Proof. We have [g, x_{j}] = gx_{j} − x_{j}g = (g ∗ x_{j} − x_{j})g = λyg, λ ∈ K, g ∈ G. Remark that all other pairs of generating elements for R commute. Therefore, any commutator [a, b], a, b ∈ R contains as a factor the central element y. Recall that y^{p} = 0, therefore R satisfies the claimed identity.

Let us check the second claim. For g = (n_{1}, n_{2}, …, n_{i}, …) ∈ G one has g ∗ x_{j} = x_{j} + n_{j}y, so that G acts faithfully on L.

This example fits into the wording of the theorem by setting A = G, H = L, Q = ⟨y⟩_{K}. This example shows that the action of A on Q may be nontrivial. Moreover, one can check that we cannot avoid this by taking a somewhat smaller subgroup A_{1} ⊂ A of finite index and a subalgebra H_{1} ⊂ H of finite codimension.

Let us change the p-mapping on y in the previous example.

Example 3.4. Let L = ⟨y, x_{j}, x_{j}^{[p]}, …, x_{j}^{[p^{k}]}, … |j = 1, 2, …, y^{[p]} = y⟩_{K} be an abelian restricted Lie algebra and the group G = (ℤ_{p})ℕ act on L by

g_{i} ∗ x_{j}^{[p^{k}]} = x_{j}^{[p^{k}]} + δ_{ij}y, k ≥ 0; g_{i} ∗ y = y.

Then R = u(L)#K[G] does not satisfy any nontrivial identity.

Proof. If R is PI then it must satisfy the identity given below in Lemma 4.2 and so we have

F_{2}(x_{1}, …, x_{n}, g_{1}, …, g_{n}) = ∑_{π∈S_{n}} α_{π}(g_{1} − 1) ∗ x_{π(1)} ⋯ (g_{n} − 1) ∗ x_{π(n)} = y^{n} ≠ 0,

because only the summand for the identity permutation is nontrivial. This contradiction proves that R is not PI.

The same argument applies also for the following example.

Example 3.5. Let L = ⟨y, e_{1}, e_{2}, … |e_{j}^{[p]} = e_{j}, j ∈ ℕ; y^{[p]} = y⟩_{K} be an abelian restricted Lie algebra and the group G = (ℤ_{p})ℕ act on L by

g_{i} ∗ e_{j} = e_{j} + δ_{ij}y, g_{i} ∗ y = y.

Then R = u(L)#K[G] does not satisfy any identity.

By ωK[G] we denote the augmentation ideal of the group ring ωK[G] = {∑_{i} α_{i}g_{i}| ∑_{i} α_{i} = 0; α_{i} ∈ K, g_{i} ∈ G}. If L is a Lie p-algebra then by ωu(L) we denote also the augmentation ideal of the restricted enveloping algebra ωu(L) = u(L)L = Lu(L).

Next we prove the sufficiency in Theorem 3.1.

Proof. We set R = u(L)#K[G], R_{1} = u(H)#K[A], R_{0} = u(Q)#K[A′].

Let I be the subring of R_{0} generated by Q and {h − 1|h ∈ A′}; this is an ideal of codimension 1 in R_{0}. First, let us prove that I is nilpotent. We have (ωu(Q))^{q} = 0 for some q since Q is abelian finite dimensional with a nilpotent p-mapping. Also (ωK[A′])^{t} = 0 for some number t because A′ is an abelian finite p-group. Thus Q = Q_{0} ⊃ Q_{1} ⊃ ⋯ ⊃ Q_{t} = 0, where Q_{i} = (ωK[A′])^{i}Q. Now let us look at the commutators of the nilpotent elements of ωu(Q) and ωK[A′].

(h−1)z = hzh^{−1}h−z = (h∗z)h−z = ((h−1)∗z+z)h−z = z(h−1)+((h−1)∗z)h,

where h ∈ A′ and z ∈ Q_{i}, in this case (h − 1) ∗ z ∈ Q_{i+1}. This relation yields that these two commutative nilpotent subrings generate a nilpotent subring. Indeed, consider a product consisting of z_{i} ∈ Q and (h_{j} − 1) where h_{j} ∈ A′. By the above relation the number of z’s is bounded by s_{1} = q − 1. Also, the number of factors (h_{j} − 1) is bounded by s_{2} = (t − 1) + (t − 1)(q − 1). Hence, I^{s} = 0 for s = 1 + s_{1} + s_{2} = qt.

Second, we claim that R_{1} is a PI-algebra. We consider the left ideal J = R_{1}I = R_{1}(Q+ωK[A′]). The following commutator relations hold for arbitrary x ∈ H, z ∈ Q, g ∈ A, and h ∈ A′

zx = xz + [z, x],  [z, x] ∈ Q;
zg = gg^{−1}zg = g(g^{−1} ∗ z),  g^{−1} ∗ z ∈ Q;
(h − 1)g = g(g^{−1}hg − 1),  g^{−1}hg ∈ A′;
(h − 1)x = h(x − h^{−1}xh) + x(h − 1)
= h((1 − h^{−1}) ∗ x) + x(h − 1), (1 − h^{−1}) ∗ x ∈ Q,

the latter relation being true because A acts trivially on H/Q. These relations describe the commutators of all possible products of the form u · v where u, v are the generating elements of subrings I and R_{1}, respectively. It follows that IR_{1} ⊂ R_{1}I. By symmetry we have a two-sided ideal J = R_{1}I = IR_{1} ◁ R_{1} and J^{s} = (R_{1}I)^{s} ⊂ R_{1}I^{s} = 0. Next we use the fact that R_{1}/J ≅ u(H/Q)#K[A/A′]. By assumption of the theorem this is a commutative algebra. Therefore, R_{1} satisfies the identity (XY − Y X)^{s} ≡ 0.

Let f_{1}, …, f_{k} form a basis of L modulo H. Let g_{1}, …, g_{k_{0}} be the right coset representatives of A in G. Now _{R_{1}}R is a free R_{1}-module with the finite basis {f_{1}^{α_{1}} ⋯ f_{k}^{α_{k}}g_{j}|0 ≤ α_{t} < p, 1 ≤ j ≤ k_{0}}, the rank being r = p^{k}k_{0}. Indeed, suppose that we have a relation

∑_{αj} r_{αj}f_{1}^{α_{1}} ⋯ f_{k}^{α_{k}}g_{j} = 0, r_{αj} ∈ R_{1}.  (5)

Let A = {a_{λ}|λ ∈ Λ}, then

∑_{αj} (∑_{λ} u_{αjλ}a_{λ}) f_{1}^{α_{1}} ⋯ f_{k}^{α_{k}}g_{j} = 0, u_{αjλ} ∈ u(H);

∑_{jλ} ( ∑_{α} u_{αjλ}(a_{λ} ∗ f_{1})^{α_{1}} ⋯ (a_{λ} ∗ f_{k})^{α_{k}})a_{λ}g_{j} = 0;

∑_{α} u_{αjλ}(a_{λ} ∗ f_{1})^{α_{1}} ⋯ (a_{λ} ∗ f_{k})^{α_{k}} = 0.  (6)

Since H is G-invariant the elements {a_{λ}∗f_{1}, …, a_{λ}∗f_{k}} form a basis of L modulo H. We apply PBW-Theorem to (6) and obtain that u_{αjλ} = 0. Hence, (5) is trivial.

If we identify any x ∈ R with the right multiplication by x we obtain an embedding of R into a matrix ring over the PI-ring R_{1}:

R ⊂ End(_{R_{1}}R) ≅ M_{r}(R_{1}) ≅ M_{r}(K) ⊗_{K} R_{1}.

By Regev’s Theorem on the tensor product of PI-rings [17] we conclude that R = U(L)#K[G] is a PI-ring.

4. Useful identities

We start with constructing a special identity. It is similar to the identities used in the study of identical relations of restricted enveloping algebras [3].

Lemma 4.1. Suppose that R is a PI-algebra over an arbitrary field. Then it satisfies a nontrivial identity of the form

F (X_{1}, …, X_{n}, Y_{1}, …, Y_{n}) = ∑_{π∈S_{n}} α_{π}Y_{1}X_{π(1)} ⋯ Y_{n}X_{π(n)} ≡ 0, α_{π} ∈ K, α_{e} = 1.

Proof. Let A = A(X_{1}, …, X_{m}, …, Y_{1}, …, Y_{m}, …,) be the free associative algebra. For any permutation π ∈ S_{n} we define a monomial

f_{π} = Y_{1}X_{π(1)} ⋯ Y_{n}X_{π(n)} ∈ A.  (7)

We denote by P_{m}(Z_{1}, …, Z_{m}) the subspace of all multilinear polynomials in m variables Z_{1}, …, Z_{m} in the free associative algebra Ā = Ā(Z_{1}, …, Z_{m}, …) in a countable set of variables Z_{1}, …, Z_{m}, . . .. By P′_{m}(Z_{1}, …, Z_{m}) we denote the subspace of elements in P_{m}(Z_{1}, …, Z_{m}) that are the left hand sides of identities for R. Let R satisfy a nontrivial identity of degree d. The following estimate is well-known [1]

dim P_{m}(Z_{1}, …, Z_{m})/P′_{m}(Z_{1}, …, Z_{m}) < d^{2m}, m ∈ ℕ.  (8)

Let us apply this estimate to A. We consider the subspace P_{2n}(X_{1}, …, X_{n}, Y_{1}, …, Y_{n}) ⊂ A of multilinear polynomials of degree 2n depending on the variables X_{1}, …, X_{n}, Y_{1}, …, Y_{n}. This subspace contains n! monomials of the form (7) which are linearly independent. We apply (8)

dim P_{2n}(X_{1}, …, X_{n}, Y_{1}, …, Y_{n})/P′_{2n}(X_{1}, …, X_{n}, Y_{1}, …, Y_{n}) < d^{4n}, n ∈ ℕ.

If n! > d^{4n} then f_{π}, π ∈ S_{n} are linearly dependent modulo P′_{2n}(X_{1}, …, X_{n}, Y_{1}, …, Y_{n}), thus yielding the desired identity. Since n! > (n/e)^{n} > (n/3)^{n}, the number n = 3d^{4} is sufficiently large, and the result follows.

Next we construct some special weak identities for the smash products. A relation is called a weak identity of an algebra R if it vanishes whenever the selected indeterminates are replaced by the elements from the selected subsets of R. Let Ã be the free associative algebra generated by the set of symbols {z_{i} ∗ X_{j}|i, j ∈ ℕ}. We consider weak identities as the elements from Ã. We say that a weak identity is nontrivial if it is a nonzero element of Ã.

Lemma 4.2. Let R = u(L)#K[G] be a PI-algebra. Then it satisfies the following weak identities

1. \begin{aligned}& F_{1}(X_{1}, …, X_{n}, z_{1}, …, z_{n}) \\ = & ∑_{π∈S_{n}} α_{π}(z_{1} ∗ X_{π(1)}) ⋯ (z_{n} ∗ X_{π(n)}) \\ ≡ & 0; X_{1}, …, X_{n} ∈ u(L); z_{1}, …, z_{n} ∈ K[G]. \end{aligned}

2. \begin{aligned}& F_{2}(X_{1}, …, X_{n}, g_{1}, …, g_{n}) \\ = & ∑_{π∈S_{n}} α_{π}(g_{1} ∗ X_{π(1)} − X_{π(1)}) ⋯ (g_{n} ∗ X_{π(n)} − X_{π(n)}) \\ ≡ & 0; X_{1}, …, X_{n} ∈ u(L); g_{1}, …, g_{n} ∈ G. \end{aligned}

where α_{π} ∈ K, and α_{e} = 1.

Proof. We take the identity of the previous lemma, set Y_{1} = g_{1}, Y_{2} = g_{1}^{−1}g_{2}, …, Y_{n} = g_{n−1}^{−1}g_{n}, g_{i} ∈ G, and multiply on the right by g_{n}^{−1}. The for all X_{1}, …, X_{n} ∈ u(L) and g_{1}, …, g_{n} ∈ G we get

F_{1}(X_{1}, …, X_{n}, g_{1}, …, g_{n}) = ∑_{π∈S_{n}} α_{π}(g_{1} ∗ X_{π(1)}) ⋯ (g_{n} ∗ X_{π(n)}) ≡ 0.

By linearity we can substitute elements of the group ring for g_{1}, …, g_{n}. Thus we derive the first identity.

We decompose F_{2}(X_{1}, …, X_{n}, g_{1}, …, g_{n}), g_{1}, …, g_{n} ∈ G, into 2^{n} summands, each being the result of substitution of the identity element e ∈ G into F_{1}(X_{1}, …, X_{n}, g_{1}, …, g_{n}) on some places 1 ≤ i_{1} < i_{2} < ⋯ < i_{s} ≤ n:

F_{2}(X_{1}, …, X_{n}, g_{1}, …, g_{n}) =

= ∑_{s=0}^{n} ∑_{1≤i_{1}<i_{2}<···<i_{s}≤n}(−1)^{s}F_{1}(X_{1}, …, X_{n}, g_{1}, …, g_{n})|_{g_{i_{1}}=···=g_{is}=e} ≡ 0.

We remark that the following decomposition holds

F_{1}(X_{1}, …, X_{n}, z_{1}, …, z_{n})
= ∑_{i=1}^{n}(z_{1} ∗ X_{i})H_{i}(X_{1}, …, X̂_{i}, …, X_{n}, z_{2}, …, z_{n});  (9)
H_{i}(X_{1}, …, X̂_{i}, …, X_{n}, z_{2}, …, z_{n})
= ∑_{π(1)=i} α_{π}(z_{2} ∗ X_{π(2)}) ⋯ (z_{n} ∗ X_{π(n)}),

where H_{i} are the polynomials of the same type as F_{1}. Moreover, if F_{1} is nontrivial then some H_{i} is also nontrivial. Similar decompositions also hold true for polynomials of type F and F_{2}.

Suppose that L is a restricted Lie algebra. Fix some basis in L. Then we have the standard PBW-basis for the restricted enveloping algebra u(L) [6]. Let u_{n}(L) denote the span of all basis monomials for u(L) of length not greater than n. Now we have the standard filtration u_{0}(L) ⊂ u_{1}(L) ⊂ ⋯ ⊂ u_{n}(L) ⊂ · · ·. It induces the filtration for R = u(L)#K[G]

R_{0} ⊂ R_{1} ⊂ ⋯ ⊂ R_{n} ⊂ ⋯; R_{n} = u_{n}(L)K[G], n = 0, 1, 2, …

gr R = ⊕_{i=0}^{∞} R̄_{i}, R̄_{i} = R_{i}/R_{i−1}, i = 0, 1, …

Observe that gr R ≅ gr{u_{n}(L)|n = 0, 1, 2 … }#K[G] where the action of G on the vector space L̄ ≅ u_{1}(L)/u_{0}(L) is the same as the action of G on L. Recall also that gr{u_{n}(L)|n = 0, 1, 2 … } is isomorphic to the ring of truncated polynomials.

Lemma 4.3. Let R = u(L)#K[G] be a PI-algebra. Then gr R satisfies a weak identity

F_{3}(X_{1}, …, X_{n}, z_{1}, …, z_{n}) = ∑_{π∈S_{n}} α_{π}(z_{π(n)} ∗ X_{n}) ⋯ (z_{π(1)} ∗ X_{1}) ≡ 0;

X_{1}, …, X_{n} ∈ u(L), z_{1}, …, z_{n} ∈ K[G];

where α_{π} ∈ K, and α_{e} = 1.

Proof. We rewrite F_{1}(X_{1}, …, X_{n}, z_{1}, …, z_{n}) using the commutativity of gr u(L).

The nontrivial elements of the form F_{3} can be also decomposed as

F_{3}(X_{1}, …, X_{n}, z_{1}, …, z_{n})
= ∑_{i=1}^{n}(z_{i} ∗ X_{n})H_{i}(X_{1}, …, X_{n−1}, z_{1}, … ẑ_{i}, …, z_{n});  (10)
H_{i}(X_{1}, …, X_{n−1}, z_{1}, … ẑ_{i}, …, z_{n})
= ∑_{π(n)=i} α_{π}(z_{π(n−1)} ∗ X_{n−1}) ⋯ (z_{π(1)} ∗ X_{1}),

where H_{i} are the polynomials of the same type as F_{3} and some H_{i} is also nontrivial.

5. Delta-theory for smash products

Recall that there are some delta-sets inside G and L that have been effectively used in the study of the inner structure of G and L; the notation δ, Δ was used for these sets in [3], [11], [14]. A definition of delta-sets for Hopf algebras in terms of their inner actions can be found in [4].

Suppose that a group G acts on a Lie algebra L by automorphisms. In this section we introduce four more families of delta-sets defined with respect to this action. We specially reserve the symbol D for the pairing between K[G] and L.

First, we define a series of delta-sets inside L:

D_{m,G}(L) = {x ∈ L | dim K[G] ∗ x ≤ m}, m ∈ ℕ;
D_{G}(L) = ∪_{m=1}^{∞} D_{m,G}(L) ⊂ L.

Bahturin and Petrogradsky 381

Another series of delta-sets lives inside G:

D_{m,L}(G) = {g ∈ G | dim(g − 1) ∗ L ≤ m}, m = 0, 1, 2 …;
D_{L}(G) = ∪_{m=0}^{∞} D_{m,L}(G) ⊂ G.

Finally, we define families of delta-sets inside the group ring K[G] and the restricted enveloping algebra u(L):

D_{m,L}(K[G]) = {a ∈ K[G] | dim(a ∗ L) ≤ m}, m = 0, 1, 2, …;
D_{L}(K[G]) = ∪_{m=0}^{∞} D_{m,L}(K[G]) ⊂ K[G].
D_{m,G}(u(L)) = {v ∈ u(L) | dim K[G] ∗ v ≤ m}, m ∈ ℕ;
D_{G}(u(L)) = ∪_{m=1}^{∞} D_{m,G}(u(L)) ⊂ u(L).

In our study of identities in smash products we shall use essentially three families except those inside u(L). For convenience we often omit subscripts L, G and simply write, for example, D(L) instead of D_{G}(L).

One can easily check the following properties of these sets. Lemma 5.1. The sets D_{i}(L) satisfy

1. if x ∈ D_{i}(L), y ∈ D_{j}(L) then αx + βy ∈ D_{i+j}(L), α, β ∈ K;

2. if x ∈ D_{i}(L), y ∈ D_{j}(L) then [x, y] ∈ D_{i·j}(L); and x^{[p]} ∈ D_{i^{p}}(L);

3. D_{i}(L) are G-invariant;

4. D(L) is a restricted G-invariant subalgebra in L. Proof. For example, let us prove the second claim. In this case K[G] ∗ [x, y] = ⟨g ∗ [x, y]|g ∈ G⟩_{K} = ⟨[g ∗ x, g ∗ y]|g ∈ G⟩_{K} ⊂ [K[G] ∗ x, K[G] ∗ y], so dim K[G] ∗ [x, y] ≤ dim([K[G] ∗ x) · dim([K[G] ∗ y) ≤ ij.

Lemma 5.2. The sets D_{i}(G) satisfy 1. if g ∈ D_{i}(G), h ∈ D_{j}(G) then gh ∈ D_{i+j}(G);

2. if g ∈ D_{i}(G), then g^{−1} ∈ D_{i}(G);

3. if g ∈ D_{i}(G), h ∈ G then h^{−1}gh ∈ D_{i}(G);

4. 1 ∈ D_{i}(G) for all i ≥ 0;

5. D(G) is a normal subgroup in G.

Proof. In order to prove claims 1), 2), and 3) we observe that for arbitrary g, h ∈ G one has

(gh − 1) ∗ L = ((g − 1)h + h − 1) ∗ L ⊂ (g − 1) ∗ L + (h − 1) ∗ L;
(g^{−1} − 1) ∗ L = g^{−1} ∗ ((1 − g) ∗ L);
(h^{−1}gh − 1) ∗ L = (h^{−1}(g − 1)h) ∗ L ⊂ h^{−1} ∗ ((g − 1) ∗ L).

Other claims are obvious.

Lemma 5.3. The sets D_{i}(K[G]) satisfy

1. if a ∈ D_{i}(K[G]), b ∈ D_{j}(K[G]) then αa + βb ∈ D_{i+j}(K[G]), α, β ∈ K;

2. K[G] · D_{i}(K[G]) · K[G] ⊂ D_{i}(K[G]);

3. D(K[G]) is a two-sided ideal in K[G].

Proof. Let us check the second claim. Suppose that a ∈ D_{i}(K[G]), so dim(a ∗ L) ≤ i. Then for arbitrary x, y ∈ K[G] we have (xay) ∗ L ⊂ x ∗ (a ∗ L) and dim((xay) ∗ L) ≤ i. Hence, xay ∈ D_{i}(K[G]).

Lemma 5.4. The sets D_{i}(u(L)) satisfy

1. if v ∈ D_{i}(u(L)), w ∈ D_{j}(u(L)) then αv + βw ∈ D_{i+j}(u(L)), α, β ∈ K;

2. if v ∈ D_{i}(u(L)), w ∈ D_{j}(u(L)) then vw ∈ D_{i+j}(u(L));

3. D(u(L)) is a subalgebra in u(L).

Proof. is similar to that of Lemma 5.1.

Let us establish the relationship between D(G) and D(K[G]).

Lemma 5.5. 1. 1 + (D(K[G]) ∩ (G − 1)) = D(G);

2. K[G] · ωK[D(G)] = ωK[D(G)] · K[G] ⊂ D(K[G]).

Proof. The first claim follows directly from definitions. Let us prove the second one. We consider v = α_{1}g_{1} + ⋯ + α_{m}g_{m} ∈ ωK[D(G)], α_{i} ∈ K, g_{i} ∈ D(G). Then α_{1} + ⋯ + α_{m} = 0 and v = α_{1}(g_{1} − 1) + ⋯ + α_{m}(g_{m} − 1). There exists a number s such that {g_{1}, …, g_{m}} ⊂ D_{s}(G). Let w, u ∈ K[G] then

(wvu) ∗ L ⊂ w ∗ ((g_{1} − 1) ∗ L + ⋯ + (g_{m} − 1) ∗ L),

and wvu ∈ D_{ms}(K[G]).

But we lack any bounds in this lemma. We only suggest the following conjecture that would make Theorem 6.5 below unnecessary.

Conjecture Let G = D_{s}(G) then ωK[G] = D_{t}(K[G]) for some number t = f(s).

On the other hand, the next example shows that the inclusion in Lemma 5.5 is strict: K[G] · ωK[D(G)] ≠ D(K[G]).

Example 5.6. Let L = ⟨x_{1}, x_{2}, …, y_{1}, y_{2}, … |x_{i}^{[p]} = y_{i}^{[p]} = 0, i ∈ ℕ⟩_{K} be an abelian restricted Lie algebra. We consider the group G = (ℤ_{p})ℕ and set g_{i} = (0, …, 0, 1, 0, …), with 1 on i-th place, i ∈ ℕ. Suppose that G acts on L by

g_{i} ∗ x_{j} = x_{j} + ξ_{i}y_{j}, g_{i} ∗ y_{j} = y_{j}, i, j ∈ ℕ,

where {ξ_{i} ∈ K|i ∈ ℕ} are the scalars linearly independent over ℤ_{p}. We consider the smash product R = u(L)#K[G]. Then

1. D(G) = {e};

2. D_{L}(K[G]) = D_{0,L}(K[G]) and dim K[G]/ D(K[G]) = 2;

3. R is not PI;

4. R satisfies some weak identities of the type

F_{2}(X_{1}, …, X_{n}, h_{1}, …, h_{n})
= ∑_{π∈S_{n}} α_{π}(h_{1} ∗ X_{π(1)} − X_{π(1)}) ⋯ (h_{n} ∗ X_{π(n)} − X_{π(n)})
≡ 0; X_{1}, …, X_{n} ∈ L, h_{1}, …, h_{n} ∈ G.

Proof. Let e ≠ g = (n_{1}, n_{2}, …) ∈ G, then (g − 1) ∗ x_{j} = λy_{j}, where λ = ∑_{i} n_{i}ξ_{i} ≠ 0 by assumption. Now the first claim follows by the definition of the delta-sets D_{i,L}(G).

To prove the second claim we take z = ∑_{i} α_{i}g_{i} ∈ K[G] and consider

z ∗ x_{j} = ( ∑_{i} α_{i})x_{j} + ( ∑_{i} α_{i}ξ_{i})y_{j}, j ∈ ℕ.

Then z ∈ D_{m,L}(K[G]) for some m ≥ 0 if and only if ∑_{i} α_{i} = 0 and ∑_{i} α_{i}ξ_{i} = 0. Hence, D_{L}(K[G]) = D_{0,L}(K[G]) and dim K[G]/ D_{L}(K[G]) = 2.

Suppose that R is PI. We apply the first identity of Lemma 4.2 and substitute the values a_{1} = x_{1}, a_{2} = x_{2}x_{3}, …, a_{n} = x_{(n−1)n/2+1} ⋯ x_{n(n+1)/2}, and g_{i_{1}}, …, g_{i_{n}} ∈ G:

F_{1}(a_{1}, a_{2}, …, a_{n}, g_{i_{1}}, …, g_{i_{n}}) = ∑_{π∈S_{n}} α_{π}(g_{i_{1}} ∗ a_{π(1)}) ⋯ (g_{i_{n}} ∗ a_{π(n)})

= ( ∑_{π∈S_{n}} α_{π}ξ_{i_{1}}^{π(1)}ξ_{i_{2}}^{π(2)} ⋯ ξ_{i_{n}}^{π(n)})y_{1} ⋯ y_{n(n+1)/2} + terms with x-s.

Therefore, 0 ≠ f(X_{1}, …, X_{n}) = ∑_{π∈S_{n}} α_{π}X_{1}^{π(1)}X_{2}^{π(2)} ⋯ X_{n}^{π(n)} ∈ K[X_{1}, …, X_{n}] is annihilated by any substitution from the countable set of scalars X_{i} ∈ {ξ_{j}|j ∈ ℕ}, i = 1, …, n. This contradiction proves that R is not PI.

By linearity, the last claim can be checked for the action on x-s only. We remark that any element h ∈ G acts as the generators g_{i}, namely h∗x_{i} = x_{i}+μ_{h}y_{i}, μ_{h} ∈ K, i ∈ ℕ. Then

F_{2}(x_{1}, …, x_{n}, h_{1}, …, h_{n}) = ∑_{π∈S_{n}} α_{π}(h_{1} ∗ x_{π(1)} − x_{π(1)}) ⋯ (h_{n} ∗ x_{π(n)} − x_{π(n)})

= ( ∑_{π∈S_{n}} α_{π})μ_{1}μ_{2} ⋯ μ_{n}y_{1}y_{2} ⋯ y_{n}.

If ∑_{π∈S_{n}} α_{π} = 0, then we obtain a weak identity.^{3}

Also, this example shows that it is not enough to study the action of G on L, but we also need to take into account the action of G on u(L) as well (see the proof of Theorem 6.2 below).

Let us also establish the relationship between D(L) and D(u(L)).

Lemma 5.7. 1. D(u(L)) ∩ L = D(L);

2. ωu(D(L)) ⊂ D(u(L)).

Proof. The first claim follows from definitions. The subalgebra ωu(D(L)) is generated by the elements from D(L). These elements have finite width by Lemma 5.4.

We suggest to study whether there exists some bounds similar to Conjecture above. Another interesting question is to investigate if the inclusion in Lemma 5.7 is strict.

6. Structure of delta-sets

In this section we establish crucial facts about the structure of three families of delta-sets, provided that the smash product R = u(L)#K[G] satisfies a nontrivial polynomial identity. ¿From now on we assume that the number n is fixed in Lemma 4.2 (remark that all lemmas of Section 4 fix the same number n).

Let us consider the associated graded algebra gr R ≅ (gr u(L))#K[G]. It satisfies the same weak identities of Lemmas 4.2, 4.3. We observe that the action of G on the space u_{1}(L)/u_{0}(L) ≅ L inside gr u(L) remains the same. Therefore, we may assume that L is abelian with the trivial p-mapping, while studying the action of G on L in this section.

Let L be a restricted Lie algebra and some basis L = {e_{i}|i ∈ I} be fixed. Suppose that the decomposition of v ∈ u(L) via the standard basis for the restricted enveloping algebra depends on the elements e_{i_{1}}, …, e_{i_{s}}. Then we denote the support of v by supp v = {e_{i_{1}}, …, e_{i_{s}}}.

Let G be a group and T a subset of G. We say that T has finite index in G if there exist g_{1}, g_{2}, …, g_{m} ∈ G with

G = g_{1}T ∪ g_{2}T ∪ ⋯ ∪ g_{m}T.

We then define the index |G : T | to be the minimum possible integer m with the above property[11], [12]. If T is a subgroup of G, then this agrees with the usual definition of index.

Lemma 6.1. ([11]) Let T be a subset of G with |G : T | ≤ m. We set T^{∗} = T ∪ {1} ∪ T^{−1}. Then

(T^{∗})^{4^{m}} = \underbrace{T^{∗} · T^{∗} ⋯ T^{∗}}_{4^{m} times}

is a subgroup of G.

3This example was produced in collaboration with M. Kotchetov

Theorem 6.2. Let R = u(L)#K[G] be a PI-algebra. Then there exists a subgroup G_{0} ⊂ G with |G : G_{0}| < n and G_{0} ⊂ D_{n^{3}4^{n},L}(G).

Proof. Let n be the number fixed in Lemma 4.3. Let us prove that |G : D_{n^{3}}(G)| < n. We fix arbitrary g_{1}, …, g_{n} ∈ G. It suffices to prove that there exist i ≠ j such that g_{i}^{−1}g_{j} ∈ D_{n^{3}}(G). We apply the weak identity of Lemma 4.3

F_{3}(X_{1}, …, X_{n}, g_{1}, …, g_{n}) = ∑_{π∈S_{n}} α_{π}(g_{π(n)} ∗ X_{n}) ⋯ (g_{π(1)} ∗ X_{1}) ≡ 0;

X_{1}, …, X_{n} ∈ u(L).

In the identical relations that follow we denote by X′s, and Y′s the variables that range over some sets of elements inside u(L).

Let us prove by induction on m the following. Suppose that g_{1}, …, g_{m} ∈ G are fixed and satisfy

F_{3}(X_{1}, …, X_{m}, g_{1}, …, g_{m}) ≡ 0; X_{i} ∈ u_{i}(L), i = 1, …, m,

where F_{3} is nontrivial. Then we claim that there exist i ≠ j such that g_{i}^{−1}g_{j} ∈ D_{m^{3}}(G).

We consider m = 1, we have g_{1} ∗ X_{1} ≡ 0, X_{1} ∈ L. Then L = 0 and the assertion is trivial.

Suppose that the statement is valid for m − 1, m > 1. We apply (10):

F_{3}(X_{1}, …, X_{m}, g_{1}, …, g_{m})
= ∑_{i=1}^{m}(g_{i} ∗ X_{m})H_{i}(X_{1}, …, X_{m−1}, g_{1}, … ĝ_{i}, …, g_{m})
≡ 0; X_{i} ∈ u_{i}(L), 1 ≤ i ≤ m.  (11)

Without loss of generality we may assume that H_{m} is a non-trivial polynomial. Now we consider two cases. First, suppose that

H_{m}(X_{1}, …, X_{m−1}, g_{1}, …, g_{m−1}) ≡ 0; X_{i} ∈ u_{i}(L), i = 1, …, m − 1.

Then by the inductive assumption there exist i ≠ j, 1 ≤ i, j ≤ m − 1 such that g_{i}^{−1}g_{j} ∈ D_{(m−1)^{3}}(G) ⊂ D_{m^{3}}(G).

Second, there exist a_{1} ∈ u_{1}(L), a_{2} ∈ u_{2}(L), …, a_{m−1} ∈ u_{m−1}(L) such that h_{m} = H_{m}(a_{1}, …, a_{m−1}, g_{1}, …, g_{m−1}) ≠ 0. Let {e_{i}|i ∈ I} be an ordered basis for L. Since H_{m} is linear in X′s, we can consider a_{1}, …, a_{m−1} to be monomials in {e_{α_{1}}, …, e_{α_{t}}}, where t ≤ 1 + 2 + ⋯ + m − 1 < m(m − 1). We set

V_{0} = ⟨g_{i} ∗ e_{α_{j}}|1 ≤ i ≤ m − 1, 1 ≤ j ≤ t⟩_{K}.

Then dim V_{0} ≤ t(m − 1) ≤ m(m − 1)^{2}. Now we substitute a_{1}, …, a_{m−1} for X_{1}, …, X_{m−1} in (11), set X = X_{m}, and obtain the relation

∑_{i=1}^{m}(g_{i} ∗ X)h_{i} ≡ 0, X ∈ u_{m}(L);  (12)
h_{m} ≠ 0, supp h_{m} ⊂ V_{0}.  (13)

By substituting XY for X we get

∑_{i=1}^{m}(g_{i} ∗ X)(g_{i} ∗ Y)h_{i} ≡ 0, X ∈ u_{m−1}(L), Y ∈ L.  (14)

Multiplying (12) by g_{1} ∗ Y and subtracting from (14) we obtain

∑_{i=2}^{m}(g_{i} ∗ X)(g_{i} ∗ Y − g_{1} ∗ Y)h_{i} ≡ 0, X ∈ u_{m−1}(L), Y ∈ L.  (15)

Here we have two possibilities. First, (g_{m} − g_{1}) ∗ y ∈ V_{0} for all y ∈ L. Then (g_{1}^{−1}g_{m} − 1) ∗ L ⊂ g_{1}^{−1} ∗ V_{0}, therefore g_{1}^{−1}g_{m} ∈ D_{b}(G), where b = dim V_{0} ≤ m(m − 1)^{2} < m^{3}. Second, there exists y_{0} ∈ L such that (g_{m} − g_{1}) ∗ y_{0} = v_{1} ∉ V_{0}. We set V_{1} = V_{0} + ⟨v_{1}⟩_{K} and change the basis of L outside V_{0} so that v_{1} coincides with some basis element. We fix Y = y_{0} in (15), and by the construction of v_{1} and (13) we obtain

∑_{i=2}^{m}(g_{i} ∗ X)h_{i}^{(1)} ≡ 0, X ∈ u_{m−1}(L);  (16)
h_{i}^{(1)} = (g_{i} ∗ y_{0} − g_{1} ∗ y_{0}) h_{i}, i = 2, …, m;
h_{m}^{(1)} = v_{1}h_{m} ≠ 0, supp h_{m}^{(1)} ⊂ V_{1}.

We continue this process further by deleting in (16) the term for i = 2.

∑_{i=3}^{m}(g_{i} ∗ X)(g_{i} ∗ Y − g_{2} ∗ Y)h_{i}^{(1)} ≡ 0, X ∈ u_{m−2}(L), Y ∈ L.

Similarly, either (g_{m} − g_{2}) ∗ L ⊂ V_{1} and we are done (see below), or there exists y_{1} ∈ L such that (g_{m} − g_{2}) ∗ y_{1} = v_{2} ∉ V_{1}. In the latter case we change the basis of L outside V_{1} so that v_{2} is one of the basis elements, set V_{2} = V_{1} + ⟨v_{2}⟩_{K}, h_{i}^{(2)} = (g_{i} ∗ y_{1} − g_{2} ∗ y_{1})h_{i}^{(1)}, i = 3, …, m and obtain the relation

∑_{i=3}^{m}(g_{i} ∗ X)h_{i}^{(2)} ≡ 0, X ∈ u_{m−2}(L);

h_{m}^{(2)} = v_{2}h_{m}^{(1)} ≠ 0, supp h_{m}^{(2)} ⊂ V_{2}.

This process terminates with the relation

(g_{m} ∗ X)h_{m}^{(m−1)} ≡ 0, X ∈ u_{1}(L);  (17)
h_{m}^{(m−1)} ≠ 0;
supp h_{m}^{(m−1)} ⊂ V_{m−1} = V_{0} + ⟨v_{1}, …, v_{m−1}⟩_{K}.

If we substitute X = 1 in (17) then we obtain a contradiction. This contradiction proves that this process had to stop even before, and the desired relation holds: g_{i}^{−1}g_{m} ∈ D_{c}(G), where c = dim V_{i−1} ≤ dim V_{m−1} ≤ m(m − 1)^{2} + m − 1 ≤ m^{3}.

If we set T = D_{n^{3}}(G), then we have proved that |G : T | ≤ n. Remark that, by Lemma 5.2, 1 ∈ T and T = T^{−1}. We set G_{0} = T^{4^{n}} and conclude that G_{0} is a subgroup by Lemma 6.1. Of course, |G : G_{0}| ≤ n and G_{0} ⊂ D_{n^{3}4^{n}}(G) by Lemma 5.2.

Suppose that W is a subset in a vector space V . We say that W has finite codimension in V if there exist v_{1}, …, v_{m} ∈ V with V = W +⟨v_{1}, …, v_{m}⟩_{K}. If m is the minimum possible integer with such property then we set dim V/W = m. We also introduce the notation m · W = {w_{1} + ⋯ + w_{m}|w_{i} ∈ W }, m ∈ ℕ.

Lemma 6.3. Let L be a vector space. Suppose that a subset T ⊂ L is stable under multiplication by scalars and such that dim L/T ≤ m. Then ⟨T ⟩_{K} = 4^{m} ·T.

Proof. We prove this statement by induction on m. If m = 0 then the assertion is trivial. Suppose that dim L/T = m. Then there exist h_{1}, …, h_{m} such that

L = T + ⟨h_{1}, …, h_{m}⟩_{K}.  (18)

If 2·T = T then T is a subspace. Otherwise there exist t_{1}, t_{2} ∈ T with t_{1}+t_{2} ∉ T. By (18) t_{1} + t_{2} = t_{3} + α_{1}h_{1} + ⋯ + α_{m}h_{m}, t_{3} ∈ T, α_{i} ∈ K, where one of scalars is nonzero. Let α_{m} ≠ 0, then h_{m} ∈ 3 · T + ⟨h_{1}, …, h_{m−1}⟩_{K}. We substitute in (18) and obtain L = 4 · T + ⟨h_{1}, …, h_{m−1}⟩_{K}. By the inductive assumption ⟨4 · T ⟩_{K} = 4^{m−1} · (4 · T) ⊂ 4^{m} · T. Lemma is proved.

Theorem 6.4. Let R = u(L)#K[G] be a PI-algebra. Then there exists a Ginvariant restricted subalgebra L_{0} ⊂ L with dim L/L_{0} < n and L_{0} ⊂ D_{p^{n}4^{n}n^{5},G}(L).

Proof. Let n be the number fixed above. While studying the action of G on L, we temporarily assumethat L is abelian with the trivial p-mapping. First, let us prove that dim L/ D_{n^{2}}(L) < n.

We fix arbitrary a_{1}, …, a_{n} ∈ L. By Lemma 4.2 we have

F_{1}(a_{1}, …, a_{n}, g_{1}, …, g_{n}) = ∑_{π∈S_{n}} α_{π}(g_{1} ∗ a_{π(1)}) ⋯ (g_{n} ∗ a_{π(n)}) ≡ 0; g_{1}, …, g_{n} ∈ G.

In this theorem g_{i}’s, g are the variables that range over G. Let us prove by induction on m the following. Suppose that a_{1}, …, a_{m} ∈ L are fixed and satisfy the condition

F_{1}(a_{1}, …, a_{m}, g_{1}, …, g_{m}) = ∑_{π∈S_{m}} α_{π}(g_{1} ∗ a_{π(1)}) ⋯ (g_{m} ∗ a_{π(m)}) ≡ 0,  (19)
g_{1}, …, g_{m} ∈ G;

where F_{1}(X_{1}, …, X_{m}, g_{1}, …, g_{m}) is some nontrivial polynomial. Then a_{1}, …, a_{m} are linearly dependent modulo D_{m^{2}}(L).

In the case m = 1 we have g_{1} ∗ a_{1} ≡ 0, g_{1} ∈ G. Since G acts by automorphisms we have a_{1} = 0 ∈ D_{1}(L).

Suppose that the statement is valid for m−1, m > 1. We apply (9) to (19)

F_{1}(a_{1}, …, a_{m}, g_{1}, …, g_{m})
= ∑_{i=1}^{m}(g_{1} ∗ a_{i})H_{i}(a_{1}, …, â_{i}, …, a_{m}, g_{2}, …, g_{m})
≡ 0; g_{1}, …, g_{m} ∈ G.  (20)

Without loss of generality we assume that H_{1}(X_{2}, …, X_{m}, g_{2}, …, g_{m}) is a nontrivial polynomial. Now two cases are possible. In the first case H_{1}(a_{2}, …, a_{m}, g_{2}, …, g_{m}) ≡ 0 for all g_{2}, …, g_{m} ∈ G. Then by the inductive hypothesis a_{2}, …, a_{m} are linearly dependent modulo D_{(m−1)^{2}}(L) and we are done. In the second case there exist h_{2}, …, h_{m} ∈ G such that f_{1} = H_{1}(a_{2}, …, a_{m}, h_{2}, …, h_{m}) ≠ 0. We substitute these values into (20) and set f_{i} = H_{i}(a_{1}, …, â_{i}, …, a_{m}, h_{2}, …, h_{m}), i = 1, …, m:

∑_{i=1}^{m}(g_{1} ∗ a_{i})f_{i} ≡ 0, g_{1} ∈ G.

We discard the summands with f_{i} = 0, set g = g_{1}, and rewrite our relation as follows

∑_{i=1}^{r}(g ∗ a_{i})f_{i} ≡ 0, g ∈ G; f_{i} ≠ 0, i = 1, …, r.  (21)

We set V = ⟨g_{i} ∗ a_{j}|2 ≤ i ≤ m, 1 ≤ j ≤ m⟩_{K}, then dim V < m^{2} and f_{1}, …, f_{r} belong to the subalgebra of u(L) generated by V . Next we prove by induction on r that (21) implies linear dependence of a_{1}, …, a_{r} modulo D_{m^{2}}(L).

If r = 1 then

(g ∗ a_{1})f_{1} ≡ 0, g ∈ G.  (22)

Let us prove that G ∗ a_{1} ⊂ V , so a_{1} ∈ D_{m^{2}}(L). By way of contradiction suppose that there exists d ∈ G with d ∗ a_{1} = e_{0} ∉ V . Choose an ordered basis for L whose first elements is e_{0}, followed by a basis of V = ⟨v_{1}, …, v_{t}⟩_{K}. Now f_{1} is the sum of products, each product consists of m − 1 factors of the type g_{i} ∗ a_{j} ∈ V . Using the standard basis of the restricted enveloping algebra we have

f_{1} = ∑_{j} α_{j}v_{j_{1}} ⋯ v_{j_{m−1}}, α_{j} ∈ K.  (23)

Multiplying f_{1} by e_{0} we obtain a nonzero element. Thus, seting g = d we arrive at a contradiction with (22).

We consider r > 1. Suppose that G ∗ a_{r} ⊂ V in (21). Then a_{r} ∈ D_{m^{2}}(L) and the result follows. So, we assume that e_{0} = d ∗ a_{r} ∉ V for some d ∈ G. By analogy with the preceding argument we choose an ordered basis for L. We set d ∗ a_{j} = α_{j}e_{0} + w_{j}, j = 1, …, r − 1, α_{j} ∈ K and each w_{j} being a linear combination of the basis elements of L except e_{0}. By setting g = d in (21) we obtain

e_{0}(α_{1}f_{1} + ⋯ + α_{r−1}f_{r−1} + f_{r}) + w_{1}f_{1} + ⋯ + w_{r−1}f_{r−1} = 0.  (24)

We set f = α_{1}f_{1} + ⋯ + α_{r−1}f_{r−1} + f_{r}. Suppose that f ≠ 0. By analogy with the preceding argument, f is of the form (23), which means that the first summand in (24) has degree m and may be written as

e_{0}f = ∑_{j} α_{j}e_{0}v_{j_{1}} ⋯ v_{j_{m−1}}, α_{j} ∈ K.  (25)

Other nonzero summands in (24), being written in the form (25), do not contain e_{0} as their factor. This is a contradiction. So, f = α_{1}f_{1} + ⋯ + α_{r−1}f_{r−1} + f_{r} = 0. We express f_{r} from this relation and substitute into (21):

∑_{i=1}^{r−1}(g ∗ (a_{i} − α_{i}a_{r}))f_{i} ≡ 0, g ∈ G;

f_{i} ≠ 0, i = 1, …, r − 1.

By the inductive assumption a_{1} − α_{1}a_{r}, …, a_{r−1} − α_{r−1}a_{r} are linearly dependent modulo D_{m^{2}}(L), therefore a_{1}, …, a_{r} are also linearly dependent modulo this set.

Thus we have proved that dim L/ D_{n^{2}}(L) < n. Now we return to the original structure of a Lie p-algebra on L. By Lemmas 6.3, 5.1 we construct a subspace W_{1} = ⟨D_{n^{2}}(L)⟩_{K} = 4^{n} · D_{n^{2}}(L) ⊂ D_{4^{n}n^{2}}(L). We consider the chain W_{i} = W_{i−1} +[W_{i−1}, W_{1}], i = 2, 3 . . .. Due to the finiteness of codimension we have stabilization ∪_{i=1}^{∞}W_{i} = W_{n} and by Lemma 5.1 W_{n} ⊂ D_{4^{n}n^{4}}(L). By construction, W_{n} is a G-invariant subalgebra. To obtain a restricted subalgebra we consider its p-hull L_{0} = (W_{n})_{p} = ⟨w^{[p^{s}]}|w ∈ W_{n}, s ≥ 0⟩_{K}. Again by the codimension argument, we can assume here that s ≤ n. By Lemma 5.1 we get L_{0} ⊂ D_{p^{n}4^{n}n^{5}}(L), also dim L/L_{0} < n. The theorem follows.

Theorem 6.5. Let R = u(L)#K[G] be a PI-algebra. Then there exists a twosided ideal J ⊂ K[G] with dim K[G]/J < n and J ⊂ D_{4^{n}n^{2},L}(K[G]).

Proof. Similar to that for the previous theorem. Let n be the number fixed above. We may assume that L is abelian with the trivial p-mapping. First, let us prove that dim K[G]/ D_{n^{2}}(K[G]) < n.

We fix arbitrary z_{1}, …, z_{n} ∈ K[G]. By Lemma 4.3 we have

F_{3}(a_{1}, …, a_{n}, z_{1}, …, z_{n}) = ∑_{π∈S_{n}} α_{π}(z_{π(n)} ∗ a_{n}) ⋯ (z_{π(1)} ∗ a_{1}) ≡ 0; a_{1}, …, a_{n} ∈ L.

In this theorem a_{i}’s, a denote the variables that range over L. We prove by induction on m the following. Let z_{1}, …, z_{m} ∈ K[G] be fixed and satisfy the condition

F_{3}(a_{1}, …, a_{m}, z_{1}, …, z_{m}) = ∑_{π∈S_{m}} α_{π}(z_{π(m)} ∗ a_{m}) ⋯ (z_{π(1)} ∗ a_{1}) ≡ 0,  (26)
a_{1}, …, a_{m} ∈ L;

where F_{3}(X_{1}, …, X_{m}, z_{1}, …, z_{m}) is some nontrivial polynomial. Then z_{1}, …, z_{m} are linearly dependent modulo D_{m^{2}}(K[G]).

In the case m = 1 we have z_{1} ∗ a_{1} ≡ 0, a_{1} ∈ L. Then z_{1} ∈ D_{0}(K[G]) and the result follows.

Suppose that the statement is valid for m − 1, m > 1. We apply decomposition (10) to the relation (26)

F_{3}(a_{1}, …, a_{m}, z_{1}, …, z_{m})
= ∑_{i=1}^{m}(z_{i} ∗ a_{m})H_{i}(a_{1}, …, a_{m−1}, z_{1}, …, ẑ_{i}, …, z_{m})
≡ 0; a_{1}, …, a_{m} ∈ L.  (27)

Without loss of generality we may assume that H_{1}(X_{1}, …, X_{m−1}, z_{2}, …, z_{m}) is a nontrivial polynomial. We have two cases. 1) H_{1}(a_{1}, …, a_{m−1}, z_{2}, …, z_{m}) ≡ 0 for all a_{1}, …, a_{m−1} ∈ L. Then by the inductive hypothesis z_{2}, …, z_{m} are linearly dependent modulo D_{(m−1)^{2}}(K[G]) and we are done. 2) There exist b_{1}, …, b_{m−1} ∈ L such that H_{1}(b_{1}, …, b_{m−1}, z_{2}, …, z_{m}) ≠ 0. We substitute these values into (27) and set f_{i} = H_{i}(b_{1}, …, b_{m−1}, z_{1}, …, ẑ_{i}, …, z_{m}), i = 1, …, m:

∑_{i=1}^{m}(z_{i} ∗ a_{m})f_{i} ≡ 0, a_{m} ∈ L.

We omit the summands with f_{i} = 0, denote a = a_{m}, and obtain

∑_{i=1}^{r}(z_{i} ∗ a)f_{i} ≡ 0, a ∈ L; f_{i} ≠ 0, i = 1, …, r.  (28)

We set V = ⟨z_{i} ∗ b_{j}|1 ≤ i ≤ m, 1 ≤ j < m − 1⟩_{K}, then dim V < m^{2} and f_{1}, …, f_{r} belong to the subalgebra of u(L) generated by V . Next we prove by induction on r that (28) implies linear dependence of z_{1}, …, z_{r} modulo D_{m^{2}}(K[G]).

If r = 1 then

(z_{1} ∗ a)f_{1} ≡ 0, a ∈ L.  (29)

Let us prove that z_{1}∗L ⊂ V , so z_{1} ∈ D_{m^{2}}(K[G]). By way of contradiction suppose that there exists y ∈ L with z_{1} ∗ y = e_{0} ∉ V . We choose an ordered basis for L whose first element is e_{0}, followed by a basis of V = ⟨v_{1}, …, v_{t}⟩_{K}. In the standard basis of the restricted enveloping algebra we have

f_{1} = ∑_{j} α_{j}v_{j_{1}} ⋯ v_{j_{m−1}} ≠ 0, α_{j} ∈ K.

Multiplying f_{1} by e_{0} we obtain a nonzero element. Thus, if we set a = y then (29) leads to contradiction.

We consider r > 1. Suppose that z_{r} ∗L ⊂ V in (28). Then z_{r} ∈ D_{m^{2}}(K[G]) and the result follows. So, we assume that e_{0} = z_{r} ∗ y ∉ V for some y ∈ L. By analogy with the preceding argument we choose an ordered basis for L. We set z_{j} ∗y = α_{j}e_{0}+w_{j}, j = 1, …, r−1, α_{j} ∈ K and each w_{j} being a linear combination of the basis elements of L except e_{0}. By setting a = y in (28) we obtain

e_{0}(α_{1}f_{1} + ⋯ + α_{r−1}f_{r−1} + f_{r}) + w_{1}f_{1} + ⋯ + w_{r−1}f_{r−1} = 0.  (30)

Let us set f = α_{1}f_{1} + ⋯ + α_{r−1}f_{r−1} + f_{r}. Suppose that f ≠ 0. By analogy with the preceding theorem the first summand in (30) has degree m and is written in the standard basis as

e_{0}f = ∑_{j} α_{j}e_{0}v_{j_{1}} ⋯ v_{j_{m−1}}, α_{j} ∈ K.

Other nonzero summands in (30), being written in the standard basis, have no e_{0} as their factor. This is a contradiction. Therefore, f = α_{1}f_{1} +⋯+α_{r−1}f_{r−1} +f_{r} = 0. We express f_{r} from this relation and substitute into (28):

∑_{i=1}^{r−1}((z_{i} − α_{i}z_{r}) ∗ a)f_{i} ≡ 0, a ∈ L;

f_{i} ≠ 0, i = 1, …, r − 1.

By the inductive assumption z_{1} − α_{1}z_{r}, …, z_{r−1} − α_{r−1}z_{r} are linearly dependent modulo D_{m^{2}}(K[G]), therefore z_{1}, …, z_{r} are also linearly dependent modulo this set.

Thus we have proved that dim K[G]/ D_{n^{2}}(K[G]) < n. By Lemma 6.3, we form a subspace by J = ⟨D_{n^{2}}(K[G])⟩_{K} = 4^{n} · D_{n^{2}}(K[G]). Due to Lemma 5.3 J is an ideal and J ⊂ D_{4^{n}n^{2}}(K[G]). Of course, also dim K[G]/J < n. The theorem is proved.

We can also describe the properties of the delta-sets as follows, but in this case it is not possible to evaluate the numbers n_{1}, n_{2}, n_{3} below.

Corollary 6.6. Let R = u(L)#K[G] be a PI-algebra and n be the number fixed above. Then there exist numbers n_{1}, n_{2}, n_{3} such that

\begin{matrix}D_{L}(G) = D_{n_{1},L}(G), & |G : D_{L}(G)| < n; \\ D_{G}(L) = D_{n_{2},G}(L), & dim L/ D_{G}(L) < n; \\ D_{L}(K[G]) = D_{n_{3},L}(K[G]), & dim K[G]/ D_{L}(K[G]) < n. \end{matrix}

Proof. Let us check the first equality. For T = D_{n^{3}}(G), we have |G : T | < n (Theorem 6.2). We consider T_{i} = (D_{i}(G))^{4^{n}}, i ≥ n^{3}. By Lemma 6.1, we obtain the chain of subgroups and T_{i} ⊂ D_{4^{n}i}(G) by Lemma 5.2. Since |G : T_{i}| < n, this chain must stabilize. In other two cases similar chains also stabilize by the codimension argument.

Next we are going to use the next result on bilinear maps.

Theorem 6.7. (P. M. Neumann, [1]) Let φ : U × V → W be a bilinear map, where U, V, W are vector spaces over a field K. Suppose that dim φ(u, V) ≤ m for each u ∈ U and dim φ(U, v) ≤ l for all v ∈ V . Then dim⟨φ(U, V)⟩_{K} ≤ ml.

The goal of three previous theorems has been to establish the following result.

Theorem 6.8. Let R = u(L)#K[G] be a PI-algebra. Then there exist a Ginvariant restricted subalgebra of finite codimension L_{0} ⊂ L and a subgroup of finite index G_{0} ⊂ G such that dim(ωK[G_{0}] ∗ L_{0}) < ∞.

Proof. We apply Theorems 6.2, 6.4, and 6.5. These theorems yield us the subgroup of finite index G_{0} ⊂ G, the G-invariant restricted subalgebra L_{0} with dim L/L_{0} < n, and the ideal J ◁K[G] with dim K[G]/J < n. Also, their elements have finite width with respect to the action of G on L. Namely,

dim((g − 1) ∗ L) ≤ 4^{n}n^{3}, g ∈ G_{0};
dim(K[G] ∗ x) ≤ p^{n}4^{n}n^{5}, x ∈ L_{0};
dim(z ∗ L) ≤ 4^{n}n^{2},  z ∈ J.

(31)

We have dim ωK[G_{0}]/(J ∩ ωK[G_{0}]) < n. Hence, there exist g_{1}, …, g_{n} ∈ G_{0} such that

ωK[G_{0}] = ⟨(g_{1} − 1), …, (g_{n} − 1)⟩_{K} + J ∩ ωK[G_{0}].

Then

ωK[G_{0}] ∗ L_{0} ⊂ (g_{1} − 1) ∗ L_{0} + ⋯ + (g_{n} − 1) ∗ L_{0} + J ∗ L_{0}.

We use the conditions of finite width (31) and apply Theorem 6.7 to the bilinear mapping φ : J × L_{0} → L_{0}, φ(z, x) = z ∗ x, z ∈ J, x ∈ L_{0}. Finally, we have

dim(ωK[G_{0}] ∗ L_{0}) ≤ dim(g_{1} − 1) ∗ L_{0} + ⋯ + dim(g_{n} − 1) ∗ L_{0} + dim(J ∗ L_{0})
≤ n · 4^{n}n^{3} + 4^{n}n^{2} · p^{n}4^{n}n^{5} < ∞.

7. Proof of the main result

If H is a subalgebra in a restricted Lie algebra L then by H_{p} we denote the p-hull. This is a minimal restricted subalgebra containing H, and in fact H_{p} = ⟨h^{[p^{i}]} | h ∈ H, i = 0, 1, 2, … ⟩_{K}.

Suppose that Z is a finite-dimensional abelian p-algebra. Then its structure is determined by the p-mapping, which satisfies the relation

(λ_{1}x_{1} + λ_{2}x_{2})^{[p]} = λ_{1}^{p}x_{1}^{[p]} + λ_{2}^{p}x_{2}^{[p]}, x_{1}, x_{2} ∈ Z; λ_{1}, λ_{2} ∈ K.

For shortness we often write x^{p} instead of x^{[p]}. By N (Z) we denote the set of p-nilpotent elements of Z. Over an algebraically closed field there exists the following decomposition

Z = ⟨e_{1}, …, e_{q}|e_{i}^{p} = e_{i}⟩_{K} ⊕ N (Z).  (32)

Suppose that a group G acts on a space V and W is a G-invariant subspace. We set

V^{G} = {v ∈ V |g ∗ v = v, g ∈ G};
V^{G}(mod W) = {v ∈ V |g ∗ v = v(mod W), g ∈ G}.

Proposition 7.1. Let R = u(L)#K[G] be a PI-algebra, L has a finite - dimensional central restricted ideal Z with [L, L] ⊂ N (Z), and ωK[G] ∗ L ⊂ Z. Then there exists a subgroup of finite index B ⊂ G such that

dim L/(L^{B}(mod N (Z))) < ∞.

Proof. We suppose that the ground field K is algebraically closed. Let us factor out N (Z) and for simplicity keep the same notations. By (32) we have

Z = ⟨e_{1}, …, e_{q}|e_{i}^{p} = e_{i}⟩_{K}.  (33)

Recall that x is called a p-element if x^{p} = x. An easy check shows that there are only finitely many p-elements in (33), namely {n_{1}e_{1} +⋯+n_{q}e_{q}|n_{i} ∈ {0, 1, …, p− 1}}. Then Aut(Z) ⊂ S_{p^{q}}. Hence, there exists a subgroup of finite index B ⊂ G acting trivially on Z.

Let us prove by induction on q = dim Z that dim L/L^{B} < ∞. Let q = 1, then Z = ⟨e_{0}⟩_{K} where e_{0}^{p} = e_{0}. We have

g ∗ x = x + β(g, x)e_{0}, β(g, x) ∈ K, g ∈ B, x ∈ L.

We observe that β : B × L → (K, +) is the mapping into the additive group of the field, which is K -linear by the second argument. We set L_{1} = L. If β(B, L_{1}) ≠ 0 then there exist g_{1} ∈ B, x_{1} ∈ L_{1} such that β(g_{1}, x_{1}) = 1. We consider L_{2} = {x ∈ L_{1}|β(g_{1}, x) = 0}, then dim L_{1}/L_{2} = 1. If again β(B, L_{2}) ≠ 0 then there exist g_{2} ∈ B, x_{2} ∈ L_{2} such that β(g_{2}, x_{2}) = 1. Then we define L_{3} = {x ∈ L_{2}|β(g_{2}, x) = 0} and dim L_{2}/L_{3} = 1, etc. Suppose that we can make n steps. Then we have the elements g_{1}, …, g_{n} ∈ B; x_{1} ∈ L_{1}, …, x_{n} ∈ L_{n} such that

(g_{i} − 1) ∗ x_{i} = e_{0}, i = 1, …, n; (g_{i} − 1) ∗ x_{j} = 0, 1 ≤ i < j ≤ n.  (34)

We apply identity of Lemma 4.2

F_{2}(x_{1}, …, x_{n}, g_{1}, …, g_{n}) = ∑_{π∈S_{n}} α_{π}(g_{1} − 1) ∗ x_{π(1)} ⋯ (g_{n} − 1) ∗ x_{π(n)} = e_{0}^{n} ≡ 0, (35)

because by (34) the only nontrivial term is given by the identity permutation.

Of course, (35) is a contradiction. This contradiction proves that we cannot make n steps. Therefore, for some i ∈ {1, …, n} we have β(B, L_{i}) = 0. We remark that L_{i} = L^{B} and dim L/L^{B} < n. Since B acts trivially on Z, we have Z ⊂ L^{B}; also we observe that L^{B} is a restricted subalgebra.

Now suppose that dim Z = q > 1. Set D = ⟨e_{2}, …, e_{q}⟩_{K}. We consider L̃ = L/D, Z̃ = Z/D. We set L_{1} = L^{B}(mod D), by the inductive assumption for q = 1, we have dim L/L_{1} < ∞. Hence, ωK[B] ∗ L_{1} ⊂ D. We apply inductive assumption for D ⊂ L_{1}, where dim D = q − 1, and obtain that dim L_{1}/L_{1}^{B} < ∞. It remains to remark that L_{1}^{B} = L^{B}. Now we have proved that if K is algebraically closed then there exist the subgroup of finite index B ⊂ G and the number l such that dim L/(L^{B}(mod N (Z))) < l.

Now we consider the case of an arbitrary field K. Note that we can extend K by adjoining finitely many roots of p-polynomials and obtain the decomposition (32) (see [6]). Let K̄ ⊃ K be such an extension and dim_{K} K̄ = m. Given a K -space V we set V̄ = K̄ ⊗_{K} V . Then dim_{K} L̄/(L̄^{B}(mod N (Z̄))) < t = lm. Pick arbitrary x_{1}, …, x_{t} ∈ L. Then there exist α_{1}, …, α_{t} ∈ K such that α_{1}x_{1} + ⋯ + α_{t}x_{t} ∈ L̄^{B}(mod N (Z̄)). This means that for any g ∈ B we have

(g − 1) ∗ (α_{1}x_{1} + ⋯ + α_{t}x_{t}) ∈ N (Z̄) ∩ L ⊂ N (Z).

This relation reads also as dim_{K} L/(L^{B}(mod N (Z))) < t. This concludes the proof.

Now we come back to the proof Theorem 3.1.

Proof. The sufficiency was proved above. Now suppose that u(L)#K[G] satisfies a nontrivial polynomial identity.

First, let us prove that there exist G-invariant restricted subalgebras Q_{∗} ⊂ H_{∗} ⊂ L satisfying conditions 1) of Theorem. We recall the steps of the proof of Theorem 1.3 in [15] (see also this construction in [16] and [3]).

1) The existence of a nontrivial identity in u(L) implies that for some number m we have δ(L) = δ_{m}(L) and dim L/δ(L) < ∞. We set D = δ(L).

2) We apply Theorem 6.7 on bilinear maps and conclude that the commutator subalgebra D^{2} = [D, D] is finite dimensional. We set

C = C_{D}(D^{2}) = {x ∈ D|[x, D^{2}] = 0}.

Then dim D/C < ∞, and C^{3} = 0.

3) We have dim C^{2} < ∞ and (C^{2})_{p} is a finitely generated abelian p-algebra. By the structure of such algebras Q_{∗} = N ((C^{2})_{p}) is finite dimensional. Then the proof shows that H_{∗} = C(C, Q_{∗}) = {x ∈ C|[x, C] ⊂ Q_{∗}} has finite codimension in C.

We trace these steps and see by Lemma 2.4 that D, C, Q_{∗}, and H_{∗} are restricted invariant subalgebras. Hence, we obtain the required G-invariant restricted subalgebras Q_{∗} ⊂ H_{∗} ⊂ L, where H_{∗} is nilpotent of step 2.

Second, we apply results of Section 6.. By Theorem 6.8 there exist the Ginvariant restricted subalgebra of finite codimension L_{0} ⊂ L and the subgroup of finite index G_{0} ⊂ G such that dim(ωK[G_{0}] ∗ L_{0}) < ∞. Without loss of generality we may assume that G_{0} to be normal in G. We set Q_{0} = Q_{∗} ∩ L_{0}, H_{0} = H_{∗} ∩ L_{0}.

Next we consider Q_{1} = Q_{0} + ωK[G_{0}] ∗ H_{0} ⊂ H_{0}, by this construction dim Q_{1} < ∞. We observe that Q_{1} is a subalgebra, since [H_{0}, H_{0}] ⊂ Q_{0}. Let us check that Q_{1} is restricted. By the axioms of the p-map [6]

(x + y)^{[p]} = x^{[p]} + y^{[p]} + ∑_{i=1}^{p−1} s_{i}(x, y), x, y ∈ H_{0},  (36)

where each s_{i}(x, y) is the linear span of commutators in x, y of length p. Since H_{0} is nilpotent of step 2, all s_{i}(x, y) are equal to zero in the case p > 2 or belong to Q_{0} for p = 2. We compute

((g − 1) ∗ x)^{[p]} = (g ∗ x − x)^{[p]} = (g ∗ x)^{[p]} − x^{[p]}
= g ∗ (x^{[p]}) − x^{[p]} = (g − 1) ∗ (x^{[p]}), g ∈ G_{0}, x ∈ H_{0},

where in the case p = 2 we have some additional summands from Q_{0}. For arbitrary x ∈ Q_{1} we treat x^{p} again by (36) and conclude that Q_{1} is restricted. We also easily observe that Q_{1} is G-invariant because H_{0}, Q_{0} are G-invariant and G_{0} is a normal subgroup of G in which case G · ωK[G_{0}] = ωK[G_{0}] · G.

We set H_{1} = C_{H_{0}}(Q_{1}) = {x ∈ H_{0}|[x, Q_{1}] = 0}. Then H_{1} is G-invariant, dim H_{0}/H_{1} < ∞, Q_{1} is central in H_{1}, [H_{1}, H_{1}] ⊂ Q_{0} ⊂ N (Q_{1}), and ωK[G_{0}]∗H_{1} ⊂ Q_{1}. Now we can apply Proposition 7.1 to the smash product K[G_{0}]#H_{1} where Z = Q_{1}. We obtain the subgroup of finite index G_{1} ⊂ G_{0} and the subalgebras

Q = N (Q_{1}), H = H_{1}^{G_{1}}(mod Q),  (37)

where dim H_{1}/H < ∞. The proof of Proposition 7.1 allows us to assume that G_{1} is normal in G. The construction (37) shows that Q, H are G-invariant and that G_{1} acts trivially on H/Q.

Finally, we use Theorem 1.1 and find the subgroup A ⊂ G_{1} such that the conditions 2) of Theorem are satisfied.

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Yu. Bahturin Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, NF A1C 5S7 Canada bahturin@mun.ca V. M. Petrogradsky Faculty of Mathematics Ulyanovsk State University Lev Tolstoy 42 432700 Ulyanovsk Russia vmp@mmf.ulsu.ru

Received March 5, 2001 and in final form June 11, 2001