Polynomial identities of Lie subalgebras of M(n).

Abstract

Let $K\sb{n}(F)$ denote the Lie algebra of skew symmetric matrices with coefficients in a field F. For 1 $\le\ k \le\ m$ + 1, define $a\sb{m}(k)(x\sb1,\cdots,x\sb{m};y)\:=\ \sum\sb{\sigma\in{\cal S}\sb{m}} (-1)\sp{\sigma}x\sb{\sigma(1)}\cdots x\sb{\sigma(k-1)}yx\sb{\sigma(k)}\cdots x\sb{\sigma(m)},$ and let $a\sbsp{m}{+}$ (respectively $a\sbsp{m}{-}$) be the sum of all $a\sb{m}(k)$ with k even (respectively odd). This dissertation contains two parts. In the first part (Chapter 2), we prove that the almost standard polynomial $f(x\sb{1},\cdots, x\sb{m};y)\:=\ \alpha a\sbsp{m}{+} + \beta a\sbsp{m}{-} + \sum\sb{k\le m-2n+4\atop or k\ge 2n-1}$ $\gamma\sb{k} a\sb{m}(k)$ is a weak identity of $K\sb{n},$ for any $\alpha, \beta, \gamma \in\ F$ and any integer $n \ge 2$ if and only if $m \ge 2n - 2.$ As a consequence, the Lie standard polynomial identity of degree 4$n -\ 7$ on the Lie algebra $K\sb{n}$ is obtained. The sharpness of the results is also examined in this part, in the case n = 7, a positive answer is given. In the second part (Chapter 3), the similar problem on $sp\sb{2n}(F),$ the Lie subalgebra of all skew-symmetric matrices with respect to the symplectic involution, is discussed.