On matrix wreath products of algebras

. We introduce a new construction of matrix wreath products of algebras that is similar to the construction of wreath products of groups intro- duced by L. Kaloujnine and M. Krasner [17]. We then illustrate its usefulness by proving embedding theorems into ﬁnitely generated algebras and construct- ing nil algebras with prescribed Gelfand-Kirillov dimension.


Matrix Wreath Products
Let F be a field and let A, B be two associative F -algebras. Let Lin(A, B) denote the vector space of all F -linear transformations A → B.
We will define multiplication on Lin(B, B ⊗ F A). Let f, g ∈ Lin(B, B ⊗ F A). For an arbitrary element b ∈ B, let g Choose a basis {b i } i∈I of the algebra B. For a linear transformation f : Consider the I × I matrix A f = (a ij ) I×I . Each column of this matrix contains only finitely many nonzero entries. Let M I×I (A) denote the algebra of I × I matrices over A having finitely many nonzero entries in each column. Then f → A f , f ∈ Lin(B, B ⊗ F A), is an isomorphism Lin(B, B ⊗ F A) → M I×I (A).
The wreath product G 1 G 2 of two groups G 1 and G 2 embeds in the multiplicative group of the matrix wreath product F G 1 F G 2 of group algebras.
Indeed, let Fun(G 2 , G 1 ) be the group of mappings from G 2 to G 1 with pointwise multiplication: For a mapping f : If B M is a left module over the algebra B, then we can define Different constructions of wreath products of Lie algebras were introduced by A. L. Smel'kin [27] and V. Petrogradsky, Y. Razmyslov, E. Shishkin [24] and L. Bartholdi [3].
In what follows, we will always assume that the algebra B is finitely generated, infinite dimensional, and, moreover, {b ∈ B | dim bB < ∞} = (0).
Along with the algebra of matrices M I×I (A), we will consider two important subalgebras: (1) M ∞ (A) that consists of I × I matrices having finitely many nonzero entries, and (2) the subalgebra S(A, B) that consists of matrices having finitely many nonzero rows. In the language of linear transformations ϕ : B → B⊗ F A, the subalgebra S(A, B) consists of such ϕ for which there exists a finite dimensional subspace be a subalgebra such that BS + SB ⊆ S. Then (1) the algebra B + S is prime if and only if the algebra A is prime, and (2) the algebra B + S is (left) primitive if and only if the algebra A is primitive.
We say that a linear transformation γ : B → A is a generating linear transformation if γ(B) generates the algebra A. Suppose that 1 ∈ B. Let γ : B → A be a generating linear transformation. Consider the element Consider the subalgebra B, c γ generated in A B by B and the element c γ .
For an element a ∈ A and two indices i, j ∈ I, let e ij (a) denote the matrix whose (i, j)-entry is a and all other entries are equal to zero. For a fixed element u ∈ A, we consider also the subalgebra B, c γ , e 11 (u) . Clearly, B, c γ , e 11 (u) lies in B + S(A, B). If u = 1, then M ∞ (A) ⊆ B, c γ , e 11 (1) .
Since we always assume that the algebra B is finitely generated, the algebras B, c γ , B, c γ , e 11 (u) are finitely generated as well. Our immediate goal now is to estimate growth of these algebras.
We start with some general definitions. Consider an F -algebra R generated by a finite dimensional subspace V . Let Then dim F V n < ∞ and R is the union of the ascending chain V 1 ⊆ V 2 ⊆ · · · . The function g(V, n) = dim F V n is called the growth function of the algebra R that corresponds to the generating subspace V .
Let N denote the set of positive integers. Given two functions f, g : N → [1, ∞), we say that f g (f is asymptotically less than or equal to g) if there exists a constant c ∈ N such that f (n) ≤ cg(cn) for all n ∈ N. If f g and g f , then f and g are said to be asymptotically equivalent, i.e., f ∼ g. If V and W are finite dimensional generating subspaces of A, then g(V, n) ∼ g(W, n). We will denote the class of equivalence of g(V, n) as g A .
A finitely generated algebra R has polynomially bounded growth if there exists α > 0 such that g R (n) n α . Then If the growth of R is not polynomially bounded, then we let GKdim(R) = ∞. If the algebra R is not finitely generated, then the Gelfand-Kirillov dimension of R is defined as the supremum of Gelfand-Kirillov dimensions of all finitely generated subalgebras of R.
For n ≥ 1, consider the vector space We call w γ (n) the growth function of the linear transformation γ. A linear transformation γ : B → A is said to be dense if for arbitrary linearly independent elements b 1 , . . . , b n ∈ B and an arbitrary nonzero element a ∈ A, there exists an

Embedding Theorems
G. Higman, H. Neumann, and B. H. Neumann [15] proved that every countable group embeds in a finitely generated group. The papers [4], [23], [25], and [30] show that some important properties can be inherited by these embeddings. Much of this work relies on wreath products of groups.
Following [15], A. I. Malcev [21] showed that every countable dimensional associative algebra over a field is embeddable in a finitely generated algebra, and A. I. Shirshov [26] showed that every countable dimensional Lie algebra is embeddable in a finitely generated Lie algebra.
Let A be an associative algebra, and let I be a countable set. As above, we consider the algebra M ∞ (A) of I × I matrices having finitely many nonzero entries. Clearly, the algebra A is embeddable in M ∞ (A) in many ways. We say that an algebra A is M ∞ -embeddable in an algebra B if there exists an embedding ϕ : Observe that [1] extended the theorem of Malcev in the following way: every countable dimensional associative algebra over a field is M ∞ -embeddable in a finitely generated algebra as an ideal.

Radical Algebras
S. Amitsur [2] asked if a finitely generated algebra can have a non-nil Jacobson radical. The first examples of such algebras were constructed by K. Beidar [9]. J. Bell [6] constructed examples having finite Gelfand-Kirillov dimension. Finally, L. Bartholdi and A. Smoktunowicz [29] constructed a finitely generated Jacobson radical non-nil algebra of Gelfand-Kirillov dimension two.
Theorem 4. An arbitrary countable dimensional Jacobson radical algebra is embeddable in a finitely generated Jacobson radical algebra.
Theorem 5. An arbitrary countable dimensional Jacobson radical algebra of Gelfand-Kirillov dimension d over a countable field is embeddable in a finitely generated Jacobson radical algebra of Gelfand-Kirillov dimension ≤ d + 6.
We will start with the following lemma. Sketch of the proof of Theorem 4. Let B be a finitely generated infinite dimensional nil algebra of E. S. Golod [11]. Let B = B + F · 1 be its unital hull. Let A be the Jacobson radical algebra of Lemma 1, u ∈ A, u 3 = 0, A ≤ Au. Consider a generating linear transformation γ : B → A and the element c γ ∈ Lin( B, B ⊗ F A). Then the algebra B, c γ , e 11 (u) is finitely generated and Jacobson radical. Hence, the algebra A is embeddable in a finitely generated Jacobson radical algebra B, c γ , e 11 (u) .
To prove Theorem 5, we will need the following lemma.
Lemma 2. Let A be a countable dimensional algebra of Gelfand-Kirillov dimension ≤ d. Let B be an arbitrary finitely generated algebra. Then there exists a generating linear transformation γ : B → A such that w γ (n) ≤ n d+ n where n > 0, n → 0 as n → ∞.
Instead of the Golod nil algebra B, we will consider a finitely generated nil algebra B of polynomially bounded growth. Existence of such algebras was established by T. Lenagan and A. Smoktunowicz in [19] under the assumption that the ground field is countable. In [20], T. Lenagan, A. Smoktunowicz, and A. Young refined the argument of [19] and obtained a finitely generated nil algebra of Gelfand-Kirillov dimension ≤ 3.
Let A → Au, u 3 = 0, be the embedding of Lemma 1, and let B be the nil algebra of [20]. Arguing as above, we embed the algebra A in the finitely generated subalgebra B, c γ , e 11 (u) of A B, where γ is a generating linear transformation of Lemma 2. By Theorem 3(1), we have which implies GKdim B, c γ , e 11 (u) ≤ d + 6.

Nil Algebras
We say that a nil algebra A is stable nil (resp. stable algebraic) if all matrix algebras M n (A) are nil (resp. algebraic).
Theorem 6. An arbitrary stable nil algebra A is embeddable in a finitely generated stable nil algebra. If GKdim A = d < ∞ and the ground field is countable, then A is embeddable in a finitely generated nil algebra of Gelfand-Kirillov dimension ≤ d + 6.
To use the wreath product constructions as above, we will need a finitely generated infinite dimensional stable nil algebra. Existence of such algebras can be established using methods from E. S. Golod [11] based on Golod-Shafarevich inequalities [12].
More precisely, let F x 1 , · · · , x m be the associative algebra on m free generators, m ≥ 2. We consider the free algebra without 1, i.e., it consists of formal linear combinations of nonempty words in x 1 , . . . , x m . Assigning degree 1 to all variables x 1 , . . . , x m , we make F x 1 , . . . , x m a graded algebra. The degree deg(a) of an arbitrary element a ∈ F x 1 , . . . , x m is defined as the minimal degree of a nonzero homogeneous component of a.
Let R ⊂ F x 1 , . . . , x m be a subset containing finitely many elements of each degree. For a stable nil algebra A and its extension A ⊂ Au, u 3 = 0, of Lemma 1 and an algebra B of Lemma 3, the finitely generated algebra B, c γ , e 11 (u) is stable nil. It implies the first part of Theorem 6. Now let F be a countable field, let B be the Lenagan-Smoktunowicz-Young algebra [20], and let A be a countable dimensional stable nil algebra of GKdim A ≤ d. Then the algebra B, c γ , e 11 (u) is nil and has Gelfand-Kirillov dimension ≤ d + 6. We do not know if this finitely generated algebra is stable nil.

Primitive Algebras
I. Kaplansky [18] asked if there exists an infinite dimensional finitely generated algebraic primitive algebra, a particular case of the celebrated Kurosh Problem. Such examples were constructed by J. Bell and L. Small in [7]. Then J. Bell, L. Small, and A. Smoktunowicz [8] constructed finitely generated algebraic primitive algebras of finite Gelfand-Kirillov dimension provided that the ground field is countable.
Theorem 7. An arbitrary countable dimensional stable algebraic primitive algebra is M ∞ -embeddable as a left ideal in a 2-generated algebraic primitive algebra.
This theorem answers the first part of question 7 from [8].
Theorem 8. Let F be a countable field. An arbitrary countable dimensional stable algebraic primitive algebra of Gelfand-Kirillov dimension ≤ d is M ∞ -embeddable as a left ideal in a finitely generated algebraic primitive algebra of Gelfand-Kirillov dimension ≤ d + 6.
Without loss of generality, we assume that a countable dimensional stable algebraic algebra A is unital. As above, we start with Golod's finitely generated infinite dimensional nil algebra B and a generating linear transformation γ : B → A. Then the algebra B, c γ , e 11 (1) is primitive by Theorem 2(2) and contains M ∞ (A) as a left ideal.
The same argument with the Lenagan-Smoktunowicz-Young algebra B and a linear transformation of Lemma 2 implies Theorem 8.

Algebras of Locally Subexponential Growth
Recently, L. Bartholdi and A. Erschler [4] proved that a countable group of locally subexponential growth embeds in a finitely generated group of subexponential growth. We prove the analog of Bartholdi-Erschler theorem for algebras and semigroups and establish some related results.
Given two functions f, g : N → [1, ∞), we say that f is weakly asymptotically less than or equal to g if for arbitrary α > 0, we have f gn α (denoted f w g).
A function f is subexponential if lim n→∞ f (n) e αn = 0 for any α > 0. In the seminal paper [14], R. I. Grigorchuk constructed the first example of a group with an intermediate growth function: subexponential but growing faster than any polynomial. Finitely generated associative algebras with intermediate growth functions come as universal enveloping algebras of certain Lie algebras (see [28]).
A not necessarily finitely generated algebra A is of locally subexponential growth if every finitely generated subalgebra of A has a subexponential growth function.
The growth of A is locally (resp. weakly) bounded by a function f (n) if for every finitely generated subalgebra of A its growth function is f (n) (resp. w f (n)).
Theorem 9. Let f (n) be an increasing function. Let A be a countable dimensional associative algebra whose growth is locally weakly bounded by f (n). Let h(n) be a superlinear function. Then the algebra A is M ∞ -embeddable as a left ideal in a 2-generated algebra whose growth is weakly bounded by f (h(n))n 2 .
We then use Theorem 9 to derive an analog of the Bartholdi-Erschler Theorem ( [4]).
Theorem 10. A countable dimensional associative algebra of locally subexponential growth is M ∞ -embeddable in a 2-generated algebra of subexponential growth as a left ideal.
The idea of the proofs of Theorems 9 and 10 is the same as in previous sections. We consider the matrix wreath product A F [t −1 , t] with the algebra F [t −1 , t] of Laurent polynomials and choose a generating linear transformation γ : F [t −1 , t] → A with appropriate subexponential growth function w γ (n). The algebra A is then M ∞ -embeddable as a left ideal in the finitely generated algebra C = F [t −1 , t], c γ , e 11 (1) . By V. T. Markov's theorem [22], the matrix algebra M n (C) is 2-generated for a sufficiently large n, which yields the result.
Using [28] and Theorem 10, we can prove an embedding theorem for countable dimensional Lie algebra of locally subexponential growth.
Theorem 11. Let F be a field of characteristic = 2. Every countable dimensional Lie F -algebra of locally subexponential growth is embeddable in a finitely generated Lie algebra of subexponential growth.

Gelfand-Kirillov Dimension of Nil Algebras
In this section, we assume that the ground field F is countable. Question 1 from [8] asks if an arbitrary sufficiently big α ≥ 2 is the Gelfand-Kirillov dimension of some finitely generated nil algebra.
Theorem 12. Let F be a countable field. For an arbitrary d ≥ 8, there exists a finitely generated nil F -algebra of Gelfand-Kirillov dimension d.
For an arbitrary α ≥ 2, W. Bohro and H. P. Kraft [10] constructed a graded R i , generated by two elements x, y ∈ R 1 , such that for any > 0 we have for all sufficiently large n.
Using the Bohro-Kraft algebra, we construct a countable dimensional locally nilpotent algebra A and a dense generating linear transformation γ : B → A of growth w γ (n) such that for an arbitrary 0 < < α, we have n ln n n− w γ (n) n α+ (ln(ln n)) 2 .
Question. Let g : N → N be an increasing function such that n 2 g(n) and g(m + n) ≤ g(m)g(n) for all m, n ∈ N. Is g(n) asymptotically equivalent to the growth function of some finitely generated associative algebra?
Conjecture. For all sufficiently large functions g : N → N, the following assertions are equivalent: (1) g is asymptotically equivalent to the growth function of some finitely generated associative algebra, (2) g is asymptotically equivalent to the growth function of some finitely generated primitive algebra, (3) g is asymptotically equivalent to the growth function of some finitely generated nil algebra.
L. Bartholdi and A. Smoktunowicz [5] proved that if g is an increasing submultiplicative function such that g(Cn) ≥ ng(n) for some C ∈ N and all n ∈ N then g is asymptotically equivalent to the growth function of a finitely generated associative algebra. Moreover, B. Greenfeld [13] showed that in this case there exists a finitely generated primitive monomial algebra with the growth function equivalent to g. This partially answers the questions above.