Kähler differentials and coverings of complex simple lie algebras extended over a commutative algebra
Kähler differentials and coverings of complex simple lie algebras extended over a commutative algebra
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DOI：
10.1016/0022-4049(84)90040-9
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发表时间：
1984-12
影响因子：
0.8
通讯作者：
Christian Kassel
中科院分区：
文献类型：
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作者：
Christian Kassel
令 g 为有限维复简单李代数。给定 Chevalley 基以及 g 的 Z 形式，我们可以在任何交换环 k 上定义李代数 g (k)。对于ka域，R. Steinberg[12]给出了g(k)的生成元和关系，并证明了g(k)一般是单连通的，这意味着它与其通用中心外延（或覆盖）同构。 k 是任意交换环的情况后来在 van der Kallen 的论文中得到了研究 [6]。这里，除了交换环 k 之外，我们还确定了一个交换 k 代数 A。然后我们将 g (A) 视为不在 A 上的李代数，而是在基环 k 上的李代数。在 k 上的李代数范畴中，g (A) 不再有任何理由是单连通的。实际上，g (A) 的“Schur 乘子”是一个 k 模，我们证明它与 Q'/A/kd/l 同构，Q'/A/kd/l 是 A 在 k 模精确形式上的卡勒微分的模。从这个事实，我们推导出 (i) k-李代数 g (A) 及其通用中心扩展的生成器和关系的表示，(ii) g (A) 通过伴随表示对 k 及其自身作用的同调计算，即：
Let g be a finite-dimensional complex simple Lie algebra. Given a Chevalley basis and hence a Z-form of g, we may define a Lie algebra g (k) over any commutative ring k. For ka field, R. Steinberg [12] gave a presentation by generators and relations of g (k) and proved that g (k) is in general simply-connected, which means that it is isomorphic to its universal central extension (or covering). The case of k an arbitrary commutative ring was investigated later in van der Kallen’s thesis [6]. Here, beside the commutative ring k, we fix a commutative k-algebra A. We then consider g (A) as a Lie algebra not over A, but over the ground ring k. In the category of Lie algebras over k, there is no longer any reason for g (A) to be sirnplyconnected. Actually, the ‘Schur multiplier’of g (A) is a k-module which we prove to be isomorphic to Q’/A/kd/l, the module of Kahler differentials of A over k modulo exact forms. From this fact, we derive (i) a presentation by generators and relations of the k-Lie algebra g (A) and of its universal central extension,(ii) homological computations for g (A) acting trivially on k and on itself by the adjoint representation, namely: