Journal of Lie Theory Volume 13 (2003) 573–578 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

The Automorphisms of Generalized Witt Type Lie Algebras

Naoki Kawamoto, Atsushi Mitsukawa, Ki-Bong Nam, and Moon-Ok Wang

Communicated by E. Vinberg

Abstract. We find the Lie automorphisms of generalized Witt type Lie algebras W [x, e^{x}] and W [x, e^{±x}].

1. Introduction

Simplicity of several generalized Witt type Lie algebras have been considered by many authors over a field F of characteristic zero. Kac [3] studied the generalized Witt algebra on the F -algebra in the formal power series algebra F [[x_{1}, ⋯, x_{n}]]. There exist many generalized Witt type simple Lie algebras using the algebras stable under the action of derivations ([1], [3], [4], [6]). We consider one-variable cases based on using the exponential functions. Let ∂ = \frac{d}{dx}, F [x^{±1}, e^{±x}] = F [x, x^{−1}, e^{x}, e^{−x}], and let F [a_{1}, ..., a_{n}] be a subalgebra of F [x^{±1}, e^{±x}] generated by a_{1}, ..., a_{n}. If F [a_{1}, ..., a_{n}] is ∂ -stable we put W [a_{1}, ..., a_{n}] = {f∂ | f ∈ F [a_{1}, ..., a_{n}]}. Then W [a_{1}, ..., a_{n}] is a Lie algebra over F with the usual product

[f∂, g∂] = f∂ ∘ g∂ − g∂ ∘ f∂ = (f(∂g) − (∂f)g)∂ (f, g ∈ F [a_{1}, ..., a_{n}]).

The Lie algebras W [x], W [x^{±1}], W [e^{±x}], W [x, e^{±x}], and W [x^{±1}, e^{±x}] are simple, while W [x, e^{x}] and W [x^{±1}, e^{x}] are not simple. The automorphisms of W [x] is considered in [7] (cf. also [2]). The automorphisms of generalized Witt type Lie algebras of Laurent polynomials are considered in [1], [5]. In this paper we find the Lie automorphisms of W [x, e^{x}] and W [x, e^{±x}] containing polynomials and exponential functions. The automorphism group of W [x, e^{x}] is isomorphic to F^{∗} × F, while the automorphism group of W [x, e^{±x}] is isomorphic to ℤ/2ℤ ⋉ (F^{∗} × F).

2. Preliminaries

Let ℤ be the set of integers, ℤ_{+} the set of positive integers, ℤ_{−} the set of negative integers, and ℕ the set of non-negative integers. For the field F we denote by

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag F^{∗} the set of non-zero elements of F. Recall that W [x, e^{x}] = ⨁_{n∈ℕ} W_{n} and W [x, e^{±x}] = ⨁_{n∈ℤ} W_{n} are graded Lie algebras, where W_{n} = {fe^{nx}∂ | f ∈ F [x]} is a homogeneous component of degree n. Let α = α_{n} +α_{n−1} +⋯+α_{m}, where α_{i} ∈ W_{i} and α_{n}, α_{m} ≠ 0. Then we denote by \overline{α} the non-zero homogeneous component of α of highest degree α_{n}, and by \underline{α} the non-zero homogeneous component of lowest degree α_{m}. Hence α = \overline{α} + ⋯ + \underline{α}. Let W_{+} = ⨁_{n∈ℤ_{+}} W_{n} and W_{−} = ⨁_{n∈ℤ_{−}} W_{n}. Then W [x, e^{±x}] = W_{+} + W_{0} + W_{−}, and α = α_{+} + α_{0} + α_{−} for some α_{+} ∈ W_{+}, α_{0} ∈ W_{0}, and α_{−} ∈ W_{−}. For α, ..., β ∈ W [x, e^{±x}] we denote by ⟨α, ..., β⟩ the subalgebra of W [x, e^{±x}] generated by α, ..., β. We denote by sp{α, ..., β} the subspace of W [x, e^{±x}] spanned by α, ..., β. Hence, ⟨α⟩ = sp{α} = F α. For a ∈ F^{∗}, b ∈ F we define

φ_{a} : x^{n}e^{mx}∂ ⟼ a^{m}x^{n}e^{mx}∂,
ψ_{b} : x^{n}e^{mx}∂ ⟼ (x + b)^{n}e^{mx}∂,
τ : x^{n}e^{mx}∂ ⟼ (−1)^{n−1}x^{n}e^{−mx}∂.  (1)

Then it is easy to see that φ_{a}, ψ_{b} ∈ Aut_{F} W [x, e^{x}], and that φ_{a}, ψ_{b}, τ ∈ Aut_{F} W [x, e^{±x}]. Here we use the same symbols to denote the same type of the automorphisms in (1). Note that W [x, e^{x}] = ⟨∂, x^{3}∂, e^{x}∂⟩ and W [x, e^{±x}] = ⟨∂, x^{3}∂, e^{x}∂, e^{−x}∂⟩.

Note 2.1. Let φ be a Lie automorphism of W [x, e^{x}] (resp. W [x, e^{±x}]). If φ(x^{n}∂) = x^{n}∂ (n ∈ ℕ) and φ(e^{mx}∂) = e^{mx}∂ (m ∈ ℕ (resp. ℤ)), then φ = 1_{W [x,e^{x}]} (resp. 1_{W [x,e^{±x}]}). ■

Note 2.2. The Lie algebra W [x, e^{±x}] is self-centralizing, that is, if [α, β] = 0 and α, β are non-zero elements of W [x, e^{±x}], then ⟨α⟩ = ⟨β⟩. ■

Note 2.3. Let β ∈ W [x, e^{±x}]. If [∂, β] = aβ for some a ∈ F^{∗}, then β ∈ ⟨e^{ax}∂⟩ where a ∈ ℤ. ■

Note 2.4. Let I be one of ℕ, ℕ ∪ {−1}, and ℤ. Let a_{n} ∈ F^{∗} (n ∈ I) satisfy the condition a_{n+m} = a_{n}a_{m} for any n ≠ m. Then a_{n} = a_{1}^{n} for any n ∈ I. ■

3. Stabilizers

We determine the elements α, β satisfying the condition [α, β] = β in some generalized Witt type Lie algebras.

Proposition 3.1. Let α, β be non-zero elements of W [x] such that [α, β] = β. Then α − \frac{1}{n−1}(x + c)∂, β ∈ ⟨(x + c)^{n}∂⟩ for some c ∈ F and n ∈ ℕ \ \{1\}.

Proof. Let α = f∂, β = g∂ and let f = a_{m}x^{m} + ⋯ + a_{0}, g = b_{n}x^{n} + ⋯ + b_{0}, where m, n ≥ 0 and a_{m}, b_{n} ≠ 0. If m ≠ n, then from [α, β] = β we have m = 1 and

f = \frac{1}{n − 1}(x + c), g = b_{n}(x + c)^{n}

for some c ∈ F. If m = n, then it follows by taking h = f −ag, where a = \frac{a_{n}}{b_{n}} ≠ 0, that f = a_{n}(x + c)^{n} + \frac{1}{n−1}(x + c), g = b_{n}(x + c)^{n}.

Proposition 3.2. Let α, β be non-zero elements of W [x, e^{x}] and [α, β] = β. Then β_{+} = 0 or β_{0} = 0 and one of the following statements holds : (1) α− \frac{1}{n−1}(x+ c)∂, β ∈ ⟨(x + c)^{n}∂⟩ for some c ∈ F and n ∈ ℕ \ \{1\}, or (2) α − \frac{1}{n}∂, β ∈ ⟨e^{nx}∂⟩ for some n ∈ ℤ_{+}.

Proof. Let α = (f_{m}e^{mx} + ⋯ + f_{0})∂, β = (g_{n}e^{nx} + ⋯ + g_{0})∂, where f_{m}, ..., f_{0}, g_{n}, ..., g_{0} ∈ F [x], f_{m}, g_{n} ≠ 0, and m, n ∈ ℕ. If m ≠ n, then by some computation we deduce from [α, β] = β that m = 0, n > 0 and that

α = \frac{1}{n}∂, β = b_{n}e^{nx}∂ (b_{n} ≠ 0).

Let m = n. If n = 0, then we can apply Proposition 3.1. If n > 0, then we have f_{n}g′_{n} − f′_{n}g_{n} = 0, (\frac{f_{n}}{g_{n}})′ = 0, and g_{n} = cf_{n} for some constant c ≠ 0, where we write simply f′ instead of ∂f. From [α− \frac{1}{c}β, β] = β we have α = ae^{nx}∂ + \frac{1}{n}∂, β = be^{nx}∂ for some a, b ∈ F.

We continue to characterize the elements α, β satisfying the condition [α, β] = β in W [x, e^{±x}].

Lemma 3.3. Let α, β be non-zero elements of W [x, e^{±x}] and [α, β] = β. Then (1) For \overline{α} and \overline{β} we have either

(i) \overline{α − kβ} − \frac{1}{n−1}(x + c)∂, \overline{β} ∈ ⟨(x + c)^{n}∂⟩ for some k, c ∈ F, n ∈ ℕ \ \{1\}, or (ii) \overline{α − kβ} = \frac{1}{n}∂ and \overline{β} ∈ ⟨e^{nx}∂⟩ for some k ∈ F, n ∈ ℤ \ \{0\}.

(2) For \underline{α} and \underline{β} we have either

(i) \underline{α − lβ} − \frac{1}{m−1}(x + d)∂, \underline{β} ∈ ⟨(x + d)^{m}∂⟩ for some l, d ∈ F, m ∈ ℕ \ \{1\}, or (ii) \underline{α − lβ} = \frac{1}{m}∂ and \underline{β} ∈ ⟨e^{mx}∂⟩ for some l ∈ F, m ∈ ℤ \ \{0\}.

Proof. We show Case (1), since Case (2) will be proved similarly. Since [α − kβ, β] = β for any k ∈ F, if necessary we can replace α with α − kβ. Hence we may assume ⟨\overline{α}⟩ ≠ ⟨\overline{β}⟩. Then by Note 2.2 we have [\overline{α}, \overline{β}] ≠ 0. Therefore [\overline{α}, \overline{β}] = \overline{β} and \overline{α} ∈ W_{0} = W [x] since W [x, e^{±x}] is ℤ-graded. We determine \overline{α} and \overline{β}. Apply the automorphism τ if necessary. Then by Proposition 3.2 we have \overline{α}− \frac{1}{n−1}(x+c)∂, \overline{β} ∈ ⟨(x+c)^{n}∂⟩ for some c ∈ F, n ∈ ℕ\\{1\}, or \overline{α}− \frac{1}{n}∂, \overline{β} ∈ ⟨e^{nx}∂⟩ for some n ∈ ℤ \ \{0\}. In the later case \overline{α} = \frac{1}{n}∂ + be^{nx}∂ for some b ∈ F, and \overline{α} = \frac{1}{n}∂ since \overline{α} is homogeneous.

Lemma 3.4. Let α, β be non-zero elements of W [x, e^{±x}] and [α, β] = β. Then we have the following statments :

(1) If α_{+} ≠ 0, then β_{+} ≠ 0, ⟨\overline{α}⟩ = ⟨\overline{β} ⟩ ⊆ W_{n} for some n ∈ ℤ_{+}, and also β = \frac{1}{k}α_{+} + \frac{1}{k} (α_{0} − \frac{1}{n}∂) + β_{−} for some k ∈ F^{∗}.

(2) If α_{−} ≠ 0, then β_{−} ≠ 0, ⟨\underline{α}⟩ = ⟨\underline{β} ⟩ ⊆ W_{m} for some m ∈ ℤ_{−}, and β = β_{+} + \frac{1}{l} (α_{0} − \frac{1}{m}∂) + \frac{1}{l}α_{−} for some l ∈ F^{∗}.

(3) If α_{+}, α_{−} ≠ 0, then β ∈ sp{α_{+}, α_{−}, α_{0}, ∂}.

Proof. (1) Let α_{+} ≠ 0. Then \overline{α} ∈ W_{n} for some n ∈ ℤ_{+}. Assume that [\overline{α}, \overline{β}] ≠ 0. Then [\overline{α}, \overline{β}] = \overline{β}. If \overline{β} ∈ W_{m}, then \overline{β} = [\overline{α}, \overline{β}] ∈ W_{n+m}, a contradiction. Hence [\overline{α}, \overline{β}] = 0, and by Note 2.2 we have ⟨\overline{α}⟩ = ⟨\overline{β} ⟩ and β_{+} ≠ 0. Hence \overline{β} ∈ W_{n}, and we have \overline{α − kβ} = \frac{1}{n}∂ for some non-zero k ∈ F by Lemma 3.3. Then α − kβ = \frac{1}{n}∂ + α_{−} − kβ_{−} and β = \frac{1}{k}α_{+} + \frac{1}{k} (α_{0} − \frac{1}{n}∂) + β_{−}.

(2) Let α_{−} ≠ 0. Then ⟨\underline{α}⟩ ∈ W_{m} for some m ∈ ℤ_{−}, and we have β = β_{+} + \frac{1}{l} (α_{0} − \frac{1}{m}∂) + \frac{1}{l}α_{−} for some l ∈ F^{∗}.

(3) Let α_{+}, α_{−} ≠ 0. Then from (1) and (2) we have

β = \frac{1}{k}α_{+} + \frac{1}{k} (α_{0} − \frac{1}{n}∂) + β_{−} = β_{+} + \frac{1}{l} (α_{0} − \frac{1}{m}∂) + \frac{1}{l}α_{−}

for some k, l ∈ F^{∗}, n ∈ ℤ_{+}, m ∈ ℤ_{−}. Thus β ∈ sp{α_{+}, α_{−}, α_{0}, ∂}.

Lemma 3.5. Let α be a non-zero element of W [x, e^{±x}], and let {β_{i} | i ∈ I} be an infinite and linearly independent subset of W [x, e^{±x}]. If [α, β_{i}] = a_{i}β_{i} and a_{i} ≠ 0 for any i ∈ I, then α_{0} ≠ 0 and either α_{+} = 0 or α_{−} = 0.

Proof. Assume that α_{+} ≠ 0 and α_{−} ≠ 0. Since [\frac{1}{a_{i}} α, β_{i}] = β_{i}, by Lemma 3.4(3) the set {β_{i} | i ∈ I} is contained in the finite dimensional subspace sp{α_{+}, α_{−}, α_{0}, ∂}, a contradiction. Hence α_{+} = 0 or α_{−} = 0. If both α_{+} = 0 and α_{−} = 0, then clearly α_{0} ≠ 0. Let β = β_{i}. If α_{−} ≠ 0, then we apply the automorphism τ. Hence we may assume that α = α_{+} + α_{0}. By Lemma 3.4 we have

β = \frac{1}{ka_{i}} α_{+} + \frac{1}{k} (\frac{1}{a_{i}} α_{0} − \frac{1}{n}∂) + β_{−}

for some k ∈ F^{∗} and n ∈ ℤ_{+} such that \overline{β} ∈ W_{n}. Hence β_{+} ≠ 0. If β_{−} = 0, then by Proposition 3.2 we have \frac{1}{a_{i}} α_{0} − \frac{1}{n}∂ = kβ_{0} = 0 and α_{0} ≠ 0. If β_{−} ≠ 0, then [\frac{1}{a_{i}} \underline{α}, \underline{β}] = \underline{β} since ⟨\underline{α}⟩ ≠ ⟨\underline{β} ⟩. Hence α_{0} = \underline{α} ≠ 0.

4. Automorphisms

We determine the automorphisms of W [x, e^{x}] and W [x, e^{±x}] in this section.

Lemma 4.1. Let φ be an injective homomorphism of W [x]. Then φ(x^{n}∂) = a^{n−1}(x + b)^{n}∂ (n ∈ ℕ) for some a ∈ F^{∗}, b ∈ F.

Proof. Let φ be an injective homomorphism of W [x]. Since

[φ(x^{m}∂), φ(x^{n}∂)] = (n − m)φ(x^{m+n−1}∂),  (2)

we have [\frac{1}{n−1}φ(x∂), φ(x^{n}∂)] = φ(x^{n}∂) (n ≠ 1). Since φ(x^{n}∂) (n ∈ ℕ) are linearly independent it follows easily from Proposition 3.1 that

φ(x^{n}∂) = a_{n−1}(x + b)^{n}∂ (n ∈ ℕ)

for some a_{n−1} ∈ F^{∗}, b ∈ F. Then from (2) we have a_{m−1}a_{n−1} = a_{m+n−2} = a_{m−1+n−1} (n, m ∈ ℕ, n ≠ m), that is, a_{m}a_{n} = a_{n+m} (n, m ∈ ℕ ∪ {−1}, n ≠ m). By Note 2.4, a_{n} = a_{1}^{n} and φ(x^{n}∂) = a^{n−1}(x + b)^{n}∂, where a = a_{1} ∈ F^{∗}.

Let ρ_{a} : x^{n}∂ ⟼ a^{n−1}x^{n}∂. Then it is easy to see that ρ_{a} (a ∈ F^{∗}) is an automorphism of W [x]. By Lemma 4.1 we note that the automorphism group of W [x] is isomorphic to F^{∗} ⋉ F, where F^{∗} is the multiplicative group and F is the additive group (cf. [2],[7]).

Proposition 4.2. Let φ be an automorphism of W [x, e^{x}] or W [x, e^{±x}]. Then φ(W [x]) ⊆ W [x].

Proof. It holds that [φ(x∂), φ(x^{n}∂)] = (n−1)φ(x^{n}∂) (n ∈ ℕ). Let α = φ(x∂). Then by Lemma 3.5 we have α_{0} ≠ 0, and α_{+} = 0 or α_{−} = 0. Let β = φ(∂). Then similary from [φ(∂), φ(e^{mx}∂)] = mφ(e^{mx}∂) (m ∈ ℕ) we have β_{0} ≠ 0, and β_{+} = 0 or β_{−} = 0. Assume that α_{+} ≠ 0. Then from [−α, β] = β and Lemma 3.4(1) we have β_{+} ≠ 0 and α, β ∈ W [x, e^{x}]. Hence β_{0} ≠ 0 and β_{+} ≠ 0, but this contradicts to Proposition 3.2. Assume that α_{−} ≠ 0. Then applying τ we have a contradiction similar to the above. Therefore α = α_{0} ∈ W [x]. Then the case β_{+} ≠ 0 and the case β_{−} ≠ 0 cause similar contradictions. Thus β = β_{0} ∈ W [x]. From [β, φ(x^{n}∂)] = (n−1)φ(x^{n−1}∂) we have φ(x^{n}∂) ∈ W [x] (n ∈ ℕ) by induction.

Theorem 4.3. Let φ be an automorphism of W [x, e^{x}]. Then φ is a product of φ_{a} and ψ_{b} for some a ∈ F^{∗}, b ∈ F.

Proof. Let φ be an automorphism of W [x, e^{x}]. Then by Proposition 4.2 and Lemma 4.1 we have φ(∂) = a^{−1}∂ and φ(x^{n}∂) = a^{n−1}(x+b)^{n}∂ for some a ∈ F^{∗}, b ∈ F. Since [φ(∂), φ(e^{mx}∂)] = mφ(e^{mx}∂), we have [∂, φ(e^{mx}∂)] = amφ(e^{mx}∂). Then by Note 2.3 we have φ(e^{mx}∂) ∈ ⟨e^{amx}∂⟩ and am ∈ ℕ. Since φ is surjective, it follows that a = 1, φ(x^{n}∂) = (x + b)^{n}∂ (n ∈ ℕ), and φ(e^{mx}∂) = c_{m}e^{mx}∂ (m ∈ ℕ). Then from [φ(e^{mx}∂), φ(e^{kx}∂)] = (k − m)φ(e^{(m+k)x}∂) (m, k ∈ ℕ) and Note 2.4, we have c_{m} = c^{m} for some c ∈ F^{∗}. Thus φ(e^{mx}∂) = c^{m}e^{mx}∂. Hence (φ_{c} ∘ ψ_{b})^{−1} ∘ φ = 1_{W [x,e^{x}]} by Note 2.1, and therefore φ = φ_{c} ∘ ψ_{b}.

Corollary 4.4. The automorphism group of W [x, e^{x}] is isomorphic to F^{∗} ×F.

Proof. This is clear from φ_{a} ∘ ψ_{b} = ψ_{b} ∘ φ_{a} for any a ∈ F^{∗}, b ∈ F.

Theorem 4.5. An automorphism of W [x, e^{±x}] is a product of φ_{a}, ψ_{b}, and τ for some a ∈ F^{∗}, b ∈ F.

Proof. Let φ be an automorphism of W [x, e^{±x}]. Then as in the proof of Theorem 4.3 φ(x^{n}∂) = a^{n−1}(x + b)^{n}∂ (n ∈ ℕ) and φ(e^{mx}∂) = c_{m}e^{amx}∂ (m ∈ ℤ). Since φ is surjective, we have a = ±1, and applying τ if necessary we may assume a = 1. Then it follows that φ(e^{mx}∂) = c^{m}e^{mx}∂ for some c ∈ F^{∗} and that (φ_{c} ∘ ψ_{b})^{−1} ∘ φ = 1_{W [x,e^{±x}]}.

Corollary 4.6. The automorphism group of W [x, e^{±x}] is isomorphic to ℤ/2ℤ⋉ (F^{∗} × F).

Proof. This is clear from φ_{a}∘ψ_{b} = ψ_{b}∘φ_{a}, τ ∘φ_{a}∘τ = φ_{a^{−1}} , and τ ∘ψ_{b}∘τ = ψ_{−b}.

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Naoki Kawamoto Japan Coast Guard Academy 5-1 Wakaba Kure 737-8512 Japan kawamoto@jcga.ac.jp Atsushi Mitsukawa Department of Management Information Fukuyama Heisei University 117-1 Miyuki Fukuyama 720-0001 Japan mitukawa@fuhc.fukuyama-u.ac.jp

Ki-Bong Nam Department of Mathematics and Computer Science University of Wisconsin-Whitewater 800 West Main Street Whitewater WI 53190 USA namk@mail.uww.edu Moon-Ok Wang Department of Mathematics Hanyang University Ansan Kyunggi 425-791 Korea wang@mail.hanyang.ac.kr

Received September 28, 2002 and in final form March 3, 2003