Journal Fur Die Reine Und Angewandte Mathematik Finite Dimensional Algebras and Highest Weight Categories ')
Journal Fur Die Reine Und Angewandte Mathematik Finite Dimensional Algebras and Highest Weight Categories ')
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E. Cline At Worcester;B. Pm
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作者：
E. Cline At Worcester;B. Pm
本文继续我们在[8j2]，[9](另见[15]，[18])中开始的程序，在该程序中，作者已开始在半单代数群的模表示理论中利用派生范畴理论的一些强大技巧。正如上述文献所指出的，这项工作的灵感既来自几何(以Bernstein-Beilinson-Deligne[L]关于奇异空间和斜层的经典代数工作的形式)，也来自有限维代数的倾斜理论[2]、[3]、[13]、[14]。本文将这种与有限维代数表示理论的联系扩展为一个中心主题。我们从第一节开始，完善了[9]，$1的结果，它在[L]的意义下讨论了三角范畴的“回忆”。我们将第2节中的工作应用于模范畴的情况。文献[9]，9，3讨论了当B是A的商环时，衍生范畴的自然正合函数DB(mod-B)-+DB(mod-A)的情况，本文考虑A是中心环A=end(Eb)g Ebe，E B是幂等元的“对偶”情形。我们注意到，我们对这种情况的兴趣最初是由格林在[12]，9，6中对Schur代数的处理引起的。而且，它被证明与[9]中开始的分层理论非常好地照亮了。在第三节中，我们定义了最高权范畴的统一概念。虽然我们是从半单群(或李代数)的经典表示理论中抽象得到这个概念，但文献[16]中给出的其他例子表明，这种范畴在许多(可能令人惊讶的)情况下都会出现，包括箭图代数和可构造的倒叶。定理3.4和3.6将最高权范畴理论与拟遗传代数理论(在[18]中引入)联系起来。这些结果特别提供了有限维代数的表示理论与半单群和李代数的表示理论之间的强联系。定理3.5和3.9指出，对于与最高权重类别相关联的派生类别，重新收集设置是如何工作的。尤其是定理3.9‘)部分得到了美国国家科学基金会的支持。)本文推广了Happl[14]的一些结果，并把它们的范围扩大到派生范畴的Morita理论的开端。我们在这里录制…
This paper continues the program begun by us in [8j2), [9] (see also [15], [18]) in which the authors have begun to exploit in the modular representation theory of semisimple algebraic groups some of the powerful techniques of the theory of derived categories. As noted in the above references, the inspiration for this work comes both from geometry, in the form of the classic algebraic work of Bernstein-Beilinson-Deligne [l] on singular spaces and perverse sheaves, and from the tilting theory of finite dimensional algebras [2], [3], [13], [14]. The present paper broadens and extends this connection with finite dimensional algebra representation theory into a central theme. We begin in Section 1 by completing the results of [9], $ 1, which dealt with " recollement " of triangulated categories in the sense of [l]. We apply this work in Section 2 to the situation of module categories. While [9], 9 3, treats the case of the natural exact functor Db(mod-B)-+ Db(mod-A) of derived categories arising when B is a quotient ring of A, we consider in this paper the " dual " situation in which A is a centralizer ring A = End(eB) g eBe, e E B an idempotent. We remark that our interest in this situation was first kindled by Green's treatment of the Schur algebra in [12], 9 6. Also, it turns out to lit very well with the stratification theory begun in [9]. In Section 3, we define the unifying concept of a highest weight category. Although we obtain this notion by abstracting from the classical representation theory of semisimple groups (or Lie algebras), other examples given in [16], ?$j 5, 6, indicate that such categories arise in many (perhaps surprising) situations, including quiver algebras and constructible and perverse sheaves. Theorems 3. 4 and 3. 6 relate the theory of highest weight categories to the theory of quasi-hereditary algebras (introduced in [18]). Th ese results especially appear to provide a strong link between the representation theory of finite dimensional algebras and that of semisimple groups and Lie algebras. Theorems 3. 5 and 3. 9 indicate how the recollement setup works for the derived categories associated to highest weight categories. In particular, Theorem 3. 9 ') Research supported in part by N.S.F. ') This paper extended some results of Happel [14] and broadened their scope into the beginnings of a Morita theory for derived categories. We record here …