  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.
