Classification of Noncommuting Quadrilaterals of Factors

A quadrilateral of factors is an irreducible inclusion of factors $N \subset M$ with intermediate subfactors $P$ and $Q$ such that $P$ and $Q$ generate $M$ and the intersection of $P$ and $Q$ is $N$. We investigate the structure of a non-commuting quadrilateral of factors with all the elementary inclusions $P\subset M$, $Q\subset M$, $N\subset P$, and $N\subset Q$ 2-supertransitive. In particular we classify such quadrilaterals with the indices of the elementary subfactors less than or equal to 4. We also compute the angles between $P$ and $Q$ for quadrilaterals coming from $\alpha$-induction and asymptotic inclusions.