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Category of $$\mathfrak g$$ modules. (English. Russian original) Zbl 0353.18013
Funct. Anal. Appl. 10, 87-92 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 1-8 (1976).
Let $$A$$ be a finite-dimensional associative algebra with identity over a field $$K$$, and let $$\mathfrak A$$ be the category of finite-dimensional $$A$$-modules. Let $$L_1, \ldots, L_k$$ be a complete collection of irreducible $$A$$-modules. For each $$L_i$$ there exists, up to isomorphism, a unique indecomposable projective $$A$$-module $$P_i$$ such that $$\operatorname{Hom}(P_i,L_i)\neq 0$$. If $$c_{ij} = (P_i :L_j)$$ is the number of occurrences of $$L_j$$ in the Jordan-Hölder series of $$P_i$$, then the integral matrix $$C =\| c_{ij}\|$$, $$i,j=1,\ldots, k$$ is an important invariant of $$A$$. The matrix $$C$$ is called the Cartan matrix of $$A$$. In certain cases $$C$$ is symmetric, positive-definite and can be represented in the form $$C=D^t\cdot D$$, where $$D$$ is some other integral matrix. This fact means that there exists a class of $$A$$-modules $$M_1,\ldots, M_k$$, such that each $$P_i$$ has a composition series with factors isomorphic to $$M_j$$, and for any $$i,j$$ the number of occurrences of $$M_j$$ in the series for $$P_i$$ is equal to $$(M_j:L_i)$$.
The purpose of this article is to construct a category of modules over a semisimple complex Lie algebra $$\mathfrak g$$, having the same “intermediate position”-property. Simple and projective objects of this category can be indexed by the elements of the Weyl group $$W$$ of $$\mathfrak g$$, and to each $$w\in W$$ corresponds some $$\mathfrak g$$-module $$M_i$$ such that if $$C =\| (P_w :L_{w'})\|$$ and $$D =\| (M_w:L_{w'})\|$$, then $$C =D^t\cdot D$$.
Show Scanned Page ##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B56 Cohomology of Lie (super)algebras 17B55 Homological methods in Lie (super)algebras 18G05 Projectives and injectives (category-theoretic aspects)
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