论文标题
谎言代数保存归化双线性形式
The Lie algebra preserving a degenerate bilinear form
论文作者
论文摘要
让$ k $为任意字段，$ d $ a正整数。对于$ k^{d} $上的每个退化对称或反对称双线性双线性形式$ m $，我们确定保留$ m $的矩阵的Lie代数的结构，以及保留矩阵的Lie代数，这些矩阵保留了$ m $ M $。我们表明，这些谎言代数是经典谎言代数和某些表示形式的半程产物，并确定它们的激进分子，衍生的序列和半神经商。我们的主要动机和应用是确定用希尔伯特多项式$ 1+dt+t+t^{2} $的每个交换性或分级的交换代数的衍生的分级谎言代数的结构。我们的一些结果适用于更通用的双线性形式和分级代数。
Let $k$ be an arbitrary field and $d$ a positive integer. For each degenerate symmetric or antisymmetric bilinear form $M$ on $k^{d}$ we determine the structure of the Lie algebra of matrices that preserve $M$, and of the Lie algebra of matrices that preserve the subspace spanned by $M$. We show that these Lie algebras are semidirect products of classical Lie algebras and certain representations, and determine their radicals, derived series and semisimple quotients. Our main motivation and application is to determine the structure of the graded Lie algebra of derivations of each commutative or graded commutative algebra with Hilbert polynomial $1+dt+t^{2}$. Some of our results apply to more general bilinear forms and graded algebras.
