Algebraic Bol loops

In this paper, we study the category of algebraic Bol loops over an algebraically closed field of definition. On the one hand, we apply techniques from the theory of algebraic groups in order to prove structural theorems for this category. On the other hand, we present some examples showing that these loops lack some nice properties of algebraic groups; for example, we construct local algebraic Bol loops which are not birationally equivalent to global algebraic loops.

An algebraic theory for generalized Jordan chains and partial signatures in the Lagrangian Grassmannian

We discuss an algebraic theory for generalized Jordan chains and partial signatures, that are invariants associated to sequences of symmetric bilinear forms on a vector space. We introduce an intrinsic notion of partial signatures in the Lagrangian Grassmannian of a symplectic space that does not use local coordinates, and we give a formula for the Maslov index of arbitrary real analytic paths in terms of partial signatures.