Algebraic Methods in Analysis and Topology

Abstract

This thesis consists of two parts. Part I concerns the classification of unital minimal A∞-structures on graded vector spaces concentrated in degree 0, 1 and 2 up to equivalence. Here we give a complete description of every possible unital A∞-structure on such vector spaces, and partial results on the question of equivalence between such structures. We find an invariant of equivalent minimal A∞-structures, namely we prove that the first non-zero multiplications have to be equal up to automorphism. In particular, the integer where this happens is a discrete invariant of an A∞-structure, and it can take any value from 2 to ∞. We also give an example where this invariant gives a complete answer to the question of equivalence. In part II we show how to solve partial differential equations in the representation coefficients of a projective unitary representation of a Lie group. To do this we build up a theory which allows us to differentiate the representation itself. We then apply this to the case when the Lie group has a left-invariant complex structure to get holomorphic representation coefficients.