Stable representation theory: beyond the classical groups

The orthogonal groups are a series of simple Lie groups associated to symmetric bilinear forms. There is no analogous series associated to symmetric trilinear forms. We introduce an infinite dimensional group-like object that can be viewed as the limit of this non-existent series, were it to exist. We show that the representation theory of this object is well-behaved, and similar to the stable representation theory of orthogonal groups. Our theory is not specific to symmetric trilinear forms, and applies to any kind of tensorial forms. Our results can be also be viewed from the perspective of semi-linear representations of the infinite general linear group, and are closely related to twisted commutative algebras.