The largest class of algebraic hyper structures satisfying the module like axioms is the v H module. In this paper, we consider the category of v H -modules and prove that the direct limit always exists in this category. Direct limits are defined by a universal property, and so are unique. The most powerful tool in order to obtain a module from a given v H module is the quotient out procedure. ...

If Γ1, . . . ,Γn are limit groups and S ⊂ Γ1 × · · · × Γn is of type FPn(Q) then S contains a subgroup of finite index that is itself a direct product of at most n limit groups. This answers a question of Sela.

We combine the geometric realization of principal series representations of [28] with the Bott–Borel–Weil Theorem for direct limits of compact groups found in [22], obtaining limits of principal series representations for direct limits of real reductive Lie groups. We introduce the notion of weakly parabolic direct limits and relate it to the conditions that the limit representations are norm–p...

In this article we initiate a systematic study of irreducible weight modules over direct limits of reductive Lie algebras, and in particular over the simple Lie algebras A(∞), B(∞), C(∞) and D(∞). Our main tool is the shadow method introduced recently in [DMP]. The integrable irreducible modules are an important particular class and we give an explicit parametrization of the finite integrable m...

A direct limit G = lim G~ of (finite-dimensional) Lie groups has Lie algebra 9 = H m g~ and exponential map exp6 : g ~ G. Both G and g carry natural topologies. G is a topological group, and g is a topological Lie algebra with a natural structure of real analytic manifold. In this Letter, we show how a special growth condition, natural in certain physical settings and satisfied by the usual dir...