## Project

# Cohomological invariants of structurable algebras (3G004018)

A common theme in algebra is to understand algebraic structures over arbitrary fields by first studying them over their algebraic closure and then investigating the possible ways to "descend" to the base field again. A typical example occurs in the theory of quadratic forms over an arbitrary field. In order to decide when two given quadratic forms are non-isometric, a useful tool is to define invariants for quadratic forms; typical (easy) examples of such invariants are the discriminant and the Clifford algebra. These are examples of cohomological invariants (of degree 1 and 2, respectively).

Our goal is to study cohomological invariants for non-associative algebras and their related linear algebraic groups. Examples that have been well studied are octonion algebras (certain 8-dimensional algebras) and Albert algebras (certain 27-dimensional Jordan algebras); these are connected to groups of type G2 and F4, respectively.

We will study invariants of structurable algebras, a class of algebras with involution, simultaneously generalizing Jordan algebras and associative algebras with involution. We will mainly be interested in the exceptional structurable algebras: the 35-dimensional Smirnov algebras; tensor products of two composition algebras (of dimension 16, 32, 64); structurable algebras of skew-dimension one arising from hermitian cubic norm structures (of dimension 8, 20, 32, 56). These examples are related to groups of type 3D4, E6, E7 and E8.