Compact Kac Algebras and Commuting Squares

We consider commuting squares of finite dimensional von Neumann algebras having the algebra of complex numbers in the lower left corner. Examples include the vertex models, the spin models (in the sense of subfactor theory), and the commuting squares associated to finite dimensional Kac algebras. To any such commuting square we associate a compact Kac algebra and we compute the corresponding subfactor and its standard invariant in terms of it.