Gelfand-Kirillov Dimension and Local Finiteness of Special Jordan Superalgebras ^{1}

Esther García ^{2}

Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo, Spain

Abstract: In this work we extend to superalgebras a result of Skosyrskii [Algebra and Logic, 18 (1) (1979) 49-57, Lemma 2] relating associative and Jordan structures. As an application, we use it to show that the Gelfand-Kirillov dimension of an associative superalgebra coincides with that of its symmetrization, and that local finiteness is equivalent in associative superalgebras and in their symmetrizations.

2000 Math. Subj. Class.: 17C70, 17A70. Introduction

Several concepts are used to study the complexity of an algebra, and among them we can highlight local finiteness, which is the natural generalization of finite dimensionality, and the Gelfand-Kirillov dimension.

A big part of the research on local finiteness of algebras derives from Kurosh Problem (1941) on local finiteness of algebraic associative algebras, which turned up to have a negative answer in general but holds for many classes of algebras. Since then, many people have got interested in several classes of locally finite algebras such as algebraic PI-Jordan algebras (studied by Zelmanov [12]), or compact alternative or Jordan algebras (studied by Slinko [11]). It is also natural to ask which conditions are necessary to get locally finite algebras, and this is one of the goals of this paper for the particular case of Jordan superalgebras obtained by symmetrization of associative superalgebras.

Symmetrizations of associative algebras arise as the most natural way of building examples of Jordan algebras and constitute one of the fundamental models in the structure theory of Jordan algebras [8, 15.5]. Analogously, one can consider symmetrizations of associative superalgebras (see [4, 9]), and this provides a way of classifying Jordan superalgebras in special superalgebras – when they can be seen as subalgebras of the symmetrization of some associative superalgebra –, or exceptional ones – when that is not possible.

Another way of measuring how “big” an algebra is is the Gelfand-Kirillov dimension. In a sense, this invariant measures the deviation of an algebra from local finiteness since the Gelfand-Kirillov dimension of an associative algebra is zero if and only if it is locally finite, while non-locally finite associative algebras have Gelfand-Kirillov dimension greater than or equal to one. Indeed, it can be shown that there exist algebras whose Gelfand-Kirillov dimension equals any given real number ≥ 2, and that this number can be even infinite. The Gelfand-Kirillov dimension has been widely considered for Lie algebras, and concerning Jordan algebras the main advances in this area are the work on Jordan algebras of Gelfand-Kirillov dimension one by Martínez-Zelmanov [7], based on earlier work by Martínez [6], and the recent classification of graded simple Jordan superalgebras of growth one by Kac-Martínez-Zelmanov [3]. It is therefore interesting to study how this property behaves with respect to symmetrization, and that is another purpose of this work.

1 Partially supported by the Ministerio de Ciencia y Tecnología BFM2001-1938-C02-02 and the Universidad de Oviedo AYP-01-06.

2 Partially supported by a F.P.I. Grant (Ministerio de Ciencia y Tecnología).

The study of local finiteness and the Gelfand-Kirillov dimension of symmetrizations of associative superalgebras involves finding relations between associative and Jordan elements in the underlying supermodule. In the non-supersetting, this has been solved by Skosyrskii in [10, Lemma 2], who, indeed, worked in the more general case of right alternative algebras. For our purposes it is enough to consider associative algebras, and his result can be rewritten as follows:

Any monomial x in the free associative algebra FA[a_{1}, …, a_{n}] can be written as a homogeneous linear combination of monomials of the form za_{i_{1}} ⋯ a_{i_{s}}, where z is a Jordan monomial in the free special Jordan algebra FJS[a_{1}, …, a_{n}] and 1 ≤ i_{1} < i_{2} < … < i_{s} ≤ n. This result is used in [6, Cor. 1.2] to show that the Gelfand-Kirillov dimension

of an associative algebra coincides with that of its symmetrization, and a similar argument also proves that an associative algebra A is locally finite if and only if A^{(+)} is so.

Our aim is to get analogues of these results for associative superalgebras A, but unfortunately Skosyrskii’s lemma does not hold in the supercase. Moreover, due to the fact that squares and U operators are not defined on odd elements of A^{(+)}, Skosyrskii’s arguments are not valid in the superalgebra case.

After a preliminary section, we obtain in Section 1 a suitable extension to superalgebras of [10, Lemma 2]. Although the result is weaker than the original one, it still allows us to relate the Gelfand-Kirillov dimension and the local finiteness of associative superalgebras and their symmetrizations, as shown in Sections 2 and 3, respectively.

0. Preliminaries

0.1 We will deal with algebraic structures defined over an arbitrary ring of scalars Φ. A Φ-supermodule is a ℤ_{2}-graded Φ-module M = M¯0 ⊕ M¯1, where M¯0 will be called the even part of M and M¯1 will be called its odd part. For m ∈ M_{μ}, μ ∈ ℤ_{2} = {¯0, ¯1}, we put |m| = μ and call it the degree (or parity) of m. Elements in M_{μ}, μ ∈ ℤ_{2}, are called homogeneous elements.

A superalgebra over Φ is a Φ-algebra A with a ℤ_{2}-grading, i.e., A = A¯0 ⊕ A¯1 and A_{i}A_{j} ⊆ A_{i+j}, for i, j ∈ ℤ_{2}. Therefore, A¯0 is a Φ-algebra itself and A¯1 is an A¯0-module.

In general we will assume that all substructures (submodules, subalgebras and ideals) are also ℤ_{2}-graded.

We denote by G the Grassmann algebra over Φ in a countable number of generators. It is ℤ_{2}-graded: G = G¯0 ⊕ G¯1, where G¯0 – respectively G¯1 – is spanned by monomials in an even – respectively odd – number of generators.

Let A = A¯0⊕A¯1 be a superalgebra. The Grassmann envelope G(A) = (G¯0⊗A¯0)⊕ (G¯1 ⊗ A¯1) ⊆ G ⊗ A is then an algebra with respect to the product (g_{a} ⊗ a)(g_{b} ⊗ b) = g_{a}g_{b} ⊗ab for homogenous elements g_{a}, g_{b} ∈ G, a, b ∈ A satisfying |g_{a}| = |a| and |g_{b}| = |b|. One can then define an alternative, associative, commutative, or Lie superalgebra by requiring that its Grassmann envelope is, respectively, an alternative, associative, commutative, or Lie algebra. Note that associative superalgebras coincide with ℤ_{2}- graded associative algebras. For a supermodule M the Grassmann envelope G(M) is defined as for superalgebras.

This approach to defining varieties of superalgebras also works for Jordan superalgebras in case \frac{1}{2} ∈ Φ, because then Jordan (super)algebras can be defined via a bilinear product.

However, since our results do not need the assumption \frac{1}{2} ∈ Φ, we will use the general approach to Jordan supersystems via quadratic maps ([4, 9]).

0.2 Suppose we have a Φ-supermodule J together with quadratic maps

U : J → End_{Φ}(J) and (.)^{2} : J → J.

By definition (see [9, 2.8]), we therefore have supersymmetric Φ-bilinear maps of degree 0

U(., .) : J × J → End_{Φ}(J) and ∘ : J × J → J,

and Φ-quadratic maps

U¯0 : J¯0 → End_{Φ}(J)¯0 and (.)^{2} : J¯0 → J¯0,

which are related by

\begin{aligned}U(u, w) & = U¯0(u + w) − U¯0(u) − U¯0(w) and \\ u ∘ v & = (u + v)^{2} − u^{2} − v^{2}, \end{aligned}

for u, w ∈ J¯0.

These quadratic maps induce canonical G-quadratic maps [9, 2.10]

Ũ : G(J) → End_{Φ}(G(J)) and ˜(.)^{2} : G(J) → G(J),

and one calls J a Jordan superalgebra if its Grassmann envelope G(J) together with Ũ and ˜(.)^{2} is a Jordan algebra in the usual sense [2, 5].

Subalgebras of Jordan superalgebras are defined in the obvious way.

Note that the quadratic map U induces a Φ-trilinear map {., ., .} : J ×J ×J → J related to U by {u, v, w} = (−1)^{|v||w|}U(u, w)v .

0.3 Let A be an associative superalgebra. For a¯0 ∈ A¯0 and homogeneous a, b, c ∈ A define

\begin{matrix}a2 ¯0 = a¯0a¯0 & a ∘ b = ab + (−1)^{|a||b|}ba, and \\ U¯0(a¯0)b = a¯0ba¯0 & \{a, b, c\} = abc + (−1)^{|a||b|+|a||c|+|b||c|}cba. \end{matrix}  (1)

Then the supermodule underlying A together with the products above is a Jordan superalgebra, called the (super)symmetrization of A and denoted by A^{(+)} (cf. [1]).

1. A lemma of Skosyrskii for superalgebras

1.1 Let a_{1}, …, a_{s}, a_{s+1}, …, a_{t} be a set of generators and let FA[a_{1}, …, a_{t}] be the free associative algebra in those generators. In FA[a_{1}, …, a_{t}] we define a ℤ_{2}- grading in the following way: a monomial x = a_{i_{1}} ⋯ a_{i_{n}} is even if the number of variables from {a_{s+1}, …, a_{t}} appearing in x is zero or even; otherwise x is odd. Therefore, FA[a_{1}, …, a_{t}] is an associative superalgebra with even part spanned by even monomials and odd part spanned by odd monomials. In particular, a_{1}, …, a_{s} are even monomials, while a_{s+1}, …, a_{t} are odd monomials. Thus a_{1}, …, a_{s} will be said even generators and a_{s+1}, …, a_{t} will be said odd generators.

We will denote by FJS_{s}[a_{1}, …, a_{t}] the subalgebra of the (super)symmetrization FA[a_{1}, …, a_{t}]^{(+)} generated by a_{1}, …, a_{t}, which is a Jordan superalgebra.

Assigning to each homogeneous generator a_{1}, …, a_{t} the degree 1 yields a natural degree function on FA[a_{1}, …, a_{t}] and on FJS_{s}[a_{1}, …, a_{t}] (besides the just defined ℤ_{2}- grading). For a monomial w ∈ FA[a_{1}, …, a_{t}] we will say that deg_{a_{i}}(w) = n_{i} if w contains a_{i} exactly n_{i} times (if w does not contain any a_{i} we will write deg_{a_{i}}(w) = 0). We define the length of a monomial w ∈ FA[a_{1}, …, a_{t}] as length(w) := ∑_{i=1}^{t} deg_{a_{i}}(w).

1.2 Lemma. Let {a_{1}, …, a_{s}} be a set of even generators, {a_{s+1}, …, a_{t}} be a set of odd generators and let z be a new (even or odd) variable. If w = w(a_{i_{1}}, …, a_{i_{r}}, z) is an associative monomial in a_{i_{1}}, …, a_{i_{r}}, z in which i_{j} ∈ {1, …, t}, deg_{a_{i_{j}}} (w) = n_{j} ≥ 0, for j = 1, …, r, and deg_{z}(w) = 1, then w can be written as a homogeneous linear combination of terms of the form x = x̃z̃ where x̃ = a_{j_{1}} ⋯ a_{j_{v}} for a_{j_{1}}, …, a_{j_{v}} ∈ {a_{i_{1}}, …, a_{i_{r}}}, deg_{a_{i_{j}}} (x̃) ≤ n_{j} for j = 1, …, r, and z̃ ∈ FJS_{s}[a_{i_{1}}, …, a_{i_{r}}, z].

Proof: We will use induction on the length of w. If length(w) = 1 then w = z hence it satisfies the claim. Suppose that the claim of the lemma is true for monomials of length less than n and let w be a monomial of length n > 1. If w finishes with a z we have nothing to prove. Otherwise, w = w̃a_{i_{k}}, n_{k} ≥ 1 and length(w̃) = n − 1 and deg_{a_{i_{k}}} (w̃) = n_{k} − 1. Applying the induction hypothesis to w̃ we get that w can be written as a homogeneous linear combination of monomials of the form x̃z̃a_{i_{k}} = a_{j_{1}} ⋯ a_{j_{v}} z̃a_{i_{k}} where a_{j_{1}}, …, a_{j_{v}} ∈ {a_{i_{1}}, …, a_{i_{r}}}, deg_{a_{i_{1}}} (x̃) ≤ n_{1}, …, deg_{a_{i_{k}}} (x̃) ≤ n_{k} − 1, …, deg_{a_{ir}} (x̃) ≤ n_{r} and z̃ ∈ FJS_{s}[a_{i_{1}}, …, a_{i_{r}}, z].

Moreover x̃z̃a_{i_{k}} = x̃(z̃ ∘ a_{i_{k}}) − (−1)^{|z̃||a_{i_{k}} |}x̃a_{i_{k}} z̃ and these last two terms satisfy the claim of the lemma (notice that deg_{a_{i_{k}}} (x̃a_{i_{k}}) = deg_{a_{i_{k}}} (x̃) + 1 ≤ n_{k}).

1.3 Proposition. Let {a_{1}, …, a_{s}} be a set of even generators and let y be a new variable (either even or odd). Then any homogeneous element of FA[a_{1}, …, a_{s}, y] of the form x = a_{j_{1}} ⋯ a_{j_{r}} ŷ, where a_{j_{1}}, …, a_{j_{r}} ∈ {a_{1}, …, a_{s}} and ŷ ∈ FJS_{s}[a_{1}, …, a_{s}, y], can be expressed as a homogeneous linear combination of terms of the form za_{i_{1}} ⋯ a_{i_{t}}, where a_{i_{1}}, …, a_{i_{t}} ∈ {a_{j_{1}}, …, a_{j_{r}}}, t ≥ 0, 1 ≤ i_{1} < i_{2} < … < i_{t} ≤ s (hence t ≤ r), and z ∈ FJS_{s}[a_{1}, …, a_{s}, y].

Proof: We will use induction on r (where r denotes the number of a′s appearing before the Jordan element in the expression x = a_{j_{1}} ⋯ a_{j_{r}} ŷ). If r = 0 then x = ŷ ∈ FJS_{s}[a_{1}, …, a_{s}, y], t = 0, and we have the result. Suppose then that r > 0 and that the claim is true when the number of a′s before the Jordan element is less than r. Since r > 0, x = a_{j_{1}} x̃ where the number of a′s appearing in x̃ before the Jordan element is r − 1, hence applying the induction hypothesis to x̃ we get that x can be written as a homogeneous linear combination of terms of the form a_{j_{1}}za_{i_{1}} ⋯ a_{i_{t}}, where a_{i_{1}}, …, a_{i_{t}} ∈ {a_{j_{2}}, …, a_{j_{r}}}, t ≥ 0, 1 ≤ i_{1} < i_{2} < … < i_{t} ≤ s, t ≤ r − 1 and z ∈ FJS_{s}[a_{1}, …, a_{s}, y].

The following two equalities will be useful:

a_{j_{1}}za_{i_{1}} ⋯ a_{i_{t}} = \{ \begin{matrix}(U_{a_{i_{1}}} z)a_{i_{2}} ⋯ a_{i_{t}}, if j_{1} = i_{1} \\ (a_{j_{1}} ∘ z)a_{i_{1}} ⋯ a_{i_{t}} − za_{j_{1}}a_{i_{1}} ⋯ a_{i_{t}}, for any j_{1}. \end{matrix}  (1)

We will now use a second induction on j_{1} to show that an element of the form a_{j_{1}}za_{i_{1}} ⋯ a_{i_{t}} with z ∈ FJS_{s}[a_{1}, …, a_{s}, y], 0 ≤ t < r, a_{j_{1}}, a_{i_{1}}, …, a_{i_{t}} ∈ {a_{1}, …, a_{s}}, and 1 ≤ i_{1} < … < i_{t} ≤ s satisfies the assertion of the proposition. The case j_{1} = 1 is immediate from (1) because j_{1} = 1 ≤ i_{1}. We can therefore suppose that j_{1} > 1 and that the claim is true for those elements beginning with a_{k}, k < j_{1}. If j_{1} ≤ i_{1} we can use (1) again. For j_{1} > i_{1},

a_{j_{1}}za_{i_{1}} ⋯ a_{i_{t}} = (a_{j_{1}} ∘ z)a_{i_{1}} ⋯ a_{i_{t}} − {z, a_{j_{1}}, a_{i_{1}}}a_{i_{2}} ⋯ a_{i_{t}} + a_{i_{1}}a_{j_{1}}za_{i_{2}} ⋯ a_{i_{t}}. (2)

The first two terms in (2) have the proper form. Let us study the third one. Call w = a_{j_{1}}za_{i_{2}} ⋯ a_{i_{t}}, which is a monomial in a_{j_{1}}, z, a_{i_{2}}, …, a_{i_{t}} with deg_{z}(w) = 1. Using (1.2) w can be written as a homogeneous linear combination of terms of the form w̃z̃ = a_{l_{1}} ⋯ a_{l_{v}} z̃, where a_{l_{1}}, …, a_{l_{v}} ∈ {a_{j_{1}}, a_{i_{2}}, …, a_{i_{t}}}, z̃ ∈ FJS_{s}[a_{j_{1}}, a_{i_{2}}, …, a_{i_{t}}, z] ⊆ FJS_{s}[a_{1}, …, a_{s}, y], and the number of a′s appearing before z̃ is less than or equal to t < r. By our induction hypothesis on the number of a′s before the Jordan element, each w̃z̃ can be written as a homogeneous linear combination of terms of the form pa_{m_{1}} ⋯ a_{m_{q}}, where p ∈ FJS_{s}[a_{1}, …, a_{s}, y] and a_{m_{1}}, …, a_{m_{q}} ∈ {a_{l_{1}}, …, a_{l_{v}}} ⊆ {a_{j_{1}}, a_{i_{2}}, …, a_{i_{t}}}, q ≥ 0 and 1 ≤ m_{1} < m_{2} < … < m_{q} ≤ s. Therefore a_{i_{1}}w is a homogeneous linear combination of terms of the form a_{i_{1}}pa_{m_{1}} ⋯ a_{m_{q}}, and since i_{1} < j_{1} we can use the second induction to get that a_{i_{1}}pa_{m_{1}} ⋯ a_{m_{q}} can be written as a homogeneous linear combination of terms satisfying the claim of our proposition.

1.4 Lemma. Let {a_{1}, …, a_{s}} be a set of even generators and {b_{1}, …, b_{r}} be a set of odd generators. Let A = FA[a_{1}, …, a_{s}, b_{1}, …, b_{r}] be the free associative superalgebra on those generators and let J be the Jordan superalgebra (subalgebra of A^{(+)}) generated by a_{1}, …, a_{s}, b_{1}, …, b_{r}, i.e., J = FJS_{s}[a_{1}, …, a_{s}, b_{1}, …, b_{r}]. Then any monomial x ∈ A can be written as a homogeneous linear combination of terms of the form b_{i_{1}} ⋯ b_{i_{t}}a_{j_{1}} ⋯ a_{j_{u}}y where t ≥ 0, u ≥ 0 and y ∈ J is a Jordan monomial.

Proof: We will use induction on the length of x. If length(x) = 1 then either x = a_{i} or x = b_{j}, thus it has the form of the claim since x ∈ J. Suppose that the result is true for monomials of length less than n and let x be a monomial of length n > 1.

(a) If x finishes with an a_{i} then x = x̃a_{i} for a monomial x̃ ∈ A such that length(x̃) < n. Applying the induction hypothesis to x̃, x can be written as a homogeneous linear combination of terms of the form b_{i_{1}} ⋯ b_{i_{t}}a_{j_{1}} ⋯ a_{j_{u}}ya_{i} = b_{i_{1}} ⋯ b_{i_{t}}a_{j_{1}} ⋯ a_{j_{u}}(y∘a_{i})− b_{i_{1}} ⋯ b_{i_{t}}a_{j_{1}} ⋯ a_{j_{u}}a_{i}y, so that we have the result in this case.

(b) If x finishes with a b_{i} then x = x̃b_{i} for a monomial x̃ ∈ A such that length(x̃) < n. Applying the induction hypothesis to x̃, x can be written as a homogeneous linear combination of terms of the form b_{i_{1}} ⋯ b_{i_{t}}a_{j_{1}} ⋯ a_{j_{u}}yb_{i}. –If u = 0 then

b_{i_{1}} ⋯ b_{i_{t}}yb_{i} = b_{i_{1}} ⋯ b_{i_{t}}(y ∘ b_{i}) − (−1)^{|y|}b_{i_{1}} ⋯ b_{i_{t}}b_{i}y

and both terms are under the requirements of the claim.

–If u > 0 then

\begin{aligned}b_{i_{1}} ⋯ b_{i_{t}}a_{j_{1}} ⋯ a_{j_{u}}yb_{i} & = b_{i_{1}} ⋯ b_{i_{t}}a_{j_{1}} ⋯ a_{j_{u−1}}\{a_{j_{u}}, y, b_{i}\} \\ & − (−1)^{|y|}b_{i_{1}} ⋯ b_{i_{t}}a_{j_{1}} ⋯ a_{j_{u−1}}b_{i}ya_{j_{u}}, \end{aligned}

where t ≥ 0, u > 0 and y ∈ J is a Jordan monomial. Clearly, the first summand in this last expression has the form of the claim and the second one can be written in the desired way by (a).

1.5 Remark: In the situation described in (1.4), considering all products b_{i}b_{j}, i, j = 1, …, r, we get r^{2} even elements. Let us denote them by a_{s+1}, …, a_{t} (here t − s = r^{2}), and set b_{0} = 1. Therefore, Lemma (1.4) can be read as follows: any monomial x ∈ A can be expressed as a homogeneous linear combination of terms of the form b_{i}a_{k_{1}} ⋯ a_{k_{u}}y, where i ∈ {0, 1, …, r}, a_{k_{1}}, …, a_{k_{u}} ∈ {a_{1}, …, a_{s}, a_{s+1}, …, a_{t}}, u ≥ 0 and y is a Jordan monomial. Indeed, by (1.4) x is a homogeneous linear combination of terms of the form b_{i_{1}} ⋯ b_{i_{w}}a_{j_{1}} ⋯ a_{j_{v}}y, where b_{i_{1}} ⋯ b_{i_{w}}a_{j_{1}} ⋯ a_{j_{v}} is a monomial in FA[a_{1}, …, a_{s}, b_{1}, … b_{r}] and y is a Jordan monomial. Then set b_{i} = b_{i_{1}} if w is odd and b_{i} = 1 if w is even, and group the rest of b′s two by two, getting new a′s from {a_{s+1}, …, a_{t}}.

1.6 Proposition. Let {a_{1}, …, a_{s}} be a set of even generators, {b_{1}, …, b_{r}} be a set of odd generators, and consider A = FA[a_{1}, …, a_{s}, b_{1}, …, b_{r}], which is in fact an associative superalgebra. Consider all possible products of two odd generators b_{i}b_{j} and call them a_{s+1}, …, a_{t}. Then any monomial x ∈ A can be written as a homogeneous linear combination of terms of the form b_{k}za_{i_{1}} ⋯ a_{i_{p}}, where k ∈ {0, 1, …, r} and b_{0} = 1, a_{i_{1}}, …, a_{i_{p}} ∈ {a_{1}, …, a_{t}}, p ≥ 0 and 1 ≤ i_{1} < i_{2} < … < i_{p} ≤ t, and z is a homogeneous Jordan polynomial in a_{1}, …, a_{t}, b_{1}, …, b_{r}.

Proof: By (1.5) any monomial x ∈ A can be written as a homogeneous linear combination of terms of the form b_{k}a_{j_{1}} ⋯ a_{j_{u}}y, where k ∈ {0, 1, …, r}, a_{j_{1}}, …, a_{j_{u}} ∈ {a_{1}, …, a_{t}}, u ≥ 0, and y ∈ FJS_{s}[a_{1}, …, a_{s}, b_{1}, …, b_{r}]. Set x̃ = a_{j_{1}} ⋯ a_{j_{u}}y. By (1.3) x̃ can be written as a homogeneous linear combination of terms of the form za_{i_{1}} ⋯ a_{i_{p}}, where z is a homogeneous Jordan polynomial in a_{1}, …, a_{t}, y (hence in a_{1}, …, a_{t}, b_{1}, …, b_{r}), a_{i_{1}}, …, a_{i_{p}} ∈ {a_{1}, …, a_{t}}, p ≥ 0, and 1 ≤ i_{1} < i_{2} < … < i_{p} ≤ t. Therefore, x satisfies the claim.

2. Gelfand-Kirillov dimension of symmetrizations of associative superalgebras

From now on we will assume that all superstructures are defined over a field k of arbitrary characteristic.

In this section we will study the Gelfand-Kirillov dimension (GK-dimension for short) of Jordan superalgebras obtained by symmetrization of associative superalgebras.

2.1 Let J be a Jordan superalgebra over a field k. For any subspace B = B¯0⊕B¯1 of J we define B^{j(n)} and B^{j[n]} for n ∈ ℕ by

\begin{aligned}B^{j(1)} = & B \\ B^{j(2)} = & B ∘ B + (B¯0)^{2} \\ B^{j(n)} = & ∑_{l+k+m=n}\{B^{j(l)} , B^{j(k)} , B^{j(m)}\} + ∑_{2l+k=n} U¯0(Bj(l) ; ¯0)B^{j(k)} \\ & + ∑_{l+k=n} B^{j(l)} ∘ B^{j(k)} + ∑_{2l=n}(Bj(l) ; ¯0)^{2} \\ B^{j[n]} = & ∑_{1≤i≤n} B^{j(i)}. \end{aligned}  (1)

The Gelfand-Kirillov dimension of a Jordan superalgebra J is defined as

GKdim J = sup_{B} (lim sup_{n} \frac{ln(dim B^{j[n]})}{ln n}) ,  (2)

where the supremum is taken over all finite dimensional subspaces B of J. Here and in the following we write dim for dim_{k}. Obviously, the GK-dimension of any subalgebra of J is less than or equal to the GK-dimension of J.

2.2 Let A denote a (non)associative Φ-superalgebra with bilinear product denoted by juxtaposition. For any subspace B of A and any n ∈ ℕ we define

B^{(1)} = B, B^{(n)} = ∑_{i+j=n} B^{(i)}B^{(j)}, B^{[n]} = ∑_{1≤i≤n} B^{(i)},  (1)

and the Gelfand-Kirillov dimension of A as

GKdim A = sup_{B} (lim sup_{n} \frac{ln(dim B^{[n]})}{ln n}) ,  (2)

where the supremum is taken over all finite dimensional subspaces B of A.

2.3 Proposition. Let A be an associative superalgebra and let A^{(+)} be the Jordan superalgebra obtained by symmetrization of A. Then GKdim A = GKdim A^{(+)}.

Proof: Let B be a finite dimensional subspace of A. By (0.3) it is clear that B^{j[n]} ⊆ B^{[n]} for any n ∈ ℕ, hence

GKdim A^{(+)} = sup_{B} (lim sup_{n} \frac{ln(dim B^{j[n]})}{ln n})
≤ sup_{B} (lim sup_{n} \frac{ln(dim B^{[n])})}{ln n}) = GKdim A,

where the supreme is taken over all finite dimensional subspaces B of A.

To get the converse, we take a basis {x_{1}, …, x_{m}} of a finite dimensional subspace B of A and let {a_{1}, …, a_{s}, b_{1}, …, b_{r}} be the homogeneous components of x_{1}, …, x_{m} (here a_{i} ∈ A¯0, i = 1, …, s and b_{j} ∈ A¯1, j = 1, …, r). Hence B is contained in the finite dimensional subspace C of A spanned by {a_{1}, …, a_{s}, b_{1}, …, b_{r}}. Let us denote by a_{s+1}, …, a_{t} all possible products b_{i}b_{j}, i, j = 1, …, r, and set b_{0} = 1. By (1.6)

C^{[n]} ⊆ ∑ b_{k}C̃^{j[n]}a_{i_{1}} ⋯ a_{i_{p}},

where k = 0, 1, …, r, C̃ = C^{[2]}, p ≥ 0 and if p > 0 then 1 ≤ i_{1} < i_{2} < … < i_{p} ≤ t. Therefore

lim sup_{n} \frac{ln(dim B^{[n]})}{ln n} ≤ lim sup_{n} \frac{ln(dim C^{[n]})}{ln n}

≤ lim sup_{n} \frac{ln((r + 1)2^{t} dim C̃^{j[n]})}{ln n}

= lim sup_{n} \frac{ln(dim C̃^{j[n]})}{ln n} ≤ GKdim A^{(+)},

hence GKdim A ≤ GKdim A^{(+)}.

3. Local finiteness of symmetrizations of associative superalgebras

In this section we will show that local finiteness is equivalent in an associative superalgebra A and in its symmetrization A^{(+)}.

3.1 A Jordan superalgebra J over k is called finite dimensional if the underlying k-superspace J is finite dimensional. We will say that J is locally finite if every subalgebra of J generated by a finite number of elements is finite dimensional.

3.2 Proposition. Let A be an associative superalgebra and let A^{(+)} be the Jordan superalgebra obtained from A by symmetrization. Then A is locally finite if and only if A^{(+)} is locally finite.

Proof: Suppose that A is locally finite and let B be a finitely generated subalgebra of A^{(+)}. Then B is contained in the associative subalgebra B̃ of A generated by the same set, which is finite dimensional by local finiteness of A. Therefore B is finite dimensional too.

Conversely, suppose that A^{(+)} is locally finite and take a finitely generated subalgebra B of A. Then B is contained in an also finitely generated subalgebra B̃ of A such that its generators are homogeneous elements a_{1}, …, a_{n} ∈ A¯0 and b_{1}, …, b_{m} ∈ A¯1. Let a_{n+1}, …, a_{p} be all possible products of two odd generators b_{i}b_{j}, i, j = 1, …, m, and set b_{0} = 1. Consider the subalgebra C of A^{(+)} generated by {a_{1}, …, a_{p}, b_{1}, …, b_{m}}, which is finite dimensional by local finiteness of A^{(+)}. By (1.6)

B̃ ⊆ ∑ b_{k}Ca_{i_{1}} ⋯ a_{i_{u}},

where k = 0, 1, …, m, u ≥ 0 and 1 ≤ i_{1} < i_{2} < … < i_{u} ≤ p for u > 0. Hence

dim B ≤ dim B̃ ≤ (m + 1)2^{p} dim C < ∞.

Acknowledgements: The author would like to thank Prof. J. A. Anquela and Prof. T. Cortés for their careful reading of this manuscript and their valuable suggestions and comments.

References [1] E. García, E. Neher, “Gelfand-Kirillov dimension and local finiteness of

Jordan superpairs covered by grids” (to appear).

[2] N. Jacobson, Structure Theory of Jordan Algebras, University of Arkansas Lecture Notes in Mathematics, Vol. 5, Fayetteville, 1981.

[3] V.G. Kac, C. Martínez, E. Zelmanov, Graded Simple Jordan Superalgebras of Growth One, Memoirs, Vol. 711, Amer. Math. Soc., 2001.

[4] D. King,“Quadratic Jordan superalgebras”. Comm. Algebra (to appear).

[5] O. Loos, Jordan Pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, New York, 1975.

[6] C. Martínez, “Gelfand-Kirillov dimension in Jordan algebras”. Trans. Amer. Math. Soc., 348 (1996), 119-126.

[7] C. Martínez, E. Zelmanov, “Jordan algebras of Gelfand-Kirillov dimension one”. J. Algebra, 180 (1996), 211-238.

[8] K. McCrimmon, E. Zelmanov, “The Structure of Strongly Prime Quadratic Jordan Algebras”. Adv. Math. 69 (2) (1988), 133-222.

[9] E. Neher, “Quadratic Jordan Superpairs Covered by Grids” (to appear).

[10] V. G. Skosyrskii, “Nilpotency in Jordan and right alternative algebras”. Algebra and Logic, 18 (1) (1979), 49-57.

[11] A.M. Slinko, “Algebraic rings with compact topology”. Algebra and Logic 25 (1) (1986), 58-68.

[12] E. I. Zelmanov, “Absolute Zero Divisors in Jordan Pairs and Lie Algebras”. Math. USSR Sbornik 40 (4) (1981), 549-565.