Journal of Lie Theory Volume 12 (2002) 529–534 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

On Generators of Free Color Lie Superalgebras of Rank Two

Ela Aydın and Naime Ekici

Communicated by H. Schlosser

Abstract. Let L be a free color Lie superalgebra on two generators x, y. A criterion for two elements of L to be a generating set is given.

1. Introduction

P. M. Cohn proved in [3] that the t-automorphisms generate the group of all automorphisms of a free Lie algebra of finite rank. In [6], [7] Mikhalev obtained the following analogue of Cohn’s theorem: The elementary automorphisms and linear changes generate a group of automorphisms of a free color Lie superalgebra of finite rank. The freedom of the subalgebras of free color Lie algebras [6], [7] gives rise to the following analogue of Nielsen’s theorem: If n G-homogeneous elements generate a free color Lie superalgebra of rank n, then these elements are free generators of it. Now let X = {x, y} and L(X) be a free color Lie superalgebra freely generated by the set X. If two G-homogeneous elements h_{1}, h_{2} generate L(X), then they freely generate L(X). So [h_{1}, h_{2}] is a linear combination of the elements [x, x], [y, y], [x, y]. The main assertion of this note is the theorem that the subalgebra generated by h_{1}, h_{2} is equal to the free color Lie superalgebra L(X) if and only if [h_{1}, h_{2}] = α[x, x] + β[x, y] + γ[y, y], where α, β, γ ∈ K^{∗}. In [4] Dicks obtained a similar criterion for free associative algebras of rank two.

2. Preliminaries

Let K and G be a field and abelian group respectively. Assume that R = ⨁_{g∈G} R_{g} is

a G-graded K -algebra. The homogeneous elements are those from some R_{g}. For each homogeneous a ∈ R_{g} we shall write d(a) = g. The G -valued function d is called the degree map on R. Let K^{∗} be the multiplicative group of the field K, ε : G × G → K^{∗} a skew-symmetric bilinear form, and G_{−} = {g ∈ G | ε(g, g) = −1}.

Definition 2.1. We say that a G -graded algebra R is a color Lie superalgebra if [x, y] = −ε(d(x), d(y))[y, x], [x, [y, z]] = [[x, y], z] + ε(d(x), d(y))[y, [x, z]],

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag [v, [v, v]] = 0, with d(v) ∈ G_{−}, for homogeneous x, y, z, v ∈ R.

Definition 2.2. Let X = ⋃_{g∈G} X_{g} be a G-graded set, again d(x) = g for x ∈

X_{g}, d(u) = ∑_{i=1}^{n} d(x_{i}) ∈ G for u = x_{1}...x_{n} ∈ ⟨X⟩, x_{i} ∈ X, d(z) = d(γ(z)) for z ∈

V [X], (⟨X⟩)_{g} = {u ∈ ⟨X⟩ | d(u) = g}, (V [X])_{g} = {z ∈ V [X] | d(z) = g}, (F [X])_{g} is the K -linear hull of the subsets (V [X])g ∈ F [X], F [X] = ⨁_{g∈G}(F [X])_{g}, I the

G -graded ideal in F [X] generated by homogeneous elements of the form [a, b] + ε(d(a), d(b))[b, a] and [[a, b], c] − [a, [b, c]] + ε(d(a), d(b))[b, [a, c]], where a, b, c ∈ V [X] then, L[X] = F [X]/I is a free color Lie superalgebra (i.e., each G-map of degree zero from X to any color Lie superalgebra R uniquely extends to a G -homomorphism of degree zero of color Lie superalgebras. L[X] = F [X]/I → R, for z ∈ F [X] ).

If M is a G-graded associative algebra over K, then M with the operation [a, b] = ab − ε(d(b), d(a))ba for homogeneous elements a, b ∈ M is a color Lie superalgebra denoted by [M].

Let X = {x_{1}, ..., x_{n}} be a G-graded set, and A(X) be a free G-graded associative algebra with 1 over K. Then the subalgebra L(X) in [A(X)] generated by X is a free color Lie superalgebra with set X of free generators. The algebra A(X) is the enveloping algebra of L(X).

In order to proceed with the proof of our main result we have to introduce some more notation. By U(L) we denote the universal enveloping algebra of L.

There is the augmentation homomorphism ε′ : U(L) → K defined by

ε′(x_{i}) = 0, i = 1, 2, ..., n. There are mappings \frac{∂}{∂x_{i}} : U(L) → U(L), i = 1, 2, ..., n,

satisfying the following conditions whenever a, b ∈ K and u, v ∈ U(L):

1. \frac{∂(x_{j})}{∂x_{i}} = δ_{ij},

2. \frac{∂}{∂x_{i}} (au + bv) = a\frac{∂u}{∂x_{i}} + b\frac{∂v}{∂x_{i}},

3. \frac{∂}{∂x_{i}} (uv) = ε′(u)\frac{∂v}{∂x_{i}} + \frac{∂u}{∂x_{i}} v.

For any a ∈ K, \frac{∂a}{∂x_{i}} = 0. We will call these mappings Fox derivatives [5]. We

need some lemmas to be used throughout 3.

Lemma 2.3. Let L(X) be a color Lie superalgebra and v_{1}, ..., v_{m}, u be some elements of L(X). Suppose u belongs to the left ideal of U(L) generated by v_{1}, ..., v_{m}. Then u belongs to the subalgebra of L(X) generated by v_{1}, ..., v_{m}. The assertion of the lemma is similar to the corresponding assertion for Lie algebras [9]. The proof follows by using analogue of the Poincare’-Birkhoff -Witt theorem for free color Lie superalgebras.

Definition 2.4. Suppose S = {s_{α} | α ∈ I} is a G -homogeneous subset of the free color Lie superalgebra L(X). By an elementary transformation of the set S we mean a mapping ω : S → L(X) such that ω(s_{α}) = s_{α} for all α ∈ I \β, ω(s_{β}) = λs_{β} + ω(s_{α_{1}}...s_{α_{t}}), where λ ∈ K, λ ≠ 0, α_{1,}..., α_{t} ≠ β, d(ω(s_{α_{1}}...s_{α_{t}})) = d(s_{β}).

Lemma 2.5. The elementary transformations and linear changes of a set of free generators induce automorphisms of a free color Lie superalgebra.

Proof. The proof is analogous to that of Lemma 2.7.2 of [1].

In [7] Mikhalev obtained the following analogue of Cohn’s theorem:

Theorem 2.6. The elementary automorphisms and linear changes generate a group of automorphisms of a free color Lie superalgebra of finite rank.

The following theorem is an analog of the Birman’s result [2] for groups.

Theorem 2.7. Let X = {x_{1}, ..., x_{n}} and let f_{1}, ..., f_{n} be G -homogeneous elements in L(X), d(x_{i}) = d(f_{i}). Then the endomorphism φ : L(X) → L(X),

where φ(x_{i}) = f_{i}, 1 ≤ i ≤ n is an automorphism if and only if the matrix (\frac{∂f_{i}}{∂x_{j}} ),

1 ≤ i, j ≤ n is invertible over A(X).

Proof. The proof is along the lines of the proof of the theorem in [9].

Now let K be a field and L be the free color Lie superalgebra on two generators x, y. Clearly the group of automorphisms of L generated by the automorphisms of the form

φ : \begin{matrix}x → & y \\ y → & x \end{matrix} , ψ : \begin{matrix}x → & x \\ y → & αy + x \end{matrix}

α ∈ K, α ≠ 0, d(x) = d(y).

Lemma 2.8. Let h_{1}, h_{2} ∈ L, h_{1} = αx + βy, h_{2} = γx + δy, where α, β, γ, δ ∈ K. Then h_{1} and h_{2} freely generate L if and only if αδ − βγ ≠ 0.

The lemma follows immediately from the Theorem 2.7. We now come to our main result.

3. Main Theorem

Suppose K is a field of characteristic zero. Let X = {x, y} and L be a free color Lie superalgebra on X.

Theorem 3.1. Let h_{1}, h_{2} be G -homogeneous elements of L and H the subalgebra they generate. Then H = L if and only if [h_{1}, h_{2}] = α[x, y]+β[x, x]+γ[y, y], where α, β, γ ∈ K^{∗}.

Proof. If H = L then the set {h_{1}, h_{2}} can be transform into the set {x, y} by the automorphisms of L. Then we can write h_{1}, h_{2} as h_{1} = ax + by and h_{2} = cx + dy, where a, b, c, d ∈ K, d(x) = d(y), ad − bc ≠ 0.

If d(x), d(y) ∉ G_{−}, then [h_{1}, h_{2}] = α[x, y], where α = ad − bcε(d(y), d(x)). If d(x), d(y) ∈ G_{−}, then [h_{1}, h_{2}] = α[x, y] + β[x, x] + γ[y, y], where α =

ad + bc, γ = bd, β = ac.

Now we prove ”if” part. We will consider four cases:

Case I. Let d(x), d(y) ∉ G_{−}. In this case [h_{1}, h_{2}] is a nonzero scalar multiple of [x, y].

Let [h_{1}, h_{2}] = α[x, y], α ∈ K^{∗}. Take Fox derivative of both sides:

\frac{∂[h_{1}, h_{2}]}{∂x} = α\frac{∂[x, y]}{∂x} and \frac{∂[h_{1}, h_{2}]}{∂y} = α\frac{∂[x, y]}{∂y}.

By the chain rule for Fox derivativations [5],

\frac{∂[h_{1}, h_{2}]}{∂x} = \frac{∂h_{1}}{∂x} \frac{∂[h_{1}, h_{2}]}{∂h_{1}} + \frac{∂h_{2}}{∂x} \frac{∂[h_{1}, h_{2}]}{∂h_{2}}

and

\frac{∂[h_{1}, h_{2}]}{∂y} = \frac{∂h_{1}}{∂y} \frac{∂[h_{1}, h_{2}]}{∂h_{1}} + \frac{∂h_{2}}{∂y} \frac{∂[h_{1}, h_{2}]}{∂h_{2}}.

Therefore,

\frac{∂[h_{1}, h_{2}]}{∂h_{1}} = \frac{∂(h_{1}h_{2} − ε(d(h_{1}), d(h_{2}))h_{2}h_{1})}{∂h_{1}} = h_{2}, \frac{∂[h_{1}, h_{2}]}{∂h_{2}} = −ε(d(h_{1}), d(h_{2}))h_{1}

and

\frac{∂h_{1}}{∂x} h_{2} − ε(d(h_{1}), d(h_{2}))\frac{∂h_{2}}{∂x} h_{1} = αy,
\frac{∂h_{1}}{∂y} h_{2} − ε(d(h_{1}), d(h_{2})\frac{∂h_{2}}{∂y} h_{1} = −ε(d(x), d(y))αx.

Set k = −ε(d(x), d(y)). Then we have

α^{−1} \frac{∂h_{1}}{∂x} h_{2} − α^{−1}ε(d(h_{1}), d(h_{2}))\frac{∂h_{2}}{∂x} h_{1} = y,
k^{−1}α^{−1} \frac{∂h_{1}}{∂y} h_{2} − k^{−1}α^{−1}ε(d(h_{1}), d(h_{2}))\frac{∂h_{2}}{∂y} h_{1} = x.

We see that x and y are belong to the left ideal of A generated by h_{1} and h_{2}. From the Lemma 2.3. we conclude that x and y are belong to the subalgebra H generated by h_{1} and h_{2}. Hence H = L.

Case II. Let d(x) ∈ G_{−} and d(y) ∉ G_{−}. In this case

[h_{1}, h_{2}] = α[x, y] + β[x, x],

where α, β ∈ K^{∗}. It follows that

\frac{∂[h_{1}, h_{2}]}{∂x} = α\frac{∂[x, y]}{∂x} + β \frac{∂[x, x]}{∂x}

and

\frac{∂[h_{1}, h_{2}]}{∂y} = α\frac{∂[x, y]}{∂y} + β \frac{∂[x, x]}{∂y}.

Then

\frac{∂h_{1}}{∂x} h_{2} − ε(d(h_{1}), d(h_{2}))\frac{∂h_{2}}{∂x} h_{1} = αy + 2βx,
\frac{∂h_{1}}{∂y} h_{2} − ε(d(h_{1}), d(h_{2}))\frac{∂h_{2}}{∂y} h_{1} = −ε(d(x), d(y))αx.

By the Lemma 2.8. the elements αy + 2βx and −ε(d(x), d(y))αx freely generate L. These elements belong to the left ideal of A generated by h_{1} and h_{2}. So by the Lemma 2.3. they belong to H.

Case III. Let d(x) ∉ G_{−} and d(y) ∈ G_{−}. Since d(x) ∉ G_{−}, [x, x] = 0 and [h_{1}, h_{2}] = α[x, y] + γ[y, y], where α, γ ∈ K − {0}. If we take Fox derivative of both sides and we replace the roles of x and y we obtain the result as in Case II.

Case IV. Let d(x) and d(y) ∈ G_{−}. In this case

[h_{1}, h_{2}] = α[x, y] + β[x, x] + γ[y, y].

If we take Fox derivative of both sides we get

\frac{∂h_{1}}{∂x} h_{2} − ε(d(h_{1}), d(h_{2}))\frac{∂h_{2}}{∂x} h_{1} = αy + 2βx,
\frac{∂h_{1}}{∂y} h_{2} − ε(d(h_{1}), d(h_{2}))\frac{∂h_{2}}{∂y} h_{1} = −ε(d(x), d(y))αx + 2γy.

If α^{2}ε(d(x), d(y)) + 4βγ ≠ 0, then by the Lemma 2.8. the elements αy + 2βx and −ε(d(x), d(y))αx + 2γy freely generate L and they belong to the free color Lie superalgebra H generated by h_{1} and h_{2}. So H = L as claimed.

References

[1]  Bakhturin, Yu. A., “Identities in Lie Algebras,” Nauka, Moskow, 1985.

[2] Birman, J. S., An inverse function theorem for free groups, Proc. Amer. Math. Soc. 41 (1973), 634–638.

[3] Cohn, P. M., Subalgebras of free associative algebras, Proc. London Math. Soc. 3 (1964), 618–632.

[4] Dicks, W., A commutator test for two elements to generate the free algebra of rank two, Bull. London Math. Soc. 14 (1982), 48–51.

[5] Fox, R. H., Free differental calculus.I. Derivations in free group rings, Ann. of Math. 57 (1953), 547–560.

[6] Mikhalev, A. A., Subalgebras of free colored Lie superalgebras, Mat. Zametki. 37 (1985), 653–661.

[7] Mikhalev, A. A., Subalgebras of free Lie p-superalgebras, Mat. Zametki. 43 (1988), 178–191.

[8] Mikhalev, A. A., On rights ideals of a free associative algebra generated by free colour Lie superalgebras and p-superalgebras, Comm. of Moskow Math. Soc.

534 Aydin and Eki̇ci̇

[9] Umirbaev U. U., Sixth All Union School on Theory of Manifolds and Algebraic Systems, Magnitogorsk 1990, Abstract of Communications, p.32.

Ela Aydın Cukurova University Faculty of Arts and Sciences Department of Mathematics 01330 Adana - TURKEY eaydin@mail.cu.edu.tr Naime Ekici Cukurova University Faculty of Arts and Sciences Department of Mathematics 01330 Adana - Turkey nekici@mail.cu.edu.tr

Received July 10, 2001 and in final form October 24, 2001