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Journal of Algebra A class of zero product determined Lie algebras
A class of zero product determined Lie algebras
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331
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2011
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english
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10.1016/j.jalgebra.2010.10.037
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Journal of Algebra 331 (2011) 145–151 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra A class of zero product determined Lie algebras Dengyin Wang a,∗,1 , Xiaoxiang Yu b , Zhengxin Chen c a b c China University of Mining and Technology, Department of Mathematics, Zhaishan, Xuzhou, Jiangsu, China School of Mathematics and Computer Science, Xuzhou Normal University, Xuzhou, China School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, China a r t i c l e i n f o Article history: Received 24 May 2010 Available online 8 December 2010 Communicated by Nicolás Andruskiewitsch MSC: 17B20 17B30 17B40 Keywords: Simple Lie algebras Parabolic subalgebras Zero product determined algebras Zero product derivations Commutativitypreserving maps a b s t r a c t Let L be a Lie algebra over a ﬁeld F . We say that L is zero product determined if, for every F linear space V and every bilinear map ϕ : L × L → V , the following condition holds. If ϕ (x, y ) = 0 whenever [x, y ] = 0, then there exists a linear map f from [ L , L ] to V such that ϕ (x, y ) = f ([x, y ]) for all x, y ∈ L. This article shows that every parabolic subalgebra p of a (ﬁnitedimensional) simple Lie algebra deﬁned over an algebraically closed ﬁeld is always zero product determined. Applying this result, we present a method different from that of Wang et al. (2010) [9] to determine zero product derivations of p, and we obtain a deﬁnitive solution for the problem of describing twosided commutativitypreserving maps on p. © 2010 Elsevier Inc. All rights reserved. 1. Introduction The concept of zero product determined Lie (resp., associative, Jordan) algebras was recently introduced by [3] and further studied by [5]. The original motivation for introducing these concepts emerges from the discovery that certain problems concerning linear maps on algebras, such as describing linear maps preserving commutativity or zero products, can be effectively treated by ﬁrst examining bilinear ma; ps satisfying certain related conditions [4]. Let us recall the following deﬁnition. Let F be a (ﬁxed) ﬁeld, and let A be an algebra over F . Let A 2 denote the F linear span of all elements of the form xy for x, y ∈ A. The algebra A is called zero product determined if for every linear space X over F and every bilinear map {·,·} : A × A → X , the following holds. If {x, y } = 0 whenever * 1 Corresponding author. Email address: wdengyin@126.com (D. Wang). Supported by the Fundamental Research Funds for the Central Universities. 00218693/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2010.10.037 146 D. Wang et al. / Journal of Algebra 331 (2011) 145–151 xy = 0, then there exists a linear map f : A 2 → X such that {x, y } = f (xy ) for all x, y ∈ A. If the ordinary product is replaced by the Lie product, then it is said that A is zero Lie product determined. It should be noted that the problem of studying zero product determined algebras is nontrivial in the sense that there do exist associative and Lie algebras that are not zero product determined (see [5] for examples). The authors proved in [3] that the matrix algebra M n ( R ), n 2, where R is any unital algebra, is always zero product determined. Moreover, if R is zero Lie product determined, then so is M n ( R ). In [5], M. Grašič showed that the Lie algebra of all n × n skewsymmetric matrices over an arbitrary ﬁeld F of characteristic not 2 is zero product determined, as is the simple Lie algebra of the symplectic type over the above ﬁeld F . The purpose of this paper is to extend the results from [5] to all parabolic subalgebras of the ﬁnitedimensional simple Lie algebras over algebraically closed ﬁelds of characteristic 0. We also show the applicability of our main theorem to the study of zero product derivations and commutativitypreserving maps. 2. Basic Theorem In this paper, the notation concerning Lie algebras mainly follows [6]. Let F be an algebraically closed ﬁeld of characteristic 0. We denote by g a (ﬁnitedimensional) simple Lie algebra over F of rank l. By h we denote a ﬁxed Cartan subalgebra of g, and by Φ we denote the corresponding root system of g. Let be a ﬁxed base of Φ , and let Φ + (resp., Φ − ) be the set of positive (resp., negative) roots relative to . The roots in are called simple. For the base of Φ , let d = {dα  α ∈ } be the value 0 when β = α ∈ , and it takes dual basis of h relative to . Namely, β(dα ) takes the the value 1 when β = α ∈ . Each root β can be written as β = α ∈ kα α with kα ∈ Z. The integer α ∈ kα is β , which we denote by ht β . We denote by ker α , for α ∈ Φ , the kernel of α in h. called the height of Let g = h + β∈Φ gβ be the root space decomposition of g, where gβ = {x ∈ g  [h, x] = β(h)x, ∀h ∈ h} is the root space relative to β ∈ Φ . For each α ∈ Φ + , let e α be a nonzero element of gα . Then, there is a unique element e −α ∈ g−α such that e α , e −α , hα = [e α , e −α ] span a threedimensional 0 1 0 0 1 0 , e −α → , hα → 0 −1 . The set simple subalgebra of g isomorphic to sl(2, F ) via e α → 00 10 {hα , e β , e −β  α ∈ , β ∈ Φ + } forms the basis of g. If α , β, α + β ∈ Φ , then [e α , e β ] is a nonzero scalar multiple of e α +β , because [gα , gβ ] = gα +β . Deﬁne N α ,β so that [e α , e β ] = N α ,β e α +β , which we call the structure constants of g. If x, y ∈ g, deﬁne k(x, y ) = Tr(ad x · ad y ). Then k is a symmetric bilinear form of g called the Killing form of g. It is well known that the restriction of the Killing form of g to h is nondegenerate. Thus for each φ ∈ h∗ there exists a unique t φ ∈ h such that k(t φ , h) = φ(h) for all h ∈ h, and the map from h∗ to h, deﬁned by φ → t φ , is an isomorphic map. A symmetric bilinear form ( , ) is deﬁned on the ldimensional real vector space spanned by Φ , which is dual to the Killing 2t β . A subform of h. For β ∈ Φ , we know that hβ is a nonzero multiple of t β . More deﬁnitely, hβ = (β,β) algebra p of g is called parabolic if it includes some Borel subalgebra of g. For a given subset π of , deﬁne p (relative to π ) to be the subalgebra of g generated by all g α , α ∈ ∪ {−π } along with h. Let Φπ = Zπ ∩ Φ , Φπ+ = Φπ ∩ Φ + , Φπ− = Φπ ∩ Φ − . In fact, p = h + α ∈Φ + ∪Φπ− gα . It is evident that if π = , then p is g itself. If π = ∅, then p is a Borel subalgebra of g, which we denote by b. It is well known that every parabolic subalgebra of g is conjugate under an automorphism to some p. From this point of view, to determine a bilinear map on an arbitrary parabolic subalgebra, we only need to determine those maps on p. In the following, we always denote by p the parabolic subalgebra of g relative to a ﬁxed subset π of . We now present the Basic Theorem as follows. Basic Theorem. Let p be a parabolic subalgebra of g relative to a subset π of . Then, p is zero product determined. More concretely, if a bilinear map ϕ from p × p to a linear space V over F satisﬁes the property that ϕ (x, y ) = 0 whenever [x, y ] = 0, then there exists a linear map f from [p, p] to V such that ϕ (x, y ) = f ([x, y ]) for all x, y ∈ p. One should note ﬁrst that the bilinear map ϕ mentioned in the Basic Theorem is skew symmetric. In fact, by ϕ (x + y , x + y ) = ϕ (x, x) = ϕ ( y , y ) = 0, ϕ (x, y ) = −ϕ ( y , x) for all x, y ∈ p. Because the set {hα , e β  α ∈ π , β ∈ Φ + ∪ Φπ− } forms the basis of [p, p], to deﬁne a linear map f from [p, p] to V , we only need to deﬁne its action on hα for α ∈ π as well as on e β for β ∈ Φ + ∪ Φπ− and then extend it D. Wang et al. / Journal of Algebra 331 (2011) 145–151 147 linearly. We now use ϕ to induce a linear map f from [p, p] to V by deﬁning the action of f on the basis of [p, p] as follows. • f (hα ) = ϕ (e α , e −α ) for α ∈ π ; • For each β ∈ Φ + ∪ Φπ− , we choose dβ ∈ h (depending on β ) such that β(dβ ) = 1, and we deﬁne the action of f on e β as f (e β ) = ϕ (dβ , e β ). Lemma 2.1. ϕ (h, e β ) = f ([h, e β ]) for all h ∈ h and β ∈ Φ + ∪ Φπ− . Proof. By [h − β(h)dβ , e β ] = 0, then f ([h, e β ]). 2 ϕ (h − β(h)dβ , e β ) = 0, which immediately implies that ϕ (h, e β ) = Lemma 2.2. For α , γ ∈ Φ + ∪ Φπ− , if α + γ = 0, then ϕ (e α , e γ ) = f ([e α , e γ ]). Proof. If α + γ is not a root, then [e α , e γ ] = 0, and thus, the assertion obviously holds. Suppose that α + γ is a root β . Then, either β + α or β + γ fails to be a root. Assume, without loss of / Φ . Then, [e β , e α ] = 0. Choose h ∈ h such that γ (h) = 0 and β(h) = − N α ,γ . generality, that β + α ∈ Then, [h + e α , e β + e γ ] = 0, which implies that ϕ (h + eα , eβ + eγ ) = 0. Applying Lemma 2.1, we note that ϕ (eα , eγ ) = −ϕ (h, eβ ) = − f [h, eβ ] = −β(h) f (e β ) = N α ,γ f (e β ) = f [e α , e γ ] . 2 Lemma 2.3. Let β ∈ Φπ+ . Then, ϕ (e β , e −β ) = f ([e β , e −β ]) if there exist two distinct roots that β, γ , α satisfy the following three conditions. γ , α ∈ Φπ+ such (i) The set {β, γ , α } is linearly dependant; (ii) β + α , β + γ , γ − α all are not roots; (iii) ϕ (e γ , e −γ ) = f ([e γ , e −γ ]) and ϕ (e α , e −α ) = f ([e α , e −α ]). Proof. By (i), we may assume that β = aα + bγ with a, b ∈ F . Because the map from h∗ to h, as 2t β , deﬁned by φ → t φ , is an isomorphic map, we note that t β = at α + bt γ . Recalling that hβ = (β,β) (γ ,γ ) (α ,α ) hβ = a1 hα + b1 hγ , where a1 = (β,β) a, b1 = (β,β) b. By (ii), we have that [e β , e α ] = [e −β , e −α ] = 0, [e β , e γ ] = [e −β , e −γ ] = 0, and [e γ , e −α ] = [e −γ , e α ] = 0. Thus, one can easily see that [e β + a1 e −α + b1 e −γ , e −β + e α + e γ ] = 0, from which we note that ϕ (eβ + a1 e−α + b1 e−γ , e−β + eα + eγ ) = 0. Applying condition (iii) of this lemma, we note that 148 D. Wang et al. / Journal of Algebra 331 (2011) 145–151 ϕ (eβ , e−β ) = a1 ϕ (eα , e−α ) + b1 ϕ (eγ , e−γ ) = f (a1 hα + b1 hγ ) = f (hβ ) 2 = f [e β , e −β ] . Lemma 2.4. ϕ (e β , e −β ) = f ([e β , e −β ]) for all β ∈ Φπ . Proof. Because ϕ is skew symmetric, we need only prove the result for the case that β ∈ Φπ+ . The proof is divided into three parts for Φ of different types. Case 1. Φ is type G 2 . In this case, we arrange the basis of the root system as = {α1 , α2 }, where α1 is a long root and α2 is a short root. If π is the empty set or π has only one element, then the result obviously holds. Now, we consider the case that π = . Then Φπ+ = Φ + = {α1 , α2 , α1 + α2 , α1 + 2α2 , α1 + 3α2 , 2α1 + 3α2 }. • If β is α1 or α2 , then the deﬁnition of f implies that ϕ (e β , e −β ) = f ([e β , e −β ]). • If β is the maximal root 2α1 + 3α2 , choose α to be α1 and choose γ to be α2 . Then, β , α , and γ satisfy the three conditions of Lemma 2.3. Thus, ϕ (e β , e −β ) = f ([e β , e −β ]). • If β is the root α1 + 3α2 , set α to be α2 and set γ to be 2α1 + 3α2 . Then β , α and γ satisfy the three conditions of Lemma 2.3. We also have that ϕ (e β , e −β ) = f ([e β , e −β ]). • Either β takes the root α1 + 2α2 , or it takes the root α1 + α2 . We choose α to be α1 and choose γ to be α1 + 3α2 . Then, β , α and γ satisfy the three conditions of Lemma 2.3. The assertion holds. So ϕ (e β , e −β ) = f ([e β , e −β ]) for all β ∈ Φπ+ . Case 2. All roots in Φ have the same length. In this case, we provide a proof by induction on ht β . If ht β = 1, then the assertion holds by deﬁnition of f . Suppose that the assertion holds for γ ∈ Φπ+ with height k. Consider root β ∈ Φπ+ with height k + 1. There exists some α ∈ π such that β − α ∈ Φπ+ . Denote β − α by γ . Then ϕ (e γ , e −γ ) = f ([e γ , e −γ ]), because of the induction assumption. Because all roots in Φ have the same length, we know that β + α , β + γ , γ − α all are not roots. Thus, β, α and γ satisfy the conditions of Lemma 2.3, such that ϕ (e β , e −β ) = f ([e β , e −β ]). Case 3. Φ has two root lengths and is not of type G 2 . We again use induction on ht β to prove that ϕ (e β , e −β ) = f ([e β , e −β ]) for β ∈ Φπ+ . If ht β = 1, then the assertion holds. Assume the assertion holds for γ ∈ Φπ+ with height not larger than k. Consider root β ∈ Φπ+ with height k + 1. There exists some α ∈ π such that β − α ∈ Φπ+ . Denote β − α by γ . If γ − α , denoted by γ1 , is a root, then β + α , β + γ1 and γ1 − α all are not roots. Then, β, α and γ1 satisfy the conditions of Lemma 2.3. Thus, ϕ (e β , e −β ) = f ([e β , e −β ]). Now suppose that γ − α is not a root. Note that β + α and β + γ cannot both be roots. If β + α and β + γ both are not roots, then β, α and γ satisfy the conditions of Lemma 2.3. Thus, ϕ (eβ , e−β ) = f [eβ , e−β ] . If β + α is a root but β + γ is not a root, then one may verify that β + α , α , γ satisfy the conditions D. Wang et al. / Journal of Algebra 331 (2011) 145–151 149 of Lemma 2.3, so ϕ (eβ+α , e−β−α ) = f [eβ+α , e−β−α ] . Moreover, one may verify that β, β + α and Lemma 2.3, we also note that γ also satisfy the conditions of Lemma 2.3. Applying ϕ (eβ , e−β ) = f [eβ , e−β ] . If β + γ is a root and β + α is not a root, by an analogous process, we note that ϕ (eβ , e−β ) = f [eβ , e−β ] . 2 Combining Lemma 2.2 with Lemma 2.4, we note that Lemma 2.5. ϕ (e β , e γ ) = f ([e β , e γ ]) for all β, γ ∈ Φ + ∪ Φπ− . With Lemma 2.1 and Lemma 2.5, we are now ready to prove the Basic Theorem. Proof of the Basic Theorem. Let f be deﬁned as above. We now show that x, y ∈ p. Express x and y as x=h+ y =d+ aβ e β , β∈Φ + ∪Φπ− ϕ (x, y ) = ϕ h + + bγ eγ bγ ϕ (h, e γ ) − γ ∈Φ + ∪Φπ− = f [h, d] + γ ∈Φ + ∪Φπ− = ϕ (h, d) + ϕ is skew symmetric and applying Lemma 2.1 and aβ e β , d + β∈Φ + ∪Φπ− γ ∈Φ + ∪Φπ− β∈Φ + ∪Φπ− β∈Φ + ∪Φπ− = f [x, y ] . 2 aβ e β , d + γ ∈Φ + ∪Φπ− bγ eγ aβ f [d, e β ] aβ bγ f [e β , e γ ] h+ β,γ ∈Φ + ∪Φπ− bγ f [h, e γ ] − aβ ϕ (d, e β ) + β∈Φ + ∪Φπ− β,γ ∈Φ + ∪Φπ− =f bγ eγ , γ ∈Φ + ∪Φπ− where h, d ∈ h and aβ , bγ ∈ F . Recalling that Lemma 2.5, we note that ϕ (x, y ) = f ([x, y ]) for all aβ bγ ϕ (e β , e γ ) 150 D. Wang et al. / Journal of Algebra 331 (2011) 145–151 3. Application to zero product derivations Recently, some researchers have become interested in generalizing derivation of Lie algebras. Leger and Luks introduced the concept of quasiderivation of Lie algebras in [7]. Let L be a Lie algebra. A linear map f on L is called a quasiderivation of L if there exits a linear map f on [ L , L ] such that f (x), y + x, f ( y ) = f [x, y ] , ∀x, y ∈ L . In [7], it was shown that Q Der( L ) = Der( L ) + C ( L ) if L is generated by special weight spaces, where Q Der( L ) denotes the set of all quasiderivations of L, and C ( L ) indicates the centroid of L. In particular, for a parabolic subalgebra p of a simple Lie algebra of characteristic 0, each quasiderivation of p was shown to be the sum of an inner derivation and a scalar multiplication map on p if rank(g) 2. A linear map f on a Lie algebra L is called a zero product derivation of L if [x, y ] = 0 implies that [ f (x), y ] + [x, f ( y )] = 0. It is easy to verify that a quasiderivation is a zero product derivation of L. As such the concept of zero product derivation is slightly more general than that of quasiderivation. The problem of describing zero product Lie derivations for certain rings was ﬁrst described by [1] (see Theorem 4 in that paper). In [9] we studied zero product derivations for parabolic subalgebras of simple Lie algebras and obtained the following theorem using a method that mainly depends on direct calculation. To apply the Basic Theorem of this article, we now prove this result in a different way. Theorem 3.1. (See [9].) If rank(g) = 1, then every linear map on p is a zero product derivation of p. If rank(g) 2, then a zero product derivation of p is simply the sum of an inner derivation and a scalar multiplication map. Proof. Let ψ be a zero product derivation of p. Using ψ , we deﬁne ϕ : p × p → p so that ϕ (x, y ) = [ψ(x), y ] + [x, ψ( y )] for x, y ∈ p. Then note that ϕ is bilinear, and if [x, y ] = 0 for x, y ∈ p, then ϕ (x, y ) = [ψ(x), y ] + [x, ψ( y )] = 0, recalling that ψ is a zero product derivation of p. Applying the Basic Theorem, we can ﬁnd a linear map f from [p, p] to p such that ϕ (x, y ) = [ψ(x), y ] + [x, ψ( y )] = f ([x, y ]) for all x, y ∈ p. This implies that ψ is exactly a quasiderivation of p. Applying Corollary 4.13 in [7], we note that ψ is the sum of an inner derivation and a scalar multiplication map in the case that rank(g) 2. In the case that rank(g) = 1, the result obviously holds. 2 4. Application to commutativitypreserving maps Much attention has been paid to commutativitypreserving problems on associative F algebras, particularly matrix algebras. The earliest paper on such problems dates back to 1976, when Watkins [10] studied commutativitypreserving maps on the full matrix algebra M n over a ﬁeld F . To review the rather long and rich history of commutativity preserver problems, the reader is referred to the historic remarks in the book [2] by Brešar, Chebotar and Martindale. For a Lie algebra L, we say that x commutes with y if [x, y ] = 0. An invertible linear map ψ on L is called a twosided commutativitypreserving map if [ψ(x), ψ( y )] = 0 ⇔ [x, y ] = 0 for x, y ∈ L. An invertible linear map φ on L is called a quasiautomorphism of L if there exists an invertible linear map φ̄ on [ L , L ] such that [φ(x), φ( y )] = φ̄([x, y ]) for all x, y ∈ L. Surveying the literature, we ﬁnd that in 1981 Wong [11] studied invertible linear maps on Lie algebras that preserve commutativity. However, he only studied such maps on simple Lie algebras of linear types. We extend Wong’s result to parabolic subalgebras of simple Lie algebras. Our goal is to reduce twosided commutativitypreserving maps on p to quasiautomorphisms on p. Obviously, a quasiautomorphism of a Lie algebra L must be a twosided commutativitypreserving map on L. Using the Basic Theorem presented in this article, we prove in Theorem 4.2 that if L is taken to be the parabolic subalgebra p of g, then the converse statement D. Wang et al. / Journal of Algebra 331 (2011) 145–151 151 also holds. To provide a deﬁnite description of twosided commutativitypreserving maps on p, we must ﬁrst characterize quasiautomorphisms of p. We note that this work has been done in our other article [8]. The main result of this article is as follows. Theorem 4.1. (See [8].) Let g be a simple Lie algebra of rank l over an algebraically closed ﬁeld F of characteristic 0, and let p be an arbitrary parabolic subalgebra of g. (i) If l = 1, then every invertible linear map on p is a quasiautomorphism; (ii) If l 2, then every quasiautomorphism of p is a composition of an automorphism and a nonzero scalar multiplication map on p. Applying Theorem 4.1 and the Basic Theorem of the present article, we now describe deﬁnitely twosided commutativitypreserving maps on p. Theorem 4.2. An invertible linear map on p is a twosided commutativitypreserving map if and only if it is a quasiautomorphism of p. More precisely, (i) if l = 1, then every invertible linear map on p is a twosided commutativitypreserving map; (ii) if l 2, then every twosided commutativitypreserving map on p is a composition of an automorphism and a nonzero scalar multiplication map on p. Proof. A quasiautomorphism of p is a twosided commutativitypreserving map on p. Let ψ be a twosided commutativitypreserving map on p. ψ −1 is also such a map. Using ψ , we deﬁne ϕ : p × p → p so that ϕ (x, y ) = [ψ(x), ψ( y )] for all x, y ∈ p. Note that ϕ is bilinear, and if [x, y ] = 0 for x, y ∈ p, then ϕ (x, y ) = [ψ(x), ψ( y )] = 0. Applying the Basic Theorem, we ﬁnd a linear map f from [p, p] to p such that ϕ (x, y ) = [ψ(x), ψ( y )] = f ([x, y ]) for all x, y ∈ p. Similarly, there exists a linear map f 1 from [p, p] to p such that [ψ −1 (x), ψ −1 ( y )] = f 1 ([x, y ]) for all x, y ∈ p. Thus, f is invertible, with f 1 as its inverse. Therefore, ψ is precisely a quasiautomorphism of p. Applying Theorem 4.1, we obtain the deﬁnite description of ψ . 2 Acknowledgment We are grateful to the referee for providing us with substantial help, particularly in referring us to the related paper [5]. References [1] M. Brešar, Commuting traces of biadditive mappings, commutativitypreserving mappings, and Lie mappings, Trans. Amer. Math. Soc. 335 (1993) 525–546. [2] M. Brešar, M.A. Chebotar, W.S. Martindale, Functional Identities, Birkhäuser Verlag, 2007. [3] M. Brešar, M. Grašič, J.S. Ortega, Zero product determined matrix algebras, Linear Algebra Appl. 430 (2009) 1486–1498. [4] M. Brešar, P. Šemrl, On bilinear maps on matrices with applications to commutativity preservers, J. Algebra 301 (2006) 803–837. [5] M. Grašič, Zero product determined classical Lie algebras, Linear Multilinear Algebra 58 (8) (2010) 1007–1022. [6] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972. [7] G.F. Leger, E.M. Luks, Generalized derivations of Lie algebras, J. Algebra 228 (2000) 165–203. [8] D.Y. Wang, Z.X. Chen, Quasiautomorphisms of Lie algebras, Comm. Algebra, in press. [9] D.Y. Wang, W. Zhang, Z.X. Chen, Product zero derivations of parabolic subalgebras of simple Lie algebras, J. Lie Theory 20 (2010) 167–174. [10] W. Watkins, Linear maps that preserve commuting pairs of matrices, Linear Algebra Appl. 14 (1976) 29–35. [11] W.J. Wong, Maps on simple algebras preserving zero products, II: Lie algebras of linear type, Paciﬁc J. Math. 92 (1981) 469–487.