Linear Algebra and its Applications

Linear Algebra and its Applications 437 (2012) 2719 2726 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Lie derivations of generalized matrix algebras Yiqiu Du a,yuwang b, a College of Mathematics, Jilin Normal University, Siping 136000, PR China b Department of Mathematics, Shanghai Normal University, Shanghai 200234, PR China ARTICLE INFO Article history: Received 12 February 2012 Accepted 11 June 2012 Available online 30 July 2012 SubmittedbyJ.F.Queiro ABSTRACT The aim of this paper is to give a description of Lie derivations of generalized matrix algebras. As a consequence Lie derivations of full matrix algebras are determined. 2012 Elsevier Inc. All rights reserved. AMS classification: 15A78 16W25 47L35 Keywords: Generalized matrix algebras Triangular algebras Lie derivations Full matrix algebras Throughout the paper, by an algebra we shall mean an algebra over a fixed unital commutative ring R, and we assume without further mentioning that 1 2 R. Let A be an algebra with center Z(A).Set[x, y] =xy yx for x, y A.AnR-linear map δ of A is called derivation if δ(xy) = δ(x)y + xδ(y) for all x, y A. For example, for a fixed element b A,wedefinea map ad(b) : x [x, b] for all x A.Thenad(b) is a derivation of A, which is called inner derivation of A induced by b.anr-linear map δ of A is said to be a Lie derivation if δ([x, y]) =[δ(x), y]+[x,δ(y)] for all x, y A. Lie derivations have been discussed by many authors (see [1,2,4,6,7,10,12]). Let A and B be two unital algebras with unit 1 and 1, respectively. A Morita context consists of A, B, two bimodules A M B and B N A, and two bimodule homomorphisms called the pairings MN : M B N A and MN : N A M B satisfying the following commutative diagrams: Corresponding author. E-mail addresses: duyiqiu-2006@163.com (Y. Du), ywang2004@126.com (Y. Wang). 0024-3795/$ - see front matter 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2012.06.013

2720 Y. Du, Y. Wang / Linear Algebra and its Applications 437 (2012) 2719 2726 and M B N A M I M NM M B B MN I M A A M = = M N A M B N NM I N B B N I N MN = = N A A N. If (A, B, M, N, MN,ψ NM ) is a Morita context, then the set G = A M = a m a A, m M, n N, b B N B n b forms an R-algebra under matrix-like addition and multiplication, where at least one of the two bimodules M and N is distinct from zero. Such an R-algebra is called a generalized matrix algebra (see [11,13] for details). We further assume that M is faithful as an (A, B)-bimodule. If N = 0, then G becomes a triangular algebra (see [5] for details). Consider a generalized matrix algebra G. Any element of the form a 0 G 0 b will be denoted by a b. Let us define two natural projections π A : G A and π B : G B by π A : a m a and π B : a m b. n b n b The center of G is Z(G) = {a b am = mb, na = bn for all m M, n N}. Furthermore, π A (Z(G)) Z(A) and π B (Z(G)) Z(B), and there exists a unique algebra isomorphism τ from π A (Z(G)) to π B (Z(G)) such that am = mτ(a) and τ(a)n = na for all m M, b N (see [13, Lemma 3.1 and Lemma 3.2]). Cheung [5] initiated the study of mapping problems on triangular algebras. Cheung [6] investigated Lie derivations of triangular algebras. Recently, Ji and Qi [7] discussed Lie derivations on some subsets of triangular algebras. Recently, some results on maps of triangular algebras have been extended to generalized matrix algebras (see [8,9,13]). The aim of this paper is to describe Lie derivations of generalized matrix algebras. More precisely, we will prove the following result. Theorem 1. Let G be a generalized matrix algebra. Suppose that (i) Z(A) = π A (Z(G)) and Z(B) = π B (Z(G)); (ii) either A or B does not contain nonzero central ideals.

Y. Du, Y. Wang / Linear Algebra and its Applications 437 (2012) 2719 2726 2721 If δ is an R-linear map of G such that δ([x, y]) =[δ(x), y]+[x,δ(y)] for all x, y G with xy = 0, then there exists a derivation d of G and an R-linear map τ : G Z(G) vanishing at commutators [x, y] with xy = 0 such that for all x G. Proof. Set δ(x) = d(x) + τ(x) U = AM. 0 B Then U is a triangular algebra itself. We first claim that Z(U) = Z(G). Trivially,Z(G) Z(U). Wenow show that Z(U) Z(G). Ifa b Z(U), thenam = mb for all m M [5, Proposition 3]. Since a Z(A) = π A (Z(G)) we get that there exists τ(a) π B (Z(G)) such that a τ(a) Z(G) and am = mτ(a) for all m M. Thus,m(b τ(a)) = 0forallm M. Recall that M is faithful as a right B-module. Hence b τ(a) = 0 and so b = τ(a). This implies that a b Z(G).HenceZ(U) = Z(G). Since δ is an R-linear map, we get that there exist R-linear maps f 1 : A M and g 1 : A N such that δ a 0 = f 1(a) 00 g 1 (a) for all a A. Similarly, there exist R-linear maps f 2 : B M, f 3 : M M, f 4 : N M, g 2 : B N, g 3 : M N, and g 4 : N N such that δ 00 = f 2(b), 0 b g 2 (b) δ 0 m = f 3(m), 0 0 g 3 (m) δ 00 = f 4(n) 0 n g 4 (n) for all b B, m M and n N.Hence δ a m f = 1 (a) + f 2 (b) + f 3 (m) + f 4 (n) n b g 1 (a) + g 2 (b) + g 3 (m) + g 4 (n) for all a A, b B, m M, and n N. Since 00 a m = 0 we get that 0 b 0 0

2722 Y. Du, Y. Wang / Linear Algebra and its Applications 437 (2012) 2719 2726 δ 0 mb = δ 00, a m 0 0 0 b 0 0 = δ 00, a m + 00,δ a m. 0 b 0 0 0 b 0 0 This implies that =, a m +,. g 3 (mb) g 2 (b) 0 0 0 b g 1 (a) + g 3 (m) We get from the last relation that g 3 (mb) = g 2 (b)a + bg 1 (a) + bg 3 (m) (1) for all a A, b B, and m M. Taking m = 0in(1) we get that g 2 (b)a + bg 1 (a) = 0. This implies that g 1 (1) + g 2 (1 ) = 0, g 1 (a) = g 1 (1)a, and g 2 (b) = bg 1 (1). Thus,wegetfrom(1) that g 3 (mb) = bg 3 (m). Taking b = 1 in the last relation we get that 2g 3 (m) = 0 and then g 3 = 0. Similarly, computing a 0, 00 we can obtain that f1 (a) = af 1 (1), f 2 (b) = f 1 (1)b, 00 nb and f 4 = 0. Hence δ a m af = 1 (1) f 1 (1)b + f 3 (m). n b g 1 (1)a bg 1 (1) + g 4 (n) 0 f Replacing δ by δ + ad 1 (1) we may assume that g 1 (1) 0 δ a m = f 3(m). n b g 4 (n) In particular, we have δ a m = f 3(m). 0 b 0 (2) This implies that δ induces an R-linear map of U such that δ([x, y]) =[δ(x), y]+[x,δ(y)] for all x, y U with xy = 0. By [7, Theorem 2.1] we have that there exists a derivation d 1 of U and an R-linear map μ : U Z(G) vanishing at commutators [x, y] with xy = 0 such that δ(x) = d 1 (x) + μ(x) (3) for all x U. Inviewof[6, Lemma 5] we have that d 1 a m = p A(a) as sb + f (m) (4) 0 b 0 p B (b)

Y. Du, Y. Wang / Linear Algebra and its Applications 437 (2012) 2719 2726 2723 for all a A, b B, and m M, where s M and (i) p A is a derivation of A, f (am) = p A (a)m + af (m); and (ii) p B is a derivation of B, f (mb) = mp B (b) + f (m)b. Substituting both (2) and (4)into(3) we get that in particular f 3 (m) = as sb + f (m) for all a A, b B, and m M. Thisimpliess = 0 and f 3 = f.hence δ a m = p A(a) f (m) + μ a m (5) 0 b 0 p B (b) 0 b for all a A, b B, and m M.Sincef 4 = 0, we have that there exist R-linear maps h 1 : N A and h 2 : N B such that δ 00 = h 1(n) 0 n 0 g 4 (n) h 2 (n) for all n N. Since a 0 00 = 0wegetthat 00 n 0 δ 0 0 = δ a 0, 00 na 0 00 n 0 = δ a 0, 00 + a 0,δ 00. 00 n 0 00 n 0 It follows from (5) that h 1(na) 0 = A(a) 0, 00 +, h 1(n) 0. g 4 (na) h 2 (na) 0 0 n 0 00 g 4 (n) h 2 (n) Thus, we get from the last relation that h 1 (na) = ah 1 (n) + h 1 (n)a; h 2 (na) = 0; g 4 (na) = np A (a) + g 4 (n)a for all a A and n N. Taking a = 1 in the last relations we obtain that h 1 = 0 and h 2 = 0. Thus, we get that δ 00 = 0 0. n 0 g 4 (n) 0

2724 Y. Du, Y. Wang / Linear Algebra and its Applications 437 (2012) 2719 2726 Similarly, computing δ 00, 00 we can obtain that 0 b n 0 g 4 (bn) = p B (b)n + bg 4 (n) for all b B and n N.Setg = g 4 and μ a m = μ 1(a, b, m) 0. 0 b 0 μ 2 (a, b, m) Considering the above relations we have that δ a m = d 1 a m + δ 00 + μ a m n b 0 b n 0 0 b = p A(a) + μ 1 (a, b, m) f (m). (6) g(n) p B (b) + μ 2 (a, b, m) Since 00 0 m = 0 we see that n 1 0 nm δ mn m = δ 00, 0 m nmn nm n 1 0 nm = δ 00, 0 m + 00,δ 0 m. n 1 0 nm n 1 0 nm Substituting (6) into the last relation we can get that p A (mn) = mg(n) + f (m)n μ 1 (mn, nm, m), (7) p B (nm) = g(n)m + nf (m) + μ 2 (mn, nm, m) (8) for all m M, n N. Assume without loss of generality that A does not contain nonzero central ideals. It follows from (7) that p A (amn) amg(n) f (am)n = μ 1 (amn, nam, am) for all a A, m M, and n N. Extending the last relation we can get p A (a)mn + ap A (mn) amg(n) p A (a)mn af (m)n = μ 1 (amn, nam, am) It further follows from (7) that aμ 1 (mn, nm, m) = μ 1 (amn, nam, am) for all a A, m M, and n N. This implies that for each pair of elements m M, n N the set Aμ 1 (mn, nm, m) is a central ideal of A. Hence,μ 1 (mn, nm, m) = 0. Since μ 1 (mn, nm, m) μ 2 (mn, nm, m) Z(G)

Y. Du, Y. Wang / Linear Algebra and its Applications 437 (2012) 2719 2726 2725 for all m M and n N, we get that μ 2 (mn, nm, m) = 0. Thus, we get from both (7) and (8) that p A (mn) = mg(n) + f (m)n and p B (nm) = g(n)m + nf (m) for all m M and n N. Define d a m = p A(a) f (m) n b g(n) p B (b) for all a A, b B, and m M. It is easy to check that d is a derivation of G.Set Then τ a m = μ a m. n b 0 b δ(x) = d(x) + τ(x) for all x G. The proof is now complete. We say that M is loyal if amb = 0impliesa = 0orb = 0foranya A and b B (see [3, Section 2]). Applying Theorem 1 we can obtain the following result. Theorem 2. Let G be a generalized matrix algebra. Suppose that (i) Z(A) = π A (Z(G)) and Z(B) = π B (Z(G)); (ii) M is loyal as an (A, B)-bimodule; (iii) either A or B is noncommutative. If δ is an R-linear map of G such that δ([x, y]) =[δ(x), y]+[x,δ(y)] for all x, y G with xy = 0, then there exists a derivation d of G and an R-linear map τ : G Z(G) vanishing at commutators [x, y] with xy = 0 such that for all x G. δ(x) = d(x) + τ(x) Proof. We assume without loss of generality that A is noncommutative. We claim that A does not contain nonzero central ideals. Indeed, suppose that I is a nonzero central ideal of A. Taking 0 = a I we see that Aa is a nonzero central ideal of A.Thisresultsin[A, A]a = 0 and so [A, A]Mτ(a) = 0. Since M is loyal and τ(a) = 0 we get that [A, A] =0, contradicting our assumption. The result now follows from Theorem 1. Note that the main examples of generalized matrix algebras are full matrix algebras (an inflated algebra can be viewed as a full matrix algebra) and triangular algebras (see [11,13]). Let A be a unital algebra and M n (A) be the algebra of all n n matrices over A (n 2). Then the full matrix algebra M n (A) can be represented as a generalized matrix algebra of the form A M 1 (n 1) (A). M (n 1) 1 (A) M (n 1) (n 1) (A)

2726 Y. Du, Y. Wang / Linear Algebra and its Applications 437 (2012) 2719 2726 Lemma 1. M n (A) does not contain nonzero central ideals. Proof. Denote Z(M n (A)) as the center of M n (A). Note that Z(M n (A)) = Z(A) I n, where I n is the unit of M n (A). Suppose that a is a nonzero central element of M n (A). Thena = c I n for some nonzero element c Z(A).Setb = n i,j=1 e ij, where e ij denotes the usual matrix unit of M n (A). It is clear that ba = ij ce ij Z(M n (A)). That is, M n (A)a Z(M n (A)). This implies that M n (A) does not contain nonzero central ideals. As a consequence of Lemma 1 and Theorem 1 we have the following result. Corollary 1. Let M n (A) be a full matrix algebra with n 3.Ifδ : M n (A) M n (A) is an R-linear map such that δ([x, y]) =[δ(x), y]+[x,δ(y)] for all x, y M n (A) with xy = 0, then there exists a derivation d of M n (A) and an R-linear map τ : M n (A) Z(A) I n,wherei n is the unit of M n (A), vanishing at commutators [x, y] with xy = 0 such that δ(x) = d(x) + τ(x) for all x M n (A). Acknowledgments The authors would like to express their sincere thanks to the referee for his/her careful reading of the manuscript. The valuable suggestions have clarified the paper greatly. The first author is supported in part by the Natural Science Foundation Grants of Jilin Province (20125220) and the second author is supported in part by the innovation program of Shanghai Municipal Education Commission (11ZZ119). References [1] K.I. Beidar, M. Brešar, M.A. Chebotar, W.S. Martindale 3rd, On Herstein s Lie maps conjectures, J. Algebra 238 (2001) 239 264. [2] K.I. Beidar, M.A. Chebotar, On Lie derivations of Lie ideals of prime rings, Israel J. Math. 123 (2001) 131 148. [3] D. Benkovič, D. Eremita, Commuting traces and commmutativity preserving maps on triangular algebras, J. Algebra 280 (2004) 797 824. [4] M. Brešar, M.A. Chebotar, A. Mikhail, W.S. Martindale 3rd, Functional Identities, Birkhäuser Verlag, 2007. [5] W.-S. Cheung, Commuting maps of triangular algebras, J. London Math. Soc. 63 (2001) 117 127. [6] W.-S. Cheung, Lie derivations of triangular algebras, Linear and Multilinear Algebra 51 (2003) 299 310. [7] P.S. Ji, W.Q. Qi, Charactrizations of Lie derivations of triangular algebras, Linear Algebra Appl. 435 (2011) 1137 1146. [8] Y.B. Li, F. Wei, Semi-centralizing maps of generalized matrix algebras, Linear Algebra Appl. 436 (2012) 1122 1153. [9] Y.B. Li, Z.K. Xiao, Additivity of maps on generalized matrix algebras, Electron. J. Linear Algebra 22 (2011) 743 757. [10] F.Y. Lu, W. Jing, Characterizations of Lie derivations of B(X), Linear Algebra Appl. 432 (2010) 89 99. [11] A.D. Sands, Radicals and Morita contexts, J. Algebra 24 (1973) 335 345. [12] A.R. Villena, Lie derivations on Banach algebras, J. Algebra 226 (2000) 390 409. [13] Z.K. Xiao, F. Wei, Commuting mappings of generalized matrix algebras, Linear Algebra Appl. 433 (2010) 2178 2197.