PDF《数学杂志》

296 数学杂志Vol.

36 于是 dim V = r1 rt ≤ tn ? (i1 it) ? (1 t ? 1) ≤ tn ? (1 t) ? (1 t ? 1) = t(n ? t) ≤ n2 /4 . 另一方面, 考虑到前面定义的 A2m或B2m+1, B2m+1 的结构, 可知 dim V ≥ n2 /4 . 故dim V = n2 /4 . 从而 ik = k, rk = n ? t , 其中 k = 1,t. (2.6) 若n=2m, 则t=m. 由(2.6) 式知当 k = 1,m 时, 有rk = m. 由(2.5) 与(2.6) 式, 知前 面出现的广义矩阵单位的水平分别为 (1, m + 1) (1, m + 2)1, n) (2, m + 1) (2, m + 2)2, n) (m, m + 1) (m, m + 2)m, n). (2.7) 若n=2m + 1, 则t=m或m+1. 由(2.6) 式知当 k = 1,m 时, 有rk = m + 1;

或者当 k = 1,m +

1 时, 有rk = m. 由于 (2.5) 与(2.6) 式, 所以前面出现的广义矩阵单位的水 平分别为 (1, m + 1) (1, m + 2)1, n) (2, m + 1) (2, m + 2)2, n) (m, m + 1) (m, m + 2)m, n) 或(1, m + 2) (1, m + 3)1, n) (2, m + 2) (2, m + 3)2, n) (m + 1, m + 2) (m + 1, m + 3)m + 1, n). 首先假设 n = 2m. 由(2.2)C(2.4) 及(2.7) 式, 知ht(Vm?i) = m ? i +

1 及dim Vm?i = im,

1 ≤ i ≤ m, 其中 V0 := V. 对i归纳且由引理 , 知Vm?i 由如下形式矩阵组成 Om*m B Om*m Om*m , 其中 ht(Vm?i) = m ? i +

1 ≤ ht(B) ≤ m,

1 ≤ i <

m. 关于任意取定的 i, 若1≤j≤m, 则Um?i,m+j 与Vm?i 中每个矩阵都弱交换. 进而, 通过简 单的计算, 知Um?i,m+j = Om*m A Om*m Om*m , 其中 A 的前 m ? i 行构成 m ? i * j 的矩阵 单位 Em?i,j. 即Vm?i ∪ {Um?i,m+j |

1 ≤ j ≤ m} ? A2m. No.

2 王淑娟等: 弱交换空间的极大维数

297 特别的, 当i=m?1时, 有V1 ∪ {U1,m+j |

1 ≤ j ≤ m} ? A2m, 进而 T?1

1 V T1 = FU1,m+1 FU1,m+n ? V1 ? A2m, 因为 dim A2m = m2 = n2 /4 = dim V = dim T?1

1 V T1, 所以 T?1

1 V T1 = A2m. 若n=2m + 1, 可作类似的讨论, 此处略. 参考文献[1] Schur I. Zur Theorie vertauschbaren Matrizen[J]. J. Reine Angew. Math., 1905, 130: 66C76. [2] Burde D. On a re?nement of Ado'

s theorem[J]. Arch. Math., 1998, 70: 118C127. [3] Burde D, Moens W. Minimal faithful representation of reductive Lie algebras[J]. Arch. Math., 2008, 89: 513C523. [4] Jacobson N. Schur'

s theorems on commutative matrices[J]. Bull. Amer. Math. Soc., 1944, 50: 431C 436. [5] Liu W, Wang S. Minimal faithful representations of abelian Jordan algebras and Lie superalgebras[J]. Linear Algebra Appl., 2012, 437: 1293C1299. THE MAXIMAL DIMENSION FOR WEAKLY COMMUTATIVE SPACES WANG Shu-juan, LIU Wen-de (School of Mathematical Sciences, Harbin Normal University, Harbin 150025, China) Abstract: In this paper, we study the maximal dimension of weakly commutative spaces over algebraically closed ?elds of characteristic zero. By similar operations for matrices, we obtain the classi?cation for the maximal weakly commutative spaces in the sense of conjugation and generalize Schur'

s theorem. Keywords: weakly commutative;

maximal dimension;

similar operation

2010 MR Subject Classi?cation: 17B05;

17B10 ........