Classification of pointed weak torsion free representations for classical Lie algebras (Q1346108)

Let \({\mathfrak g}\) be a classical Lie algebra over an algebraically closed field of characteristic \(p>3\) defined by a Cartan matrix, Serre's relations, and generators \(E_ i\), \(H_ i\), \(F_ i\). A representation \(V\) of \({\mathfrak g}\) is called pointed weak torsion free if \(V\) is irreducible with a one-dimensional weight space and the generators \(E_ i\), \(F_ i\) act injectively on \(V\). Assuming further that \({\mathfrak g}\) is not of type \(A_{ip -1}\), the author classifies the pointed weak torsion free representations of \({\mathfrak g}\) up to isomorphism. Such representations exist if and only if \({\mathfrak g}\) is of type \(A_ \ell\) or \(C_ \ell\). Explicit constructions of these representations are given by specifying the actions of \(E_ i\), \(F_ i\), \(H_ i\) on a certain basis. It is also shown that all of these representations have modular Weyl algebra realizations.