Groupoids and Asymptotic Actions
by
Vagif A. Kasimov
Baku State University

In this work the relation between asymptotic actions and groupoids is established. Also, a new definition of groupoid is given and it is proved that the category generated by asymptotic actions of groups at a point formes a groupoid. The relation between the Poincaré group and the stabilizer of points is an isomorphism.

Let K=(Ob K, Mor K) be any category, where the class morphisms Mor K posses the following property: Any morphism from Mor K is invertible. Such categories are called groupoids (see [1], [2]). Naturally, the problem arises as to how to define groupoids. Suppose the partial operation P is defined on the non-empty set H, that is, P is a given mapping P : H₂ --> H, \emptyset =/= H₂ subset H×H. Then P is called associative, if the following conditions are true:

(x.y) in H2 and (y, z) in H2 ===> (p(x, y), z) in H2 and (x, p(y, z)) in H2, (x, y) in H2 and (p(x, y), z) in H2 ===> (y, z) in H2 and (x, p(y, z)) in H2, (y, z) in H2 and (x, p(y, z)) in H2 ===> (x, y) in H2 and (p(x, y), z) in H2, \Downarrow p(x, p(y, z)) = p(p(x, y), z)      (1)

If the associative partial operation P : H₂ --> H satisfies the next conditions:

1) Every element x in H has left and right units, that is, there are elements e_x and _xe, such that

ex·x = x and x·xe = x,                   (2)

2) Every element x in H is inevitable, that is there is an element x^-1 in H, such that

x-1·x = xe and x·x-1 = ex,             (3)

then the triple (H, H₂, P) is called a groupoid.

Let the family j = {j_x : G×M --> M}_{x in N} be an asymptotic action of the group G in the Hilbert module M ([3], [4]). Consider j for the orbit O_\nu(m) of the point m in M. Construct the new category whose objects are the points of the orbit O_\nu. To every pair of points (m₁, m₂) in O_\nu(m)×O_\nu(m) corresponds the set G(m₁, m₂) = {g in G : limj_n(g, m₁) = m₂}. Let H_m be the stabilizer of m according to the asymptotic action j.

Theorem 1. For every point m₁ in O_\nu(m) the set G (m, m₁) is a class of co-sets for the subgroup H_m.

Every element g in G(m₁, m₂) we will call a morphism from m₁ to m₂. Suppose G_m is a disconnected sum of class G(m₁, m₂), that is

Gm = Õ m1, m2 in O\nu(m)  G(m1, m2).

Proposition 1. The pair of classes H(m) = (O_\nu(m), G_m) forms a category.

Proposition 2. The category H(m) is a groupoid.

Denote by \pi(G, x) the Poincaré group of the groupoid H(m) for the object x in O_\nu(m).

Theorem 2. The Poincaré group \pi(G, x) is isomorphic to the stabiliser of the point x, that is \pi(G, x) \approx H_x.

Literature.

1. P. Gabriel and M. Zisman, Calcules of fractions and homotopy theory, 1967.
2. Y. A. Neretin, Category of symmetry and infinite dimensional groups (in Russian), 1998.
3. V. A. Kasimov, Asymptotic G-bundles on Hilbert modules, Academy of Sciences of Azerbayjan, Proceedings of Institute of Mathematics and Mechanics, Vol.5, Baku 1996.
4. V. A. Kasimov, Bundle associated with the asymptotic Hilbert G-modul (in Russian), News of Baku State University, 2, 1998.

Date received: August 3, 2001