Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy
Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy
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DOI：
10.1007/bf02100608
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发表时间：
1992-12
影响因子：
3.1
通讯作者：
S. Berman;R. Moody
中科院分区：
文献类型：
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作者：
S. Berman;R. Moody
本文讨论了环面李代数，Slodowy定义的若干交矩阵李代数，以及它们之间的相互关系和与Steinberg群的若干李代数类似物的关系。本文的主要结果是在若干Steinberg李代数和环面李代数（命题5.9和5.10）下，对A、D、E型的多重仿射Cartan矩阵所产生的交矩阵代数的辨识。论文的主要部分是研究和分类由有限根分级的李代数。它们成为我们分析交矩阵代数的主要工具。每一个由秩为bbbb2的简单系有限根系统分级的李代数，都有一个根据其类型和秩，要么是交换结合的代数，要么是唯一结合的代数，要么是可选的代数。所有这些可能性都出现在我们对交矩阵代数的描述中。设R是在特征为0的域k上，具有恒等式的任意结合代数，不一定是有限维的。对于每一个正整数n， n•n个系数为R的矩阵的结合代数M，(R)在对易子积下形成k上的李代数。我们用l (R)表示这个李代数。设Eis为m (R)的（i, j）矩阵单位，设n> 2。OI的子代数e (R)(R)由元素rEis生成，R ~ R, i 4= j，是91的理想，(R)是完美的，即它是自己的派生代数。
This paper is about toroidal Lie algebras, certain intersection matrix Lie algebras defined by Slodowy, and their relationship to one another and to certain Lie algebra analogues of Steinberg groups. The main result of the paper is the identification of the intersection matrix algebras arising from multiply-affinized Cartan matrices of types A, D and E with certain Steinberg Lie algebras and toroidal Lie algebras (Propositions 5.9 and 5.10). A major part of the paper studies and classifies Lie algebras graded by finite root systems. These become the principal tool in our analysis of intersection matrix algebras. Each Lie algebra graded by a simply-laced finite root system of rank> 2 has attached to it an algebra which, according to the type and rank, is either commutative and associative, only associative, or alternative. All these possibilities occur in our description of intersection matrix algebras.Let R be any associative algebra with identity, not necessarily finite dimensional, over a field k of characteristic 0. For each positive integer n the associative algebra M,(R) of n• n matrices with coefficients in R forms a Lie algebra over k under the commutator product. We denote this Lie algebra by ol,(R). Let Eis be the (i, j) matrix unit of M.(R) and assume that n> 2. The subalgebra e.(R) of OI.(R) generated by the elements rEis, r~ R, i 4= j, is an ideal of 91,(R) and is perfect, ie it is its own derived algebra.