A-infinity structures on the algebra of extensions of Verma modules in the parabolic category O
This is the author's diploma thesis. In the first part of the thesis the algebra structure on the Ext-spaces Ext^k(M(x), M(y)) of Verma modules M(x) and M(y) in the parabolic category O for the case of the parabolic subalgebras gl(n) x gl(m) for n=1 and n=2 is computed and expressed in terms of quivers. For arbitrary n, more general results about Hom-spaces of projective modules are achieved. The second part of the thesis deals with A-infinity structures on the algebras described above. An explicit construction for a minimal model is given and all higher multiplications are determined in the cases n=1 and n=2. In the first case the algebra turns out to be formal. The main result of the thesis is presented in the general vanishing theorem. It says that for arbitrary n we get a minimal model with vanishing m_k for k > n^2+1. The tools used for this proof are developed throughout the entire thesis.