Finite dimensional representations of algebras
Finite dimensional representations of algebras
复制标题
代数的有限维表示
DOI：
10.1007/bf02756630
复制
发表时间：
1974
影响因子：
1
通讯作者：
C. Procesi
中科院分区：
文献类型：
--
作者：
C. Procesi
令r为戒指，k一个场，n自然数字。我们将关注以下类型的问题：（a）对r的表示（ho，（或n维表示）中的表示形式。（b）将n维表示为自然等效性。 r〜（k），我们说〜Bx等于〜b 2，如果有（k）的k自动形态7，因此7q〜1 = 4）2。 （c）对不可还原和半圣经表示的等价类别进行分类。为了简单起见，我们将主要局限于k是代数关闭的情况，而r是有限生成的k代数。实际上，对于本文的大多数，我们将假设r = k {xa，...，x，）是一个自由代数。最后，我们将从结果中推断出更普遍的定理，而不一定是自由代数。在描述我们将获得的定理之前，让我们离题以激发我们将使用多项式身份作为攻击之前所述问题的工具来激发我们对环理论的使用。因此，让我们回想起该理论的一些基本结构结果，并根据表示理论来解释它们。
Let R be a ring, K a field, and n a natural number. We will be concerned with the following type of questions: (a) Classify the representations of R in (hO, (or n dimensional representations). (b) Classify n dimensional representations up to the natural equivalence. If ~bt, ~b2 : R ~ (K), we say ~bx is equivalent to ~b 2 if there is a K automorphism 7 of (K), such that 7q~1 = 4)2. (c) Classify equivalence classes of irreducible and semisimple representations. For the sake of simplicity we will restrict ourselves mainly to the case that K is algebraically closed and R is a finitely generated K algebra. In fact, for most of this paper, we will assume that R = K{xa, ..., x,) is a free algebra; at the end we will deduce from the results in this ease the more general theorems for not necessarily free algebras. Before describing the theorems that we will obtain, let us digress in order to motivate the use that we will make of the theory of rings with polynomial identities as a tool for attacking the problems stated before. Let us recall, therefore, some of the basic structural results of this theory and interpret them in terms of representation theory.