Category of \(\mathfrak g\) modules (Q1236176)

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 \(\Hom(P_i,L_i)\neq 0\). 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 \(C =D^t\cdot D\).