Maps on simple algebras preserving zero products. II. Lie algebras of linear type.
Maps on simple algebras preserving zero products. II. Lie algebras of linear type.
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DOI：
10.2140/pjm.1981.92.469
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发表时间：
1981-02
影响因子：
0.6
通讯作者：
W. J. Wong
中科院分区：
文献类型：
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作者：
W. J. Wong
引言 * 在[8]中，我们开始研究的半线性映射的代数在一个域k保持零产品，一个问题所产生的最近的调查特征的线性变换的n × n矩阵代数Mn（k）在k保持各种性质，特别是工作沃特金斯地图保持交换对矩阵[7]。如果L是一个李代数，这意味着我们关注L上的双射半线性映射f使得[f（x），f（y）] - 0，对于L的所有元素对x，y使得[x，y] - 0。我们说/保持零李积。如果L是有限维的，则这些映射f形成群G（L）[8]。显然G（L）包含所有半线性自同构和反自同构的群G±（半线性映射，它们是L的乘法结构的自同构或反自同构），L的质心的单位群G2（与L中的左乘法交换的线性变换的代数），以及形式为f（x）= x + g（x）的所有双射变换f的群Gs，其中g是L到其中心Z（L）的线性映射。设GQ（L）= GXG 2GZ。本文确定了一类单李代数G（L），这是通过取域k上的有限维单结合代数A，构成李代数L - [A，A]l[A，A]<$Z（A）而得到的，其中[A，A]是所有扩张子[x，y] = xy-yx所张成的子空间，Z（A）是A的中心.如果A是非交换的，则L是单李代数，除非A的特征为2并且是Z（A）上的4维[1，p. 17]。除了“小长度l”的情形外，我们证明了对这样的李代数L，G（L）= G 0（L）.事实上，我们可以处理更广泛的类别，
Introduction* In [8] we began the study of the semilinear maps on an algebra over a field k which preserve zero products, a problem arising from recent investigations characterizing the linear transformations on the n x n matrix algebra Mn(k) over k which preserve various properties, particularly the work of Watkins on maps preserving commuting pairs of matrices [7]. If L is a Lie algebra, this means that we are concerned with the bijective semilinear maps f on L such that [f(x), f(y)] — 0 for all pairs of elements x, y of L such that [x, y] — 0. We say that / preserves zero Lie products. If L is finite-dimensional, these maps / form a group G(L) [8]. Clearly G(L) contains the group G± of all semilinear automorphisms and anti-automorphisms (semilinear maps which are automorphisms or anti-automorphisms of the multiplicative structure of L), the group of units G2 of the centroid of L (the algebra of linear transformations which commute with left multiplications in L), and the group Gs of all bijective transformations / of the form f(x) = x + g(x), where g is a linear map of L into its center Z(L). Let GQ(L) = GXG2GZ. In this paper we determine G(L), for a class of simple Lie algebras L, These are obtained by taking finite-dimensional simple associative algebras A over a field k and forming the Lie algebra L — [A, A]l[A, A] Π Z(A), where [A, A] is the subspace spanned by all the commutators [x, y] = xy — yx, and Z(A) is the center of A. If A is noncommutative, then L is a simple Lie algebra, except when A has characteristic 2 and is 4-dimensional over Z{A) [1, p. 17]. Except for cases of "small length/' we show that G(L) = G0(L) for such a Lie algebra L. In fact, we can deal with a wider class of